diff --git a/.gitattributes b/.gitattributes index 8b511d2a693092acf7efe36a592ce0f120680d6b..3fffc576de7a96b1aed69f19d1c3202bb9f87929 100644 --- a/.gitattributes +++ b/.gitattributes @@ -136,3 +136,8 @@ aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf filter=lfs diff=lfs merge=lfs -tex xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf filter=lfs diff=lfs merge=lfs -text Q9FKT4oBgHgl3EQfiC5d/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text 6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf filter=lfs diff=lfs merge=lfs -text +89FST4oBgHgl3EQfajh_/content/2301.13796v1.pdf filter=lfs diff=lfs merge=lfs -text +BdE2T4oBgHgl3EQfRge6/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf filter=lfs diff=lfs merge=lfs -text +zNFLT4oBgHgl3EQfni-P/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/3tFKT4oBgHgl3EQf8i6_/content/tmp_files/2301.11950v1.pdf.txt b/3tFKT4oBgHgl3EQf8i6_/content/tmp_files/2301.11950v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..04bf25f5d0b9e7b9ea0fe6638636632488221b21 --- /dev/null +++ b/3tFKT4oBgHgl3EQf8i6_/content/tmp_files/2301.11950v1.pdf.txt @@ -0,0 +1,1654 @@ +TUM-EFT 173/22 +Strong decays of T + +cc at NLO in an effective field theory +Lin Dai,1, ∗ Sean Fleming,2, † Reed Hodges,3, ‡ and Thomas Mehen3, § +1Physik Department, Technische Universit¨at M¨unchen, 85748 Garching, Germany +2Department of Physics and Astronomy, +University of Arizona, Tucson, Arizona 85721, USA +3Department of Physics, Duke University, +Durham, North Carolina 27708, USA +Abstract +The T + +cc exotic meson, discovered by the LHCb collaboration in 2021, can be interpreted as a +molecular state of D(∗)0 and D(∗)+ mesons. We compute next-leading order (NLO) contributions to +the strong decay of T + +cc in an effective field theory for D mesons and pions, considering contributions +from one-pion exchange and final state rescattering. Corrections to the total width, as well as the +differential distribution in the invariant mass of the final state D meson pair are computed. The +results remain in good agreement with LHCb experimental results when the NLO contributions +are added. The leading uncertainties in the calculation come from terms which depend on the +scattering length and effective range in D meson scattering. +∗Electronic address: lin.dai@tum.de +†Electronic address: spf@email.arizona.edu +‡Electronic address: reed.hodges@duke.edu +§Electronic address: mehen@phy.duke.edu +1 +arXiv:2301.11950v1 [hep-ph] 27 Jan 2023 + +I. +INTRODUCTION +The LHCb collaboration has observed a narrow resonance, the exotic tetraquark T + +cc, +in the final state D0D0π+ [1–5]. The resonance is close to both the D∗0D+ and D∗+D0 +thresholds. When using a unitarized Breit-Wigner profile appropriate for a coupled channel +problem, LHCb finds the difference between the resonance mass and the D∗+D0 threshold, +δm, and the decay width, Γ, to be: [5] +δm = −360 ± 40+4 +−0 keV , +Γ = 48 ± 2+0 +−14 keV . +(1) +The D∗0D+ threshold is 1.7 MeV above the resonance. The closeness of the resonance to +the two thresholds suggests the possibility that T + +cc has a molecular nature. +After the announcement of the discovery of T + +cc, many theory papers attempted to under- +stand various aspects of the exotic meson [6–26]. Several papers tried to predict its decay +width and differential decay width, with considerable success [6, 7, 10, 13, 14, 20, 21]. In one +of these papers [6], we wrote down an effective field theory for T + +cc considering it a molecular +state of two D mesons treated nonrelativistically, and computed leading-order strong and +electromagnetic decays. Special attention was paid to the coupled channel nature of the +problem. We found a decay width of 52 keV when the tetraquark is in an isospin-0 state, +using a value of δm = −273 keV, which arises from using a relativistic P-wave two-body +Breit-Wigner function with a Blatt-Weisskopf form factor. This was in good agreement with +the LHCb experiment. The predicted differential spectra as a function of the invariant mass +of the final state charm meson pair were also in good agreement with the binned experimen- +tal data. In this paper we investigate how these conclusions are affected by next-to-leading +order (NLO) strong decays. +The effective theory we will use is similar to XEFT for the χc1(3872) [27–41]. +Refs. +[27, 42, 43] have considered NLO XEFT diagrams for χc1(3872) decays. One-pion exchange +was found to have a negligible contribution to the decay width [27, 43], while final state +rescattering led to uncertainty in the decay rate of +50% +−30% when the binding energy of the +χc1(3872) is 0.2 MeV [43]. The differential spectrum dΓ[χc1(3872) → D0 ¯D0π0]/dEπ was +found to have a curve whose peak location and overall shape are insensitive to NLO correc- +tions; only the normalization is affected [43]. The sharply peaked nature of the differential +2 + +spectrum can inform about the molecular nature of the χc1(3872): since it is a function of +the virtual D∗0 propagator (p2 +D + γ2)−1, where γ is the binding momentum, as the binding +energy goes to zero the distribution becomes sharply peaked as pD → 0. +By analogy with this earlier work on χc1(3872), in this paper we compute NLO contri- +butions to the decay of T + +cc to find the uncertainties due to one-loop one-pion exchange and +final state rescattering diagrams. We calculate the uncertainty in the decay width, as well +as in the shape, peak location, and normalization of differential spectra. The calculation is +complicated by the presence of a coupled channel, which is not present for χc1(3872). We +find the decay width including NLO corrections to be 47+53% +−25% keV, which is consistent with +XEFT [43]. We also discuss the physical significance of several of the parameters in the +effective theory, and their effect on the decay width. +In Section II we write down the effective Lagrangian to NLO. The required Feynman +diagrams and their amplitudes, along with the explicit formulae for the partial widths are +shown in Section III. Plots of the differential distribution are shown in Section IV, followed +by concluding remarks in Section V. +II. +EFFECTIVE LAGRANGIAN +The leading-order effective Lagrangian for strong decays of T + +cc is [6] +LLO = H∗i† +� +i∂0 + +∇2 +2mH∗ − δ∗ +� +H∗i + H† +� +i∂0 + ∇2 +2mH +− δ +� +H ++ g +fπ +H†∂iπH∗i + H.c. +−C(0) +0 (H∗Tτ2H)†(H∗Tτ2H) − C(1) +0 (H∗Tτ2τaH)†(H∗Tτ2τaH) . +(2) +Here H and H∗ are isodoublets of the pseudoscalar and vector charm meson fields, respec- +tively, and π is the usual matrix of pion fields. The diagonal matrices δ and δ∗ contain the +residual masses, which are the difference between the mass of the charm meson D(∗)i, where +i = 0, +, and that of the D0. The coupling g = 0.54 is the heavy hadron chiral perturbation +theory (HHχPT) axial coupling [44–46] and fπ = 130 MeV is the pion decay constant. The +terms on the last two lines are contact interactions mediating D∗D scattering, where C(n) +0 +mediates S-wave scattering in the isospin-n channel, and τa are Pauli matrices acting in +isospin space. +3 + +Several new classes of terms appear at NLO in the effective theory. There are new contact +interactions involving two derivatives: +LC2 = C(0) +2 +4 (H∗Tτ2H)†(H∗Tτ2 +←→ +∇ 2H) + C(1) +2 +4 (H∗Tτ2τaH)†(H∗Tτ2τa +←→ +∇ 2H) . +(3) +These interactions occur in XEFT and are proportional to the effective range [27]. We can +also write down Dπ interaction terms by constructing isospin invariants out of the fields. +LCπ = C(1/2) +π +(πH)†(πH) + C(3/2) +π +� +vaH − 1 +3τaπH +�†� +vaH − 1 +3τaπH +� +. +(4) +Here v = +� +π1 π2 π0 +�T +/ +√ +2 is a vector of pion fields, with π± ≡ (π1 ∓ iπ2)/ +√ +2, such that +vaτa = π. C(1/2) +π +and C(3/2) +π +mediate scattering in the isospin-1/2 and isospin-3/2 channels, +respectively. The interactions which are relevant to our calculation are: +LCπ → C(1) +π D0†π0†D+π− − C(1) +π D+†π0†D0π+ + H.c. ++C(2) +π D0†π0†D0π0 + C(2) +π D+†π0†D+π0 ++C(3) +π D0†π+†D0π+ , +(5) +where the couplings C(1) +π , C(2) +π , and C(3) +π +are particular linear combinations of C(1/2) +π +and C(3/2) +π +as governed by Eq. (4). These interactions can be matched onto the chiral Lagrangian [47]. +The values we use for these Cπ couplings are computed from lattice data; see Appendix C +for details. +We can write down D∗D → DDπ interactions by using the same strategy of constructing +isospin invariants out of the fields. That would lead to: +LB1 = B(I=0) +1 +εαβ(H∗ +αHβ)†(Hτ2τiH∇vi) ++B(I=1) +1 +(H∗τ2τkH)†(εijkHτ2τiH∇vj) + H.c. . +(6) +However, we need isospin-breaking terms in order to fully renormalize the theory at NLO, +so ultimately we have four unique B1 couplings, one for each possible channel. Written in +terms of the charm meson fields, the interactions become: +LB1 → B(1) +1 (D+D∗0)†(D+D0∇π0) + B(2) +1 (D0D∗+)†(D+D0∇π0) ++B(3) +1 +2 (D0D∗+)†(D0D0∇π+) + B(4) +1 +2 (D+D∗0)†(D0D0∇π+) . +(7) +4 + +Relations between the B(i) +1 +implied by Eq. (6) are given in the Appendix. We can construct +DD contact terms out of the isospin invariants. There are only interactions in the isospin-1 +channel, +LC0D = C(1) +0D(Hτ2τaH)†(Hτ2τaH) +→ C(1) +0D +2 (D0D0)†(D0D0) + C(1) +0D(D+D0)†(D+D0) , +(8) +where in the second line we have restricted to terms that are relevant to our calculation. +The authors in Ref. [43] chose to vary their C(1) +0D coupling, which described D ¯D scattering +as opposed to DD, over a range of [−1, 1] fm2. We test several different values for it within +that range. Lastly, we need a kinetic term for the pions; in contrast to XEFT, we treat them +relativistically, +Lπ = tr(∂µπ†∂µπ − m2 +ππ†π) . +(9) +The full NLO Lagrangian is then LNLO = LC2 + LCπ + LB1 + LC0D + Lπ. +III. +FORMULAE FOR DECAY WIDTHS +Writing down the decay width for the T + +cc at NLO requires care due to the coupled channel +nature of the problem. We define a two-point correlation function matrix ˆG as +ˆG = +� +d4x e−iEt ⟨0|T[X(x)XT(0)]|0⟩ = iΣ(1 + CΣ)−1 , +(10) +where the interpolating field is +X = +� +� +� +D0D∗+ +D+D∗0 +� +� +� . +(11) +The right-hand side of Eq. (10) arises from expressing ˆG to all orders as an infinite sum +of the C0-irreducible two-point function Σ, in a manner similar to that in Appendix A of +Ref. [48], but here C0 and Σ are matrices due to the presence of a coupled channel. −iΣ is +given by the sum of D∗D self-energy diagrams in Fig. 1. Its diagonal elements correspond +to those two-point diagrams which do not swap channels, and the off-diagonal elements to +those which do swap channels. We can then project out the isospin-0 and isospin-1 channels, +5 + +−iΣ += ++ ++ +C2 ++ ++ +C0D ++ +Cπ ++ +B1 +FIG. 1: Some of the D∗D self-energy diagrams contributing to −iΣ. Bold solid lines represent D∗ +mesons, regular solid lines represent D mesons, and dashed lines represent pions. The first row is +LO, the second row is NLO, and the third and fourth rows are NNLO. There are also other NNLO +diagrams not shown which are C0-reducible combinations of the NLO diagrams. +and tune the parameters of the two-point correlators so that there is a pole corresponding +to the location of the T + +cc bound state. Near the vicinity of the pole, the Green’s function +can be written as +G0/1 = +� +� +� +1 +∓1 +� +� +� +T +ˆG +� +� +� +1 +∓1 +� +� +� ≈ 1 +2 +iZ0/1 +E + ET + +iΓ0/1 +2 +, +(12) +where Γ0/1 is the decay width and the residue Z0/1 is the wave function renormalization. We +find for the decay width in the isospin-0 channel +ΓNLO +0 +≈ −ΓLO Re Σ′NLO +0 +(−ET) +Re tr Σ′LO(−ET) + 2 Im ΣNLO +0 +(−ET) +Re tr Σ′LO(−ET) , +(13) +where Σ0 ≡ Σ11 + Σ22 − Σ12 − Σ21 is a particular combination of the elements of the Σ +matrix appropriate for isospin-0. The first term of Eq. (13) is a correction to the LO decay +width from NLO D∗D self-energy corrections, i.e., diagrams on the second row of Fig. 1. +The second term of Eq. (13) consists of NLO decay diagrams, from various cuts of diagrams +6 + +on the third and fourth rows of Fig. 1. Note that Im ΣNLO is from Σ diagrams of one +higher order than in Re ΣNLO because the LO self-energy graph has no imaginary part +below threshold. The derivatives of Σ are with respect to E and evaluated at E = −ET. +For a more detailed derivation of Eq. (13) refer to Appendix A. +Three diagrams in Fig. 1 contribute to Re Σ to NLO. They are the LO self-energy dia- +gram (−iΣ1), the one-pion exchange diagram (−iΣ2), and the C2 contact diagram (−iΣ3). +They are evaluated in the power divergence subtraction (PDS) scheme [49]. This scheme +corresponds to using MS to handle logarithmic divergences as well as subtracting poles in +d = 3 to keep track of linear divergences. A 1/ϵ pole appears in Σ2, but the dependence +on the renormalization scale drops out when the derivative with respect to E is taken. We +neglect terms in the propagators that go as p4/m2 +H or (δm)p2/mH, where δm is of the order +of the pion mass, compared to p2. In Σ2 and Σ3 we use a Fourier transform to evaluate +the integrals over three-momentum, using a procedure outlined in Ref. [50]. We define a +reduced mass µ(m1, m2) ≡ m1m2/(m1 + m2) and the binding momenta are defined to be +γ2(m1, m2) = 2µ(m1, m2)(m1 + m2 − mT). The expressions for the self energy diagrams are: +−iΣ1(m, m∗) = −iµ(m, m∗) +2π +[ΛPDS − γ(m, m∗)] , +(14) +−iΣ2(m1, m∗ +1, m2, m∗ +2, mπ, g1, g2) = −4ig1g2 +3 +µ(m1, m∗ +1)µ(m2, m∗ +2) +× +� +1 +16π2[ΛPDS − γ(m1, m∗ +1)][ΛPDS − γ(m2, m∗ +2)] ++(m∗ +2 − m1)2 − m2 +π +(8π)2 +�1 +ϵ + 2 +−4 log +� +γ(m1, m∗ +1) + γ(m2, m∗ +2) +−i(m∗ +2 − m1)2 + im2 +π +� +− 4 log µ +�� +, +(15) +−iΣ3(m1, m∗ +1, m2, m∗ +2, C2) = − i +4π2C2[γ2(m1, m∗ +1) + γ2(m2, m∗ +2)]µ(m1, m∗ +1) +×µ(m2, m∗ +2)[ΛPDS − γ(m1, m∗ +1)][ΛPDS − γ(m2, m∗ +2)] . (16) +To be consistent with the implementation of the PDS scheme in the decay diagrams (see +Appendix B), for the double integral in Σ2 we have used rotational symmetry to replace +7 + +p +m +(a) +g1 +p +pπ +g2 +g3 +m∗ +1 +mext +mπ +m +m∗ +2 +(b) +pπ +p +Cπ +mπ +m +(c) +C2 +m∗ +1 +m +p +m∗ +2 +mext +(d) +B1 +m +(e) +C0D +m∗ +m1 +(f) +FIG. 2: Feynman diagrams at LO and NLO contributing to the decay of T + +cc. We label the vertices +and lines whose naming might be ambiguous. These diagrams arise from cuts of the diagrams on +the third and fourth lines of Fig. 1. +the tensor structure in the numerator with δij/3 and not δij/(d − 1). +This choice does +not affect the derivative of Σ2 as it only changes the constant terms which drop out upon +differentiation with respect to E. +The decay diagrams that contribute to 2 Im ΣNLO +0 +(−ET) are shown in Fig. 2. By the +optical theorem the square of these diagrams are given by the sum over the cuts of the +NNLO diagrams in Fig. 1. If there is only one pion/charm meson vertex in a diagram, its +coupling is labeled gπ. If there are more than one such vertex, the couplings are numbered +gi. Depending on the type of pion and charm meson, these couplings will be either g/fπ or +±g/( +√ +2fπ).The expressions are written in terms of the basis integrals given in Appendix B. +These basis integrals depend on parameters b, c1, and c2, the definitions for c1 and c2 are +provided where appropriate, b = 1 unless otherwise specified, and the momentum arguments +for the integrals are p unless otherwise specified. +8 + +iA(2a)(p, m, m∗, gπ) = 2igπϵT · pπµ(m, m∗) +p2 + γ2(m, m∗) +. +(17) +iA(2b)(p, m, mext, mπ, m∗ +1, m∗ +2, g1, g2, g3) = 4iµ(m, m∗ +1)µ(mext, m∗ +2)g1g2g3 +p2 + γ2(mext, m∗ +2) +× +� +ϵT · p pπ · p +� +I(2) +0 +− 2I(1) + I +� ++ϵT · pπp2I(2) +1 +� +, +(18) +c1 = γ2(m, m∗ +1) , +c2 = p2 − (mT − m − mext)2 + m2 +π . +iA(2c)(m, mext, mπ, m∗, gπ, Cπ) = 2iµ(m, m∗)gπCπϵT · p[I(1) − I] , +(19) +c1 = γ2(m, m∗) , +c2 = p2 − (mT − m − mext)2 + m2 +π . +iA(2d)(m, mext, m∗ +1, m∗ +2, gπ, C2) = 1 +πiC2gπϵT · pπµ(m, m∗ +1)µ(mext, m∗ +2) +× p2 − γ2(m, m∗ +1) +p2 + γ2(mext, m∗ +2)[γ(m, m∗ +1) − ΛPDS] . +(20) +iA(2e)(m, m∗, B1) = −iB1 +2π ϵT · pπµ(m, m∗)[γ(m, m∗) − ΛPDS] . +(21) +iA(2f)(m1, m2, m∗, p0 +π, gπ, C0D) = 4iµ(m1, m2)µ(m2, m∗)gπC0DϵT · pπI(pπ) , (22) +c1 = γ2(m2, m∗) , +c2 = −2µ(m1, m2) +� +mT − m1 − m2 − p0 +π − p2 +π +2m1 +� +, +b = µ(m1, m2) +m1 +. +Following Eq. (13) and using the amplitudes defined above, the decay widths for the two +strong decays of T + +cc are +9 + +dΓNLO +0 +(T + +cc → D+D0π0) +dp2 +0dp2 ++ += +2 +Re tr Σ′LO(−ET)Re +� +A(2a)(p+, m+, m∗ +0, −g/ +√ +2fπ) +× +� +A(2b)(p0, m+, m0, mπ0, m∗ +0, m∗ ++, −g/ +√ +2fπ, g/ +√ +2fπ, g/ +√ +2fπ) ++A(2b)(p+, m+, m+, mπ−, m∗ +0, m∗ +0, g/fπ, g/fπ, −g/ +√ +2fπ) +−A(2b)(p0, m0, m0, mπ+, m∗ ++, m∗ ++, g/fπ, g/fπ, g/ +√ +2fπ) +−A(2b)(p+, m0, m+, mπ0, m∗ ++, m∗ +0, g/ +√ +2fπ, −g/ +√ +2fπ, −g/ +√ +2fπ) ++A(2c)(p0, m+, m0, mπ0, m∗ +0, −g/ +√ +2fπ, C(2) +π ) +−A(2c)(p0, m0, m0, mπ+, m∗ ++, g/fπ, C(1) +π ) ++A(2f)(m0, m+, m∗ +0, −g/ +√ +2fπ, C(1) +0D) +−A(2f)(m+, m0, m∗ ++, g/ +√ +2fπ, C(1) +0D) +�∗ ++ (D0 ↔ D+, π+ ↔ π−) +� +− +1 +Re tr Σ′LO(−ET) +� +[β1(p2 ++ + γ2 ++) + β2] +���A(2a)(p+, m+, m∗ +0, −g/ +√ +2fπ) +��2 +−A(2a)(p0, m0, m∗ ++, g/ +√ +2fπ)A∗ +(2a)(p+, m+, m∗ +0, −g/ +√ +2fπ) +� ++[β3(p2 +0 + γ2 +0) + β4] +���A(2a)(p0, m0, m∗ ++, g/ +√ +2fπ) +��2 +−A(2a)(p+, m+, m∗ +0, −g/ +√ +2fπ)A∗ +(2a)(p0, m0, m∗ ++, g/ +√ +2fπ) +�� +−dΓLO +0 (T + +cc → D+D0π0) +dp2 +0dp2 ++ +Re Σ′NLO +0 +Re tr Σ′LO +���� +C2→0,E=−ET +(23) +10 + +dΓNLO +0 +(T + +cc → D0D0π+) +dp2 +1dp2 +2 += +1 +Re tr Σ′LO(−ET)Re +� +A(2a)(p2, m0, m∗ ++, g/fπ) +× +� +A(2b)(p1, m0, m0, mπ+, m∗ ++, m∗ ++, g/fπ, g/fπ, g/fπ) ++A(2b)(p2, m0, m0, mπ+, m∗ ++, m∗ ++, g/fπ, g/fπ, g/fπ) +−A(2b)(p1, m+, m0, mπ0, m∗ +0, m∗ ++, −g/ +√ +2fπ, g/ +√ +2fπ, g/fπ) +−A(2b)(p2, m+, m0, mπ0, m∗ +0, m∗ ++, −g/ +√ +2fπ, g/ +√ +2fπ, g/fπ) ++A(2c)(p1, m0, m0, mπ+, m∗ ++, g/fπ, C(3) +π ) +−A(2c)(p1, m+, m0, mπ0, m∗ +0, −g/ +√ +2fπ, C(1) +π ) ++A(2f)(m0, m0, m∗ ++, g/fπ, C(1) +0D/2) +�∗ ++ (p1 ↔ p2) +− +�2gµ0 +fπ +�2p2 +π +3 β5 +� +1 +p2 +1 + γ2 +0 ++ +1 +p2 +2 + γ2 +0 +�� +−dΓLO +0 (T + +cc → D0D0π+) +dp2 +1dp2 +2 +� +β4 + Re Σ′NLO +0 +Re tr Σ′LO +���� +C2→0,E=−ET +� +(24) +In the previous formulae we have used subscripts on µ and γ to indicate which charm +meson is a pseudoscalar in that particular channel, e.g., µ0 = µ(m0, m∗ ++). The combinations +of self-energy diagrams that we need are Re tr Σ′LO(−ET) and Re Σ′NLO +0 +(−ET, C2 → 0). In +terms of the functions defined above, these are given by: +Re tr Σ′LO = Re Σ′ +1(m0, m∗ ++) + Re Σ′ +1(m+, m∗ +0) , +Re Σ′NLO +0 +|C2→0 = Re +� +Σ′ +2(m+, m∗ +0, m+, m∗ +0, mπ+, g/fπ, g/fπ) ++Σ′ +2(m0, m∗ ++, m0, m∗ ++, mπ+, g/fπ, g/fπ) ++Σ′ +2(m+, m∗ +0, m0, m∗ ++, mπ0, −g/ +√ +2fπ, g/ +√ +2fπ) ++Σ′ +2(m0, m∗ ++, m+, m∗ +0, mπ0, g/ +√ +2fπ, −g/ +√ +2fπ) +� +(25) +The expressions for βi are given in Appendix C. The terms dependent on A(2b) and Re Σ′ +2 +have linear divergences that must cancel against each other. They cancel exactly in the limit +µ0 = µ+. We make that approximation in those terms only to ensure the cancellation; it +is a reasonable approximation as µ0/µ+ ≈ 0.99948. See Appendix B for more discussion of +these linear divergences. +11 + +3730 +3732 +3734 +3736 +3738 +0 +20 +40 +60 +80 +100 +FIG. 3: A plot of the differential decay width as a function of the invariant mass of the final state +D meson pair. Solid lines represent the LO calculation; the dashed lines represent the addition +of non-analytic and NLO self-energy corrections. Overlaid is the binned experimental data from +LHCb, with the background subtracted. +IV. +DIFFERENTIAL DECAY DISTRIBUTIONS AND PARTIAL WIDTHS +Once we have formulae for the T + +cc → DDπ partial widths, we can numerically integrate +over part of three-body phase space in Mathematica and plot the differential distribution +dΓ/dmDD. It is insightful to compare our predicted curves to the LHCb experimental data +for the total yield. This will inform us about the effect and importance of the different +interactions in the effective theory. We normalize our distributions by performing a least- +squares fit of the LO distribution to the data, and using the same normalization factor for +the NLO distributions. The Cπ decay diagrams, individually and as a whole, contribute +negligibly to the distributions. The parameters β1, β3, and β5 also have a small impact on +the distributions over the range in which we vary them. We therefore do not show plots +varying these parameters individually. +The contributions from the non-C2-dependent NLO self-energy corrections (i.e. the first +12 + +3730 +3732 +3734 +3736 +3738 +0 +20 +40 +60 +80 +100 +FIG. 4: A plot of the differential decay width as a function of the invariant mass of the final state +D meson pair. Solid lines represent the LO calculation; The dashed and dotted lines represent +two different ranges for C0D. +Overlaid is the binned experimental data from LHCb, with the +background subtracted. +diagram on the second line of Fig. 1), as well as the contributions from Fig. 2b, serve to +increase the partial widths by a small but noticeable amount (Fig. 3). The effect of the C0D, +β2, and β4 terms on the distributions can be significant. In the following we will investigate +their impact by setting all other contributions to dΓNLO/dmDD to zero and varying them +individually. +The C0D interaction has a sizeable contribution to the partial widths, as evidenced in +Fig. 4, where we plot the differential distributions and vary this coupling in two possible +ranges: C0D ∈ [−1, 1] fm2 and ∈ [−0.25, 0.25] fm2. Its effect on the neutral pion decay is +twice as large as on the charged pion decay, because the coupling of charged pions to D +mesons is bigger by a factor of +√ +2. Clearly the differential distributions are sensitive to the +coupling’s magnitude. If C0D is +1 fm2 the peak of theD+D0 mass distribution is too high, +and if it is −1 fm2 three higher data points are underpredicted. It would be interesting to +13 + +3730 +3732 +3734 +3736 +3738 +0 +50 +100 +150 +FIG. 5: A plot of the differential decay width as a function of the invariant mass of the final state +D meson pair. Solid lines represent the LO calculation. The dashed and dotted lines represent +two different values of β2 and β4. Overlaid is the binned experimental data from LHCb, with the +background subtracted. +do a more careful analysis of the constraints this data puts on C0D but that is beyond the +scope of this paper. C0D is directly proportional to the I = 1 D meson scattering length, +so more precise knowledge of C0D from lattice simulations or experiments would allow us to +sharpen our predictions for T + +cc. +We can glean the significance of β2 and β4 by taking the isospin limit m0 = m+. In +Appendix C we see that in this limit: +β2 = β4 = −γr0 , +(26) +where γ is the binding momentum and r0 is the effective range in the I = 0 channel. The +effective range is positive and we expect γr0 < 1. In Fig. 5, we plot the distribution with all +other NLO interactions turned off, and for two values of β2 = β4 ≡ β: −0.1 and −0.59, along +with the LO curve (β = 0). We get γr0 = 0.59 if we use the largest binding momentum +(γ+) and r0 = 1/(100 MeV). +For nucleons, r0 ≈ 1/(100 MeV); since charm mesons are +14 + +LO result NLO lower bound NLO upper bound +Γ[T + +cc → D0D0π+] +28 +21 +44 +Γ[T + +cc → D+D0π0] +13 +7.8 +21 +Γstrong[T + +cc] +41 +29 +66 +Γstrong[T + +cc] + ΓLO +EM[T + +cc] +47 +35 +72 +TABLE I: Partial and total widths in units of keV at LO and NLO. +considerably more compact objects one might expect the effective range for charm mesons +to be smaller. We can see that the distribution is highly sensitive to the choice of β. A +β of −0.59 greatly increases the differential distribution, and is in much poorer agreement +with the experimental data. This suggests that the effective range for T + +cc is smaller than +for nucleons. +Clearly the partial widths and their differential distributions can vary substantially de- +pending on the choice of parameters in the effective field theory. However, the availability +of experimental data for the decays presents the possibility of performing fits of the dis- +tributions to the data to obtain estimates for these parameters. This could improve the +predictive power of the effective theory. We save such a careful statistical analysis for a +future publication. +We can use these plots that show the effect of a subset of the NLO contributions to inform +which ranges for the parameters to use when estimating the total NLO contribution to the +differential distribution (Fig. 6). The upper and lower bounds in the figure reflect varying +C0D from −1 fm2 to 0.25 fm2. The parameters β1, β3, and β5 are varied from −1/(100 MeV)2 +to +1/(100 MeV)2. The parameters β2 and β4, which reduce to −γr0 in the isospin limit, +are varied between 0 and −0.26. The latter value corresponds to a binding momentum for +the D∗+D0 channel, γ0, and r0 = 1/(100 MeV). While the uncertainty in the total width +of the T + +cc can be significant depending on the values of the NLO couplings, the qualitative +aspects of the plots of the differential decay widths in Fig. 6 are consistent between LO and +NLO. The overall shape and location of the peaks are unchanged by pion exchange and final +state rescattering. +When integrating over the full phase space to get the partial widths, we use the same +15 + +3730 +3732 +3734 +3736 +3738 +0 +20 +40 +60 +80 +100 +120 +140 +FIG. 6: A plot of the differential decay width as a function of the invariant mass of the final state +D meson pair. Solid lines represent LO calculation; the dashed lines represent the lower and upper +bounds of the NLO corrections. Here, we vary −1 fm2 ≤ C0D ≤ 0.25 fm2 and −0.26 ≤ β2/4 ≤ 0. +Overlaid is the binned experimental data from LHCb, with the background subtracted. +ranges for the parameters as in Fig. 6. The partial widths are given in Table I. Note that +the LO numbers differ from those in our original paper [6] because here we use the binding +energy from the unitarized Breit-Wigner fit, whereas in Ref. [6] we used the value from the +P-wave two-body Breit Wigner fit with a Blatt-Weisskopf form factor. This has the effect +of slightly increasing the prediction for the width compared to the initial paper, bringing +it closer to the experimental value. When adding the LO electromagnetic decay width of +6.1 keV (which is only slightly affected by the different binding energy) the total LO width +predicted by our effective theory is 47 keV which is already in excellent agreement with the +LHCb experimental value of 48 keV. Adding in the NLO contribution to the strong decay +widths, the total width of the T + +cc can range from 35 keV to 72 keV. So we can establish +an uncertainty in the width due to NLO strong decays of Γ[T + +cc] = 47+53% +−25% keV. This is +comparable to the uncertainty from similar operators contributing to the decay of χc1(3872) +16 + +3730 +3732 +3734 +3736 +3738 +0 +2.×10-7 +4.×10-7 +6.×10-7 +8.×10-7 +1.×10-6 +1.2×10-6 +1.4×10-6 +FIG. 7: Comparing our LO differential decay width to one where the D∗ propagators are taken to +be constant. The curves are fixed to have the same normalization. Note the lack of a sharp peak +in the constant propagator curves. +in XEFT [43]. +We did not consider NLO corrections to the electromagnetic decay, because the LO +electromagnetic decay was already a small contribution to the total width. In particular, +the differential distribution for the electromagnetic decay was negligible compared to the +strong decays’ distributions. +To illustrate why these differential decay width plots are good tests of the molecular +nature of the T + +cc, in Fig. 7 we can compare the LO differential curves to those which would +arise if we replaced the virtual D∗ propagators with a constant. The latter do not have +sharp peaks and thus would be in poor agreement with the experimental data. +V. +CONCLUSIONS +In this paper we have determined the effects of NLO strong decays on the total width +and differential decay width of the exotic meson T + +cc. We considered pion exchange and +final state rescattering diagrams, from similar operators to those in XEFT for the χc1(3872) +17 + +[43]. We arrive at similar conclusions as Ref. [43]. The differential decay width plots have +shapes and peaks that are relatively unchanged by the NLO effects, but the total width has +significant uncertainty: Γ[T + +cc] = 47+53% +−25% keV. The central value (the LO result) is in good +agreement with data. +We varied the parameters in the NLO calculation to get a sense of the uncertainty in +the predictions and determine which parameters in the NLO calculation give the biggest +corrections. Nonanalytic corrections for pion loops are not important. The parameter C0D, +which is proportional to the I = 1 D meson scattering length, and β2 and β4, which in the +isospin limit are equal and proportional to the I = 0 D meson effective ranges, significantly +affect the decay width and normalization of the differential distribution. It would be inter- +esting to fit the NLO differential curves to the experimental data and obtain bounds on the +undetermined couplings, thereby learning more about these physical quantities. Alterna- +tively, one might hope to get information about these parameters from lattice simulations or +other experiments. Any improvement in our understanding of these parameters in D meson +scattering would increase the predictive power of the effective field theory. +Acknowledgments - L. D. is supported by the Alexander von Humboldt Foundation. +S. F. is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear +Physics, under award number DE-FG02-04ER41338. T. M. and R. H. are supported by +the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under grant +Contract Numbers DE-FG02-05ER41367. +Appendix A: Coupled channel decay width +The full expression for the isospin-0 two-point correlator is +−iG0 = 1 +2 +−Σ0 − 4C(1) +0 det Σ +1 + C(0) +0 Σ0 + C(1) +0 Σ1 + 4C(0) +0 C(1) +0 det Σ +, +(A1) +where Σ0/1 ≡ Σ11 + Σ22 ∓ Σ12 ∓ Σ21 are the isospin-0 and isopsin-1 combinations of the +elements of Σ. Since we expect T + +cc to be an isospin-0 state we treat C(1) +0 +perturbatively and +expand to NLO in C(1) +0 . +−iG0 ≈ 1 +2 +−Σ0 +1 + C(0) +0 Σ0 ++ 1 +2 +C(1) +0 (ΣLO +11 − ΣLO +22 )2 +(1 + C(0) +0 Σ0)2 +. +(A2) +18 + +We see that the real numerator of the C(1) +0 +term is the residue of a double pole at 1+C(0) +0 Σ0 = +0. That can be interpreted physically as a small shift in the location of the bound state, +which can be seen from expanding the right-hand side of Eq. (12) about ENLO +T += ET −ELO +T . +But since we are already tuning ET to be the location of the T + +cc bound state, we can set +C(1) +0 +to zero to remove the double pole from the amplitude. +−iG0 → 1 +2 +−Σ0 +1 + C(0) +0 Σ0 +. +(A3) +At this stage the problem is identical to the single-channel problem in XEFT [27], with the +single-channel two-point function replaced by our isospin-0 combination of coupled-channel +two-point functions. The wave function renormalization and decay width are therefore: +Z0 = +1 +� +C(0) +0 +�2Re Σ′ +0(−ET) +, +Γ0 = 2 Im Σ0(−ET) +Re Σ′ +0(−ET) . +(A4) +Σ0 has LO contributions from the diagonal elements, and NLO contributions from all ele- +ments. After expanding in the NLO terms we find our corrections to the LO decay width. +Γ0 ≈ ΓLO +� +1 − Re Σ′NLO +0 +(−ET) +Re tr Σ′LO(−ET) +� ++ 2 Im ΣNLO +0 +(−ET) +Re tr Σ′LO(−ET) . +(A5) +Appendix B: Basis integrals and the PDS scheme +The most basic integral that arises when evaluating the one-loop diagrams in the PDS +scheme is: +�ΛPDS +2 +�4−d � +dd−1l +(2π)d−1 +1 +l2 + c − iϵ = 1 +4π(ΛPDS − +√ +c − iϵ) . +(B1) +This result is obtained by subtracting the pole in d = 3 with a counterterm, then evaluating +the result in d = 4, yielding a linear divergence in ΛPDS. +The scalar integral I(p) is finite in d = 3 and d = 4, so no PDS counterterm is needed. +I(p) = +� +dd−1l +(2π)d−1 +1 +l2 + c1 − iϵ +1 +l2 − 2bl · p + c2 − iϵ += +1 +8π +1 +� +b2p2 +� +tan−1 +� c2 − c1 +2 +� +b2p2c1 +� ++ tan−1 +� +2b2p2 + c1 − c2 +2 +� +b2p2(c2 − b2p2) +�� +. +(B2) +19 + +The linear tensor integral I(1)(p) can be solved using algebraic manipulation of the nu- +merator, which yields two integrals of the form of Eq. (B1) that have opposite sign for the +divergence, and so I(1)(p) is UV finite. +piI(1)(p) = +� +dd−1l +(2π)d−1li +1 +l2 + c1 − iϵ +1 +l2 − 2bl · p + c2 − iϵ , +→ p2I(1)(p) = +1 +2b +� 1 +4π +√ +c1 − iϵ − 1 +4π +� +c2 − b2p2 − iϵ + (c2 − c1)I(p) +� +. +(B3) +The quadratic tensor integrals I(2) require care when implementing the PDS scheme. The +linear divergences which arise in the decay width can only cancel if the subtraction scheme +is implemented correctly. After using Feynman parameters to combine the propagators and +obtain an integrand like liljf(l2), the correct procedure is to replace lilj → δij/3 immediately, +and not with δij/(d − 1). The latter would cancel the factor of d − 1 that arises when +evaluating the loop momentum integral, and this results in the incorrect coefficient for the +PDS subtraction scale ΛPDS. Additionally, algebraic manipulation of the numerator of I(2) +to reduce it to integrals of the form of I(1) and I leads to yet another incorrect coefficient. +This is the method used to obtain the expressions in the appendix of Ref. [43]; as such, the +formulae for the decay width in that paper are only correct if ΛPDS = 0 and d = 4. +Using the correct procedure for the basis integrals gives the following results: +pipjI(2) +0 (p) + δijp2I(2) +1 (p) = +� +dd−1l +(2π)d−1lilj +1 +l2 + c1 − iϵ +1 +l2 − 2bl · p + c2 − iϵ , +I(2) +0 (p) = b2 +8π +� 1 +0 +dx +x2 +� +∆(x) +, +(B4) +→ p2I(2) +1 (p) = +1 +8π +�2 +3ΛPDS − +� 1 +0 +dx +� +∆(x) +� +, +(B5) +for ∆(x) = −b2p2x2 + (c2 −c1)x+c1 −iϵ. One can be reassured that this implementation of +the PDS scheme is correct because the same relative weight of the ΛPDS and +� 1 +0 dx +� +∆(x) +terms is obtained when using a hard cutoff. That does not occur when using lilj → δij/(d−1) +or algebraic manipulation of the numerator. Furthermore, unless the relative weight of the +two terms in I(2) +1 +is 2/3, the linear divergences that appear in ΓNLO +0 +as A(2b) and Re Σ′ +2 do +not cancel in the isospin limit, as they do in XEFT. For the T + +cc, they cancel when µ0 = µ+, +an approximation we make in the cutoff-dependent terms to ensure cancellation. +With algebraic manipulation of the integrand in Eq. (B4) and integration by parts in +20 + +Eq. (B5), we can rewrite these expressions in terms of I and I(1). +p2I(2) +0 += − 1 +16π +� +c2 − b2p2 − iϵ + c1 +2 I(p) + 3 +4 +c2 − c1 +b +I(1)(p) , +(B6) +p2I(2) +1 += ΛPDS +12π − +1 +16π +� +c2 − b2p2 − iϵ − c1 +2 I(p) − 1 +4 +c2 − c1 +b +I(1)(p) . +(B7) +Appendix C: Cπ couplings and βi expressions +In the isospin |I, mI⟩ basis, we use the phase convention +|π+⟩ = − |1, 1⟩ , +|π0⟩ = |1, 0⟩ , +|D+⟩ = +���� +1 +2, 1 +2 +� +, +|D0⟩ = +���� +1 +2, −1 +2 +� +. +(C1) +Then the Clebsch-Gordan decomposition of the Dπ pairs is +|D0π0⟩ = +� +2 +3 +���� +3 +2, −1 +2 +� ++ 1 +√ +3 +���� +1 +2, −1 +2 +� +, +|D+π0⟩ = +� +2 +3 +���� +3 +2, 1 +2 +� ++ 1 +√ +3 +���� +1 +2, 1 +2 +� +, +|D0π+⟩ = − +� +2 +3 +���� +1 +2, 1 +2 +� +− 1 +√ +3 +���� +3 +2, 1 +2 +� +. +(C2) +From this we can deduce +aD0π0 = aD+π0 = 2 +3a3/2 +Dπ + 1 +3a1/2 +Dπ , +aD0π+ = 1 +3a3/2 +Dπ + 2 +3a1/2 +Dπ . +(C3) +These scattering lengths are calculated on the lattice in Ref. [51] to be a1/2 +Dπ = 0.37+0.03 +−0.02 fm +and a3/2 +Dπ = −(0.100±0.002) fm. The matching from tree level scattering tells us that, for the +diagonal couplings C(2) +π +and C(3) +π , we can use Cπ = 4π(1+mπ/mD)aDπ, with the appropriate +masses and scattering lengths for each process. We can then use those two values to solve +for C(1/2) +π +and C(3/2) +π +and obtain C(1) +π . We get +C(1) +π += −3.0+0.32 +−0.40 fm , +C(2) +π += −0.76+0.14 +−0.09 fm , +C(3) +π += 2.9+0.3 +−0.2 fm . +(C4) +The expressions for the βi are given below. The subscripts on the γ and µ variables indicate +the pseudoscalar charm meson is in that channel, e.g. +γ+ = γ(m+, m∗ +0) is the binding +momentum in the channel with the D+ meson. +21 + +β1 = (ΛPDS − γ+) +� +fπ +√ +2πgB(1) +1 ++ 1 +πC(+) +2 +µ+ − 1 +πC(−) +2 +µ0 +ΛPDS − γ0 +ΛPDS − γ+ +� +, +(C5) +β2 = +� 1 +πC(+) +2 +µ+(−2γ2 ++)(ΛPDS − γ+) − 1 +πC(−) +2 +µ0(−γ2 +0 − γ2 ++)(ΛPDS − γ0) ++2π +�µ2 +0 +γ0 ++ µ2 ++ +γ+ +�−1� +− 1 +π2C(+) +2 +µ3 ++(γ+ − ΛPDS)(2γ+ − ΛPDS) +− 1 +π2C(+) +2 +µ3 +0(γ0 − ΛPDS)(2γ0 − ΛPDS) +−C(−) +2 +(γ2 ++ + γ2 +0)µ+µ0 +2π +�µ+ +γ0 +(ΛPDS − γ0) + µ0 +γ+ +(ΛPDS − γ+) +� ++C(−) +2 +µ+µ0(µ+ + µ0) +π2 +(ΛPDS − γ+)(ΛPDS − γ0) +�� +, +(C6) +β3 = (ΛPDS − γ0) +� +− +fπ +√ +2πgB(2) +1 ++ 1 +πC(+) +2 +µ0 − 1 +πC(−) +2 +µ+ +ΛPDS − γ+ +ΛPDS − γ0 +� +, +(C7) +β4 = +� 1 +πC(+) +2 +µ0(−2γ2 +0)(ΛPDS − γ0) − 1 +πC(−) +2 +µ+(−γ2 +0 − γ2 ++)(ΛPDS − γ+) ++2π +�µ2 +0 +γ0 ++ µ2 ++ +γ+ +�−1� +− 1 +π2C(+) +2 +µ3 ++(γ+ − ΛPDS)(2γ+ − ΛPDS) +− 1 +π2C(+) +2 +µ3 +0(γ0 − ΛPDS)(2γ0 − ΛPDS) +−C(−) +2 +(γ2 ++ + γ2 +0)µ+µ0 +2π +�µ+ +γ0 +(ΛPDS − γ0) + µ0 +γ+ +(ΛPDS − γ+) +� ++C(−) +2 +µ+µ0(µ+ + µ0) +π2 +(ΛPDS − γ+)(ΛPDS − γ0) +�� +, +(C8) +β5 = 1 +πC(+) +2 +µ0(ΛPDS − γ0) − 1 +πC(−) +2 +µ+(ΛPDS − γ+) ++B(3) +1 fπ +4πg (γ0 − ΛPDS) − B(4) +1 fπ +4πg (γ+ − ΛPDS)µ+ +µ0 +. +(C9) +It is instructive to take the isospin limit of these β expressions and compare to XEFT. +Referring to Eq. (6), we can write down the B1 couplings in this limit. +B(1) +1 += −B(2) +1 += − +√ +2B(I=0) +1 +, +B(3) +1 += 2(B(I=1) +1 ++ B(I=0) +1 +) , +B(4) +1 += 2(B(I=1) +1 +− B(I=0) +1 +) . +(C10) +22 + +Then taking µ+ = µ0 = µ, γ+ = γ0 = γ we find: +β1 = β3 = β5 = 1 +π(γ − ΛPDS) +�B(I=0) +1 +fπ +g +− 2C(0) +2 µ +� +, +β2 = β4 = −4C(0) +2 µγ +π +(γ − ΛPDS)2 . +(C11) +The isospin-1 couplings drop out, which is to be expected given that we have projected out +the isospin-0 state and are here dropping isospin-breaking interactions. These expressions +also match the dependence of the decay rate on C2 and B1 in XEFT [27]. Using Eq. (24) +of [27] (and adjusting for a factor of 4 in the definition of C2 in that paper) we see that +β2 = β4 = −γr0 in the isospin limit. 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D 87, 014508 +(2013), 1208.4535. +25 + diff --git a/3tFKT4oBgHgl3EQf8i6_/content/tmp_files/load_file.txt b/3tFKT4oBgHgl3EQf8i6_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..609623391177018fb4f9a03453cd974ba19f7f09 --- /dev/null +++ b/3tFKT4oBgHgl3EQf8i6_/content/tmp_files/load_file.txt @@ -0,0 +1,949 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf,len=948 +page_content='TUM-EFT 173/22 Strong decays of T + cc at NLO in an effective field theory Lin Dai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' ∗ Sean Fleming,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' † Reed Hodges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' ‡ and Thomas Mehen3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' § 1Physik Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Germany 2Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' University of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Tucson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Arizona 85721,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' USA 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Duke University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Durham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' North Carolina 27708,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' USA Abstract The T + cc exotic meson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' discovered by the LHCb collaboration in 2021,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' can be interpreted as a molecular state of D(∗)0 and D(∗)+ mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We compute next-leading order (NLO) contributions to the strong decay of T + cc in an effective field theory for D mesons and pions, considering contributions from one-pion exchange and final state rescattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Corrections to the total width, as well as the differential distribution in the invariant mass of the final state D meson pair are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The results remain in good agreement with LHCb experimental results when the NLO contributions are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The leading uncertainties in the calculation come from terms which depend on the scattering length and effective range in D meson scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' ∗Electronic address: lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='dai@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='de †Electronic address: spf@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='edu ‡Electronic address: reed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='hodges@duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='edu §Electronic address: mehen@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='11950v1 [hep-ph] 27 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' INTRODUCTION The LHCb collaboration has observed a narrow resonance, the exotic tetraquark T + cc, in the final state D0D0π+ [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The resonance is close to both the D∗0D+ and D∗+D0 thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' When using a unitarized Breit-Wigner profile appropriate for a coupled channel problem, LHCb finds the difference between the resonance mass and the D∗+D0 threshold, δm, and the decay width, Γ, to be: [5] δm = −360 ± 40+4 −0 keV , Γ = 48 ± 2+0 −14 keV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (1) The D∗0D+ threshold is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='7 MeV above the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The closeness of the resonance to the two thresholds suggests the possibility that T + cc has a molecular nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' After the announcement of the discovery of T + cc, many theory papers attempted to under- stand various aspects of the exotic meson [6–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Several papers tried to predict its decay width and differential decay width, with considerable success [6, 7, 10, 13, 14, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' In one of these papers [6], we wrote down an effective field theory for T + cc considering it a molecular state of two D mesons treated nonrelativistically, and computed leading-order strong and electromagnetic decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Special attention was paid to the coupled channel nature of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We found a decay width of 52 keV when the tetraquark is in an isospin-0 state, using a value of δm = −273 keV, which arises from using a relativistic P-wave two-body Breit-Wigner function with a Blatt-Weisskopf form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' This was in good agreement with the LHCb experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The predicted differential spectra as a function of the invariant mass of the final state charm meson pair were also in good agreement with the binned experimen- tal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' In this paper we investigate how these conclusions are affected by next-to-leading order (NLO) strong decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The effective theory we will use is similar to XEFT for the χc1(3872) [27–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [27, 42, 43] have considered NLO XEFT diagrams for χc1(3872) decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' One-pion exchange was found to have a negligible contribution to the decay width [27, 43], while final state rescattering led to uncertainty in the decay rate of +50% −30% when the binding energy of the χc1(3872) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='2 MeV [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The differential spectrum dΓ[χc1(3872) → D0 ¯D0π0]/dEπ was found to have a curve whose peak location and overall shape are insensitive to NLO correc- tions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' only the normalization is affected [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The sharply peaked nature of the differential 2 spectrum can inform about the molecular nature of the χc1(3872): since it is a function of the virtual D∗0 propagator (p2 D + γ2)−1, where γ is the binding momentum, as the binding energy goes to zero the distribution becomes sharply peaked as pD → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' By analogy with this earlier work on χc1(3872), in this paper we compute NLO contri- butions to the decay of T + cc to find the uncertainties due to one-loop one-pion exchange and final state rescattering diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We calculate the uncertainty in the decay width, as well as in the shape, peak location, and normalization of differential spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The calculation is complicated by the presence of a coupled channel, which is not present for χc1(3872).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We find the decay width including NLO corrections to be 47+53% −25% keV, which is consistent with XEFT [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We also discuss the physical significance of several of the parameters in the effective theory, and their effect on the decay width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' In Section II we write down the effective Lagrangian to NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The required Feynman diagrams and their amplitudes, along with the explicit formulae for the partial widths are shown in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Plots of the differential distribution are shown in Section IV, followed by concluding remarks in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' EFFECTIVE LAGRANGIAN The leading-order effective Lagrangian for strong decays of T + cc is [6] LLO = H∗i† � i∂0 + ∇2 2mH∗ − δ∗ � H∗i + H† � i∂0 + ∇2 2mH − δ � H + g fπ H†∂iπH∗i + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −C(0) 0 (H∗Tτ2H)†(H∗Tτ2H) − C(1) 0 (H∗Tτ2τaH)†(H∗Tτ2τaH) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (2) Here H and H∗ are isodoublets of the pseudoscalar and vector charm meson fields, respec- tively, and π is the usual matrix of pion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The diagonal matrices δ and δ∗ contain the residual masses, which are the difference between the mass of the charm meson D(∗)i, where i = 0, +, and that of the D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The coupling g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='54 is the heavy hadron chiral perturbation theory (HHχPT) axial coupling [44–46] and fπ = 130 MeV is the pion decay constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The terms on the last two lines are contact interactions mediating D∗D scattering, where C(n) 0 mediates S-wave scattering in the isospin-n channel, and τa are Pauli matrices acting in isospin space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 3 Several new classes of terms appear at NLO in the effective theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' There are new contact interactions involving two derivatives: LC2 = C(0) 2 4 (H∗Tτ2H)†(H∗Tτ2 ←→ ∇ 2H) + C(1) 2 4 (H∗Tτ2τaH)†(H∗Tτ2τa ←→ ∇ 2H) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (3) These interactions occur in XEFT and are proportional to the effective range [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We can also write down Dπ interaction terms by constructing isospin invariants out of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' LCπ = C(1/2) π (πH)†(πH) + C(3/2) π � vaH − 1 3τaπH �†� vaH − 1 3τaπH � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (4) Here v = � π1 π2 π0 �T / √ 2 is a vector of pion fields, with π± ≡ (π1 ∓ iπ2)/ √ 2, such that vaτa = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' C(1/2) π and C(3/2) π mediate scattering in the isospin-1/2 and isospin-3/2 channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The interactions which are relevant to our calculation are: LCπ → C(1) π D0†π0†D+π− − C(1) π D+†π0†D0π+ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' +C(2) π D0†π0†D0π0 + C(2) π D+†π0†D+π0 +C(3) π D0†π+†D0π+ , (5) where the couplings C(1) π , C(2) π , and C(3) π are particular linear combinations of C(1/2) π and C(3/2) π as governed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' These interactions can be matched onto the chiral Lagrangian [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The values we use for these Cπ couplings are computed from lattice data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' see Appendix C for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We can write down D∗D → DDπ interactions by using the same strategy of constructing isospin invariants out of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' That would lead to: LB1 = B(I=0) 1 εαβ(H∗ αHβ)†(Hτ2τiH∇vi) +B(I=1) 1 (H∗τ2τkH)†(εijkHτ2τiH∇vj) + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (6) However, we need isospin-breaking terms in order to fully renormalize the theory at NLO, so ultimately we have four unique B1 couplings, one for each possible channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Written in terms of the charm meson fields, the interactions become: LB1 → B(1) 1 (D+D∗0)†(D+D0∇π0) + B(2) 1 (D0D∗+)†(D+D0∇π0) +B(3) 1 2 (D0D∗+)†(D0D0∇π+) + B(4) 1 2 (D+D∗0)†(D0D0∇π+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (7) 4 Relations between the B(i) 1 implied by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (6) are given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We can construct DD contact terms out of the isospin invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' There are only interactions in the isospin-1 channel, LC0D = C(1) 0D(Hτ2τaH)†(Hτ2τaH) → C(1) 0D 2 (D0D0)†(D0D0) + C(1) 0D(D+D0)†(D+D0) , (8) where in the second line we have restricted to terms that are relevant to our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The authors in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [43] chose to vary their C(1) 0D coupling, which described D ¯D scattering as opposed to DD, over a range of [−1, 1] fm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We test several different values for it within that range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Lastly, we need a kinetic term for the pions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' in contrast to XEFT, we treat them relativistically, Lπ = tr(∂µπ†∂µπ − m2 ππ†π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (9) The full NLO Lagrangian is then LNLO = LC2 + LCπ + LB1 + LC0D + Lπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' FORMULAE FOR DECAY WIDTHS Writing down the decay width for the T + cc at NLO requires care due to the coupled channel nature of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We define a two-point correlation function matrix ˆG as ˆG = � d4x e−iEt ⟨0|T[X(x)XT(0)]|0⟩ = iΣ(1 + CΣ)−1 , (10) where the interpolating field is X = � � � D0D∗+ D+D∗0 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (11) The right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (10) arises from expressing ˆG to all orders as an infinite sum of the C0-irreducible two-point function Σ, in a manner similar to that in Appendix A of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [48], but here C0 and Σ are matrices due to the presence of a coupled channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −iΣ is given by the sum of D∗D self-energy diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Its diagonal elements correspond to those two-point diagrams which do not swap channels, and the off-diagonal elements to those which do swap channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We can then project out the isospin-0 and isospin-1 channels, 5 −iΣ = + + C2 + + C0D + Cπ + B1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 1: Some of the D∗D self-energy diagrams contributing to −iΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Bold solid lines represent D∗ mesons, regular solid lines represent D mesons, and dashed lines represent pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The first row is LO, the second row is NLO, and the third and fourth rows are NNLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' There are also other NNLO diagrams not shown which are C0-reducible combinations of the NLO diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' and tune the parameters of the two-point correlators so that there is a pole corresponding to the location of the T + cc bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Near the vicinity of the pole, the Green’s function can be written as G0/1 = � � � 1 ∓1 � � � T ˆG � � � 1 ∓1 � � � ≈ 1 2 iZ0/1 E + ET + iΓ0/1 2 , (12) where Γ0/1 is the decay width and the residue Z0/1 is the wave function renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We find for the decay width in the isospin-0 channel ΓNLO 0 ≈ −ΓLO Re Σ′NLO 0 (−ET) Re tr Σ′LO(−ET) + 2 Im ΣNLO 0 (−ET) Re tr Σ′LO(−ET) , (13) where Σ0 ≡ Σ11 + Σ22 − Σ12 − Σ21 is a particular combination of the elements of the Σ matrix appropriate for isospin-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (13) is a correction to the LO decay width from NLO D∗D self-energy corrections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=', diagrams on the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (13) consists of NLO decay diagrams, from various cuts of diagrams 6 on the third and fourth rows of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Note that Im ΣNLO is from Σ diagrams of one higher order than in Re ΣNLO because the LO self-energy graph has no imaginary part below threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The derivatives of Σ are with respect to E and evaluated at E = −ET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' For a more detailed derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (13) refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Three diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 1 contribute to Re Σ to NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' They are the LO self-energy dia- gram (−iΣ1), the one-pion exchange diagram (−iΣ2), and the C2 contact diagram (−iΣ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' They are evaluated in the power divergence subtraction (PDS) scheme [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' This scheme corresponds to using MS to handle logarithmic divergences as well as subtracting poles in d = 3 to keep track of linear divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' A 1/ϵ pole appears in Σ2, but the dependence on the renormalization scale drops out when the derivative with respect to E is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We neglect terms in the propagators that go as p4/m2 H or (δm)p2/mH, where δm is of the order of the pion mass, compared to p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' In Σ2 and Σ3 we use a Fourier transform to evaluate the integrals over three-momentum, using a procedure outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We define a reduced mass µ(m1, m2) ≡ m1m2/(m1 + m2) and the binding momenta are defined to be γ2(m1, m2) = 2µ(m1, m2)(m1 + m2 − mT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The expressions for the self energy diagrams are: −iΣ1(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗) = −iµ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗) 2π [ΛPDS − γ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (14) −iΣ2(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g2) = −4ig1g2 3 µ(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 1)µ(m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 2) × � 1 16π2[ΛPDS − γ(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 1)][ΛPDS − γ(m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 2)] +(m∗ 2 − m1)2 − m2 π (8π)2 �1 ϵ + 2 −4 log � γ(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 1) + γ(m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 2) −i(m∗ 2 − m1)2 + im2 π � − 4 log µ �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (15) −iΣ3(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' C2) = − i 4π2C2[γ2(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 1) + γ2(m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 2)]µ(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 1) ×µ(m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 2)[ΛPDS − γ(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 1)][ΛPDS − γ(m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 2)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (16) To be consistent with the implementation of the PDS scheme in the decay diagrams (see Appendix B), for the double integral in Σ2 we have used rotational symmetry to replace 7 p m (a) g1 p pπ g2 g3 m∗ 1 mext mπ m m∗ 2 (b) pπ p Cπ mπ m (c) C2 m∗ 1 m p m∗ 2 mext (d) B1 m (e) C0D m∗ m1 (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 2: Feynman diagrams at LO and NLO contributing to the decay of T + cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We label the vertices and lines whose naming might be ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' These diagrams arise from cuts of the diagrams on the third and fourth lines of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' the tensor structure in the numerator with δij/3 and not δij/(d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' This choice does not affect the derivative of Σ2 as it only changes the constant terms which drop out upon differentiation with respect to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The decay diagrams that contribute to 2 Im ΣNLO 0 (−ET) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' By the optical theorem the square of these diagrams are given by the sum over the cuts of the NNLO diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' If there is only one pion/charm meson vertex in a diagram, its coupling is labeled gπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' If there are more than one such vertex, the couplings are numbered gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Depending on the type of pion and charm meson, these couplings will be either g/fπ or ±g/( √ 2fπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='The expressions are written in terms of the basis integrals given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' These basis integrals depend on parameters b, c1, and c2, the definitions for c1 and c2 are provided where appropriate, b = 1 unless otherwise specified, and the momentum arguments for the integrals are p unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 8 iA(2a)(p, m, m∗, gπ) = 2igπϵT · pπµ(m, m∗) p2 + γ2(m, m∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (17) iA(2b)(p, m, mext, mπ, m∗ 1, m∗ 2, g1, g2, g3) = 4iµ(m, m∗ 1)µ(mext, m∗ 2)g1g2g3 p2 + γ2(mext, m∗ 2) × � ϵT · p pπ · p � I(2) 0 − 2I(1) + I � +ϵT · pπp2I(2) 1 � , (18) c1 = γ2(m, m∗ 1) , c2 = p2 − (mT − m − mext)2 + m2 π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' iA(2c)(m, mext, mπ, m∗, gπ, Cπ) = 2iµ(m, m∗)gπCπϵT · p[I(1) − I] , (19) c1 = γ2(m, m∗) , c2 = p2 − (mT − m − mext)2 + m2 π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' iA(2d)(m, mext, m∗ 1, m∗ 2, gπ, C2) = 1 πiC2gπϵT · pπµ(m, m∗ 1)µ(mext, m∗ 2) × p2 − γ2(m, m∗ 1) p2 + γ2(mext, m∗ 2)[γ(m, m∗ 1) − ΛPDS] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (20) iA(2e)(m, m∗, B1) = −iB1 2π ϵT · pπµ(m, m∗)[γ(m, m∗) − ΛPDS] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (21) iA(2f)(m1, m2, m∗, p0 π, gπ, C0D) = 4iµ(m1, m2)µ(m2, m∗)gπC0DϵT · pπI(pπ) , (22) c1 = γ2(m2, m∗) , c2 = −2µ(m1, m2) � mT − m1 − m2 − p0 π − p2 π 2m1 � , b = µ(m1, m2) m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (13) and using the amplitudes defined above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' the decay widths for the two strong decays of T + cc are 9 dΓNLO 0 (T + cc → D+D0π0) dp2 0dp2 + = 2 Re tr Σ′LO(−ET)Re � A(2a)(p+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ) × � A(2b)(p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/ √ 2fπ) +A(2b)(p+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ) −A(2b)(p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/ √ 2fπ) −A(2b)(p+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ) +A(2c)(p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' C(2) π ) −A(2c)(p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' C(1) π ) +A(2f)(m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' C(1) 0D) −A(2f)(m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' C(1) 0D) �∗ + (D0 ↔ D+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' π+ ↔ π−) � − 1 Re tr Σ′LO(−ET) � [β1(p2 + + γ2 +) + β2] ���A(2a)(p+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ) ��2 −A(2a)(p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/ √ 2fπ)A∗ (2a)(p+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ) � +[β3(p2 0 + γ2 0) + β4] ���A(2a)(p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/ √ 2fπ) ��2 −A(2a)(p+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ)A∗ (2a)(p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/ √ 2fπ) �� −dΓLO 0 (T + cc → D+D0π0) dp2 0dp2 + Re Σ′NLO 0 Re tr Σ′LO ���� C2→0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='E=−ET (23) 10 dΓNLO 0 (T + cc → D0D0π+) dp2 1dp2 2 = 1 Re tr Σ′LO(−ET)Re � A(2a)(p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ) × � A(2b)(p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ) +A(2b)(p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ) −A(2b)(p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ) −A(2b)(p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ) +A(2c)(p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' C(3) π ) −A(2c)(p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' mπ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −g/ √ 2fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' C(1) π ) +A(2f)(m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' m∗ +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' g/fπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' C(1) 0D/2) �∗ + (p1 ↔ p2) − �2gµ0 fπ �2p2 π 3 β5 � 1 p2 1 + γ2 0 + 1 p2 2 + γ2 0 �� −dΓLO 0 (T + cc → D0D0π+) dp2 1dp2 2 � β4 + Re Σ′NLO 0 Re tr Σ′LO ���� C2→0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='E=−ET � (24) In the previous formulae we have used subscripts on µ and γ to indicate which charm meson is a pseudoscalar in that particular channel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=', µ0 = µ(m0, m∗ +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The combinations of self-energy diagrams that we need are Re tr Σ′LO(−ET) and Re Σ′NLO 0 (−ET, C2 → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' In terms of the functions defined above, these are given by: Re tr Σ′LO = Re Σ′ 1(m0, m∗ +) + Re Σ′ 1(m+, m∗ 0) , Re Σ′NLO 0 |C2→0 = Re � Σ′ 2(m+, m∗ 0, m+, m∗ 0, mπ+, g/fπ, g/fπ) +Σ′ 2(m0, m∗ +, m0, m∗ +, mπ+, g/fπ, g/fπ) +Σ′ 2(m+, m∗ 0, m0, m∗ +, mπ0, −g/ √ 2fπ, g/ √ 2fπ) +Σ′ 2(m0, m∗ +, m+, m∗ 0, mπ0, g/ √ 2fπ, −g/ √ 2fπ) � (25) The expressions for βi are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The terms dependent on A(2b) and Re Σ′ 2 have linear divergences that must cancel against each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' They cancel exactly in the limit µ0 = µ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We make that approximation in those terms only to ensure the cancellation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' it is a reasonable approximation as µ0/µ+ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='99948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' See Appendix B for more discussion of these linear divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 11 3730 3732 3734 3736 3738 0 20 40 60 80 100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 3: A plot of the differential decay width as a function of the invariant mass of the final state D meson pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Solid lines represent the LO calculation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' the dashed lines represent the addition of non-analytic and NLO self-energy corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Overlaid is the binned experimental data from LHCb, with the background subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' DIFFERENTIAL DECAY DISTRIBUTIONS AND PARTIAL WIDTHS Once we have formulae for the T + cc → DDπ partial widths, we can numerically integrate over part of three-body phase space in Mathematica and plot the differential distribution dΓ/dmDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' It is insightful to compare our predicted curves to the LHCb experimental data for the total yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' This will inform us about the effect and importance of the different interactions in the effective theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We normalize our distributions by performing a least- squares fit of the LO distribution to the data, and using the same normalization factor for the NLO distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The Cπ decay diagrams, individually and as a whole, contribute negligibly to the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The parameters β1, β3, and β5 also have a small impact on the distributions over the range in which we vary them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We therefore do not show plots varying these parameters individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The contributions from the non-C2-dependent NLO self-energy corrections (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' the first 12 3730 3732 3734 3736 3738 0 20 40 60 80 100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 4: A plot of the differential decay width as a function of the invariant mass of the final state D meson pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Solid lines represent the LO calculation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The dashed and dotted lines represent two different ranges for C0D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Overlaid is the binned experimental data from LHCb, with the background subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' diagram on the second line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 1), as well as the contributions from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 2b, serve to increase the partial widths by a small but noticeable amount (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The effect of the C0D, β2, and β4 terms on the distributions can be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' In the following we will investigate their impact by setting all other contributions to dΓNLO/dmDD to zero and varying them individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The C0D interaction has a sizeable contribution to the partial widths, as evidenced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 4, where we plot the differential distributions and vary this coupling in two possible ranges: C0D ∈ [−1, 1] fm2 and ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='25] fm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Its effect on the neutral pion decay is twice as large as on the charged pion decay, because the coupling of charged pions to D mesons is bigger by a factor of √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Clearly the differential distributions are sensitive to the coupling’s magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' If C0D is +1 fm2 the peak of theD+D0 mass distribution is too high, and if it is −1 fm2 three higher data points are underpredicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' It would be interesting to 13 3730 3732 3734 3736 3738 0 50 100 150 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 5: A plot of the differential decay width as a function of the invariant mass of the final state D meson pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Solid lines represent the LO calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The dashed and dotted lines represent two different values of β2 and β4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Overlaid is the binned experimental data from LHCb, with the background subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' do a more careful analysis of the constraints this data puts on C0D but that is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' C0D is directly proportional to the I = 1 D meson scattering length, so more precise knowledge of C0D from lattice simulations or experiments would allow us to sharpen our predictions for T + cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We can glean the significance of β2 and β4 by taking the isospin limit m0 = m+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' In Appendix C we see that in this limit: β2 = β4 = −γr0 , (26) where γ is the binding momentum and r0 is the effective range in the I = 0 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The effective range is positive and we expect γr0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 5, we plot the distribution with all other NLO interactions turned off, and for two values of β2 = β4 ≡ β: −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='1 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='59, along with the LO curve (β = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We get γr0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='59 if we use the largest binding momentum (γ+) and r0 = 1/(100 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' For nucleons, r0 ≈ 1/(100 MeV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' since charm mesons are 14 LO result NLO lower bound NLO upper bound Γ[T + cc → D0D0π+] 28 21 44 Γ[T + cc → D+D0π0] 13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='8 21 Γstrong[T + cc] 41 29 66 Γstrong[T + cc] + ΓLO EM[T + cc] 47 35 72 TABLE I: Partial and total widths in units of keV at LO and NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' considerably more compact objects one might expect the effective range for charm mesons to be smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We can see that the distribution is highly sensitive to the choice of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' A β of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='59 greatly increases the differential distribution, and is in much poorer agreement with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' This suggests that the effective range for T + cc is smaller than for nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Clearly the partial widths and their differential distributions can vary substantially de- pending on the choice of parameters in the effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' However, the availability of experimental data for the decays presents the possibility of performing fits of the dis- tributions to the data to obtain estimates for these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' This could improve the predictive power of the effective theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We save such a careful statistical analysis for a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We can use these plots that show the effect of a subset of the NLO contributions to inform which ranges for the parameters to use when estimating the total NLO contribution to the differential distribution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The upper and lower bounds in the figure reflect varying C0D from −1 fm2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='25 fm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The parameters β1, β3, and β5 are varied from −1/(100 MeV)2 to +1/(100 MeV)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The parameters β2 and β4, which reduce to −γr0 in the isospin limit, are varied between 0 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The latter value corresponds to a binding momentum for the D∗+D0 channel, γ0, and r0 = 1/(100 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' While the uncertainty in the total width of the T + cc can be significant depending on the values of the NLO couplings, the qualitative aspects of the plots of the differential decay widths in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 6 are consistent between LO and NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The overall shape and location of the peaks are unchanged by pion exchange and final state rescattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' When integrating over the full phase space to get the partial widths, we use the same 15 3730 3732 3734 3736 3738 0 20 40 60 80 100 120 140 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 6: A plot of the differential decay width as a function of the invariant mass of the final state D meson pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Solid lines represent LO calculation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' the dashed lines represent the lower and upper bounds of the NLO corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Here, we vary −1 fm2 ≤ C0D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='25 fm2 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='26 ≤ β2/4 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Overlaid is the binned experimental data from LHCb, with the background subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' ranges for the parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The partial widths are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Note that the LO numbers differ from those in our original paper [6] because here we use the binding energy from the unitarized Breit-Wigner fit, whereas in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [6] we used the value from the P-wave two-body Breit Wigner fit with a Blatt-Weisskopf form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' This has the effect of slightly increasing the prediction for the width compared to the initial paper, bringing it closer to the experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' When adding the LO electromagnetic decay width of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='1 keV (which is only slightly affected by the different binding energy) the total LO width predicted by our effective theory is 47 keV which is already in excellent agreement with the LHCb experimental value of 48 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Adding in the NLO contribution to the strong decay widths, the total width of the T + cc can range from 35 keV to 72 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' So we can establish an uncertainty in the width due to NLO strong decays of Γ[T + cc] = 47+53% −25% keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' This is comparable to the uncertainty from similar operators contributing to the decay of χc1(3872) 16 3730 3732 3734 3736 3738 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='×10-7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='×10-7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='×10-7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='×10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='×10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='2×10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='4×10-6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 7: Comparing our LO differential decay width to one where the D∗ propagators are taken to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The curves are fixed to have the same normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Note the lack of a sharp peak in the constant propagator curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' in XEFT [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We did not consider NLO corrections to the electromagnetic decay, because the LO electromagnetic decay was already a small contribution to the total width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' In particular, the differential distribution for the electromagnetic decay was negligible compared to the strong decays’ distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' To illustrate why these differential decay width plots are good tests of the molecular nature of the T + cc, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 7 we can compare the LO differential curves to those which would arise if we replaced the virtual D∗ propagators with a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The latter do not have sharp peaks and thus would be in poor agreement with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' CONCLUSIONS In this paper we have determined the effects of NLO strong decays on the total width and differential decay width of the exotic meson T + cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We considered pion exchange and final state rescattering diagrams, from similar operators to those in XEFT for the χc1(3872) 17 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We arrive at similar conclusions as Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The differential decay width plots have shapes and peaks that are relatively unchanged by the NLO effects, but the total width has significant uncertainty: Γ[T + cc] = 47+53% −25% keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The central value (the LO result) is in good agreement with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We varied the parameters in the NLO calculation to get a sense of the uncertainty in the predictions and determine which parameters in the NLO calculation give the biggest corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Nonanalytic corrections for pion loops are not important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The parameter C0D, which is proportional to the I = 1 D meson scattering length, and β2 and β4, which in the isospin limit are equal and proportional to the I = 0 D meson effective ranges, significantly affect the decay width and normalization of the differential distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' It would be inter- esting to fit the NLO differential curves to the experimental data and obtain bounds on the undetermined couplings, thereby learning more about these physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Alterna- tively, one might hope to get information about these parameters from lattice simulations or other experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Any improvement in our understanding of these parameters in D meson scattering would increase the predictive power of the effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Acknowledgments - L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' is supported by the Alexander von Humboldt Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' is supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Nuclear Physics, under award number DE-FG02-04ER41338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' are supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Nuclear Physics under grant Contract Numbers DE-FG02-05ER41367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Appendix A: Coupled channel decay width The full expression for the isospin-0 two-point correlator is −iG0 = 1 2 −Σ0 − 4C(1) 0 det Σ 1 + C(0) 0 Σ0 + C(1) 0 Σ1 + 4C(0) 0 C(1) 0 det Σ , (A1) where Σ0/1 ≡ Σ11 + Σ22 ∓ Σ12 ∓ Σ21 are the isospin-0 and isopsin-1 combinations of the elements of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Since we expect T + cc to be an isospin-0 state we treat C(1) 0 perturbatively and expand to NLO in C(1) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −iG0 ≈ 1 2 −Σ0 1 + C(0) 0 Σ0 + 1 2 C(1) 0 (ΣLO 11 − ΣLO 22 )2 (1 + C(0) 0 Σ0)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (A2) 18 We see that the real numerator of the C(1) 0 term is the residue of a double pole at 1+C(0) 0 Σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' That can be interpreted physically as a small shift in the location of the bound state, which can be seen from expanding the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (12) about ENLO T = ET −ELO T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' But since we are already tuning ET to be the location of the T + cc bound state, we can set C(1) 0 to zero to remove the double pole from the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' −iG0 → 1 2 −Σ0 1 + C(0) 0 Σ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (A3) At this stage the problem is identical to the single-channel problem in XEFT [27], with the single-channel two-point function replaced by our isospin-0 combination of coupled-channel two-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The wave function renormalization and decay width are therefore: Z0 = 1 � C(0) 0 �2Re Σ′ 0(−ET) , Γ0 = 2 Im Σ0(−ET) Re Σ′ 0(−ET) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (A4) Σ0 has LO contributions from the diagonal elements, and NLO contributions from all ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' After expanding in the NLO terms we find our corrections to the LO decay width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Γ0 ≈ ΓLO � 1 − Re Σ′NLO 0 (−ET) Re tr Σ′LO(−ET) � + 2 Im ΣNLO 0 (−ET) Re tr Σ′LO(−ET) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (A5) Appendix B: Basis integrals and the PDS scheme The most basic integral that arises when evaluating the one-loop diagrams in the PDS scheme is: �ΛPDS 2 �4−d � dd−1l (2π)d−1 1 l2 + c − iϵ = 1 4π(ΛPDS − √ c − iϵ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (B1) This result is obtained by subtracting the pole in d = 3 with a counterterm, then evaluating the result in d = 4, yielding a linear divergence in ΛPDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The scalar integral I(p) is finite in d = 3 and d = 4, so no PDS counterterm is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' I(p) = � dd−1l (2π)d−1 1 l2 + c1 − iϵ 1 l2 − 2bl · p + c2 − iϵ = 1 8π 1 � b2p2 � tan−1 � c2 − c1 2 � b2p2c1 � + tan−1 � 2b2p2 + c1 − c2 2 � b2p2(c2 − b2p2) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (B2) 19 The linear tensor integral I(1)(p) can be solved using algebraic manipulation of the nu- merator, which yields two integrals of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (B1) that have opposite sign for the divergence, and so I(1)(p) is UV finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' piI(1)(p) = � dd−1l (2π)d−1li 1 l2 + c1 − iϵ 1 l2 − 2bl · p + c2 − iϵ , → p2I(1)(p) = 1 2b � 1 4π √ c1 − iϵ − 1 4π � c2 − b2p2 − iϵ + (c2 − c1)I(p) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (B3) The quadratic tensor integrals I(2) require care when implementing the PDS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The linear divergences which arise in the decay width can only cancel if the subtraction scheme is implemented correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' After using Feynman parameters to combine the propagators and obtain an integrand like liljf(l2), the correct procedure is to replace lilj → δij/3 immediately, and not with δij/(d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The latter would cancel the factor of d − 1 that arises when evaluating the loop momentum integral, and this results in the incorrect coefficient for the PDS subtraction scale ΛPDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Additionally, algebraic manipulation of the numerator of I(2) to reduce it to integrals of the form of I(1) and I leads to yet another incorrect coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' This is the method used to obtain the expressions in the appendix of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [43];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' as such, the formulae for the decay width in that paper are only correct if ΛPDS = 0 and d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Using the correct procedure for the basis integrals gives the following results: pipjI(2) 0 (p) + δijp2I(2) 1 (p) = � dd−1l (2π)d−1lilj 1 l2 + c1 − iϵ 1 l2 − 2bl · p + c2 − iϵ , I(2) 0 (p) = b2 8π � 1 0 dx x2 � ∆(x) , (B4) → p2I(2) 1 (p) = 1 8π �2 3ΛPDS − � 1 0 dx � ∆(x) � , (B5) for ∆(x) = −b2p2x2 + (c2 −c1)x+c1 −iϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' One can be reassured that this implementation of the PDS scheme is correct because the same relative weight of the ΛPDS and � 1 0 dx � ∆(x) terms is obtained when using a hard cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' That does not occur when using lilj → δij/(d−1) or algebraic manipulation of the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Furthermore, unless the relative weight of the two terms in I(2) 1 is 2/3, the linear divergences that appear in ΓNLO 0 as A(2b) and Re Σ′ 2 do not cancel in the isospin limit, as they do in XEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' For the T + cc, they cancel when µ0 = µ+, an approximation we make in the cutoff-dependent terms to ensure cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' With algebraic manipulation of the integrand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (B4) and integration by parts in 20 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (B5), we can rewrite these expressions in terms of I and I(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' p2I(2) 0 = − 1 16π � c2 − b2p2 − iϵ + c1 2 I(p) + 3 4 c2 − c1 b I(1)(p) , (B6) p2I(2) 1 = ΛPDS 12π − 1 16π � c2 − b2p2 − iϵ − c1 2 I(p) − 1 4 c2 − c1 b I(1)(p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (B7) Appendix C: Cπ couplings and βi expressions In the isospin |I, mI⟩ basis, we use the phase convention |π+⟩ = − |1, 1⟩ , |π0⟩ = |1, 0⟩ , |D+⟩ = ���� 1 2, 1 2 � , |D0⟩ = ���� 1 2, −1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (C1) Then the Clebsch-Gordan decomposition of the Dπ pairs is |D0π0⟩ = � 2 3 ���� 3 2, −1 2 � + 1 √ 3 ���� 1 2, −1 2 � , |D+π0⟩ = � 2 3 ���� 3 2, 1 2 � + 1 √ 3 ���� 1 2, 1 2 � , |D0π+⟩ = − � 2 3 ���� 1 2, 1 2 � − 1 √ 3 ���� 3 2, 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (C2) From this we can deduce aD0π0 = aD+π0 = 2 3a3/2 Dπ + 1 3a1/2 Dπ , aD0π+ = 1 3a3/2 Dπ + 2 3a1/2 Dπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (C3) These scattering lengths are calculated on the lattice in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [51] to be a1/2 Dπ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='02 fm and a3/2 Dπ = −(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='100±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='002) fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The matching from tree level scattering tells us that, for the diagonal couplings C(2) π and C(3) π , we can use Cπ = 4π(1+mπ/mD)aDπ, with the appropriate masses and scattering lengths for each process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We can then use those two values to solve for C(1/2) π and C(3/2) π and obtain C(1) π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' We get C(1) π = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='40 fm , C(2) π = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='09 fm , C(3) π = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='2 fm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (C4) The expressions for the βi are given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' The subscripts on the γ and µ variables indicate the pseudoscalar charm meson is in that channel, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' γ+ = γ(m+, m∗ 0) is the binding momentum in the channel with the D+ meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 21 β1 = (ΛPDS − γ+) � fπ √ 2πgB(1) 1 + 1 πC(+) 2 µ+ − 1 πC(−) 2 µ0 ΛPDS − γ0 ΛPDS − γ+ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (C5) β2 = � 1 πC(+) 2 µ+(−2γ2 +)(ΛPDS − γ+) − 1 πC(−) 2 µ0(−γ2 0 − γ2 +)(ΛPDS − γ0) +2π �µ2 0 γ0 + µ2 + γ+ �−1� − 1 π2C(+) 2 µ3 +(γ+ − ΛPDS)(2γ+ − ΛPDS) − 1 π2C(+) 2 µ3 0(γ0 − ΛPDS)(2γ0 − ΛPDS) −C(−) 2 (γ2 + + γ2 0)µ+µ0 2π �µ+ γ0 (ΛPDS − γ0) + µ0 γ+ (ΛPDS − γ+) � +C(−) 2 µ+µ0(µ+ + µ0) π2 (ΛPDS − γ+)(ΛPDS − γ0) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (C6) β3 = (ΛPDS − γ0) � − fπ √ 2πgB(2) 1 + 1 πC(+) 2 µ0 − 1 πC(−) 2 µ+ ΛPDS − γ+ ΛPDS − γ0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (C7) β4 = � 1 πC(+) 2 µ0(−2γ2 0)(ΛPDS − γ0) − 1 πC(−) 2 µ+(−γ2 0 − γ2 +)(ΛPDS − γ+) +2π �µ2 0 γ0 + µ2 + γ+ �−1� − 1 π2C(+) 2 µ3 +(γ+ − ΛPDS)(2γ+ − ΛPDS) − 1 π2C(+) 2 µ3 0(γ0 − ΛPDS)(2γ0 − ΛPDS) −C(−) 2 (γ2 + + γ2 0)µ+µ0 2π �µ+ γ0 (ΛPDS − γ0) + µ0 γ+ (ΛPDS − γ+) � +C(−) 2 µ+µ0(µ+ + µ0) π2 (ΛPDS − γ+)(ΛPDS − γ0) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (C8) β5 = 1 πC(+) 2 µ0(ΛPDS − γ0) − 1 πC(−) 2 µ+(ΛPDS − γ+) +B(3) 1 fπ 4πg (γ0 − ΛPDS) − B(4) 1 fπ 4πg (γ+ − ΛPDS)µ+ µ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (C9) It is instructive to take the isospin limit of these β expressions and compare to XEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Referring to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (6), we can write down the B1 couplings in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' B(1) 1 = −B(2) 1 = − √ 2B(I=0) 1 , B(3) 1 = 2(B(I=1) 1 + B(I=0) 1 ) , B(4) 1 = 2(B(I=1) 1 − B(I=0) 1 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (C10) 22 Then taking µ+ = µ0 = µ, γ+ = γ0 = γ we find: β1 = β3 = β5 = 1 π(γ − ΛPDS) �B(I=0) 1 fπ g − 2C(0) 2 µ � , β2 = β4 = −4C(0) 2 µγ π (γ − ΛPDS)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (C11) The isospin-1 couplings drop out, which is to be expected given that we have projected out the isospin-0 state and are here dropping isospin-breaking interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' These expressions also match the dependence of the decay rate on C2 and B1 in XEFT [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (24) of [27] (and adjusting for a factor of 4 in the definition of C2 in that paper) we see that β2 = β4 = −γr0 in the isospin limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' It is an important check on our calculation that in the isospin limit the theory can be properly renormalized with isospin respecting counterterms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' When isospin breaking in the masses and binding momentum is included, isospin breaking in the B1 operators needs to be included as we have done in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Muheim (2021), the European Physical Society Conference on High Energy Physics, URL https://indico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='de/event/28202/contributions/102717/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [2] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Polyakov (2021), the European Physical Society Conference on High Energy Physics, URL https://indico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='de/event/28202/contributions/105627/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' An (2021), 19th International Conference on Hadron Spectroscopy and Struc- ture, URL https://indico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='nucleares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='mx/event/1541/session/4/contribution/ 35/material/slides/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (LHCb) (2021), 2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='01038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' (LHCb) (2021), 2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='01056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Fleming, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Hodges, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Mehen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' D 104, 116010 (2021), 2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='02188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Meng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Wang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Zhu (2021), 2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='14784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Agaev, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Azizi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Sundu (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='00188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Pan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Luo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Liu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Geng (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='00923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [10] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Ling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Geng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Wang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Xie (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='00947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Chen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Liu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Zhu (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='01911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [12] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Dong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Guo, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Zou (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='02673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Feijoo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Liang, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Oset (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='02730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 23 [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Yan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Valderrama (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='04785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Dai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Kang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Szczepaniak, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Yu (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='06002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [16] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Weng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Deng, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Zhu (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='07242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [17] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Zhu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Geng, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Wang (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='13028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Chen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Liu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Zhu (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='12730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [19] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Xin and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Wang (2021), 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='12597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Albaladejo (2021), 2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='02944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Du, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Baru, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Dong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Filin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Hanhart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Nefediev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Nieves, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Wang (2021), 2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='13765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [22] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Jin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Qin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Si, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Yu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' D 104, 114009 (2021), 2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='05678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Abreu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Navarra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Nielsen, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Vieira (2021), 2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='11145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' [24] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Hanhart, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Meissner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' D 87, 014508 (2013), 1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content='4535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQf8i6_/content/2301.11950v1.pdf'} diff --git a/4NAyT4oBgHgl3EQfo_jf/content/tmp_files/2301.00519v1.pdf.txt b/4NAyT4oBgHgl3EQfo_jf/content/tmp_files/2301.00519v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1835add0e05bdf491649c35ce1c31d82433cd953 --- /dev/null +++ b/4NAyT4oBgHgl3EQfo_jf/content/tmp_files/2301.00519v1.pdf.txt @@ -0,0 +1,3979 @@ +1 +Holistic Network Virtualization and Pervasive +Network Intelligence for 6G +Xuemin (Sherman) Shen, Fellow, IEEE, Jie Gao, Senior Member, IEEE, Wen Wu, Member, IEEE, +Mushu Li, Member, IEEE, Conghao Zhou, Student Member, IEEE, and Weihua Zhuang, Fellow, IEEE +(Invited Paper) +Abstract—In this tutorial paper, we look into the evolution +and prospect of network architecture and propose a novel +conceptual architecture for the 6th generation (6G) networks. +The proposed architecture has two key elements, i.e., holistic +network virtualization and pervasive artificial intelligence (AI). +The holistic network virtualization consists of network slicing and +digital twin, from the aspects of service provision and service +demand, respectively, to incorporate service-centric and user- +centric networking. The pervasive network intelligence integrates +AI into future networks from the perspectives of networking +for AI and AI for networking, respectively. Building on holistic +network virtualization and pervasive network intelligence, the +proposed architecture can facilitate three types of interplay, i.e., +the interplay between digital twin and network slicing paradigms, +between model-driven and data-driven methods for network +management, and between virtualization and AI, to maximize +the flexibility, scalability, adaptivity, and intelligence for 6G +networks. We also identify challenges and open issues related +to the proposed architecture. By providing our vision, we aim +to inspire further discussions and developments on the potential +architecture of 6G. +Index Terms—6G, network architecture, network virtualiza- +tion, digital twin, AI for networking, networking for AI. +I. INTRODUCTION +A. Background +With the ongoing worldwide deployment of the 5th genera- +tion (5G) networks, the technical community in wireless com- +munications and networking has started looking into the 6th +generation (6G) networks for 2030 and beyond. While the ex- +act concepts and techniques that define 6G are not determined +yet, visions, requirements, use cases, and candidate techniques +are discussed in an increasing amount of works, e.g., [1]– +[3]. Among these discussions, some preliminary consensus +regarding 6G emerges. For instance, in terms of main require- +ments of 6G, the urgency of improving security [4] and energy +efficiency [5] is understood unanimously. For use cases of +6G, the combination of enhanced mobile broadband (eMBB), +This work was supported by research grants from the Natural Sciences and +Engineering Research Council (NSERC) of Canada. +Xuemin (Sherman) Shen, Mushu Li, Conghao Zhou, and Weihua Zhuang +are with the Department of Electrical and Computer Engineering, University +of Waterloo, Waterloo, ON, Canada, N2L 3G1 (email: {sshen, m475li, +c89zhou, wzhuang}@uwaterloo.ca). +Jie Gao is with the Department of Electrical and Computer En- +gineering, Marquette University, Milwaukee, WI, USA 53233 (email: +j.gao@marquette.edu). +Wen Wu is with the Frontier Research Center, Peng Cheng Laboratory, +Shenzhen, Guangdong, China, 518055 (email: wuw02@pcl.ac.cn). +The corresponding author of this paper is Weihua Zhuang. +ultra-reliable and low-latency communications (uRLLC), and +massive machine-type communications (mMTC) have been +brought up, despite the different terminologies used in dif- +ferent works [6], [7]. As to candidate techniques, commonly +mentioned examples include the integration of satellite, aerial, +terrestrial, and underwater networks [8], [9], (sub)terahertz and +visible light communications [10], artificial intelligence (AI) +empowered networks [11]–[13], to name a few. +One consensus deserving special attention is that 6G may +need a brand-new network architecture. Driven by cost ef- +fectiveness and efficiency, the evolution of network architec- +ture follows the evolving services provided by the networks. +For instance, to introduce data service, a packet-switched +core network component emerged in the 3G architecture as +a complement to its circuit-switched counterpart for voice +service [14]. Then, to accommodate the exponential growth of +data traffic, 4G introduced a redesigned and simplified network +architecture for a flat all-Internet protocol (IP) network with in- +creased data rate and reduced latency [15]. In the era of 5G, as +networks become more heterogeneous than ever while services +become diversified, various network architecture innovations +have been proposed towards flexible service-oriented network- +ing, including software defined networking (SDN) [16], cloud +radio access network (C-RAN) [17], and network slicing [18], +[19]. Therefore, envisioned to support unprecedentedly diverse +services with exceedingly stringent quality of service (QoS) or +quality of experience (QoE) requirements, 6G will most likely +need ground-breaking innovations in network architecture. +While conceiving an architecture for 6G, it is difficult to +overlook two key elements, i.e., virtualization and AI. Network +virtualization already plays an important role in the architec- +ture of 5G [20]. The virtualization of resources, functions, +and networks enables resource sharing, software implemen- +tation of network functions, and service-orientated network- +ing, respectively, and thereby increases resource utilization +while reducing the cost for deploying and operating networks. +Virtualization reflects a trend of softwarization for flexible, +scalable, and adaptive network management [21]. Therefore, +it is foreseeable that virtualization will remain crucial in the +architecture of 6G. As for the second key element, i.e., AI, a +growing number of research teams worldwide are investigating +AI-driven networks, and high expectation is placed on AI +for empowering 6G [1], [22]. In comparison with heuristic +or mathematical model based approaches for communications +and networking, AI based approaches can handle complicated +networking problems and obtain accurate results, provided +arXiv:2301.00519v1 [cs.NI] 2 Jan 2023 + +2 +that sufficient data are available for training. This advantage +suits the increasingly heterogeneous and dynamic networks, +where mathematical models may not exist or cannot accurately +characterize the considered problems. Therefore, it is not +difficult to predict the significance of AI in 6G. +B. Architectural Innovations Required for 6G +Recognizing the importance of virtualization and AI, we +further look into their limitations in the state-of-the-art to +comprehend the architectural innovations required for 6G. +Existing virtualization techniques mostly deal with service +provision in communication networks. For instance, network +slicing highlights available network resources, service provi- +sion capability, and QoS satisfaction for various services [23]. +While such virtualization, with a focus on service provision, +enables 5G to handle diverse coexisting services, it may not +suffice for 6G since the characteristics of end user service +demand can be the key to achieving user-centric networking. +Therefore, in the future, virtualization should focus on both +the service provision capability of a network and the service +demand of end users in the network. This will lead to the +virtualization of end users in addition to the virtualization of +networks. As for AI, existing research on AI mostly addresses +specific functions (e.g., routing [24]), layers (e.g., physical +layer [25]), network segments (e.g., access networks [26]), +or applications (e.g., autonomous driving [27]) of a network. +Meanwhile, how to integrate AI into the network architecture +across different layers or network segments needs further +investigation. The scope and extent of AI-driven networks are +yet to be determined. +As virtualization extends to cover both service provision and +service demand while AI pervades every corner of the network, +close connections between the two elements are foreseeable +and can dominate the architectural needs of 6G. The first +connection is through network and end user data [28]. Vir- +tualization facilitates the characterization of network service +provision capability, service performance, resource utilization +and, in future networks, end user service demand. As a result, a +vast amount of data will be generated, which can be exploited +to characterize the network and end users. Such data, if +properly managed, can empower both AI-driven networking +and AI applications (e.g., object detection) [29]. The second +connection is through network control. AI can be used to +make decisions pertinent to virtualization, including service +admission, slice establishment, dynamic virtual network func- +tion orchestration, and resource scheduling. In the future, AI +can also help control data collection for the virtualization of +end users and extract features of virtualized end users. Thus, +AI has the potential to improve the efficacy and adaptivity +of virtualization. The third connection is through network +resources. A main motivation of network virtualization is +to coordinate resource sharing among different services and +thereby improve network resource utilization and service sat- +isfaction. AI-driven networking can target efficient utilization +of network resources. As both virtualization and AI consume +computing, communication, and storage resources, they may +compete for network resources. However, AI has a potential +to increase the efficiency of virtualization through intelligent +network planning and operations, while virtualization may +increase the efficiency of AI through proper data provision +and management. As a result, they should work together to +enhance network resource utilization and service quality. +A rudiment of the above connections through data and +control can be observed in the existing architecture of 5G. +For instance, the 3rd Generation Partnership Project (3GPP) +introduces a network data analytics function (NWDAF) for +5G in Release 15 [30] and enablers for network automation +(eNA) in Release 16 [31]. The architecture design provides +a framework for the NWDAF to collect data from other +network functions (such as policy control and network slice +selection functions) and provide analytics (such as data traffic +statistics and predictions) back to these network functions. In +6G, the scope and level of both data collection and analytics +will expand significantly. Most likely, network data analysis, +instead of being limited to one or two specific functions, will +be AI-driven and available everywhere in a network. Similarly, +the data available for network management, instead of being +limited in type, content, or format, should provide information +of the network and end users as needed. Such expectations can +be fulfilled by extending the roles of virtualization and AI in +the network architecture. +C. Our Vision +Our vision of network architecture for 6G is based on +the importance of virtualization and AI, their limitations in +existing networks, and the essential connections between them. +Specifically, we aim to design a network architecture that i) +supports virtualization of the network and end users from +the perspectives of service provision and service demand, +respectively, ii) integrates AI in various network functions, +layers, segments, and applications under a unified architecture, +and, more importantly, iii) facilitates the interplay between +virtualization and AI, enabling their coexistence, integration, +and mutual enhancement. To consolidate the vision, we raise +the following three key questions: +• How to further advance virtualization beyond network +slicing? +• How to enable AI into every facet of a network? +• How to effectively integrate virtualization and AI through +network architecture design? +In pursuit of answering the preceding questions, we develop +the ideas of holistic network virtualization and pervasive +network intelligence for 6G network architecture. Holistic +network virtualization advances virtualization toward 6G by +incorporating network slicing and digital twin paradigms. +The former enables service-centric network management, and +the latter adds a user-centric perspective to virtualization +for future networks. Pervasive network intelligence enables +generic integration of AI into a network from the perspectives +of AI for networking and networking for AI. The former +emphasizes the role of AI in network management, while the +latter leverages network design to support AI applications. +In this tutorial paper, for both holistic network virtualization +and pervasive network intelligence, we survey existing studies, + +3 +present our network architecture designs, and illustrate their +benefits. Unifying these two components, we further introduce +an overall conceptual network architecture, which fulfills our +vision of unprecedentedly flexible, scalable, adaptive, and +intelligent networks for 6G. +This tutorial paper can provide useful information and +benefit readers from three aspects. First, for readers who +are interested in the historical and current developments of +virtualization and AI techniques, we survey the literature and +provide a review of both in the context of communication +networks. Second, for readers who are exploring future direc- +tions in virtualization and AI, we propose original ideas for +advancing them toward 6G. Specifically, we illustrate designs +and ideas, such as incorporating digital twins for holistic +network virtualization, connected AI for network management, +AI slices with training and inference separation, and hybrid +data-model driven methods, throughout this paper. Last, after +introducing our vision of holistic network virtualization and +pervasive network intelligence, we present open issues and +challenges to inspire further research. +There are a few surveys on virtualization and AI in the liter- +ature [21], [32]–[34]. Regarding virtualization, Minerva et al. +present existing digital twin based applications in the context +of IoT [32], and another survey introduces the key enabling +technologies and design principles of network slicing [21]. +Regarding AI, Boutaba et al. undertake a comprehensive +survey on AI applications in various areas of networking [33], +and another survey focuses on deep learning (DL) based +applications in wireless networking [34]. In comparison, this +tutorial paper focuses on the vision of 6G. Specifically, after +introducing state-of-the-art virtualization and AI techniques, +we propose original designs, including holistic network vir- +tualization and pervasive network intelligence, to establish a +novel conceptual architecture for 6G networks. +D. Structure of the Paper +The structure of this tutorial paper is shown in Fig. 1. +Section II illustrates our vision of 6G networks from the +aspect of holistic network virtualization. We review existing +network virtualization concepts and techniques in Subsec- +tion II-A. Then, we introduce end user virtualization with a +focus on digital twins in Subsection II-B. Lastly, we present +our idea of holistic network virtualization, highlighting a six- +layer virtualization architecture, in Subsection II-C. +Section III illustrates our vision of 6G networks from the +aspect of pervasive network intelligence. Subsection III-A +presents an overview of representative AI techniques that are +potentially useful for 6G networks. Subsection III-B introduces +the motivation for pervasive network intelligence and presents +a four-level AI architecture. Subsections III-C and III-D sum- +marize the existing research and present our ideas on AI for +networking and networking for AI, respectively. +Section IV integrates holistic network virtualization and +pervasive network intelligence and presents our overall vision +for 6G. Subsection IV-A reviews related studies on archi- +tectures for 6G. Subsection IV-B introduces a conceptual +architecture for 6G networks that incorporates holistic net- +TABLE I: List of Acronyms +3GPP +3rd Generation Partnership Project +5G +5th Generation +6G +6th Generation +AI +Artificial Intelligence +AL +AI Level +AP +Access Point +API +Application Programming Interface +ARQ +Automatic Repeat-Request +BS +Base Station +C-RAN +Cloud Radio Access Network +DL +Deep Learning +DNN +Deep Neural Network +DRL +Deep Reinforcement Learning +eMBB +Enhanced Mobile Broadband +FL +Federated Learning +IoT +Internet of Things +IP +Internet Protocol +ITU +International Telecommunication Union +LSTM +Long Short-Term Memory +LTE +Long Term Evolution +mMTC +Massive Machine-Type Communications +MIMO +Multiple-Input Multiple-Output +ML +Machine Learning +NFV +Network Function Virtualization +NN +Neural Network +NWDAF +Network Data Analytics Function +QoE +Quality of Experience +QoS +Quality of Service +RAN +Radio Access Network +SBS +Small Base Station +SDN +Software Defined Networking +SNR +Signal-to-Noise Ratio +UAV +Unmanned Aerial Vehicle +uRLLC +Ultra-Reliable and Low-Latency Communications +VL +Virtualization Layer +VM +Virtual Machine +WSN +Wireless Sensor Network +work virtualization and pervasive network intelligence. Sub- +sections IV-C and IV-D discuss the components, subsystems, +and potential implementation of the proposed architecture. +Subsections IV-E to IV-G elaborate on three types of inter- +play enabled by the proposed architecture, i.e., the interplay +between digital twin and network slicing, between data-driven +and model-driven methods, and between virtualization and AI, +respectively. +Section V identifies key challenges and open issues related +to the proposed network architecture, and Section VI con- +cludes this research. +Table I lists the acronyms used in this paper. +II. HOLISTIC NETWORK VIRTUALIZATION +In this section, we first review virtualization techniques in +existing networks and their benefits. Then, we introduce the +idea of holistic network virtualization. +A. Network Virtualization +The concept and techniques of network virtualization have +been evolving over more than three decades [35]. Early +research on network virtualization includes virtual local area +networks motivated by facilitating different types of operations +(services) in distributed systems [36], as well as providing +flexible network control and improving link utilization [37]. +Another example of network virtualization is virtual private + +4 +Section I. Introduction +Section II. Holistic Network +Virtualization +Section III. Pervasive Network +Intelligence +Section IV. A Potential Network +Architecture for 6G +Section V. Challenges and Open Issues +Section VI. Conclusions +Network Virtualization +End User Virtualization +Holistic Network Virtualization +Motivation and AI Architecture +Networking for AI +AI for Networking +Architecture Overview +Components and Subsystems +Interplay between Model-Driven and +Data-Driven Methods +Interplay between Virtualization and +AI +AI Techniques: An Overview +Implementation +Interplay between Digital Twin +Paradigm and Network Slicing +Fig. 1: The structure of this paper. +networks, which establish efficient and secure communication +links to connect geographically dispersed end users. Over time, +the desire for programmable network management extends to +the objective of enhancing network architecture. +The advancement in cloud computing has propelled re- +cent development in network virtualization, including network +function virtualization (NFV) and network slicing. With NFV, +software instances running on virtual machines at general +computing servers replace customized and proprietary hard- +ware for various network functions [38]. At the network +core, NFV applies to functions such as switching, firewall, +deep packet inspection, and session border controller [39]. +At radio access networks, NFV applies to frame generation, +modulation, carrier allocation, etc. [40]. The realization of +NFV becomes an enabler for network slicing, which is a +key network architecture innovation in 5G. Network slicing +emphasizes a service-oriented perspective in network man- +agement by creating multiple end-to-end virtual networks, i.e., +slices, for different services on top of shared physical network +infrastructure. With network slicing, network resources are +first reserved for respective services in network planning stages +and later allocated to individual users in network operation +stages [11]. The creation, adjustment, and termination of slices +are based on the varying spatiotemporal distribution of service +demands to provide a high level of flexibility and adaptivity +in network management [23].1 +Virtualization can be applied on different levels and scales +in a network. Existing techniques include virtualization at +node, link, resource, and network levels. Virtual nodes are +abstractions of substrate nodes in a network such as servers, +routers, and switches, and typical examples of node virtualiza- +tion are storage and computing server virtualization [41], [42]. +Virtual links are the logical channels that interconnect virtual +nodes. Virtual resources are abstractions of computing, mem- +ory, storage, and communication resources in a network [43], +while physical resources at different locations can form virtual +1SDN and C-RAN are also closely related to network virtualization since +virtualization significantly simplifies and expedites their realization in modern +wireless networks. +resource pools [38]. For instance, the virtualization of a +network function is the execution of a network control or +service function by running software, supported with necessary +resources. A virtual network is the combination of virtual +nodes and links with proper virtual resource allocation for +a service request to meet its QoS requirements, supported by +necessary networking protocols. Besides the aforementioned +works, more representative research works on node, link, re- +source, and network virtualization are summarized in Table II. +Regardless of its level and scale, virtualization in the context +of networking typically demonstrates the following character- +istics: +• Abstraction - Abstraction provides a high-level overview +of a network while hiding details of the underlying +physical network entities (nodes, links, or networks) or +resources [63]. This simplifies network management and +facilitates flexible service provision; +• Co-existence - Multiple virtual entities corresponding +to a shared physical entity co-exist, or multiple virtual +resource pools co-exist on the same physical resource +pool [35]. This enables service-oriented virtual networks +and improves network resource utilization efficiency; +• Isolation - Coexisting virtual entities corresponding to the +same physical entity should function independently [64]. +This is necessary for guaranteeing service reliability, +security, scalability, and QoS satisfaction. +Both academia and industry have spent a significant amount +of efforts on network virtualization. For virtualizing core +networks, some works leverage SDN techniques to separate +the control and data planes through different protocols or +application programming interface (API), e.g., OpenFlow [65]. +Furthermore, network virtualization has been extended to radio +access networks (RANs), and several frameworks for RAN +virtualization are proposed. A SoftRAN framework enables +both centralized and distributed RAN control based on the time +sensitiveness of control decisions [66]. Another framework, +FlexRAN, offers a hierarchical architecture for real-time RAN +control and incorporates a flexible API to separate control +and data planes in RANs [67]. Initiated by industry, such as + +5 +TABLE II: Some Representative Works on Node, Link, Resource, and Network Virtualization +Type +Work +Scenario +Research Focus +Objective +Node +Virtualization +[44] +Edge computing +Virtual edge node placement +Low-cost placement and fast response to user requests +[45] +Cloud computing +Virtual machine (VM) place- +ment +Reliable VM placement and routing +[46] +IP network +Virtual node/router as IP over- +lay +Practical IP-level resilience to link failures +[47] +Wireless sensor network +(WSN) +Architecture for sensor virtu- +alization in WSN +Multiple applications share the same WSN +[48] +C-RAN +Clustering of access points +Forming user-specific virtual base stations given QoS requirements +Link +Virtualization +[49] +WSN +Virtual backbone construction +Enabling low-complexity backbone construction with performance +guarantee +[50] +Generic +Virtual link embedding +Reducing congestion probability given bandwidth demands +[51] +Internet service provider +(ISP) network with SDN +Virtual link provision +Maximizing network throughput subject to QoS and robustness +constraints +Resource +Virtualization +[52] +Cloud computing +Composite +virtual +resource +mapping +Efficient mapping of computing and networking resources to substrate +resources within networked clouds +[53] +Cloud computing +VM migration +Low-cost transferring of VM storage and memory during VM migra- +tion over wide area networks +[54] +Radio +access +network +(RAN) +Radio resource virtualization +Maximizing throughput with fairness among multiple mobile network +operators +[55] +RAN +Radio resource virtualization +Delay-bounded QoS provisioning through radio resource virtualiza- +tion +[56] +Vehicular network +Resource +sharing +among +slices +Reusing communication and caching resources to support applica- +tions with different QoS requirements +Network +Virtualization +[57] +5G core network with +SDN +Network function chain em- +bedding +Minimizing embedding cost subject to network resource constraints +[58] +Core network with SDN +Network function chain em- +bedding +Minimizing total flow in the network subject to network resource +constraints +[59] +C-RAN +Slice request admission +Maximizing the revenue of the C-RAN operator subject to network +resource constraints +[60] +Heterogeneous +wireless +network +Dynamic radio resource slic- +ing +Maximizing network utility through optimal bandwidth slicing and +user association +[61] +5G RAN +Radio resource allocation in +RAN slicing +Satisfying QoS requirements by proper resource mapping and +scheduling +[62] +IoT +Service-oriented +authentication +Privacy-preserving slice selection and secure access of service data +AT&T and China Mobile, Open-RAN (O-RAN) is proposed as +an open-source and open-interface platform to support RAN +virtualization [68], which can incorporate AI and provide APIs +for data-driven networking [69]–[71]. +The adoption of virtualization techniques renders modern +networks programmable, flexible, and scalable, which signifi- +cantly increases cost effectiveness in network deployment and +operation. Due to these benefits, it is foreseeable that advanced +virtualization techniques will be essential to 6G. Meanwhile, +the existing scope of network virtualization is limited in the +sense that virtualization techniques mostly focus on network +infrastructure and resources, yet less attention is given to end +users. In 6G, end user virtualization will become necessary for +two reasons. First, with increasingly diverse end user devices, +resource-demanding services, and heterogeneous and dynamic +networks, providing QoE guarantee for end users will become +more challenging in the era of 6G. Accurate characterization +and abstraction of end users, which necessitate end user virtu- +alization, can be a precondition to QoE satisfaction. Second, as +AI will be a highlight of 6G, extensive user data are required +to fuel AI services and AI-based network management. Given +such need for data, end user virtualization can be a competitive +approach for collecting, managing, and processing data from +end users. +B. End User Virtualization +Until recently, only a few works study end user virtu- +alization in the context of networking. One early example +relevant to end user virtualization is network-hosted avatars, +i.e., virtual agents, of end users for applications such as file +downloading when the users are offline [72]. Another example +is virtual objects, proposed as a component in Internet of +things (IoT) platforms [73]. The motivation is to handle the +heterogeneity of physical objects (end users) via virtualization +and to facilitate the provision of services to end users. +As a potential paradigm to enable end user virtualization, +digital twin attracts much attention lately. The concept of +digital twin was originally conceived by Michael Grieves for +product life-cycle management in industry in 2003 [74], [75]. +Later, NASA and U.S. Air Force Vehicles developed a digital +twin paradigm for vehicles to forecast their remaining usable +life and the mission success probability [76]. A digital twin +is characterized by a full digital representation of a physical +object or a process and real-time synchronization between +the physical object or process and its corresponding digital +replica. Digital twins can contain a large volume of data from +physical objects or processes for advanced analytics, and the +analytical results can be used to improve the performance of +the corresponding physical objects or processes. Exemplary +digital twins in general application scenarios, as well as +potential requirements for the digital twins to enable big data +analytics, are discussed in [77]. Potential implementation of +digital twins representing IoT devices in industrial systems +is proposed in [78]. Other representative research works on +digital twins are summarized in Table III. +Most existing research on digital twins in the network field +focuses on applications, e.g., distributed clock synchroniza- +tion [79] and computation offloading [80]. In comparison, +the study of digital twins from the perspective of network + +6 +architecture and network management is limited at the mo- +ment.2 A digital twin based cloud-centric network architecture +is proposed in [83], where digital twins of end users hosted +at the network edge play the role of communication assistants +or network data loggers. +Digital twin appears to be an intuitive solution to end user +virtualization. Nevertheless, extending the existing network +virtualization, represented by network slicing, to end users is +not straightforward, given the target of flexible and efficient +network management and service provision. For instance, it is +trivial to simply use node-level virtualization and to represent +end users as virtual data sources or sinks in a virtual network. +Moreover, while end users may possess communication and +computing resources, resource-level virtualization does not +characterize user-specific properties, e.g., location and mobil- +ity, or service-specific properties, e.g., data traffic variations, +of end users. It is necessary to understand potential benefits, +requirements, and implementation of digital twin based end +user virtualization, with a particular focus on the integration +of digital twin and existing network virtualization frameworks. +There are potentially two-fold benefits of digital twin based +end user virtualization, i.e., extensive end user data and +powerful network emulation capability. While the virtual- +ization of network infrastructure and resources characterizes +the network status and service provision capabilities, digital +twins of end users can provide extensive data regarding +service demand and user QoS/QoE satisfaction. Such data +can play a significant role in network management through +facilitating well-informed network planning and operation +decisions. Moreover, the real-time or near real-time synchro- +nization between end users and their digital twins enables +powerful network emulations. For instance, multiple instances +of the same virtual network can be created, with real-time +end user information, e.g., location and data traffic volume, +provided to all instances through synchronized end user digital +twins.3 Different network planning or operation strategies can +be applied and emulated in different instances, while each +instance remains synchronized with the real-world network +environment through the information provided by the digital +twins of end users. +To take part in network virtualization, digital twins of end +users should satisfy the following requirements: +• Flexible: The abstraction of end users into digital twins +must be sufficiently flexible to represent heterogeneous +physical devices (such as smartphones, vehicles, and +industrial sensors) and serve various applications (such as +virtual reality gaming, autonomous driving, and industrial +automation); +• Compatible: The end user virtualization based on digital +twins should complement and enhance the state-of-the-art +network virtualization, i.e., network slicing. For instance, +digital twins of end users should provide data to support +2Some works focus on distributed networks, e.g., vehicular networks, and +adopt digital twins as an approach for network virtualization instead of end +user virtualization [81], [82]. +3The emulation can apply to a virtual network segment, e.g., the network +edge. +various network slices, while each slice may only have +access to a subset of data pertinent to that slice; +• Customizable: The attributes of digital twins should be +customized and updated based on the corresponding +service, network traffic, resource utilization, etc. For +instance, the amount and types of data included in a +digital twin should be adaptable rather than fixed. In +addition, while the focus of digital twins is placed on +end users, digital twins should be able to represent other +network entities, e.g., unmanned aerial vehicle (UAV) +mounted mobile base stations (BSs). +In addition, network resource consumption from creating and +maintaining digital twins should be taken into account. +Noting the aforementioned benefits and requirements, we +aim to answer the following key questions with respect to the +implementation of digital twins: +• Location: Where should digital twins be hosted in a +network? +• Affiliation: Should digital twins exist within or outside +network slices? +• Data: What data attributes pertinent to networking should +be included in a digital twin? How much historical +data should be included for a specific attribute? Should +predicted user information be included? +• Synchronization: How to determine the frequencies of up- +dating various data entries of a digital twin by acquiring +new data from the physical object? +• Control: Who should determine and update digital twin +models and based on what information? +In the next subsection, we propose a novel conceptual architec- +ture for holistic network virtualization, which integrates digital +twins and network slicing, and delve into the above questions. +C. Holistic Network Virtualization +We propose a novel virtualization architecture, i.e., holistic +network virtualization, for integrating digital twins into net- +work virtualization, in order to improve network management +and service provision capabilities. The proposed virtualization +architecture consists of six layers and is illustrated in Fig. 2, in +which virtualization layer (VL) 1 is the bottom layer for data +collection and VL 6 is the top layer for digital twin model +control. The outline of each layer is given as follows: +VL 1 – Data Collection: Data required for the digital +twin representation of selected end users are collected from +the corresponding physical entities following prescribed data +precision, uploading method, collection frequency, etc. The +data are collected via access points, and the data collection is +controlled by local controllers deployed at network edge; +VL 2 – Level-One Abstraction: Based on the current digital +twin model from the digital twin model control layer (i.e., VL +6), which determines the content and format of data included +in every digital twin, digital twins are formed and updated +using data collected by VL 1. The abstraction may include +the aggregation of data from different sources, the update of +historical data, and the creation of digital twins for new or +additional end users. The digital twins created in this layer + +7 +TABLE III: Some Related Works on Digital Twins +Work +Application +Type of Physical Ob- +ject +Role of Digital Twin +Target of Using Digital Twin +[84] +Underwater +network +for ocean observation +Underwater +sensors/actuators +Monitoring and testing observation sys- +tem +Visualizing an ocean observation system and enhancing simulations +[85] +Edge computing for in- +ternet of vehicles +Vehicles +Collecting +and +sharing +information +about vehicles and surroundings +Empowering computing offloading by facilitating data analytics +[86] +5G network slicing +Network slices +Predicting and monitoring slice perfor- +mance +Assisting autonomous network slicing +[87] +Smart factory +Workstations in a con- +veyor system +Evaluating and validating control strate- +gies +Implementing intelligent conveyor systems +[88] +Smart city +Road infrastructure +Monitoring roads and detecting vehi- +cles/persons +Supporting smart city applications through gathering and processing +data +[89] +IoT +Objects with sensing +capability +Storing data for detecting events and +recognizing behaviors +Facilitating synthetic sensing through situation awareness and ex- +plainability +[90] +Industry 4.0 +Industrial machinery +Generating training dataset and simula- +tions +Achieving accurate anomaly detection with limited labelled data +[91] +Smart healthcare +Patients +Handling data for analysis and develop- +ing AI models +Improving healthcare operations +[92] +Industry 4.0 +Technical assets (e.g., +machine, environment) +Integrating knowledge from model and +data for simulations +Enhancing simulation-based systems engineering +[93] +Mobile edge comput- +ing +Real-world +network +environments +Training learning algorithms and moni- +toring network environments +Enabling learning for optimizing user association, resource alloca- +tion, and offloading +[94] +Industrial 4.0 +Products, workstations, +conveyor belts +Data sharing and control of security- +critical processes +Building a security architecture based on state replication and +synchronization +[95] +Cyber-physical +systems +Generic physical de- +vices +Monitoring, diagnostics, and prognos- +tics +Supporting applications such as context aware interaction and +driving assistance +[96] +IoT +Generic physical sys- +tems +Managing context information and self- +adapting +Increasing autonomy and enhancing cooperation through autonomic +digital twins +[97] +Welding +manufactur- +ing +Human-robot +interac- +tion systems +Monitoring welding robot and enabling +simulations +Visualizing welder behavior and training welders +[98] +Smart manufacturing +Job +shop +scheduling +systems +Obtaining scheduling data and simulat- +ing scheduling strategies +Enabling timely response and reducing scheduling plan deviation +[99] +Smart manufacturing +Manufacturing systems +Predicting and verifying the system per- +formance +Increasing autonomy and enhancing fault diagnosis +[100] +Smart building +Photovoltaic +energy +conversion units +Estimating the status of photovoltaic en- +ergy conversion unit +Improving the accuracy of fault detection +[101] +Internet of vehicles +Vehicles and road side +units +Monitoring the real-time status of vehi- +cles and road side units +Supporting network resource management +[102] +Mobile edge caching +Vehicles +Capturing the social characteristics of +vehicles +Improving the effectiveness of cache management +are level-one digital twins, representing individual end users, +and hosted at servers connected to local controllers; +VL 3 – Local Processing and Control: The data from +level-one digital twins are processed at network edge for +predicting behaviors of individual users, such as user data +traffic and mobility patterns, and making user-level service +decisions, such as computing offloading, content delivery, +and link-layer protocol adaption. Local processing may also +include emulations of an edge network or a part of it based +on level-one digital twins. Local control may include further +data aggregation from level-one digital twins for VL 4, the +migration of digital twins based on user mobility, and the +selection of end users for digital twin representation. Similar +to the case of VL 2, the local processing and control occur at +servers affiliated with local controllers; +VL 4 – Level-Two Abstraction: The aggregated data from +VL 3 is sorted into service-specific data for respective net- +work slices in VL 4. Additional data that describe slice +configuration, slice resource utilization, slice service level +agreement satisfaction, etc., are generated for each slice. Then, +the aforementioned data are abstracted to form or update the +digital twins of various slices. The digital twins created in +this layer are level-two digital twins, which are associated +with virtual networks (slices). The level-two digital twins are +hosted at servers connected to the centralized controller of the +network; +VL 5 – Slice-Level Processing and Control: The data +from level-two digital twins of network slices are processed +for service-specific prediction, e.g., spatiotemporal service +demand distribution forecast, or slice-level decision making, +e.g., planning and operation decisions. Slice-level processing +may include emulations of an end-to-end slice or a part of +it based on level-two digital twins. Slice-level control may +include slice admission, resource reservation, and slice service +coverage control. Similar to the case of VL 4, the service- +level processing and control occur at servers affiliated with +the centralized controller of the network; +VL 6 – Digital Twin Model Control: This layer determines +and updates the models of level-one and level-two digital twins +based on available network resources for digital twins, the +performance of network management and service provision +decisions derived based on the current digital twins, and +the dynamic spatiotemporal service demands. For instance, +VL 6 determines data precision, synchronization frequencies +for different data attributes, the amount of historical data +contained in the digital twins for each attribute, and the +inclusion of predicted user information. In addition, this layer +decides the subset of data in level-one digital twins that each +slice can access. The digital twin model control also occurs at +servers affiliated with the centralized controller of the network; +The level-one digital twin model configured by VL 6 may +include the following data, which shall be collected by the +local controllers from end users at VL 1: (1) connectivity +and channel information, such as the AP(s) that an end user + +8 +Digital Twins +VL 6 +Digital Twin Model Control +VL 5 +Slice-Level Processing and +Control +VL 4 +Level-Two Abstraction +VL 3 + Local Processing and Control +VL 2 +Level-One Abstraction +VL 1 +Data Collection +Aggregation +Digital Twin Model Update +Level-Two Digital Twin +Slices +Level-One Digital Twin +Physical Network +Core +Network +Network Slicing +Data Flow (Physical) +Control (Physical) +Data Flow (Cyber) +Control (Cyber) +Gateway +Local Controller +Centralized Controller +Fig. 2: The conceptual six-layer virtualization architecture for holistic network virtualization. +is connected to and the channel state information for each +connection; (2) service information, such as active service +types, data traffic volume of each service, and QoS satisfaction +of each service; (3) user information, such as user profile, +user location and mobility, network resources allocated to the +user, and the local computing and caching capabilities of the +user; and (4) additional use case-specific information, such +as motion sensor readings for augmented reality interactive +gaming or operation log for industrial IoT devices. The level- +two digital twin model configured by VL 6 may include +the following data, which shall be collected or generated +by the centralized controller: (1) slice service demand, such +as the number of service requests and the spatiotemporal +service request distribution; (2) slice resource configuration, +such as the reserved communication, computing, and caching +resources for the slice; (3) slice performance, such as the slice +service level agreement satisfaction, slice resource utilization, +and slice energy consumption; (4) slicing strategy, such as the +method or algorithm used for network function deployment, +resource reservation, and resource scheduling; (5) additional +use case-specific information, such as UAV trajectory config- +uration for UAV-assisted networks. Note that different end +user digital twin models are applicable to different types +of end users, and each network slice may have a uniquely +defined slice digital twin model. For example, the digital twins +of vehicles and industrial IoT devices most likely contain +different data, and the digital twin models may differ between +slices for industrial IoT and those for smart home or between +slices of different network operators. Accordingly, the need +for customization necessitates the digital twin model control +in VL 6. +In the conceptual virtualization architecture, VL 1 to VL 3 +interface with the local controllers in the network, VL 4 and +VL 5 interface with the centralized controller of the network, +and VL 6 interfaces with both the local controllers and the +centralized controller. This architecture fully exploits the two +benefits of digital twins, i.e., providing extensive data for net- +work management and enabling powerful network emulations. +It also satisfies the aforementioned requirements for digital +twins in terms of flexibility, compatibility, and customization. +Last but not least, it answers the key questions regarding the +implementation of digital twins raised in Subsection II-B. +With the architecture design in Fig. 2, digital twins and +network slicing are integrated in the idea of holistic network +virtualization. Network slicing incorporates existing network +virtualization techniques such as NFV. Digital twins enhance +network slicing by providing organized and customized end +user data to slices and by further abstracting slices into level- +two digital twins. The design of two-level digital twins avoids +extra resource consumption from creating and maintaining +multiple digital twins of the same user for different slices and +the resulting burden of synchronizing them. Instead, each slice +has access to a subset of data from level-one digital twins +pertinent to either the corresponding service or general user +information such as location and mobility, and the pertinent +data are further aggregated to the level-two digital twins for +that slice. In this architecture, network slicing conforms to +service-centric network management, while digital twins add +a user-centric perspective to the virtualization. Specifically, +level-one digital twins characterize end users and their service +demands, and level-two digital twins characterize network ser- +vice provision capability, network performance, and network +resource utilization. Overall, the digital twin paradigm and +network slicing jointly support network management and ser- +vice provision, while the network configures digital twins and +network slices as needed, depending on network dynamics. + +9 +D. Holistic Network Virtualization: A Summary +In this section, we have reviewed the existing scope and +techniques of network virtualization, identified the insuffi- +ciency of current network virtualization, introduced the idea +of holistic network virtualization to incorporate network and +end user virtualization, and developed a six-layer virtualization +architecture for holistic network virtualization. +The virtualization of resources, network functions, and +networks in 5G is expected to remain important in 6G, since +they contribute to flexible and adaptive network management. +Meanwhile, the virtualization techniques in 5G, represented +by network slicing and NFV, mostly focus on network vir- +tualization from the perspective of service provision. In 6G, +it will be essential to extend the scope of virtualization and +incorporate end user virtualization. +The digital twin paradigm is a promising solution to end- +user virtualization. In 6G, digital twins can be used for +characterizing the status and the service demand of individual +end users. The study of digital twins in the context of 6G +networks is still in an initial stage, and various definitions or +implementations exist. In our vision of holistic network vir- +tualization, digital twins are configurable assemblage of data, +including both historical and real-time data and both collected +and generated data, for describing end users, infrastructure, or +network slices. Moreover, the corresponding data collection +and processing are also configurable. +To consolidate holistic network virtualization, we have +proposed a six-layer virtualization architecture for 6G. The +architecture provides a reference design for systematically +integrating digital twins and network slicing and answers +important questions related to digital twins in 6G networks, +including where are they hosted, what data do they contain, +and how to manage them. +III. PERVASIVE NETWORK INTELLIGENCE +Pervasive network intelligence is the second element of our +vision for 6G. In this section, we first present an overview +of existing AI techniques. Then, we introduce the motivation +and propose a four-level architecture for pervasive network +intelligence. Next, we elaborate the idea of pervasive network +intelligence from the perspectives of AI for networking and +networking for AI, and review related works. Rather than +surveying specific AI techniques, this section focuses on the +architecture and methods of pervasive network intelligence. +A. AI Techniques: An Overview +The idea of AI is to design intelligent machines or sys- +tems to demonstrate human intelligence and perform tasks +as humans do or even better [131]. The advancement of +machine learning (ML) has facilitated the success of AI in +both academia and industry. Applications supported by ML +techniques, such as computer vision and natural language +processing, can achieve beyond human-level accuracy. Lately, +for its potential in enabling intelligent networks, AI has +received significant attention in the research field of wireless +networks. +ML techniques can be categorized into three types: un- +supervised learning, supervised learning, and reinforcement +learning. In terms of learning structures, the techniques can +be subdivided into centralized and decentralized techniques. +We list common ML techniques used in wireless networks in +Table IV. +Unsupervised learning evaluates features and patterns hid- +den in data for data analysis, such as prediction, without using +a labeled dataset. One popular application of unsupervised +learning techniques is data clustering, e.g., k-means [103] and +mixture models [105], for solving network planning problems, +such as cluster-forming in wireless sensor networks [104] +and small-cell deployment [132]. Neural networks can be +adopted to facilitate novel unsupervised learning algorithms. +For example, neural network-based autoencoders can learn the +compressed features of input data with a limited number of +neurons and can be leveraged for data prediction, such as +traffic forecasting [107]. +Supervised learning exploits the mapping between the in- +put and output data via a given labeled dataset. Supervised +learning techniques can derive a mapping function, i.e., a +training model, from the input data to the labeled output +data in the dataset. Through applying a training model, the +output corresponding to a new input can be evaluated, which +can be utilized for decision making or prediction. A typical +method for supervised learning is using deep neural networks +(DNNs). DNNs use layers of artificial neurons to estimate +a non-linear correlation between the input and the output +data and iteratively improve the estimation accuracy. There +have been many successful applications of DNN techniques in +communications. For example, convolutional neural networks +(CNNs) utilize convolutional and pooling layers to identify +the correlation of multi-dimensional input data and have been +applied in modulation classification [112]; recurrent neural +networks (RNNs) explore the correlation among a sequence +of the data and have been widely adopted for traffic prediction +[113] and wireless channel modeling [133]. +Reinforcement learning iteratively learns the optimal policy +by interacting with the environment, sensing network states, +and evaluating feedback. The goal is to maximize a cumula- +tive reward in a dynamic environment. Deep reinforcement +learning (DRL), which combines DNN and reinforcement +learning techniques, is used extensively in resource manage- +ment to solve complex decision-making problems. In DRL, +neural networks play the role of approximators to store high- +dimensional states or actions, which enables DRL to solve +complex problems efficiently. DRL has been widely used for +network optimization [134], resource allocation [19], [118], +[135], and user association [116], [121], [123] in wireless +networks. +With the development of mobile edge computing, dis- +tributed AI has been developed to harvest computing resources +at network edge and reduce communication overhead due to +data collection and exchange [136]. The learning models can +be trained and evaluated at network edge in a semi- or fully- +distributed manner. Specifically, federated learning (FL), as +one of the most popular distributed learning techniques, trains +models with data distributed over network edge. A centralized + +10 +TABLE IV: Common ML algorithms. +Unsupervised Learning +Supervised Learning +Reinforcement Learning +Centralized +Learning +Algorithms +• K-means [103], [104] +• Mixture models [105], [106] +• Autoencoders [107] +• Generative adversarial +network [108], [109] +• Support-vector machine [110] +• Logistic regression [111] +• Deep neural network +[107], [112], [113] +• Deep Q-learning [114]–[116] +• Policy gradient [117]–[119] +• Actor-critic [120], [121] +• Deep deterministic policy +gradient (DDPG) [19], [122], [123] +Distributed +Learning +Algorithms +• Federated learning [124], [125] +• Split learning [126] +• Multi-agent reinforcement learning +[127]–[130] +controller aggregates locally-computed learning models and +updates parameters in the learning models. Due to such de- +centralized model training, FL is capable of preserving privacy +and can be applied in privacy-sensitive network management +scenarios [137], [138]. In addition, multi-agent reinforcement +learning has been developed to implement reinforcement learn- +ing in a distributed manner, which aims to handle scenarios +in which network agents cannot obtain sufficient information +from each other. Multi-agent reinforcement learning tech- +niques can be used, for example, to solve resource allocation +problems in heterogeneous networks [129], [130]. +B. Motivation and AI Architecture +In 6G, AI is expected to penetrate every facet of the +network including end users, the network edge, and the cloud, +resulting in pervasive network intelligence. Such trend is due +to advancements and innovations in the areas of ML, data +collection, edge and cloud computing, and programmable net- +work control in recent decades. As such, AI will fundamentally +transform modern networks in many aspects and foster a +myriad of exciting applications. +The AI applications can be categorized into management- +oriented and service-oriented applications, which are detailed +as follows: +• Management-Oriented AI Applications - In these ap- +plications, AI is used as a tool for network manage- +ment, such as transmission power allocation in cellu- +lar networks [139] and resource reservation in network +slices [19]. AI techniques, such as reinforcement learn- +ing, have the potential of handling complicated decision +making problems in a dynamic network environment. +Resorting to AI techniques, the management-oriented AI +applications can analyze a large amount of network data, +make real-time network management decisions, and then +update network management policies based on the newly +analyzed data. Hence, for such applications, the key issue +is how to leverage advanced AI techniques to manage and +enhance complex networks, which falls into the scope of +AI for networking; +• Service-Oriented AI Applications - In these applications, +AI is offered as services for end users. Fuelled by +powerful computing servers and well-curated datasets, +AI techniques, especially DL, can outperform traditional +techniques in a wide range of applications, such as +environmental perception in autonomous driving, audio +recognition in intelligent healthcare, and object detection +in mobile virtual reality [140]–[142]. For instance, an +AL 2: +Edge-Hosted AI +AL 1: +End User-Hosted AI +Management-Oriented +Management-Oriented +Service-Oriented +Service-Oriented +AL 3: +Edge-Hosted AI +AL 4: +Cloud-Hosted AI +AI for + Networking +Networking +for AI +Fig. 3: An illustration of the four-level AI architecture for +pervasive network intelligence. +AI-based YOLO algorithm can detect objects with a +high accuracy [143], [144], and the state-of-the-art DL- +based face recognition algorithm can achieve an accuracy +of 99% or higher [145]. Facilitating service-oriented AI +applications in a network consumes a large amount of +network resources, including storage and computing re- +sources for model training/inference, and communication +resources for data collection and model uploading. Hence, +for such applications, the key issue is how to design and +optimize the network to support emerging AI services, +which falls into the scope of networking for AI. +Note that the scope of AI in 6G includes AI for networking +and networking for AI, which is larger than that in 5G, as the +latter simply focuses on applying AI in communications. +An AI architecture is needed to characterize AI’s different +functionalities in different network segments. In the literature, +there are a few studies on the AI architecture. Edge intelligence +(or edge AI) is represented in six levels based on the amount +and path length of data offloading [131]. Moreover, edge +intelligence can be categorized into two parts: AI for edge +(i.e., to enhance and optimize the network edge with AI +techniques) and AI on edge (i.e., to carry out AI models +on the network edge) [146], [147]. Different from these +works on edge intelligence, our work focuses on a broader +scope of pervasive network intelligence and categorizes it into +multiple levels based on AI’s locations and functionalities in +the network. +As shown in Fig. 3, we propose a four-level AI architecture, +in which AI levels (ALs) 1 and 2 focus on service-oriented +applications, and ALs 3 and 4 aim at management-oriented +applications. We describe each level in detail as follows. +AL 1 - End User-Hosted Service-Oriented AI: Utilizing +local data and computing resources at end users, end user- +hosted service-oriented AI applications are offered as services + +11 +for end users by processing AI tasks locally, such as next +word prediction in mobile keyboards [148], user traffic de- +mand prediction [149], and vehicle trajectory prediction [150]. +When computing resources of end users are insufficient for +computation-intensive AI tasks, partial computation workloads +can be offloaded to nearby edge servers for collaborative +processing. +AL 2 - Edge-Hosted Service-Oriented AI: Residing at +network edge (e.g., Wi-Fi access points and BSs) close to +end users, edge-hosted service-oriented AI applications are +offered as low-latency services for end users, such as face +recognition in video surveillance [151] and object detection in +virtual reality [152]. To support edge-hosted service-oriented +AI applications, service demand data from end users are +collected, stored, and analyzed, and then the analytical results +are utilized for service provision. +AL 3 - Edge-Hosted Management-Oriented AI: At this +level, AI is hosted at local controllers at network edge for +network management that is executed in real time, such as +spectrum allocation, content caching [153], and computation +offloading [154]. Specifically, the edge-hosted management- +oriented AI is to allocate network resources to network nodes +for supporting services, including AI services at ALs 1 and 2. +For instance, the edge-hosted management-oriented AI can +be used to perform service migration across edge networks +to guarantee service continuity for high-mobility users, e.g., +vehicular users. +AL 4 - Cloud-Hosted Management-Oriented AI: Cloud- +hosted management-oriented AI resides at the centralized con- +troller in the cloud for network management that is executed +once every several minutes or hours, such as slice admission +control [155] and virtual network function deployment [156]. +Since the cloud possesses abundant computing and storage +resources, powerful and complex AI models can be trained +and deployed for managing large-scale networks. +Next, AI for networking is elaborated in Subsection III-C +to illustrate AI’s role in network management, and networking +for AI is discussed in Subsection III-D to illustrate AI service +provision in 6G networks. +C. AI for Networking +In this subsection, we discuss how AI techniques can +support network management. We first review existing works +on AI-based network slicing. Then, we introduce our idea of +connected AI solution for AI-based network slicing. +1) AI-Based Network Slicing: Network slicing includes two +stages: network planning stage for resource reservation and +network operation stage for resource scheduling [11]. In the +network planning stage, network resources are reserved for +network slices on a large time scale (e.g., from several minutes +to several hours). In the network operation stage, the reserved +resources of each slice are allocated to end users on a small +time scale (e.g., from several milliseconds to several seconds). +Due to network dynamics such as spatiotemporally changing +network traffic, it can be difficult for model-based solutions to +attain the optimal network slicing strategies. By contrast, AI +techniques can characterize network dynamics by analyzing +the collected network data and obtain the optimal network +slicing strategies accordingly. Next, we review AI-based net- +work slicing, taking into account the interplay between the +planning and operation stages. Representative research works +on AI-based network slicing are summarized in Table V. +On a small time scale, a local controller collects data and +provides resource scheduling strategies to allocate resources +reserved for each slice to end users. Specifically, the local +controller determines resource scheduling strategies based on +two factors: the amount of resources reserved for each slice, +which is determined by the centralized controller, and the in- +stantaneous user data from level-one digital twins pertinent to +that slice, such as service type, user location, and user mobility. +The main challenges of determining the optimal resource +scheduling strategies are two-fold: a large number of end users +and service demand dynamics. AI techniques have potentials +to cope with both challenges. First, to schedule resources for +a large number of end users, unsupervised learning methods, +such as k-means [167] and DNN based autoencoders [107], +can be utilized to classify end users according to their service +demands. Similar resource scheduling strategies can be applied +to end users with similar service demands, which facilitates +scalable network management. For instance, end users in close +proximity and with similar mobility patterns may experience +similar channel statistical behaviors, and the same power +control policy can be applicable to them. Second, to deal with +network dynamics, reinforcement learning can be applied for +generating adaptive resource scheduling strategies [168]. Rein- +forcement learning iteratively allocates resources to maximize +a long-term reward function and updates the reward function +based on feedback from the network environment. Moreover, +reinforcement learning can be combined with DNNs, such +as recurrent neural networks [27] and conventional neural +networks [123], to analyze the spatiotemporal pattern of user +data for finding the optimal resource scheduling strategies. +On a large time scale, local controllers aggregate the col- +lected user-level data to service-level information from level- +two digital twins, i.e., slice digital twins. Utilizing information +from slice digital twins, the centralized controller reserves +network resources for each slice. The challenges of resource +reservation are two-fold. First, making proactive resource +reservation that can avoid either resource over-provisioning or +under-provisioning is challenging with time-varying network +traffic. Second, the strategies for resource reservation and +scheduling are coupled, which further complicates resource +reservation. AI techniques can cope with these challenges as +follows. To address the challenge of proactive resource reser- +vation, supervised learning, such as long short-term memory +(LSTM) networks, can be used to exploit the features of +historic network traffic loads and predict traffic loads in near +future [149], [159]. The centralized controller can then use +the predicted traffic loads for proactive resource reservation. +To handle the correlation between resource reservation and +scheduling, reinforcement learning can be adopted to reserve +resources while considering network operation strategies as a +part of the dynamic network environment [19], [135], [164]. +Moreover, an option-based hierarchical reinforcement learning +technique can be a potential solution for jointly optimizing + +12 +TABLE V: Representative Works on AI-based Network Slicing +Stage +Work +Research Focus +Objective +AI Method +Network +Planning +[107] +Network capacity prediction +Forecasting the capacity for individual virtual networks +Deep neural network based autoen- +coder +[86] +Virtual representation for net- +work slices +Capturing the relationships among slices and monitoring the +end-to-end performance in dynamic network environments +Graph neural networks +[157] +Resource reservation adjust- +ment +Maximizing the overall reward obtained from the tenants of +slices +Deep dueling neural networks +[158] +Bandwidth allocation +Jointly maximizing spectrum efficiency and the QoS require- +ment satisfaction ratio +Generative adversarial network and +deep Q network +[159] +Bandwidth allocation +Jointly maximizing spectrum efficiency and overall service +level agreement satisfaction ratio of slices +Long short-term memory and advan- +tage actor-critic +[160] +Traffic +prediction +and +re- +source provisioning +Minimizing the probability of slice service level agreements +violation +Gated recurrent unit +Network +Operation +[161] +Computation offloading +Minimizing average computing time of services and maxi- +mizing user computing experience +Deep Q network +[162] +Slice selection and channel al- +location +Minimizing the power consumption of wireless transmission +for a sliced fog-RAN +Reinforcement learning +[163] +Content +caching +placement +and delivery +Managing caching resources to maximize cache hit ratio +while satisifying resource reservation constraints +Deep Q network +[164] +Inter-slice coordination +Maximizing long-term payoff from the competition among +service providers through resource orchestration +Deep Q network +[165] +Inter-slice coordination +Maximizing QoS satisfaction ratio for slices by scheduling +transmission power and sharing resources among slices +Multi-agent deep Q learning +Two-Stage +Interplay +[19] +Computing resource alloca- +tion in vehicular networks +Allocating spectrum and computing resources for slices +while minimizing computing service delay +Deep deterministic policy gradient +[166] +Cross-slice +admission +and +congestion control +Maximizing operator revenue by resource reservation and +adjust reserved resources in real time +State-action-reward-state-action +(SARSA) +resource reservation and network operation policies and ad- +dressing network dynamics in both stages. This technique +has been used to tackle complex DRL problems by grouping +decision variables according to decision time scales [169] +or decision-making agents [170] and then determining the +decision variables. Through this novel reinforcement learning +technique, the complex correlation between resource reser- +vation and scheduling strategies can be obtained iteratively. +To apply this technique in network slicing, the centralized +controller can select the resource reservation strategies on a +large time scale, and local controllers find optimal resource +scheduling strategies on a small time scale, thereby jointly +optimizing both the resource reservation and the scheduling +strategies. +2) Connected AI Solution for Network Management: Ex- +isting AI applications on network management mostly focus +on individual control functions. For instance, learning-based +autoencoders can achieve reliable transmission power control +for high-speed data transmission with limited channel state in- +formation [171], and DNNs can select medium access control +protocol parameters with low communication and processing +overhead [172], [173]. Although various AI techniques have +been proposed for network management, AI models among +network control functions are usually isolated. Such isolation +may result in inefficient and redundant data processing, which +brings up a pressing need for integrating the AI models in +AI-based network control functions. +There are three types of solutions for integrating AI mod- +els [6]. In the first type of solutions, the entire network is +viewed as a black box, where a single AI model characterizes +the entire network and generates network control policies. +Such structure simplifies decision-making processes in net- +work management. However, training the single AI model +can be extremely difficult due to high-dimensional input data. +Then, the second type of solutions adopts different AI models +in a network for different network control functions, and the AI +models are generally independent on each other to reduce the +complexity of training. However, this approach neglects the +correlation and interplay among network functions and thus +cannot obtain a global-optimal network management strategy. +Moreover, network data may be repetitively processed by +different AI models with similar network functions, which +degrades network management efficiency. For instance, AI +models for user mobility management and computing service +migration would repetitively analyze end user mobility. In +contrast, the third type of solutions, namely connected AI, +exploits the correlations among network control functions, +connects their AI models, and allows them to jointly make +network control decisions. The connected AI solution offers +benefits in integrating AI models by highlighting the interplay +among them and balancing training complexity and network +performance. Therefore, the connected AI solution has great +potential in facilitating AI-based network slicing. However, +existing research on the connected AI solution is limited. +How to apply connected AI solution to network management +requires further studies [26]. +Recent advancements in distributed learning techniques fa- +cilitate the development of a connected AI solution for network +management. Model partition, investigated in [174], can divide +a global DNN into multiple sub-neural networks (sub-NNs). +The sub-NNs can reside at different network entities, accord- +ing to the available computing and communication resources, +and communicate with each other [175], [176]. Furthermore, +the idea of nested DNN, which allows sub-NNs to have their +own functionalities while contributing to the global DNN for +model inference and training, has been proposed and evaluated +in [177] and [178]. Using the above two techniques, each sub- +NN can perform a specific network control function. Accord- +ingly, multiple sub-NNs can collaboratively fulfill common +control functions, thereby applying the connected AI solution +to network management. +Based on the above advanced DNN techniques, we present + +13 +BS Power Control +DNN-Based +Channel +Estimator +Power Allocation +Scheduler +Intelligent Module +... +... +... +... +... +... +... +... +BS Power Control +... +... +... +... +Computing Offloading +SBS 1 +... +... +... +... +BS Power Control +... +... +... +... +Computing Offloading +SBS 2 +... +... +... +... +Handover +... +... +... +... +Service Migration +MBS +Computing +Offloading +Computing +Offloading +Service Migration +SBS 1 +SBS 2 +MBS +Information +Exchange +... +... +... +... +Intelligent +Module +Fig. 4: The connected AI solution for network management. +our idea of applying the connected AI solution to network +management next. The control functions for network manage- +ment are encapsulated into intelligent modules. An intelligent +module can be implemented solely by a DNN or cooperatively +by a DNN and conventional model-based techniques. An +example is shown in the upper right corner of Fig. 4, in which +the intelligent module for power control includes a learning- +based channel estimator and a model-based power allocation +scheme, e.g., water-filling power allocation [179]. Moreover, +the DNN in each intelligent module can play the role of a +sub-NN of a global DNN. The intelligent modules connect +with each other to share information, such as their outputs +and gradient information in model training, and aggregated +user data. Via the intelligent modules, control functions can +manage the network in a divide-and-conquer manner to avoid +the complicated model training required for layer-free AI. With +model partition and nested DNN techniques, multiple network +control functions can cooperatively make control decisions to +achieve globally optimal network management. +Fig. 4 shows another example of the connected AI design, +i.e., supporting mobile edge computing. We explain the design +using the case of vehicular networks as an example. Note +that other networks can use the same or a similar design. +Small base stations (SBSs), as edge servers, can process +computation tasks offloaded by vehicles. Intelligence modules +at SBSs provide computing offloading decisions, including the +computing tasks to be offloaded, transmit power for offloading, +task scheduling, etc., based on network status and computing +offloading requests. Due to the high mobility of vehicles and +the limited communication coverage of the SBSs, computing +tasks are often migrated among the SBSs, referred to as service +migration, and migration decisions are determined by a macro +base station (MBS). Service migration and computing offload- +ing decisions are highly coupled. For example, the chance +of service migration increases when vehicles offload more +computing tasks to an SBS. In addition, service migration +requires the collaboration of multiple SBSs. In our idea of +connected AI, service migration and computing offloading +decisions are jointly determined. Specifically, we split the +DNN into multiple sub-NNs by DNN splitting and nested +DNN techniques. Some sub-NNs are deployed at the SBSs +to provide computing offloading decisions. These sub-NNs are +also connected with a sub-NN deployed at an MBS, which can +be leveraged to make migration decisions. In this example, the +input of the service migration module includes the output of +intelligent modules at the SBSs, e.g., computing offloading +decisions and the parameters of sub-NNs, and the output of +the service migration module is the service migration policy. +In this way, the intelligent modules can cooperate to make +decisions for mobile edge computing. +D. Networking for AI +In addition to managing networks, AI can function as +services, namely AI services, which reside at ALs 1 and 2 in +the proposed AI architecture in Fig. 3. Networking for AI is to +design and optimize networks to facilitate AI services. In this +subsection, we first introduce the motivation of networking for +AI. Next, existing works are reviewed, and research challenges +are presented. Finally, the idea of AI slice is proposed and +elaborated. +1) Motivation: Networking for AI is attracting great atten- +tion in both academia and industry. In academia, networking +for AI calls for extensive interdisciplinary research efforts +between networking researchers and AI researchers to de- +velop new communication standards and technologies to cater +for AI services at scale [141], [180]–[182]. In industry, the +International Telecommunication Union (ITU) is discussing +high-level architectures to integrate, orchestrate, and update +AI components for future networks, including IMT-2020 net- +works [183], [184]. Some 3GPP working groups are studying +data collection frameworks in the network for supporting AI +services [185], [186]. Notably, networking for AI is becoming +an indispensable component for facilitating AI services in +networks and is expected to be a key enabling technology +in 6G. +Networking for AI should take the following factors into +consideration: +• Distributed Data - With the wide deployment of various +IoT devices and small BSs, massive data are generated +from many distributed network nodes, e.g., end users +and the network edge. In the traditional cloud-based AI +paradigm, the cloud collects massive distributed data for +model training, and a well-trained model is deployed +at the cloud for model inference. This paradigm suffers +from spectrum resource scarcity and user privacy leakage +concerns.4 To address these issues, a potential solution is +to facilitate AI services over a large number of network +nodes in a distributed manner [148], which requires +new networking protocols to coordinate multiple network +nodes; +• Constrained Resources - Network nodes, such as end +users, have limited resources, while state-of-the-art AI +models (e.g., DNN models with dozens of neural net- +work layers) are complex. As such, running a complex +4Google’s autonomous driving vehicle can generate more than 750 MB of +data per second [187]. + +14 +AI model on a single network node can exhaust its +computing resource and energy.5 With advanced model +partition techniques (e.g., DNN partition), a complex AI +model can be partitioned into multiple sub-models and +embedded into a network with data exchange among +the sub-models [189]. Executing sub-models consumes +computing resources of network nodes, and exchanging +data between sub-models also consumes communication +resources. Hence, running AI models at multiple network +nodes in a cost-effective manner requires innovative net- +work embedding designs; +• Network Heterogeneity and Dynamics - 6G networks +will be highly heterogeneous, in which network nodes +possess different amounts of communication, computing, +and storage resources. As complex AI models need to +be deployed at multiple network nodes, executing AI +tasks requires judiciously allocating resources of these +network nodes. Moreover, network dynamics, such as +time-varying channel conditions among network nodes +and spatiotemporal service demands, further complicate +the resource allocation decision making problem. Hence, +it is necessary to design tailored resource management +algorithms to optimize AI performance, while adapting +to network dynamics. +The scope of networking for AI covers the entire lifecycle +of AI services, which consists of three stages. The first stage +is data collection for model training via communication links. +For instance, real-time service load data from end users need +to be collected to train an AI model for service demand +prediction. The second stage is model training, which is to +achieve a certain objective based on the collected data. For +instance, a large number of images are processed to train +DNN-based object detection modules until the target accuracy +requirement is satisfied. The third stage is model inference, +which is to apply well-trained models to complete specific +computation tasks. For instance, AI-based object recognition +for autonomous driving detects and classifies nearby vehicles, +pedestrians, and obstacles based on real-time images captured +by on-board cameras [143]. +2) State-of-the-Art Approaches: The research on network- +ing for AI is still at its infancy stage with only a few existing +works. In this subsection, the existing studies are categorized +into data collection, model training, and model inference +according to the lifecycle of AI services. Representative related +works are summarized in Table VI. +Data Collection - The objective is to efficiently collect the +data from end users for optimizing AI performance. Since data +are distributed across end users in the network, transmission +resource is scheduled to end users for uploading their data. +For instance, the level-one digital twins require periodical data +synchronization with the end users, and such data can be +provided for AI services. Data collection is a classic research +problem widely investigated in wireless sensor networks [203] +and UAV networks [204], and these works focus on optimizing +either the reliability of data collection or the amount of +5The energy consumption of using AlexNet to process an image on a +tailored energy-efficient Eyeriss chip is up to 0.28 W [188]. +collected data. In AI services, the collected data are used +to train AI models, and the data samples may have different +importance levels for model training. Merely maximizing the +reliability or the amount of the collected data is not optimal. +Hence, novel data collection designs taking model training into +account are required for performance optimization. +Recently, AI-centric data collection is investigated in the +following two research directions: +• Resource Allocation - Data importance-aware resource +allocation schemes have been proposed to optimize AI +model accuracy. The idea is to schedule data transmission +while taking both end users’ channel conditions and data +importance levels into account [190]. The data impor- +tance level can be captured via data uncertainty, i.e., +higher uncertainty means higher importance. The data +uncertainty can be measured by entropy [205]. Power +allocation for data collection is investigated in multi- +model training scenarios [191]. Since the number of +collected data samples impacts the model accuracy, a +learning-centric power allocation scheme can adjust the +users’ transmission power to determine the amount of col- +lected data for different AI models, thereby maximizing +the overall model accuracy given a transmission power +budget; +• Protocols - There are a few AI-centric data collection +protocols. In a network environment with poor channel +conditions, data retransmission is applied to improve data +collection reliability. Existing automatic repeat-request +(ARQ) retransmission protocols, such as hybrid ARQ +in long term evolution (LTE) networks, trigger data +retransmissions for lost packets once the end user’s +signal-to-noise ratio (SNR) threshold is satisfied. The +importance of data samples should be incorporated in +transmission protocols to speed up the model training +process. An importance-aware ARQ protocol is proposed +for CNN-based classification model training in [192]. In +the protocol, both data importance levels and channel +conditions are taken into account to determine the data +retransmission threshold, which can enhance the model +training performance. +Model Training - Due to the distributed data and user +privacy concerns, distributed training is suitable for training +AI models in a network [206]. FL is one of the most promising +distributed training paradigms, which can be applied in various +fields such as smart healthcare and financial services [138], +[207], [208]. The FL operates as follows. Each end user +iteratively trains a local model with its own data, and the +local model is uploaded to an edge server. Then, the edge +server aggregates the local models to obtain a global model. +The model training lasts multiple rounds until the global model +achieves satisfactory accuracy. +Since the model is trained locally, FL is communication- +efficient and can preserve data privacy of end users [148], +[209]. However, with the increase of data sizes in state-of- + +15 +TABLE VI: Summary of Related Works on Networking for AI +Topic +Work +Contribution +Highlight +Data +Collection +[190] +Scheduling data transmission based on users’ data importance levels and channel conditions +Data importance-aware spectrum +allocation +[191] +Allocating users’ transmission power that can adjust the amount of collected data samples for +multiple AI models to enhance the overall model accuracy +Data amount-aware power allo- +cation +[192] +Designing an importance-aware ARQ protocol, in which users’ data importance levels and channel +conditions are jointly considered to trigger data retransmission +Data importance-aware retrans- +mission protocol +Model +Training +[193] +Proposing an edge-cloud assisted FL framework, in which the edge and cloud servers alternatively +aggregate local models to reduce communication overhead +Two-tier FL framework +[194] +Proposing an over-the-air computation approach for model aggregation +Over-the-air model aggregation +[195] +Selecting users with more contribution to convergence for model aggregation based on users’ data +distribution +Data distribution-aware user se- +lection +[196] +Selecting users with low training delay considering heterogeneity among users +Training latency-aware user se- +lection +[197] +Optimizing the number of local model updates given a resource budget +Local +update +frequency +opti- +mization +[198] +Scheduling model uploading based on end users’ model importance levels and channel conditions +Model importance-aware model +uploading +Model +Inference +[144] +Optimizing video frame rate and input image resolution to balance service latency and detection +accuracy for virtual reality users +Data resolution optimization +[199] +Selecting the optimal DNN model for real-time video analytics +DNN model selection +[200] +Selecting the optimal DNN model’s cut layer to minimize inference latency for user-edge DNN +synergy +User-edge DNN model partition +[201] +Partitioning a complicated DNN model across end users, the network edge, and the cloud to reduce +communication overhead +User-edge-cloud +DNN +model +partition +[202] +Designing a collaborative DNN model inference scheme with light-weight models at IoT devices +and an uncompressed model at the network edge +Collaborative DNN model infer- +ence +the-art AI models,6 uploading local AI models still places +a growing strain on spectrum-constrained wireless networks. +In addition, end users with powerful computing servers can +conduct local model training with a low delay. As such, the +model uploading delay due to limited radio resources can +be the dominant component in the entire FL delay. Hence, +it is necessary to maximize FL performance in resource- +constrained wireless networks. +Recent research works optimize FL performance from the +following perspectives: +• FL Framework Design - A line of works focus on design- +ing innovative FL frameworks to reduce communication +overhead. A novel two-tier hierarchical FL framework +is proposed in [193], which coordinates end users, edge +servers, and the cloud server to perform FL. Each edge +server aggregates local models from end users in its cov- +erage in every FL round, and the cloud server aggregates +the models from edge servers in its coverage once in a +few FL rounds. The proposed two-tier framework can +accommodate a large number of end users for model +training due to its broad coverage and, at the same +time, reduce the backhaul data traffic between the cloud +server and edge servers due to a low model aggregation +frequency. Such framework is applied to industrial IoT +networks with geographically distributed data in [215]; +• Model Aggregation - Another line of works study ra- +dio spectrum-efficient model aggregation. Over-the-air +computation based approaches are investigated in [194], +[216], [217]. The basic idea is to exploit the superposition +property of wireless multiple-access channels to perform +model aggregation, which can reduce radio resource +consumption; +• Resource Management - The FL performance can be +6The data sizes of ResNet32 [210], Inception-v3 [211], AlexNet [212] +and VGG16 [213] models are 50 MB, 108 MB, 240 MB, and 552 MB, +respectively [214]. +optimized via efficient resource management. As FL +performance depends on multiple factors, such as end +user selection, the number of local model updates, and +local model importance levels, different resource man- +agement schemes are developed as follows: (1) User +selection - How to select participating end users in the +FL process impacts model convergence and training delay +and hence is crucial to FL performance. A few end +user selection algorithms are proposed based on princi- +ples such as training data distribution [195] and local +training latency [196]; (2) FL parameters - To alleviate +communication overhead, end users conduct a few local +model updates before model uploading. Given a resource +budget, the optimal number of local model updates is +studied in [197], which provides a theoretical guideline +for selecting the number of local model update; (3) Local +model importance level, which is a concept extended +from the idea of data importance7 - An importance-aware +model uploading strategy is proposed in [198], in which +end users with high model importance levels and good +channel conditions are scheduled with high priority, to +speed up the convergence of FL. +Model Inference - For many AI services in the network, +AI models are usually deployed at end users and edge +servers to achieve low service latency. The model inference +is computation-intensive, while end users and edge servers +usually have limited computing capabilities and battery power. +Executing model inference tasks usually results in long service +latency and high energy consumption. Hence, performing +model inference should satisfy service latency under node +energy constraints, thereby calling for innovative model in- +ference schemes. +7The model importance can be measured by layer-wise gradient norm. +Local models with larger gradient norm contribute more to global model +convergence in FL [197]. + +16 +Existing studies on model inference can be categorized as +follows: +• Data Resolution - Raw data are offloaded to edge or cloud +servers for model inference. The input data resolution +influences the inference accuracy. For instance, the ac- +curacy of object detection is related to the input image +resolution [144], which in turn affects the offloaded data +volume since the data size of high-resolution images is +usually large. Taking into account the trade-off between +the inference accuracy and the amount of offloaded data, +the input image resolution should be optimized to satisfy +the target AI service requirements. The optimal video +frame rate and input image resolution are investigated +in [144] to balance service latency and detection accuracy +for virtual reality users; +• Model Selection - An appropriate AI model is selected +to satisfy specific AI service requirements. In addition to +the data resolution, the inference accuracy depends on the +type of AI models. A DNN model with more hidden lay- +ers can usually achieve a higher inference accuracy than +a shallow DNN model. Considering multiple available +DNN models deployed at the network edge, the optimal +DNN model selection for real-time video analytics is +investigated in [199]; +• Model Partition - With advanced model partition tech- +niques, an AI model can be partitioned into multiple sub- +models and then embedded into different network nodes +to conduct model inference. For instance, leveraging the +layered structure of DNNs, the entire DNN model can be +partitioned into an end user-side model and a server-side +model at a proper DNN layer (i.e., the cut layer). As such, +the end users and the edge servers can conduct model +inference in a collaborative manner. DNN models can be +partitioned for achieving different goals. For instance, the +optimal model partition for minimizing inference latency +is studied in [200], in which an online learning algorithm +can adaptively determine the optimal cut layer. To reduce +communication overhead among network nodes, compli- +cated DNNs models can be partitioned into sub-models +for end users, edge servers, and the cloud as in [201]; +• Model Compression - Light-weight models are used +to facilitate prompt model inference at end users. +Computation-efficient compressed models can be ob- +tained via various model compression techniques, such +as weight pruning [218], knowledge distilling [219] and +fast exiting [220]. For instance, weight pruning tech- +niques remove less important model weights to reduce +the computational complexity of model inference, while +achieving inference accuracy close to that of the un- +compressed models. To enhance service performance, +a collaborative model inference scheme that deploys +light-weight models at IoT devices and uncompressed +models at the network edge is proposed in industrial +IoT networks [202]. The IoT devices dynamically make +AI task offloading decisions according to time-varying +channel conditions to minimize the service delay while +guaranteeing the accuracy requirements of DNN-based +fault diagnosis services. +3) Research Challenges: Despite the aforementioned re- +search efforts, facilitating AI services in a network faces vari- +ous challenges, some of which are discussed in the following. +Complex Implementation Option Selection - An AI service +can be implemented by various options with different model +structures, training procedures, and inference processes. For +instance, a service of object detection can be implemented +via different neural networks, such as AlexNet [212] and +SqueezeNet [221]. Even if the model structure is the same, +a model can be trained in different ways, such as centralized +training, decentralized training (e.g., FL [207]), and semi- +centralized training (e.g., split training [126]). In addition, +model inference can be conducted in various manners, such as +end user-only inference, edge-only inference, and collaborative +inference. Different implementation options consume different +amounts of computing, storage and communication resources. +Hence, it is necessary to select an implementation solution for +AI services that suits the service characteristics and network +dynamics. +Multi-Dimensional QoS Requirements - The QoS require- +ments of AI services are multi-dimensional. AI model ac- +curacy is usually a key performance metric. In addition, AI +services should be offered to end users with low latency in +many use cases. For instance, the service latency of object +detection in autonomous driving should be less than 100 ms +for safety considerations [143], whereas autonomous vehicles +require an ultra-high accuracy in 3D object detection [222]. +Moreover, these performance metrics are correlated. High- +accuracy object detection usually requires high-resolution im- +ages as input and advanced AI models to process the input +images, which can result in long service latency. How to satisfy +multi-dimensional QoS requirements of AI services requires +further investigation. +4) AI Slice: To better support AI services, we extend the +network slice concept and propose an idea of AI slice with two +subslices. The basic idea is to construct a training subslice for +model training and an inference subslice for model inference. +The two subslices are logically isolated and use their own +network resources. The rationale behind training and inference +separation is that the two stages can have different goals. +An illustration of an AI slice is given in Fig. 5. In the +AI slice, the training and inference subslices share the same +resource pool and are coordinated to jointly support the +AI service. First, the multi-dimensional QoS requirement of +the AI slice is decoupled into two separate QoS require- +ments for the two subslices. For an object detection service +in autonomous driving, both high detection accuracy (e.g., +99%) and low service latency (e.g., 100 ms) are required. +The training subslice should satisfy the detection accuracy +requirement, while the inference subslice should satisfy the +service latency requirement. Second, to satisfy the individual +QoS requirements of the two subslices, the resources reserved +for the AI slice are judiciously allocated between the two +subslices, based on the performance of the two subslices and +their QoS requirements. Then, given the allocated resources, +the two subslices are configured to satisfy their individual QoS +requirements, as described in the following: + +17 +... +Physical Network +Inference Subslice +Training Subslice +Slices +AI Slice +Pruned Model +Uncompressed Model +Partitioned Model +Inference Subslice +... +Model Deployment +BS 1 +Core +Network +BS N +Model +Distribution +Model +Aggregation +Local Model + Training +Local Model +Training +... +Training Subslice +Local Model +Uploading +Local Model +Uploading +Time +Well-Trained + Model +One Training Round +Virtual Network +Core +Network +BS +Network Slicing +Computing Resource +Communication Resource +Storage Resource +............ +............ +............ +... +... +... +... +Subslice Controller +Fig. 5: Conceptual AI slice consisting of a training subslice and an inference subslice. +• In the training subslice, based on the training data dis- +tribution in the network, a subslice controller determines +training configurations (e.g., data collection schemes and +model training methods) and schedules resources to net- +work nodes to train a model given the target accuracy. +In addition, since the training data vary over time in +a dynamic network, the AI model may need to be +retrained from time to time. Note that allocating dedicated +resources for the training subslice can effectively mitigate +the straggler effect that plagues distributed learning in +large-scale networks, thereby speeding up the model +training process; +• In the inference subslice, the subslice controller analyzes +the service demand pattern at each BS and determines +inference configurations (e.g., model inference and input +data compression schemes) to satisfy the inference la- +tency requirement. For instance, uncompressed models +can be deployed at resource-abundant BSs, and parti- +tioned and pruned models can be deployed at resource- +limited BSs. This can achieve close inference service +latency performance across different BSs. +Overall, the two logically-isolated subslices focus on satisfying +different QoS requirements and jointly support the AI service. +To elaborate the idea of AI slices, we present the fol- +lowing example on real-time video analytics in vehicular +networks [199]. Smart cameras are deployed in intersections +to provide a video surveillance service such as vehicle plate +recognition. In such service, a CNN model is trained using +the video streams collected by smart cameras, and then the +well-trained model is used to conduct video analytics tasks. +Using the proposed AI slice framework, CNN model training +is conducted in a training subslice, while real-time video +analytics is conducted in an inference subslice. Specifically, +in the training subslice, the CNN model can be trained via a +FL framework for protecting data privacy. The corresponding +computing resources at smart cameras and spectrum resources +in the network are allocated to satisfy model training require- +ments, such as training accuracy. In the inference subslice, +different user-edge orchestration schemes (e.g., DNN model +partition), input data compression schemes (e.g., frame rate +reduction), and network resource management policies can be +configured to satisfy the inference delay requirement in video +analytics services based on time-varying service demands +and network conditions due to vehicle mobility. With the AI +slice for video analytics, both training accuracy and inference +latency requirements can be satisfied in a dynamic network +environment. +E. Summary +In this section, we have reviewed some common AI tech- +niques, explored the role of AI in 6G networks, and proposed a +four-layer AI architecture for pervasive intelligence in 6G. Two +perspectives of AI in wireless networks, i.e., AI for networking +and networking for AI, have been discussed, which correspond +to using AI as a powerful tool for network management and +optimizing networks to support AI applications, respectively. +Recent advancements in ML algorithms have accelerated the +deployment of AI in wireless networks. In 5G, AI techniques +are used to address particular networking problems, whereas, +in 6G, AI will penetrate every corner of wireless networks +from network management to network services. Therefore, an +architecture for AI is needed for identifying the role of AI and +characterizing the functionalities of AI across a network. + +18 +Appropriate AI techniques should be selected to tackle +networking problems with different characteristics and on +different decision time scales when it comes to AI for network- +ing. Furthermore, the collaboration among intelligent modules +is important to implement AI-driven networks efficiently and +flexibly. The idea of connected AI is to enable cooperative +decision making among intelligent modules for network con- +trol. In terms of networking for AI, a distributed architecture +of AI algorithms connects AI models and network resources +located at network edge. The study of networking for AI is +still in its infancy but essential to supporting an expanding +group of AI services. Network slicing will remain to be an +enabler for delivering AI services, but slicing policies should +be customized according to the features of AI algorithms and +the training and inference stages of AI. +IV. A POTENTIAL NETWORK ARCHITECTURE FOR 6G +In this section, we propose a conceptual network architec- +ture for 6G, which integrates holistic network virtualization +(including digital twins and network slicing) and pervasive +network intelligence (including connected AI and AI slices). +Then, we illustrate three types of interplay enabled by the +proposed architecture, i.e., the interplay between digital twin +paradigm and network slicing, model-driven methods and data- +driven methods, and virtualization and AI, respectively. +A. Related Studies on Architecture for 6G +Several works have proposed architectures with various +focuses for 6G networks, e.g., space-air-ground integrated net- +works for global coverage [8], cell-free massive multiple-input +multiple-output (MIMO) architecture for inter-cell interference +mitigation [223], and multi-tier computing architecture for +ubiquitous computing service provisioning [224]. Pursuing the +goal of advanced network management, most of the proposed +architectures highlight AI techniques to optimize network +architecture, control, and management [12], [183]. For exam- +ple, AI-based data analytics functions, which mine historical +data for network operation troubleshooting, network resource +optimization, and network traffic prediction, are incorporated +in the network architecture in [12]. The ITU specifies a high- +level AI-based architectural framework for future networks, in +which several novel components such as ML management and +orchestration functionalities are incorporated for flexible AI- +based function placement [183]. In addition to AI techniques, +some recent conceptual network architectures start to embrace +digital twin techniques [75], [83]. For example, a digital +twin-based network architecture constructs a digital twin for +each end user to serve as its communication assistant and +data asset manager [83]. Another digital twin-enabled network +architecture adopts three categories of digital twins, i.e., edge- +based, cloud-based, and hybrid digital twins, for supporting +different types of services [75]. +Different from the existing network architectures, our pro- +posed network architecture features novel holistic network +virtualization, which incorporates network slicing and digital +twin paradigms, and pervasive network intelligence, which +integrates AI for networking and networking for AI. Moreover, +featuring the designs in Sections II and III, the proposed +architecture enables various interplay among its key elements +to empower 6G. In the following subsections, we present the +details of the proposed architecture. +B. Architecture Overview +The overall network architecture is illustrated in Fig. 6, +which consists of the physical space and the cyber space. The +physical space includes end users and network infrastructure at +the edge and the core networks. Data describing end users are +collected from the physical network to create level-one digital +twins as introduced in detail in Subsection II-C, and network +slices are created for various services. The slices are further +abstracted into level-two digital twins, which are supplemented +with service-specific information aggregated from level-one +digital twins. The six-layer virtualization architecture in Fig. 2 +applies to the network slices and the digital twins, both of +which reside in the cyber space in Fig. 6. +AI pervades the entire architecture, which supports both AI +for networking and networking for AI. First, AI is used to +manage network slices and digital twins, as shown in the logic +network control section in Fig. 6. For network management, a +connected AI solution discussed in Subsection III-C is applied +to enable intelligent modules, which in turn manage network +slices and digital twins. The connected AI solution corresponds +to AL 3 and 4 in Fig. 3. Second, the architecture supports +dedicated AI slices with training and inference separation for +AI service provisioning, as mentioned in Subsection III-D. AI +slices provide services corresponding to AL 1 and 2 in Fig. 3, +while the management of AI slices is conducted by intelligent +modules. +With the overall network architecture in Fig. 6, we integrate +holistic network virtualization and pervasive network intel- +ligence for 6G. Virtualization is supported from the aspects +of both the network and the end users, while intelligence is +reflected through both AI for networking and networking for +AI. Taking advantage of digital twin paradigm and network +slicing as well as those of virtualization and AI, the proposed +architecture aims at exceeding flexibility, scalability, adaptiv- +ity, and intelligence. +C. Components and Subsystems +In the physical space, the proposed architecture includes +both RANs and core networks. Specifically, the following +components are involved: +• Assorted APs: This component includes MBSs, SBSs, +mobile APs (such as UAVs), satellites, and other non- +cellular APs; +• Network controllers: This component includes local con- +trollers located at APs or servers on network edge and the +centralized controller located at servers in core networks +or in the cloud. Each controller can consist of computing +servers and affiliated network storage servers; +• General computing servers: This component includes +computing servers for implementing network functions, +such as routing and firewall, and hosting the VNFs; + +19 +Level-One Digital Twin +Cyber Space +Physical Space +Level-Two Digital Twin +Aggregation +Digital Twin Model Control +... +Network Control +Inference Subslice +Training Subslice +AI Slice +Slices +AI Slice +Core Network +Network +Slicing +Intelligent Modules for +Network Management +... +... +... +... +... +... +... +... +... +... +... +... +Gateway +Local Controller +Centralized Controller +Data Flow (Physical) +Control (Physical) +Data Flow (Cyber) +Control (Cyber) +Generation/ +Updating +Generation/ +Updating +Fig. 6: The proposed network architecture for 6G networks. +• Application servers: This component includes computing +and network storage servers for supporting general edge +computing and AI services. These servers are not used for +network management or implementing network functions; +• Other network devices: This component includes special- +ized network hardware that are not general computing +servers, such as baseband processing units and network +switches; +• End users: This component includes human mobile users, +sensors, vehicles, and various IoT devices, such as meters, +actuators, and robots. +In the cyber space, the proposed architecture includes three +subsystems, i.e., network slices, digital twins, and connected +AI, as follows: +• Network slices: This subsystem includes all virtual net- +works created in network slicing, including AI slices. A +network slice can involve a RAN, a core network, or both. +General slices are inherited from existing networks, while +AI slices are described in detail in Subsection III-D; +• Digital twins: This subsystem includes level-one and +level-two digital twins. The digital twin subsystem is +described in detail in Subsection II-C; +• Connected AI: This subsystem includes intelligent mod- +ules deployed across a network at both the local con- +trollers and the centralized controller. The connected AI +subsystem is described in detail in Subsection III-C. +Interconnections between different components and subsys- +tems of the proposed architecture are elaborated in Subsec- +tions IV-E to IV-G, which highlight the interplay between dig- +ital twin paradigm and network slicing, between model-driven +and data-driven methods, and between virtualization and AI, +in the proposed architecture. Some open issues and challenges +regarding the architecture are presented in Section V. +Note that the proposed conceptual architecture can apply +to various types of physical networks, such as vehicular +networks and integrated terrestrial-satellite networks, although +Fig. 6 cannot illustrate every possible network scenario. In +different physical networks, the implementation of holistic +network virtualization and pervasive network intelligence can +be different and require certain customization. For example, +the deployment of intelligent modules and the data flow among +the modules in a satellite network segment can be different +from those in a terrestrial network segment. Furthermore, the + +f(x)f(x)f(x)20 +migration of digital twins can be more important in a vehicular +network than in a static IoT network. Related discussions can +be found in Section V, where we present challenges and open +issues. Nevertheless, the basic ideas in the proposed conceptual +architecture, including the two-level digital twins, intelligent +modules, and AI slices, are applicable in various physical +networks. +D. Implementation +In this subsection, we provide a case study on a vehicular +network to demonstrate the potential implementation of the +proposed network architecture. Roadside BSs co-located with +edge computing and caching servers facilitate autonomous +driving services for vehicles on the road. To implement the +proposed network architecture, the following steps are con- +ducted. +• Network Slice Establishment: Multiple network slices are +established for autonomous driving services with different +QoS requirements, achieving network virtualization. Con- +ventional network slices are established for non-AI based +services, e.g., high-definition map downloading, while AI +slices consisting of training and inference subslices are +established for AI based services, such as deep learning +based cooperative sensing. The network slices are stored +and managed by a centralized controller. +• Digital Twin Construction: By collecting extensive data +from physical entities, digital twins are constructed for +vehicle users, roadside BSs, and the established net- +work slices, achieving the virtualization of end users +and slices. Digital twins of vehicle users and roadside +BSs are located at edge servers, while digital twins of +network slices are located at a cloud server. Due to high +vehicle mobility, digital twins of vehicle users should be +migrated across edge servers to ensure service continuity. +In addition to collected data, digital twins can include +generated user and service specific data, such as predicted +vehicle trajectory and spatial-temporal service demands, +via mining historical data. The generated vehicle data will +be used for network management and service provision. +• AI Module Deployment: AI modules with different func- +tionalities can be deployed at both the centralized and +local network controllers, achieving intelligent network +management. The AI modules at the centralized network +controller are in charge of network planning. For guar- +anteeing QoS requirements of different slices, these AI +modules can make resource reservation decisions based +on the predicted service demands from the digital twins +of roadside BSs and collected slice performance data +from the digital twins of network slices. The AI modules +at local network controllers are in charge of network +operations. For enhancing the perceived performance of +the vehicle users, the AI modules schedule on-demand +network resources based on the collected data (e.g., +vehicle users’ channel conditions) and the generated data +(e.g., predicted vehicle trajectory) from the digital twins +of vehicle users. +E. Interplay between Digital Twin Paradigm and Network +Slicing +As the two components of holistic network virtualization, +digital twin paradigm and network slicing are connected in the +following two aspects. +First, the digital twin paradigm for end user virtualization +focuses on data management, while network slicing focuses +on network management. Data may be viewed as a new type +of resources in future networks, in addition to communica- +tion, computing, caching, and sensing resources. Meanwhile, +as a resource, data has its unique features. First, data can +be considered as an application-layer resource rather than +a physical-layer resource. Second, different from computing +or communication resources, the amount of data resources +available to a network is not fixed but progressive. Last, +the collection and processing of data, which is necessary for +utilizing any data resource, consume other network resources. +On one hand, effectual utilization of the data resource will +benefit network management, and hence digital twin paradigm +can enhance network slicing. On the other hand, network +management should take into account the need and cost +of allocating other network resources for utilizing the data +resource. Hence, network slicing can facilitate digital twins. +Second, digital twins will enable user-centric networking +in future networks, while network slicing enables service- +centric networking. Creating an isolated slice for each service +and provisioning the service through managing the slice yield +a service-centric focus in network management. Meanwhile, +creating a digital copy of each end user and administrating data +that characterize the end user provide a user-centric perspective +of network management. Having a set of information, selected +by the centralized controller through digital twin model con- +trol, to describe various characteristics of the end users, such +as their location, service request profile, resource utilization, +and channel information, creates the possibility of user-specific +scheduling within each slice in the network operation stage. +For instance, access control and resource allocation decisions +for an end user may depend on the data profile from its +digital twin, while different data profiles may lead to different +scheduling policies. Accordingly, future networks may feature +service-centric network planning and user-centric network +operations, which can improve the granularity of network +management for handling highly diversified end users and +dynamic network environments. +F. Interplay between Model-Driven and Data-Driven Methods +The second interplay enabled by the proposed architecture is +the interplay between model-driven and data-driven methods in +network operation and service provision. This interplay applies +to the intelligent modules for network management shown in +Fig. 6. +Network management mostly relied on model-driven or +heuristic methods before 5G. Prior to the prevalence of +AI, mathematical tools such as optimization methods and +game theory have been widely used for network manage- +ment. Optimization methods formulate the objective and con- +straints in a closed form, and the corresponding network + +21 +management problems are solved using optimization algo- +rithms [225]–[227]. Game-theoretic approaches analyze the +interactions among network entities in either cooperative or +non-cooperative scenarios to identify the optimal strategy of +each entity [228]–[230]. Mechanism design, an analytical +framework in game theory, has also been used to coordinate +network entities with locally-held information to achieve desir- +able network-wide solutions in network utility maximization +problems [231]. +Through characterizing the relations among several key +variables, model-driven methods can lead to either closed-form +solutions or algorithms for network management problems. +Based on mathematical models, model-driven methods are +usually explainable and generalize well for different specific +problems [232].8 However, when networks become complex +(i.e., when there are a large number of variables and/or +complicated correlation among them) or highly dynamic (e.g., +when the network environment changes too rapidly for an +optimization algorithm to converge or for a game to achieve an +equilibrium), model-driven methods may no longer be accurate +or applicable. +The investigation of data-driven methods for network man- +agement has gained momentum since 5G. Through collecting +and exploiting real-world data, data-driven methods implicitly +characterize the relations among variables to generate and fine- +tune policies for network management. Given sufficient data +and a stationary network environment, data-driven methods +can provide close-to-optimal solutions to problems that are +too complicated for model-driven methods. However, when +the network environment is non-stationary so that new and +unknown situations occur from time to time, the performance +of data-driven methods can be questionable [233]. In addition, +data-driven methods may not generalize well due to their +strong dependence on data collected from a specific network +environment. +In 6G, data-driven and model-driven methods should work +in synergy. The proposed architecture enables the interplay +between data-driven and model-driven methods for creating +advanced hybrid data-model driven methods. There are differ- +ent options of hybrid data-model driven methods, as illustrated +in Fig. 7 and elaborated below. The first three options suit AI +for networking, while the last option suits networking for AI. +• Backup/Switching - Data-driven and model-driven meth- +ods can be the backup for each other. For instance, models +can be selected to back up data-driven methods, for +the case when unknown situations occur in the network +environment and degradation in the performance of data- +driven methods appears. Meanwhile, switching between +data-driven and model-driven methods, e.g., based on the +available resources, can potentially increase the adaptivity +of network management. +• Task Division - Date-driven and model-driven methods +can target different steps and solve different subprob- +lems of network management. Specifically, data-driven +8For instance, the water-filling algorithm could be applied to various +power allocation problems, and the Rayleigh fading model could characterize +channels in various network environments. +Model +Data +Solution +Solution +Backup/ +Switching +Problem +(a) Backup/switching +Model +Data +Solution Part 1 +Solution Part 2 +Subproblem 1 +Subproblem 2 +(b) Task division +Model +Data +Solution +(Initial) +Solution +(Refined) +Problem +(c) Refinement +Model +Data +Data +Model +Data +(d) Mixing +Fig. 7: Options for hybrid data-model driven methods. The +“Data” and “Model” blocks represent “data-driven methods” +and “model-driven methods”, respectively. +methods can solve the subproblems with a large number +of variables or complicated coupling relations among +variables, while model-driven methods can solve rela- +tively isolated subproblems with a few key variables. This +would allow data-driven and model-driven methods to +play to their respective strengths. +• Refinement - Model-driven methods can provide rough +solutions based on general mathematical models, and +then data-driven methods, taking the rough solutions as +input and exploiting real-world data from the network, +can refine the solutions for the specific network scenario. +Having the initial solution generated from models may +reduce either the amount of data or the amount of time +needed by data-driven methods. +• Mixing - In networking for AI, while deploying a service +function chain for an AI service, some of the function +modules can use data-driven methods, while other func- +tion modules in the same service function chain can +use model-driven methods. For example, in an AI-based +image processing service, a model-driven module can be +used for image resolution adjustment prior to a data- + +22 +driven module for object detection. The idea is similar to +task division, except that the scenario here is networking +for AI instead of AI for networking [28]. +G. Interplay between Virtualization and AI +The third interplay enabled by the proposed architecture, +i.e., the interplay between virtualization and AI, is illustrated +in Fig. 8. +First and foremost, virtualization and AI are coupled +through data. With the introduction of digital twins, a vast +amount of organized data regarding end users, i.e., level-one +digital twins, and network services, i.e., level-two digital twins, +become available. The data included in the digital twins can be +provided to the intelligent modules, the training or inference +subslice of an AI slice, or both. For instance, edge-hosted AI, +possibly collaborating with end user-hosted AI, can perform +user-specific data processing and prediction based on the data +from digital twins. The results, such as prediction results, +resource scheduling schemes, or slicing policies, can be fed +back to the digital twins to record certain predicted status, e.g., +location and mobility, of the end users. Correspondingly, data +in the digital twins of end users, network infrastructure, and +slices can be either the input or the output of AI modules, +leading to a bidirectional interaction between virtualization +and AI.9 +The second connection between virtualization and AI is +through control. Based on the data from digital twins, AI +functions hosted at the edge and core networks can make the +network management and service provisioning decisions. The +decisions may include network slice control, which are fed +back to the physical network and network slices for execution +and, at the same time, to the level-two digital twins for data +update. In addition, the decisions may include digital twin +model control for level-one and level-two digital twins. Digital +twin model control may include the determination of the type +and the amount of data to be included in digital twins, the +frequency and the method of data collection, the format and +the precision of stored data, and so on. The digital twin +models affect the availability and quality of data available +for network control, especially AI-driven network control, +and thereby impact the network performance. Therefore, from +the perspective of network control, the interaction between +virtualization and AI is also bi-directional.10 +The third and implicit connection between virtualization +and AI is through resources. Holistic network virtualization +requires extensive resources, including computing resource for +virtual network functions, caching resource for storing digital +twins, and communication resource for the synchronization +between end users and their digital twins. Similarly, perva- +sive network intelligence also requires extensive computing +resource and possibly other resources, e.g., communication +resource for distributed training as mentioned in Section III. +Therefore, the network resources need to be shared and +9Interested readers are referred to [234] for the relation between AI and +data life cycle, although the discussions therein do not involve virtualization. +10The interaction between digital twin and AI for intelligent network control +is discussed in [235]. Note that the definition of digital twins therein is +different from ours. +Physical Network +Slices +Data +Control +Digital Twins +AI for +Networking +Networking +for AI +Level-Two Digital Twin +Level-One Digital Twin +Fig. 8: Interplay between virtualization and AI. +coordinated between virtualization and AI functions. However, +this does not mean that virtualization and AI functions simply +compete for resources. Instead, they can help each other +improve resource utilization efficiency. Digital twin paradigm +may reduce the resource consumption of AI functions by +providing high-importance data only. This can be achieved +by the aforementioned digital twin model control. Meanwhile, +creating a digital twin for every end user may be too resource- +demanding for networks in the near future. Using AI to +select representative end users for generating digital twins +and optimizing digital twin models may reduce the resource +consumption of maintaining and updating digital twins. One +potential implementation is using AI to categorize end users +and select a portion of users from each category for creating +digital twins. Alternatively, since it may be more challenging +to provide QoE guarantee for some end users than others, +using AI to select such end users for creating digital twins can +potentially reduce the resource consumption on digital twins +for 6G networks. +H. Potential Network Architecture for 6G: A Summary +This section has provided a potential network architecture +for 6G, which integrates two key elements, i.e., pervasive +network intelligence and holistic network virtualization. In +the proposed network architecture, detailed network compo- +nents, subsystems, and potential implementation have been +discussed. Moreover, three types of interplay in the archi- +tecture are provided to characterize the proposed network +architecture. +The proposed network architecture holds great potential +for achieving advanced network management schemes and +supporting AI services in 6G networks. Firstly, integrating +digital twins and network slicing facilitates user-centric net- +working and improves the granularity of network management. + +23 +Secondly, integrating data-driven methods and model-driven +methods enables novel hybrid data-model driven methods, +which has the potential to outperform existing network man- +agement methods in terms of adaptivity, granularity, and so on. +Thirdly, leveraging the network slicing concept in AI services +facilitates AI services targeting QoS performance guarantees. +V. CHALLENGES AND OPEN ISSUES +Many challenges and open issues are yet to be addressed for +holistic network virtualization and pervasive network intelli- +gence in 6G. In the following, we present some key challenges +and open issues. +A. Digital Twin +The six-layer architecture in Subsection II-C provides a +high-level design for integrating the digital twin paradigm +into network virtualization. Open issues to be investigated for +practical implementation of this architecture include quantita- +tive performance characterization of digital twins, the optimal +digital twin model, digital twin migration, and data security. +First, it is necessary to quantitatively characterize the +network performance improvement from introducing digital +twins, either from the perspective of QoS/QoE satisfaction or +from the perspective of resource utilization. Second, level-one +digital twin models configured by the centralized controller +may be different for different edge networks to account for +network heterogeneity, and how to determine effective digital +twin models is a challenge. Third, the mobility of end users +such as vehicles creates a need for updating and migrating +digital twins across different edge networks, which requires +further study. Last, ensuring the security of user data in the +digital twin paradigm is yet another challenge. As the local and +centralized network controllers have access to a vast amount of +user data, developing proper security mechanisms for data col- +lection, aggregation, and migration becomes essential. Readers +are referred to [32], [236]–[238] for discussions on some of +the aforementioned challenges, such as the heterogeneity and +migration of digital twins, and more open issues related to +digital twins in 6G. +B. Network Management Oriented Data Abstraction and Pro- +cessing +While digital twins provide data to enable AI for network- +ing, including automated network slicing and AI-empowered +network control, efficient data management can be challeng- +ing. First, it is necessary to develop data abstraction methods +to aggregate the data with different levels of granularity for +making different network management decisions. For instance, +in network slicing, high-granularity data are required for +determining the optimal network operation strategies and low- +granularity data are sufficient for determining the optimal +network planning strategies [11], [239]. How to determine +the appropriate data granularity for different network man- +agement decisions is an open issue. A potential solution is +to empirically adjust data granularity and the time scale for +decision-making [240]. Meanwhile, as the number of variables +and data types in network management can be huge, more +scalable and efficient solutions are required. Second, while +applying the connected AI solution for network management, +the settings of intelligent modules, such as the selection of +algorithms, the input and output attributes, and the connections +among intelligent modules, should be configured to maximize +the utilization of data with low communication and processing +overhead, yet finding the optimal settings is challenging. The +cooperation between model-driven and data-driven methods +in intelligent modules can be a potential approach to address +the challenge, yet how to support such cooperation among +different types of intelligent modules requires further inves- +tigation. Third, as data can be generated, transmitted, and +processed at different network stakeholders, configurable and +regulation-compliant data management is also a challenge. The +integration of the blockchain and privacy-enhancing technolo- +gies can be a potential solution, while the trade-offs between +privacy preservation and processing efficiency need in-depth +investigation. Readers are referred to [4], [131], [182], [234] +for discussions on the aforementioned challenges, such as +privacy preservation, AI model selection, intelligent modules, +and more open issues about data abstraction and processing. +C. Model and Resource Orchestration +Networking for AI in Subsection III-D can facilitate AI +services in a network. One key issue is to optimize AI service +performance, which requires judicious configuration of the +network, including AI algorithm selection, data collection, +and network resource allocation. The main challenge lies in +modeling the relationship between AI performance and these +network configurations. Establishing an accurate mathematical +or empirical model requires extensive measurements in real- +world networks. Even if establishing a model is viable, the +model may be suitable only for a chosen AI algorithm. In +addition, to adapt to network dynamics (e.g., rapidly fluc- +tuating service demands), an online network configuration +scheme is desirable. Since reinforcement learning algorithms +are able to make online decisions in a dynamic environment, +developing cost-effective reinforcement learning algorithms +for high-dimensional network configuration problems can be a +promising approach. For example, a reinforcement learning al- +gorithm is developed for joint AI model selection and resource +allocation in industrial IoT [202]. For more discussions on +the above challenges, interested readers are referred to [175], +[241], [242]. +D. Training and Inference Coordination +The concept of AI slice is proposed to meet specific +QoS requirements of AI services in Subsection III-D. The +training and inference stages for an AI service consume multi- +dimensional network resources [131], [243]. In an AI slice, +two subslices share the virtualized network resource pool, +and hence resource reservation decisions for the two subslices +are closely correlated. On the one hand, reserving abundant +resources for the training subslice may help achieve a high +training accuracy but potentially render resource insufficiency +in the inference subslice, which can result in a long service + +24 +latency. On the other hand, insufficient resource provisioning +for the training subslice may yield a model with low accuracy +and consequently create a bottleneck for inference accuracy. To +optimize the performance of the AI service, resource reserva- +tion for training and inference subslices should be coordinated. +Developing an accurate mathematical model to characterize +the interplay between training and inference stages is difficult, +since a large number of system factors should be taken into +account. Hence, it is necessary to study efficient model-free +approaches to characterize the interplay. +E. Energy Efficiency of AI +With hundreds of neural network layers, thousands of +neurons, and millions of parameters, state-of-the-art AI mod- +els usually consume extensive energy and incur substantial +environmental costs.11 Improving energy efficiency has be- +come a major issue for wide deployment of AI services. In +addition, recent research shows that improving the accuracy +of an AI model may come at an exponential increase in +the computation, environmental and economic costs [245].12 +Hence, deploying energy-efficient AI services in a network +is necessary for reducing costs for the network operator and +meeting environmental standards. Several model compression +techniques, such as weight pruning [218], parameter quanti- +zation [246], and model compression [247], can be applied to +alleviate the problem. In addition, hybrid data-model driven +methods can train AI models with a reduced amount of data, +which can also decrease energy consumption. +F. Hybrid Data-Model Driven Methods +The four options listed in Subsection IV-F provide our +initial ideas for hybrid data-model driven methods. Related +open issues to be investigated include the following. First, it +is necessary to study how to determine which option to use +and how to switch among options. Designing mechanisms for +choosing and switching among options will allow networks +to flexibly and adaptively integrate data-driven and model- +driven methods. Second, for a chosen option, it is important to +understand how much the data-driven and model-driven com- +ponents affect the overall performance and how much impact +they have on each other. For instance, in the mixing option, +the AI service performance may depend on the combined +choices of data-driven and model-driven methods, and finding +a proper combination can be a challenge. Third, in addition to +the four options as introduced, there should be other potential +options for hybrid data-model driven methods, and identifying +other promising options is an open issue of great importance. +Last, due to the lack of explainability in existing data-driven +methods, careful investigations and analysis should be directed +to the management of critical network operations. The role +11The estimated carbon footprint of training a state-of-the-art natural +language processing model is about five times the life emissions of an average +car [244]. +12It is estimated that reducing the classification error probability from +11.5% to 5% over the ImageNet dataset needs to increase computation from +1014 to 1019 Gflops, carbon emissions from 106 to 1010 lbs, and economic +costs from 106 to 1011 USD [245], respectively. +of hybrid data-model driven methods in enhancing system +robustness is an open issue that deserves further investigation. +For more discussions on challenges in hybrid data-model +driven methods for networks, interested readers are referred +to [232], [248], [249].13 +VI. CONCLUSION +Designing an architecture for future networks is challenging, +especially when the use cases and defining techniques are still +beneath the surface. Nevertheless, the evolution of networks +through the previous generations demonstrates a necessity to +support increasingly heterogeneous networks, diverse services, +and stringent QoS/QoE requirements. This has been driving +the trend of virtualization and generating significant interest +in AI-driven networking. Recognizing the insufficiency of the +existing scope and level of virtualization and AI for future 6G +networks, we have presented a conceptual architecture design +that integrates holistic network virtualization and pervasive +network intelligence. To complement and solidify our over- +all network architecture, we have proposed several specific +designs, including the six-layer holistic network virtualization +based on digital twins, the connected AI solution for network +management, as well as ideas, including AI slices and hybrid +data-model driven methods. As a result, the proposed network +architecture has the potential to achieve unprecedented scala- +bility and flexibility due to the holistic network virtualization +as well as exceeding adaptivity and intelligence due to the +pervasive network intelligence. At last, we have identified +some challenges and open issues related to the proposed +architecture. We hope this study will lead to further discussions +and developments on the architecture of 6G networks. +ACKNOWLEDGEMENT +The authors would like to thank Dr. Dongxiao Liu for +helpful discussions on open issues related to data privacy and +security. +REFERENCES +[1] X. You et al., “Towards 6G wireless communication networks: Vision, +enabling technologies, and new paradigm shifts,” Sci. China Inf. Sci., +vol. 64, no. 1, pp. 1–74, Nov. 2021. +[2] W. Saad, M. Bennis, and M. 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IEEE GLOBECOM, Virtual Conference, Dec. 2020. + diff --git a/4NAyT4oBgHgl3EQfo_jf/content/tmp_files/load_file.txt b/4NAyT4oBgHgl3EQfo_jf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..40e230ef47c727eaf2ee483e7fed507a4cc3b9ec --- /dev/null +++ b/4NAyT4oBgHgl3EQfo_jf/content/tmp_files/load_file.txt @@ -0,0 +1,4290 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf,len=4289 +page_content='1 Holistic Network Virtualization and Pervasive Network Intelligence for 6G Xuemin (Sherman) Shen, Fellow, IEEE, Jie Gao, Senior Member, IEEE, Wen Wu, Member, IEEE, Mushu Li, Member, IEEE, Conghao Zhou, Student Member, IEEE, and Weihua Zhuang, Fellow, IEEE (Invited Paper) Abstract—In this tutorial paper, we look into the evolution and prospect of network architecture and propose a novel conceptual architecture for the 6th generation (6G) networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The proposed architecture has two key elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', holistic network virtualization and pervasive artificial intelligence (AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The holistic network virtualization consists of network slicing and digital twin, from the aspects of service provision and service demand, respectively, to incorporate service-centric and user- centric networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The pervasive network intelligence integrates AI into future networks from the perspectives of networking for AI and AI for networking, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Building on holistic network virtualization and pervasive network intelligence, the proposed architecture can facilitate three types of interplay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', the interplay between digital twin and network slicing paradigms, between model-driven and data-driven methods for network management, and between virtualization and AI, to maximize the flexibility, scalability, adaptivity, and intelligence for 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' We also identify challenges and open issues related to the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' By providing our vision, we aim to inspire further discussions and developments on the potential architecture of 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Index Terms—6G, network architecture, network virtualiza- tion, digital twin, AI for networking, networking for AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' INTRODUCTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Background With the ongoing worldwide deployment of the 5th genera- tion (5G) networks, the technical community in wireless com- munications and networking has started looking into the 6th generation (6G) networks for 2030 and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' While the ex- act concepts and techniques that define 6G are not determined yet, visions, requirements, use cases, and candidate techniques are discussed in an increasing amount of works, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', [1]– [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Among these discussions, some preliminary consensus regarding 6G emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, in terms of main require- ments of 6G, the urgency of improving security [4] and energy efficiency [5] is understood unanimously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For use cases of 6G, the combination of enhanced mobile broadband (eMBB), This work was supported by research grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Xuemin (Sherman) Shen, Mushu Li, Conghao Zhou, and Weihua Zhuang are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada, N2L 3G1 (email: {sshen, m475li, c89zhou, wzhuang}@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Jie Gao is with the Department of Electrical and Computer En- gineering, Marquette University, Milwaukee, WI, USA 53233 (email: j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='gao@marquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Wen Wu is with the Frontier Research Center, Peng Cheng Laboratory, Shenzhen, Guangdong, China, 518055 (email: wuw02@pcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The corresponding author of this paper is Weihua Zhuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' ultra-reliable and low-latency communications (uRLLC), and massive machine-type communications (mMTC) have been brought up, despite the different terminologies used in dif- ferent works [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As to candidate techniques, commonly mentioned examples include the integration of satellite, aerial, terrestrial, and underwater networks [8], [9], (sub)terahertz and visible light communications [10], artificial intelligence (AI) empowered networks [11]–[13], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' One consensus deserving special attention is that 6G may need a brand-new network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Driven by cost ef- fectiveness and efficiency, the evolution of network architec- ture follows the evolving services provided by the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, to introduce data service, a packet-switched core network component emerged in the 3G architecture as a complement to its circuit-switched counterpart for voice service [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Then, to accommodate the exponential growth of data traffic, 4G introduced a redesigned and simplified network architecture for a flat all-Internet protocol (IP) network with in- creased data rate and reduced latency [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the era of 5G, as networks become more heterogeneous than ever while services become diversified, various network architecture innovations have been proposed towards flexible service-oriented network- ing, including software defined networking (SDN) [16], cloud radio access network (C-RAN) [17], and network slicing [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Therefore, envisioned to support unprecedentedly diverse services with exceedingly stringent quality of service (QoS) or quality of experience (QoE) requirements, 6G will most likely need ground-breaking innovations in network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' While conceiving an architecture for 6G, it is difficult to overlook two key elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', virtualization and AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Network virtualization already plays an important role in the architec- ture of 5G [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The virtualization of resources, functions, and networks enables resource sharing, software implemen- tation of network functions, and service-orientated network- ing, respectively, and thereby increases resource utilization while reducing the cost for deploying and operating networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Virtualization reflects a trend of softwarization for flexible, scalable, and adaptive network management [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Therefore, it is foreseeable that virtualization will remain crucial in the architecture of 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As for the second key element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', AI, a growing number of research teams worldwide are investigating AI-driven networks, and high expectation is placed on AI for empowering 6G [1], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In comparison with heuristic or mathematical model based approaches for communications and networking, AI based approaches can handle complicated networking problems and obtain accurate results, provided arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='00519v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='NI] 2 Jan 2023 2 that sufficient data are available for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This advantage suits the increasingly heterogeneous and dynamic networks, where mathematical models may not exist or cannot accurately characterize the considered problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Therefore, it is not difficult to predict the significance of AI in 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Architectural Innovations Required for 6G Recognizing the importance of virtualization and AI, we further look into their limitations in the state-of-the-art to comprehend the architectural innovations required for 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Existing virtualization techniques mostly deal with service provision in communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, network slicing highlights available network resources, service provi- sion capability, and QoS satisfaction for various services [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' While such virtualization, with a focus on service provision, enables 5G to handle diverse coexisting services, it may not suffice for 6G since the characteristics of end user service demand can be the key to achieving user-centric networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Therefore, in the future, virtualization should focus on both the service provision capability of a network and the service demand of end users in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This will lead to the virtualization of end users in addition to the virtualization of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As for AI, existing research on AI mostly addresses specific functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', routing [24]), layers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', physical layer [25]), network segments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', access networks [26]), or applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', autonomous driving [27]) of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Meanwhile, how to integrate AI into the network architecture across different layers or network segments needs further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The scope and extent of AI-driven networks are yet to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As virtualization extends to cover both service provision and service demand while AI pervades every corner of the network, close connections between the two elements are foreseeable and can dominate the architectural needs of 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The first connection is through network and end user data [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Vir- tualization facilitates the characterization of network service provision capability, service performance, resource utilization and, in future networks, end user service demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As a result, a vast amount of data will be generated, which can be exploited to characterize the network and end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Such data, if properly managed, can empower both AI-driven networking and AI applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', object detection) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The second connection is through network control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI can be used to make decisions pertinent to virtualization, including service admission, slice establishment, dynamic virtual network func- tion orchestration, and resource scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the future, AI can also help control data collection for the virtualization of end users and extract features of virtualized end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Thus, AI has the potential to improve the efficacy and adaptivity of virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The third connection is through network resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A main motivation of network virtualization is to coordinate resource sharing among different services and thereby improve network resource utilization and service sat- isfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI-driven networking can target efficient utilization of network resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As both virtualization and AI consume computing, communication, and storage resources, they may compete for network resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' However, AI has a potential to increase the efficiency of virtualization through intelligent network planning and operations, while virtualization may increase the efficiency of AI through proper data provision and management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As a result, they should work together to enhance network resource utilization and service quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A rudiment of the above connections through data and control can be observed in the existing architecture of 5G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, the 3rd Generation Partnership Project (3GPP) introduces a network data analytics function (NWDAF) for 5G in Release 15 [30] and enablers for network automation (eNA) in Release 16 [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The architecture design provides a framework for the NWDAF to collect data from other network functions (such as policy control and network slice selection functions) and provide analytics (such as data traffic statistics and predictions) back to these network functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In 6G, the scope and level of both data collection and analytics will expand significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Most likely, network data analysis, instead of being limited to one or two specific functions, will be AI-driven and available everywhere in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Similarly, the data available for network management, instead of being limited in type, content, or format, should provide information of the network and end users as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Such expectations can be fulfilled by extending the roles of virtualization and AI in the network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Our Vision Our vision of network architecture for 6G is based on the importance of virtualization and AI, their limitations in existing networks, and the essential connections between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Specifically, we aim to design a network architecture that i) supports virtualization of the network and end users from the perspectives of service provision and service demand, respectively, ii) integrates AI in various network functions, layers, segments, and applications under a unified architecture, and, more importantly, iii) facilitates the interplay between virtualization and AI, enabling their coexistence, integration, and mutual enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To consolidate the vision, we raise the following three key questions: How to further advance virtualization beyond network slicing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' How to enable AI into every facet of a network?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' How to effectively integrate virtualization and AI through network architecture design?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In pursuit of answering the preceding questions, we develop the ideas of holistic network virtualization and pervasive network intelligence for 6G network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Holistic network virtualization advances virtualization toward 6G by incorporating network slicing and digital twin paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The former enables service-centric network management, and the latter adds a user-centric perspective to virtualization for future networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Pervasive network intelligence enables generic integration of AI into a network from the perspectives of AI for networking and networking for AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The former emphasizes the role of AI in network management, while the latter leverages network design to support AI applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In this tutorial paper, for both holistic network virtualization and pervasive network intelligence, we survey existing studies, 3 present our network architecture designs, and illustrate their benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Unifying these two components, we further introduce an overall conceptual network architecture, which fulfills our vision of unprecedentedly flexible, scalable, adaptive, and intelligent networks for 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This tutorial paper can provide useful information and benefit readers from three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First, for readers who are interested in the historical and current developments of virtualization and AI techniques, we survey the literature and provide a review of both in the context of communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Second, for readers who are exploring future direc- tions in virtualization and AI, we propose original ideas for advancing them toward 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Specifically, we illustrate designs and ideas, such as incorporating digital twins for holistic network virtualization, connected AI for network management, AI slices with training and inference separation, and hybrid data-model driven methods, throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Last, after introducing our vision of holistic network virtualization and pervasive network intelligence, we present open issues and challenges to inspire further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' There are a few surveys on virtualization and AI in the liter- ature [21], [32]–[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Regarding virtualization, Minerva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' present existing digital twin based applications in the context of IoT [32], and another survey introduces the key enabling technologies and design principles of network slicing [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Regarding AI, Boutaba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' undertake a comprehensive survey on AI applications in various areas of networking [33], and another survey focuses on deep learning (DL) based applications in wireless networking [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In comparison, this tutorial paper focuses on the vision of 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Specifically, after introducing state-of-the-art virtualization and AI techniques, we propose original designs, including holistic network vir- tualization and pervasive network intelligence, to establish a novel conceptual architecture for 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Structure of the Paper The structure of this tutorial paper is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Section II illustrates our vision of 6G networks from the aspect of holistic network virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' We review existing network virtualization concepts and techniques in Subsec- tion II-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Then, we introduce end user virtualization with a focus on digital twins in Subsection II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Lastly, we present our idea of holistic network virtualization, highlighting a six- layer virtualization architecture, in Subsection II-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Section III illustrates our vision of 6G networks from the aspect of pervasive network intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Subsection III-A presents an overview of representative AI techniques that are potentially useful for 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Subsection III-B introduces the motivation for pervasive network intelligence and presents a four-level AI architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Subsections III-C and III-D sum- marize the existing research and present our ideas on AI for networking and networking for AI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Section IV integrates holistic network virtualization and pervasive network intelligence and presents our overall vision for 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Subsection IV-A reviews related studies on archi- tectures for 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Subsection IV-B introduces a conceptual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='architecture for 6G networks that incorporates holistic net- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='TABLE I: List of Acronyms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='3GPP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='3rd Generation Partnership Project ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='5G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='5th Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='6G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='6th Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='AI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Artificial Intelligence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='AL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='AI Level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='AP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Access Point ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='API ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Application Programming Interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ARQ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Automatic Repeat-Request ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='BS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Base Station ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='C-RAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Cloud Radio Access Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='DL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Deep Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='DNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Deep Neural Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='DRL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Deep Reinforcement Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='eMBB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Enhanced Mobile Broadband ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='FL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Federated Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='IoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Internet of Things ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='IP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Internet Protocol ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ITU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='International Telecommunication Union ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='LSTM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Long Short-Term Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='LTE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Long Term Evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='mMTC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Massive Machine-Type Communications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='MIMO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Multiple-Input Multiple-Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Machine Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='NFV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Network Function Virtualization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='NN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Neural Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='NWDAF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Network Data Analytics Function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='QoE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Quality of Experience ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='QoS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Quality of Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='RAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Radio Access Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='SBS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Small Base Station ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='SDN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Software Defined Networking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='SNR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Signal-to-Noise Ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='UAV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Unmanned Aerial Vehicle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='uRLLC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Ultra-Reliable and Low-Latency Communications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='VL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Virtualization Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='VM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Virtual Machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='WSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Wireless Sensor Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='work virtualization and pervasive network intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Sub- sections IV-C and IV-D discuss the components, subsystems, and potential implementation of the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Subsections IV-E to IV-G elaborate on three types of inter- play enabled by the proposed architecture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', the interplay between digital twin and network slicing, between data-driven and model-driven methods, and between virtualization and AI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Section V identifies key challenges and open issues related to the proposed network architecture, and Section VI con- cludes this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Table I lists the acronyms used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' HOLISTIC NETWORK VIRTUALIZATION In this section, we first review virtualization techniques in existing networks and their benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Then, we introduce the idea of holistic network virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Network Virtualization The concept and techniques of network virtualization have been evolving over more than three decades [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Early research on network virtualization includes virtual local area networks motivated by facilitating different types of operations (services) in distributed systems [36], as well as providing flexible network control and improving link utilization [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Another example of network virtualization is virtual private 4 Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Introduction Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Holistic Network Virtualization Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Pervasive Network Intelligence Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A Potential Network Architecture for 6G Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Challenges and Open Issues Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Conclusions Network Virtualization End User Virtualization Holistic Network Virtualization Motivation and AI Architecture Networking for AI AI for Networking Architecture Overview Components and Subsystems Interplay between Model-Driven and Data-Driven Methods Interplay between Virtualization and AI AI Techniques: An Overview Implementation Interplay between Digital Twin Paradigm and Network Slicing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 1: The structure of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' networks, which establish efficient and secure communication links to connect geographically dispersed end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Over time, the desire for programmable network management extends to the objective of enhancing network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The advancement in cloud computing has propelled re- cent development in network virtualization, including network function virtualization (NFV) and network slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' With NFV, software instances running on virtual machines at general computing servers replace customized and proprietary hard- ware for various network functions [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' At the network core, NFV applies to functions such as switching, firewall, deep packet inspection, and session border controller [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' At radio access networks, NFV applies to frame generation, modulation, carrier allocation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The realization of NFV becomes an enabler for network slicing, which is a key network architecture innovation in 5G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Network slicing emphasizes a service-oriented perspective in network man- agement by creating multiple end-to-end virtual networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', slices, for different services on top of shared physical network infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' With network slicing, network resources are first reserved for respective services in network planning stages and later allocated to individual users in network operation stages [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The creation, adjustment, and termination of slices are based on the varying spatiotemporal distribution of service demands to provide a high level of flexibility and adaptivity in network management [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='1 Virtualization can be applied on different levels and scales in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Existing techniques include virtualization at node, link, resource, and network levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Virtual nodes are abstractions of substrate nodes in a network such as servers, routers, and switches, and typical examples of node virtualiza- tion are storage and computing server virtualization [41], [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Virtual links are the logical channels that interconnect virtual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Virtual resources are abstractions of computing, mem- ory, storage, and communication resources in a network [43], while physical resources at different locations can form virtual 1SDN and C-RAN are also closely related to network virtualization since virtualization significantly simplifies and expedites their realization in modern wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' resource pools [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, the virtualization of a network function is the execution of a network control or service function by running software, supported with necessary resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A virtual network is the combination of virtual nodes and links with proper virtual resource allocation for a service request to meet its QoS requirements, supported by necessary networking protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Besides the aforementioned works, more representative research works on node, link, re- source, and network virtualization are summarized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Regardless of its level and scale, virtualization in the context of networking typically demonstrates the following character- istics: Abstraction - Abstraction provides a high-level overview of a network while hiding details of the underlying physical network entities (nodes, links, or networks) or resources [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This simplifies network management and facilitates flexible service provision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Co-existence - Multiple virtual entities corresponding to a shared physical entity co-exist, or multiple virtual resource pools co-exist on the same physical resource pool [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This enables service-oriented virtual networks and improves network resource utilization efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Isolation - Coexisting virtual entities corresponding to the same physical entity should function independently [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This is necessary for guaranteeing service reliability, security, scalability, and QoS satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Both academia and industry have spent a significant amount of efforts on network virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For virtualizing core networks, some works leverage SDN techniques to separate the control and data planes through different protocols or application programming interface (API), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', OpenFlow [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Furthermore, network virtualization has been extended to radio access networks (RANs), and several frameworks for RAN virtualization are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A SoftRAN framework enables both centralized and distributed RAN control based on the time sensitiveness of control decisions [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Another framework, FlexRAN, offers a hierarchical architecture for real-time RAN control and incorporates a flexible API to separate control and data planes in RANs [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Initiated by industry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' such as 5 TABLE II: Some Representative Works on Node,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Link,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Resource,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' and Network Virtualization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Work ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Scenario ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Research Focus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Objective ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Virtualization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[44] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Edge computing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Virtual edge node placement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Low-cost placement and fast response to user requests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[45] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Cloud computing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Virtual machine (VM) place- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Reliable VM placement and routing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[46] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='IP network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Virtual node/router as IP over- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='lay ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Practical IP-level resilience to link failures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[47] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Wireless sensor network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='(WSN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Architecture for sensor virtu- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='alization in WSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Multiple applications share the same WSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[48] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='C-RAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Clustering of access points ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Forming user-specific virtual base stations given QoS requirements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Virtualization ' metadata={'source': 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with fairness among multiple mobile network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='operators ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[55] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='RAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Radio resource virtualization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Delay-bounded QoS provisioning through radio resource virtualiza- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='tion ' metadata={'source': 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caching resources to support applica- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='tions with different QoS requirements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Virtualization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[57] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='5G core network with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='SDN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Network function chain em- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='bedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Minimizing embedding cost subject to network resource constraints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[58] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Core network with SDN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Network function chain em- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='bedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Minimizing total flow in the network subject to network resource ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='constraints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[59] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='C-RAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Slice request admission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Maximizing the revenue of the C-RAN operator subject to network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='resource constraints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[60] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Heterogeneous ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='wireless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Dynamic radio resource slic- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Maximizing network utility through optimal bandwidth slicing and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='user association ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[61] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='5G RAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Radio resource allocation in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='RAN slicing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Satisfying QoS requirements by proper resource mapping and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='scheduling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[62] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='IoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Service-oriented ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Privacy-preserving slice selection and secure access of service data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='AT&T and China Mobile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Open-RAN (O-RAN) is proposed as an open-source and open-interface platform to support RAN virtualization [68],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' which can incorporate AI and provide APIs for data-driven networking [69]–[71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The adoption of virtualization techniques renders modern networks programmable, flexible, and scalable, which signifi- cantly increases cost effectiveness in network deployment and operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Due to these benefits, it is foreseeable that advanced virtualization techniques will be essential to 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Meanwhile, the existing scope of network virtualization is limited in the sense that virtualization techniques mostly focus on network infrastructure and resources, yet less attention is given to end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In 6G, end user virtualization will become necessary for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First, with increasingly diverse end user devices, resource-demanding services, and heterogeneous and dynamic networks, providing QoE guarantee for end users will become more challenging in the era of 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Accurate characterization and abstraction of end users, which necessitate end user virtu- alization, can be a precondition to QoE satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Second, as AI will be a highlight of 6G, extensive user data are required to fuel AI services and AI-based network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Given such need for data, end user virtualization can be a competitive approach for collecting, managing, and processing data from end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' End User Virtualization Until recently, only a few works study end user virtu- alization in the context of networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' One early example relevant to end user virtualization is network-hosted avatars, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', virtual agents, of end users for applications such as file downloading when the users are offline [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Another example is virtual objects, proposed as a component in Internet of things (IoT) platforms [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The motivation is to handle the heterogeneity of physical objects (end users) via virtualization and to facilitate the provision of services to end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As a potential paradigm to enable end user virtualization, digital twin attracts much attention lately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The concept of digital twin was originally conceived by Michael Grieves for product life-cycle management in industry in 2003 [74], [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Later, NASA and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Air Force Vehicles developed a digital twin paradigm for vehicles to forecast their remaining usable life and the mission success probability [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A digital twin is characterized by a full digital representation of a physical object or a process and real-time synchronization between the physical object or process and its corresponding digital replica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Digital twins can contain a large volume of data from physical objects or processes for advanced analytics, and the analytical results can be used to improve the performance of the corresponding physical objects or processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Exemplary digital twins in general application scenarios, as well as potential requirements for the digital twins to enable big data analytics, are discussed in [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Potential implementation of digital twins representing IoT devices in industrial systems is proposed in [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Other representative research works on digital twins are summarized in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Most existing research on digital twins in the network field focuses on applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', distributed clock synchroniza- tion [79] and computation offloading [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In comparison, the study of digital twins from the perspective of network 6 architecture and network management is limited at the mo- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='2 A digital twin based cloud-centric network architecture is proposed in [83], where digital twins of end users hosted at the network edge play the role of communication assistants or network data loggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Digital twin appears to be an intuitive solution to end user virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Nevertheless, extending the existing network virtualization, represented by network slicing, to end users is not straightforward, given the target of flexible and efficient network management and service provision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, it is trivial to simply use node-level virtualization and to represent end users as virtual data sources or sinks in a virtual network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover, while end users may possess communication and computing resources, resource-level virtualization does not characterize user-specific properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', location and mobil- ity, or service-specific properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', data traffic variations, of end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' It is necessary to understand potential benefits, requirements, and implementation of digital twin based end user virtualization, with a particular focus on the integration of digital twin and existing network virtualization frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' There are potentially two-fold benefits of digital twin based end user virtualization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', extensive end user data and powerful network emulation capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' While the virtual- ization of network infrastructure and resources characterizes the network status and service provision capabilities, digital twins of end users can provide extensive data regarding service demand and user QoS/QoE satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Such data can play a significant role in network management through facilitating well-informed network planning and operation decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover, the real-time or near real-time synchro- nization between end users and their digital twins enables powerful network emulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, multiple instances of the same virtual network can be created, with real-time end user information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', location and data traffic volume, provided to all instances through synchronized end user digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='3 Different network planning or operation strategies can be applied and emulated in different instances, while each instance remains synchronized with the real-world network environment through the information provided by the digital twins of end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To take part in network virtualization, digital twins of end users should satisfy the following requirements: Flexible: The abstraction of end users into digital twins must be sufficiently flexible to represent heterogeneous physical devices (such as smartphones, vehicles, and industrial sensors) and serve various applications (such as virtual reality gaming, autonomous driving, and industrial automation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Compatible: The end user virtualization based on digital twins should complement and enhance the state-of-the-art network virtualization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', network slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, digital twins of end users should provide data to support 2Some works focus on distributed networks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', vehicular networks, and adopt digital twins as an approach for network virtualization instead of end user virtualization [81], [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 3The emulation can apply to a virtual network segment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', the network edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' various network slices, while each slice may only have access to a subset of data pertinent to that slice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Customizable: The attributes of digital twins should be customized and updated based on the corresponding service, network traffic, resource utilization, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, the amount and types of data included in a digital twin should be adaptable rather than fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, while the focus of digital twins is placed on end users, digital twins should be able to represent other network entities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', unmanned aerial vehicle (UAV) mounted mobile base stations (BSs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, network resource consumption from creating and maintaining digital twins should be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Noting the aforementioned benefits and requirements, we aim to answer the following key questions with respect to the implementation of digital twins: Location: Where should digital twins be hosted in a network?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Affiliation: Should digital twins exist within or outside network slices?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Data: What data attributes pertinent to networking should be included in a digital twin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' How much historical data should be included for a specific attribute?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Should predicted user information be included?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Synchronization: How to determine the frequencies of up- dating various data entries of a digital twin by acquiring new data from the physical object?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Control: Who should determine and update digital twin models and based on what information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the next subsection, we propose a novel conceptual architec- ture for holistic network virtualization, which integrates digital twins and network slicing, and delve into the above questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Holistic Network Virtualization We propose a novel virtualization architecture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', holistic network virtualization, for integrating digital twins into net- work virtualization, in order to improve network management and service provision capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The proposed virtualization architecture consists of six layers and is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 2, in which virtualization layer (VL) 1 is the bottom layer for data collection and VL 6 is the top layer for digital twin model control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The outline of each layer is given as follows: VL 1 – Data Collection: Data required for the digital twin representation of selected end users are collected from the corresponding physical entities following prescribed data precision, uploading method, collection frequency, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The data are collected via access points, and the data collection is controlled by local controllers deployed at network edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' VL 2 – Level-One Abstraction: Based on the current digital twin model from the digital twin model control layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', VL 6), which determines the content and format of data included in every digital twin, digital twins are formed and updated using data collected by VL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The abstraction may include the aggregation of data from different sources, the update of historical data, and the creation of digital twins for new or additional end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The digital twins created in this layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='TABLE III: Some Related Works on Digital Twins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Work ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Application ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Type of Physical Ob- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ject ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Role of Digital Twin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Target of Using Digital Twin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[84] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Underwater ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='for ocean observation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Underwater ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='sensors/actuators ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Monitoring and testing observation sys- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='tem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Visualizing an ocean observation system and enhancing simulations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[85] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Edge computing for in- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ternet of vehicles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Vehicles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Collecting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='sharing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='about vehicles and surroundings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Empowering computing offloading by facilitating data analytics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[86] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='5G network slicing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Network slices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Predicting and monitoring slice perfor- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='mance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Assisting autonomous network slicing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[87] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Smart factory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Workstations in a con- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='veyor system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Evaluating and validating control strate- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='gies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Implementing intelligent conveyor systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Smart city ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Road infrastructure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Monitoring roads and detecting vehi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='cles/persons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Supporting smart city applications through gathering and processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[89] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='IoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Objects with sensing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='capability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Storing data for detecting events and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='recognizing behaviors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Facilitating synthetic sensing through situation awareness and ex- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='plainability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[90] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='0 Industrial machinery Generating training dataset and simula- tions Achieving accurate anomaly detection with limited labelled data [91] Smart healthcare Patients Handling data for analysis and develop- ing AI models Improving healthcare operations [92] Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='0 Technical assets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', machine, environment) Integrating knowledge from model and data for simulations Enhancing simulation-based systems engineering [93] Mobile edge comput- ing Real-world network environments Training learning algorithms and moni- toring network environments Enabling learning for optimizing user association, resource alloca- tion, and offloading [94] Industrial 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='0 Products,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' workstations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' conveyor belts Data sharing and control of security- critical processes Building a security architecture based on state replication and synchronization [95] Cyber-physical systems Generic physical de- vices Monitoring,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' diagnostics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' and prognos- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='tics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Supporting applications such as context aware interaction and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='driving assistance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[96] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='IoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Generic physical sys- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='tems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Managing context information and self- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='adapting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Increasing autonomy and enhancing cooperation through autonomic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='digital twins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[97] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Welding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='manufactur- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Human-robot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='interac- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='tion systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Monitoring welding robot and enabling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='simulations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Visualizing welder behavior and training welders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[98] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Smart manufacturing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Job ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='shop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='scheduling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Obtaining scheduling data and simulat- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ing scheduling strategies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Enabling timely response and reducing scheduling plan deviation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[99] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Smart manufacturing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Manufacturing systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Predicting and verifying the system per- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='formance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Increasing autonomy and enhancing fault diagnosis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[100] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Smart building ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Photovoltaic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='energy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='conversion units ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Estimating the status of photovoltaic en- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ergy conversion unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Improving the accuracy of fault detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[101] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Internet of vehicles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Vehicles and road side ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='units ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Monitoring the real-time status of vehi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='cles and road side units ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Supporting network resource management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[102] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Mobile edge caching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Vehicles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Capturing the social characteristics of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='vehicles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Improving the effectiveness of cache management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='are level-one digital twins,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' representing individual end users,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' and hosted at servers connected to local controllers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' VL 3 – Local Processing and Control: The data from level-one digital twins are processed at network edge for predicting behaviors of individual users, such as user data traffic and mobility patterns, and making user-level service decisions, such as computing offloading, content delivery, and link-layer protocol adaption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Local processing may also include emulations of an edge network or a part of it based on level-one digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Local control may include further data aggregation from level-one digital twins for VL 4, the migration of digital twins based on user mobility, and the selection of end users for digital twin representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Similar to the case of VL 2, the local processing and control occur at servers affiliated with local controllers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' VL 4 – Level-Two Abstraction: The aggregated data from VL 3 is sorted into service-specific data for respective net- work slices in VL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Additional data that describe slice configuration, slice resource utilization, slice service level agreement satisfaction, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', are generated for each slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Then, the aforementioned data are abstracted to form or update the digital twins of various slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The digital twins created in this layer are level-two digital twins, which are associated with virtual networks (slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The level-two digital twins are hosted at servers connected to the centralized controller of the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' VL 5 – Slice-Level Processing and Control: The data from level-two digital twins of network slices are processed for service-specific prediction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', spatiotemporal service demand distribution forecast, or slice-level decision making, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', planning and operation decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Slice-level processing may include emulations of an end-to-end slice or a part of it based on level-two digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Slice-level control may include slice admission, resource reservation, and slice service coverage control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Similar to the case of VL 4, the service- level processing and control occur at servers affiliated with the centralized controller of the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' VL 6 – Digital Twin Model Control: This layer determines and updates the models of level-one and level-two digital twins based on available network resources for digital twins, the performance of network management and service provision decisions derived based on the current digital twins, and the dynamic spatiotemporal service demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, VL 6 determines data precision, synchronization frequencies for different data attributes, the amount of historical data contained in the digital twins for each attribute, and the inclusion of predicted user information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, this layer decides the subset of data in level-one digital twins that each slice can access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The digital twin model control also occurs at servers affiliated with the centralized controller of the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The level-one digital twin model configured by VL 6 may include the following data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' which shall be collected by the local controllers from end users at VL 1: (1) connectivity and channel information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' such as the AP(s) that an end user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Digital Twins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='VL 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Digital Twin Model Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='VL 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Slice-Level Processing and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='VL 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Level-Two Abstraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='VL 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Local Processing and Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='VL 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Level-One Abstraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='VL 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Data Collection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Digital Twin Model Update ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Level-Two Digital Twin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Slices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Level-One Digital Twin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Physical Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Core ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Network Slicing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Data Flow (Physical) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Control (Physical) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Data Flow (Cyber) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Control (Cyber) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Gateway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Local Controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Centralized Controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 2: The conceptual six-layer virtualization architecture for holistic network virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' is connected to and the channel state information for each connection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' (2) service information, such as active service types, data traffic volume of each service, and QoS satisfaction of each service;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' (3) user information, such as user profile, user location and mobility, network resources allocated to the user, and the local computing and caching capabilities of the user;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' and (4) additional use case-specific information, such as motion sensor readings for augmented reality interactive gaming or operation log for industrial IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The level- two digital twin model configured by VL 6 may include the following data, which shall be collected or generated by the centralized controller: (1) slice service demand, such as the number of service requests and the spatiotemporal service request distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' (2) slice resource configuration, such as the reserved communication, computing, and caching resources for the slice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' (3) slice performance, such as the slice service level agreement satisfaction, slice resource utilization, and slice energy consumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' (4) slicing strategy, such as the method or algorithm used for network function deployment, resource reservation, and resource scheduling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' (5) additional use case-specific information, such as UAV trajectory config- uration for UAV-assisted networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Note that different end user digital twin models are applicable to different types of end users, and each network slice may have a uniquely defined slice digital twin model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For example, the digital twins of vehicles and industrial IoT devices most likely contain different data, and the digital twin models may differ between slices for industrial IoT and those for smart home or between slices of different network operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Accordingly, the need for customization necessitates the digital twin model control in VL 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the conceptual virtualization architecture, VL 1 to VL 3 interface with the local controllers in the network, VL 4 and VL 5 interface with the centralized controller of the network, and VL 6 interfaces with both the local controllers and the centralized controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This architecture fully exploits the two benefits of digital twins, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', providing extensive data for net- work management and enabling powerful network emulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' It also satisfies the aforementioned requirements for digital twins in terms of flexibility, compatibility, and customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Last but not least, it answers the key questions regarding the implementation of digital twins raised in Subsection II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' With the architecture design in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 2, digital twins and network slicing are integrated in the idea of holistic network virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Network slicing incorporates existing network virtualization techniques such as NFV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Digital twins enhance network slicing by providing organized and customized end user data to slices and by further abstracting slices into level- two digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The design of two-level digital twins avoids extra resource consumption from creating and maintaining multiple digital twins of the same user for different slices and the resulting burden of synchronizing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Instead, each slice has access to a subset of data from level-one digital twins pertinent to either the corresponding service or general user information such as location and mobility, and the pertinent data are further aggregated to the level-two digital twins for that slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In this architecture, network slicing conforms to service-centric network management, while digital twins add a user-centric perspective to the virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Specifically, level-one digital twins characterize end users and their service demands, and level-two digital twins characterize network ser- vice provision capability, network performance, and network resource utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Overall, the digital twin paradigm and network slicing jointly support network management and ser- vice provision, while the network configures digital twins and network slices as needed, depending on network dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 9 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Holistic Network Virtualization: A Summary In this section, we have reviewed the existing scope and techniques of network virtualization, identified the insuffi- ciency of current network virtualization, introduced the idea of holistic network virtualization to incorporate network and end user virtualization, and developed a six-layer virtualization architecture for holistic network virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The virtualization of resources, network functions, and networks in 5G is expected to remain important in 6G, since they contribute to flexible and adaptive network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Meanwhile, the virtualization techniques in 5G, represented by network slicing and NFV, mostly focus on network vir- tualization from the perspective of service provision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In 6G, it will be essential to extend the scope of virtualization and incorporate end user virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The digital twin paradigm is a promising solution to end- user virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In 6G, digital twins can be used for characterizing the status and the service demand of individual end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The study of digital twins in the context of 6G networks is still in an initial stage, and various definitions or implementations exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In our vision of holistic network vir- tualization, digital twins are configurable assemblage of data, including both historical and real-time data and both collected and generated data, for describing end users, infrastructure, or network slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover, the corresponding data collection and processing are also configurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To consolidate holistic network virtualization, we have proposed a six-layer virtualization architecture for 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The architecture provides a reference design for systematically integrating digital twins and network slicing and answers important questions related to digital twins in 6G networks, including where are they hosted, what data do they contain, and how to manage them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' PERVASIVE NETWORK INTELLIGENCE Pervasive network intelligence is the second element of our vision for 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In this section, we first present an overview of existing AI techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Then, we introduce the motivation and propose a four-level architecture for pervasive network intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Next, we elaborate the idea of pervasive network intelligence from the perspectives of AI for networking and networking for AI, and review related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Rather than surveying specific AI techniques, this section focuses on the architecture and methods of pervasive network intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI Techniques: An Overview The idea of AI is to design intelligent machines or sys- tems to demonstrate human intelligence and perform tasks as humans do or even better [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The advancement of machine learning (ML) has facilitated the success of AI in both academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Applications supported by ML techniques, such as computer vision and natural language processing, can achieve beyond human-level accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Lately, for its potential in enabling intelligent networks, AI has received significant attention in the research field of wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' ML techniques can be categorized into three types: un- supervised learning, supervised learning, and reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In terms of learning structures, the techniques can be subdivided into centralized and decentralized techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' We list common ML techniques used in wireless networks in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Unsupervised learning evaluates features and patterns hid- den in data for data analysis, such as prediction, without using a labeled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' One popular application of unsupervised learning techniques is data clustering, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', k-means [103] and mixture models [105], for solving network planning problems, such as cluster-forming in wireless sensor networks [104] and small-cell deployment [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Neural networks can be adopted to facilitate novel unsupervised learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For example, neural network-based autoencoders can learn the compressed features of input data with a limited number of neurons and can be leveraged for data prediction, such as traffic forecasting [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Supervised learning exploits the mapping between the in- put and output data via a given labeled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Supervised learning techniques can derive a mapping function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', a training model, from the input data to the labeled output data in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Through applying a training model, the output corresponding to a new input can be evaluated, which can be utilized for decision making or prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A typical method for supervised learning is using deep neural networks (DNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' DNNs use layers of artificial neurons to estimate a non-linear correlation between the input and the output data and iteratively improve the estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' There have been many successful applications of DNN techniques in communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For example, convolutional neural networks (CNNs) utilize convolutional and pooling layers to identify the correlation of multi-dimensional input data and have been applied in modulation classification [112];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' recurrent neural networks (RNNs) explore the correlation among a sequence of the data and have been widely adopted for traffic prediction [113] and wireless channel modeling [133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Reinforcement learning iteratively learns the optimal policy by interacting with the environment, sensing network states, and evaluating feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The goal is to maximize a cumula- tive reward in a dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Deep reinforcement learning (DRL), which combines DNN and reinforcement learning techniques, is used extensively in resource manage- ment to solve complex decision-making problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In DRL, neural networks play the role of approximators to store high- dimensional states or actions, which enables DRL to solve complex problems efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' DRL has been widely used for network optimization [134], resource allocation [19], [118], [135], and user association [116], [121], [123] in wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' With the development of mobile edge computing, dis- tributed AI has been developed to harvest computing resources at network edge and reduce communication overhead due to data collection and exchange [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The learning models can be trained and evaluated at network edge in a semi- or fully- distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Specifically, federated learning (FL), as one of the most popular distributed learning techniques, trains models with data distributed over network edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A centralized 10 TABLE IV: Common ML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Unsupervised Learning Supervised Learning Reinforcement Learning Centralized Learning Algorithms K-means [103],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' [104] Mixture models [105],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' [106] Autoencoders [107] Generative adversarial network [108],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' [109] Support-vector machine [110] Logistic regression [111] Deep neural network [107],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' [112],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' [113] Deep Q-learning [114]–[116] Policy gradient [117]–[119] Actor-critic [120],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' [121] Deep deterministic policy gradient (DDPG) [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' [122],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' [123] Distributed Learning Algorithms Federated learning [124],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' [125] Split learning [126] Multi-agent reinforcement learning [127]–[130] controller aggregates locally-computed learning models and updates parameters in the learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Due to such de- centralized model training, FL is capable of preserving privacy and can be applied in privacy-sensitive network management scenarios [137], [138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, multi-agent reinforcement learning has been developed to implement reinforcement learn- ing in a distributed manner, which aims to handle scenarios in which network agents cannot obtain sufficient information from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Multi-agent reinforcement learning tech- niques can be used, for example, to solve resource allocation problems in heterogeneous networks [129], [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Motivation and AI Architecture In 6G, AI is expected to penetrate every facet of the network including end users, the network edge, and the cloud, resulting in pervasive network intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Such trend is due to advancements and innovations in the areas of ML, data collection, edge and cloud computing, and programmable net- work control in recent decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As such, AI will fundamentally transform modern networks in many aspects and foster a myriad of exciting applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The AI applications can be categorized into management- oriented and service-oriented applications, which are detailed as follows: Management-Oriented AI Applications - In these ap- plications, AI is used as a tool for network manage- ment, such as transmission power allocation in cellu- lar networks [139] and resource reservation in network slices [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI techniques, such as reinforcement learn- ing, have the potential of handling complicated decision making problems in a dynamic network environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Resorting to AI techniques, the management-oriented AI applications can analyze a large amount of network data, make real-time network management decisions, and then update network management policies based on the newly analyzed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Hence, for such applications, the key issue is how to leverage advanced AI techniques to manage and enhance complex networks, which falls into the scope of AI for networking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Service-Oriented AI Applications - In these applications, AI is offered as services for end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Fuelled by powerful computing servers and well-curated datasets, AI techniques, especially DL, can outperform traditional techniques in a wide range of applications, such as environmental perception in autonomous driving, audio recognition in intelligent healthcare, and object detection in mobile virtual reality [140]–[142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, an AL 2: Edge-Hosted AI AL 1: End User-Hosted AI Management-Oriented Management-Oriented Service-Oriented Service-Oriented AL 3: Edge-Hosted AI AL 4: Cloud-Hosted AI AI for Networking Networking for AI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 3: An illustration of the four-level AI architecture for pervasive network intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI-based YOLO algorithm can detect objects with a high accuracy [143], [144], and the state-of-the-art DL- based face recognition algorithm can achieve an accuracy of 99% or higher [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Facilitating service-oriented AI applications in a network consumes a large amount of network resources, including storage and computing re- sources for model training/inference, and communication resources for data collection and model uploading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Hence, for such applications, the key issue is how to design and optimize the network to support emerging AI services, which falls into the scope of networking for AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Note that the scope of AI in 6G includes AI for networking and networking for AI, which is larger than that in 5G, as the latter simply focuses on applying AI in communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' An AI architecture is needed to characterize AI’s different functionalities in different network segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the literature, there are a few studies on the AI architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Edge intelligence (or edge AI) is represented in six levels based on the amount and path length of data offloading [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover, edge intelligence can be categorized into two parts: AI for edge (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', to enhance and optimize the network edge with AI techniques) and AI on edge (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', to carry out AI models on the network edge) [146], [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Different from these works on edge intelligence, our work focuses on a broader scope of pervasive network intelligence and categorizes it into multiple levels based on AI’s locations and functionalities in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 3, we propose a four-level AI architecture, in which AI levels (ALs) 1 and 2 focus on service-oriented applications, and ALs 3 and 4 aim at management-oriented applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' We describe each level in detail as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AL 1 - End User-Hosted Service-Oriented AI: Utilizing local data and computing resources at end users, end user- hosted service-oriented AI applications are offered as services 11 for end users by processing AI tasks locally, such as next word prediction in mobile keyboards [148], user traffic de- mand prediction [149], and vehicle trajectory prediction [150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' When computing resources of end users are insufficient for computation-intensive AI tasks, partial computation workloads can be offloaded to nearby edge servers for collaborative processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AL 2 - Edge-Hosted Service-Oriented AI: Residing at network edge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', Wi-Fi access points and BSs) close to end users, edge-hosted service-oriented AI applications are offered as low-latency services for end users, such as face recognition in video surveillance [151] and object detection in virtual reality [152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To support edge-hosted service-oriented AI applications, service demand data from end users are collected, stored, and analyzed, and then the analytical results are utilized for service provision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AL 3 - Edge-Hosted Management-Oriented AI: At this level, AI is hosted at local controllers at network edge for network management that is executed in real time, such as spectrum allocation, content caching [153], and computation offloading [154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Specifically, the edge-hosted management- oriented AI is to allocate network resources to network nodes for supporting services, including AI services at ALs 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, the edge-hosted management-oriented AI can be used to perform service migration across edge networks to guarantee service continuity for high-mobility users, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', vehicular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AL 4 - Cloud-Hosted Management-Oriented AI: Cloud- hosted management-oriented AI resides at the centralized con- troller in the cloud for network management that is executed once every several minutes or hours, such as slice admission control [155] and virtual network function deployment [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Since the cloud possesses abundant computing and storage resources, powerful and complex AI models can be trained and deployed for managing large-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Next, AI for networking is elaborated in Subsection III-C to illustrate AI’s role in network management, and networking for AI is discussed in Subsection III-D to illustrate AI service provision in 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI for Networking In this subsection, we discuss how AI techniques can support network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' We first review existing works on AI-based network slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Then, we introduce our idea of connected AI solution for AI-based network slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 1) AI-Based Network Slicing: Network slicing includes two stages: network planning stage for resource reservation and network operation stage for resource scheduling [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the network planning stage, network resources are reserved for network slices on a large time scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', from several minutes to several hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the network operation stage, the reserved resources of each slice are allocated to end users on a small time scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', from several milliseconds to several seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Due to network dynamics such as spatiotemporally changing network traffic, it can be difficult for model-based solutions to attain the optimal network slicing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' By contrast, AI techniques can characterize network dynamics by analyzing the collected network data and obtain the optimal network slicing strategies accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Next, we review AI-based net- work slicing, taking into account the interplay between the planning and operation stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Representative research works on AI-based network slicing are summarized in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' On a small time scale, a local controller collects data and provides resource scheduling strategies to allocate resources reserved for each slice to end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Specifically, the local controller determines resource scheduling strategies based on two factors: the amount of resources reserved for each slice, which is determined by the centralized controller, and the in- stantaneous user data from level-one digital twins pertinent to that slice, such as service type, user location, and user mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The main challenges of determining the optimal resource scheduling strategies are two-fold: a large number of end users and service demand dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI techniques have potentials to cope with both challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First, to schedule resources for a large number of end users, unsupervised learning methods, such as k-means [167] and DNN based autoencoders [107], can be utilized to classify end users according to their service demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Similar resource scheduling strategies can be applied to end users with similar service demands, which facilitates scalable network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, end users in close proximity and with similar mobility patterns may experience similar channel statistical behaviors, and the same power control policy can be applicable to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Second, to deal with network dynamics, reinforcement learning can be applied for generating adaptive resource scheduling strategies [168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Rein- forcement learning iteratively allocates resources to maximize a long-term reward function and updates the reward function based on feedback from the network environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover, reinforcement learning can be combined with DNNs, such as recurrent neural networks [27] and conventional neural networks [123], to analyze the spatiotemporal pattern of user data for finding the optimal resource scheduling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' On a large time scale, local controllers aggregate the col- lected user-level data to service-level information from level- two digital twins, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', slice digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Utilizing information from slice digital twins, the centralized controller reserves network resources for each slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The challenges of resource reservation are two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First, making proactive resource reservation that can avoid either resource over-provisioning or under-provisioning is challenging with time-varying network traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Second, the strategies for resource reservation and scheduling are coupled, which further complicates resource reservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI techniques can cope with these challenges as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To address the challenge of proactive resource reser- vation, supervised learning, such as long short-term memory (LSTM) networks, can be used to exploit the features of historic network traffic loads and predict traffic loads in near future [149], [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The centralized controller can then use the predicted traffic loads for proactive resource reservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To handle the correlation between resource reservation and scheduling, reinforcement learning can be adopted to reserve resources while considering network operation strategies as a part of the dynamic network environment [19], [135], [164].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' an option-based hierarchical reinforcement learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='technique can be a potential solution for jointly optimizing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='TABLE V: Representative Works on AI-based Network Slicing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Work ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Research Focus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Objective ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='AI Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Planning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[107] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Network capacity prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Forecasting the capacity for individual virtual networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Deep neural network based autoen- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='coder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[86] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Virtual representation for net- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='work slices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Capturing the relationships among slices and monitoring the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='end-to-end performance in dynamic network environments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Graph neural networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[157] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Resource reservation adjust- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Maximizing the overall reward obtained from the tenants of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='slices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Deep dueling neural networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[158] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Bandwidth allocation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Jointly maximizing spectrum efficiency and the QoS require- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='ment satisfaction ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Generative adversarial network and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='deep Q network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[159] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Bandwidth allocation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Jointly maximizing spectrum efficiency and overall service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='level agreement satisfaction ratio of slices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Long short-term memory and advan- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='tage actor-critic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[160] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Traffic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='re- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='source provisioning ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Computation offloading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Minimizing average computing time of services and maxi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='mizing user computing experience ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Deep Q network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[162] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Slice selection and channel al- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Minimizing the power consumption of wireless transmission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='for a sliced fog-RAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Reinforcement learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[163] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Content ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='caching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='placement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='and delivery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Managing caching resources to maximize cache hit ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='while satisifying resource reservation constraints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Deep Q network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[164] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Inter-slice coordination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Maximizing long-term payoff from the competition among ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='service providers through resource orchestration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Deep Q network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[165] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Inter-slice coordination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Maximizing QoS satisfaction ratio for slices by scheduling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='transmission power and sharing resources among slices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Multi-agent deep Q learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Two-Stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Interplay ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[19] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Computing resource alloca- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='tion in vehicular networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Allocating spectrum and computing resources for slices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='while minimizing computing service delay ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Deep deterministic policy gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[166] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Cross-slice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='admission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='congestion control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Maximizing operator revenue by resource reservation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='adjust reserved resources in real time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='State-action-reward-state-action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='(SARSA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='resource reservation and network operation policies and ad- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='dressing network dynamics in both stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This technique has been used to tackle complex DRL problems by grouping decision variables according to decision time scales [169] or decision-making agents [170] and then determining the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Through this novel reinforcement learning technique, the complex correlation between resource reser- vation and scheduling strategies can be obtained iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To apply this technique in network slicing, the centralized controller can select the resource reservation strategies on a large time scale, and local controllers find optimal resource scheduling strategies on a small time scale, thereby jointly optimizing both the resource reservation and the scheduling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 2) Connected AI Solution for Network Management: Ex- isting AI applications on network management mostly focus on individual control functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, learning-based autoencoders can achieve reliable transmission power control for high-speed data transmission with limited channel state in- formation [171], and DNNs can select medium access control protocol parameters with low communication and processing overhead [172], [173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Although various AI techniques have been proposed for network management, AI models among network control functions are usually isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Such isolation may result in inefficient and redundant data processing, which brings up a pressing need for integrating the AI models in AI-based network control functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' There are three types of solutions for integrating AI mod- els [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the first type of solutions, the entire network is viewed as a black box, where a single AI model characterizes the entire network and generates network control policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Such structure simplifies decision-making processes in net- work management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' However, training the single AI model can be extremely difficult due to high-dimensional input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Then, the second type of solutions adopts different AI models in a network for different network control functions, and the AI models are generally independent on each other to reduce the complexity of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' However, this approach neglects the correlation and interplay among network functions and thus cannot obtain a global-optimal network management strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover, network data may be repetitively processed by different AI models with similar network functions, which degrades network management efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, AI models for user mobility management and computing service migration would repetitively analyze end user mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In contrast, the third type of solutions, namely connected AI, exploits the correlations among network control functions, connects their AI models, and allows them to jointly make network control decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The connected AI solution offers benefits in integrating AI models by highlighting the interplay among them and balancing training complexity and network performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Therefore, the connected AI solution has great potential in facilitating AI-based network slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' However, existing research on the connected AI solution is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' How to apply connected AI solution to network management requires further studies [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Recent advancements in distributed learning techniques fa- cilitate the development of a connected AI solution for network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Model partition, investigated in [174], can divide a global DNN into multiple sub-neural networks (sub-NNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The sub-NNs can reside at different network entities, accord- ing to the available computing and communication resources, and communicate with each other [175], [176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Furthermore, the idea of nested DNN, which allows sub-NNs to have their own functionalities while contributing to the global DNN for model inference and training, has been proposed and evaluated in [177] and [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Using the above two techniques, each sub- NN can perform a specific network control function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Accord- ingly, multiple sub-NNs can collaboratively fulfill common control functions, thereby applying the connected AI solution to network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Based on the above advanced DNN techniques, we present 13 BS Power Control DNN-Based Channel Estimator Power Allocation Scheduler Intelligent Module .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' BS Power Control .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Computing Offloading SBS 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' BS Power Control .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Computing Offloading SBS 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Handover .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Service Migration MBS Computing Offloading Computing Offloading Service Migration SBS 1 SBS 2 MBS Information Exchange .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Intelligent Module Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 4: The connected AI solution for network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' our idea of applying the connected AI solution to network management next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The control functions for network manage- ment are encapsulated into intelligent modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' An intelligent module can be implemented solely by a DNN or cooperatively by a DNN and conventional model-based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' An example is shown in the upper right corner of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 4, in which the intelligent module for power control includes a learning- based channel estimator and a model-based power allocation scheme, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', water-filling power allocation [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover, the DNN in each intelligent module can play the role of a sub-NN of a global DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The intelligent modules connect with each other to share information, such as their outputs and gradient information in model training, and aggregated user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Via the intelligent modules, control functions can manage the network in a divide-and-conquer manner to avoid the complicated model training required for layer-free AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' With model partition and nested DNN techniques, multiple network control functions can cooperatively make control decisions to achieve globally optimal network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 4 shows another example of the connected AI design, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', supporting mobile edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' We explain the design using the case of vehicular networks as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Note that other networks can use the same or a similar design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Small base stations (SBSs), as edge servers, can process computation tasks offloaded by vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Intelligence modules at SBSs provide computing offloading decisions, including the computing tasks to be offloaded, transmit power for offloading, task scheduling, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', based on network status and computing offloading requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Due to the high mobility of vehicles and the limited communication coverage of the SBSs, computing tasks are often migrated among the SBSs, referred to as service migration, and migration decisions are determined by a macro base station (MBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Service migration and computing offload- ing decisions are highly coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For example, the chance of service migration increases when vehicles offload more computing tasks to an SBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, service migration requires the collaboration of multiple SBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In our idea of connected AI, service migration and computing offloading decisions are jointly determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Specifically, we split the DNN into multiple sub-NNs by DNN splitting and nested DNN techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Some sub-NNs are deployed at the SBSs to provide computing offloading decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' These sub-NNs are also connected with a sub-NN deployed at an MBS, which can be leveraged to make migration decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In this example, the input of the service migration module includes the output of intelligent modules at the SBSs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', computing offloading decisions and the parameters of sub-NNs, and the output of the service migration module is the service migration policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In this way, the intelligent modules can cooperate to make decisions for mobile edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Networking for AI In addition to managing networks, AI can function as services, namely AI services, which reside at ALs 1 and 2 in the proposed AI architecture in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Networking for AI is to design and optimize networks to facilitate AI services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In this subsection, we first introduce the motivation of networking for AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Next, existing works are reviewed, and research challenges are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Finally, the idea of AI slice is proposed and elaborated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 1) Motivation: Networking for AI is attracting great atten- tion in both academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In academia, networking for AI calls for extensive interdisciplinary research efforts between networking researchers and AI researchers to de- velop new communication standards and technologies to cater for AI services at scale [141], [180]–[182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In industry, the International Telecommunication Union (ITU) is discussing high-level architectures to integrate, orchestrate, and update AI components for future networks, including IMT-2020 net- works [183], [184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Some 3GPP working groups are studying data collection frameworks in the network for supporting AI services [185], [186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Notably, networking for AI is becoming an indispensable component for facilitating AI services in networks and is expected to be a key enabling technology in 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Networking for AI should take the following factors into consideration: Distributed Data - With the wide deployment of various IoT devices and small BSs, massive data are generated from many distributed network nodes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', end users and the network edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the traditional cloud-based AI paradigm, the cloud collects massive distributed data for model training, and a well-trained model is deployed at the cloud for model inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This paradigm suffers from spectrum resource scarcity and user privacy leakage concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='4 To address these issues, a potential solution is to facilitate AI services over a large number of network nodes in a distributed manner [148], which requires new networking protocols to coordinate multiple network nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Constrained Resources - Network nodes, such as end users, have limited resources, while state-of-the-art AI models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', DNN models with dozens of neural net- work layers) are complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As such, running a complex 4Google’s autonomous driving vehicle can generate more than 750 MB of data per second [187].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 14 AI model on a single network node can exhaust its computing resource and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='5 With advanced model partition techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', DNN partition), a complex AI model can be partitioned into multiple sub-models and embedded into a network with data exchange among the sub-models [189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Executing sub-models consumes computing resources of network nodes, and exchanging data between sub-models also consumes communication resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Hence, running AI models at multiple network nodes in a cost-effective manner requires innovative net- work embedding designs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Network Heterogeneity and Dynamics - 6G networks will be highly heterogeneous, in which network nodes possess different amounts of communication, computing, and storage resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As complex AI models need to be deployed at multiple network nodes, executing AI tasks requires judiciously allocating resources of these network nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover, network dynamics, such as time-varying channel conditions among network nodes and spatiotemporal service demands, further complicate the resource allocation decision making problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Hence, it is necessary to design tailored resource management algorithms to optimize AI performance, while adapting to network dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The scope of networking for AI covers the entire lifecycle of AI services, which consists of three stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The first stage is data collection for model training via communication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, real-time service load data from end users need to be collected to train an AI model for service demand prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The second stage is model training, which is to achieve a certain objective based on the collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, a large number of images are processed to train DNN-based object detection modules until the target accuracy requirement is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The third stage is model inference, which is to apply well-trained models to complete specific computation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, AI-based object recognition for autonomous driving detects and classifies nearby vehicles, pedestrians, and obstacles based on real-time images captured by on-board cameras [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 2) State-of-the-Art Approaches: The research on network- ing for AI is still at its infancy stage with only a few existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In this subsection, the existing studies are categorized into data collection, model training, and model inference according to the lifecycle of AI services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Representative related works are summarized in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Data Collection - The objective is to efficiently collect the data from end users for optimizing AI performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Since data are distributed across end users in the network, transmission resource is scheduled to end users for uploading their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, the level-one digital twins require periodical data synchronization with the end users, and such data can be provided for AI services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Data collection is a classic research problem widely investigated in wireless sensor networks [203] and UAV networks [204], and these works focus on optimizing either the reliability of data collection or the amount of 5The energy consumption of using AlexNet to process an image on a tailored energy-efficient Eyeriss chip is up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='28 W [188].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In AI services, the collected data are used to train AI models, and the data samples may have different importance levels for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Merely maximizing the reliability or the amount of the collected data is not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Hence, novel data collection designs taking model training into account are required for performance optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Recently, AI-centric data collection is investigated in the following two research directions: Resource Allocation - Data importance-aware resource allocation schemes have been proposed to optimize AI model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The idea is to schedule data transmission while taking both end users’ channel conditions and data importance levels into account [190].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The data impor- tance level can be captured via data uncertainty, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', higher uncertainty means higher importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The data uncertainty can be measured by entropy [205].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Power allocation for data collection is investigated in multi- model training scenarios [191].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Since the number of collected data samples impacts the model accuracy, a learning-centric power allocation scheme can adjust the users’ transmission power to determine the amount of col- lected data for different AI models, thereby maximizing the overall model accuracy given a transmission power budget;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Protocols - There are a few AI-centric data collection protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In a network environment with poor channel conditions, data retransmission is applied to improve data collection reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Existing automatic repeat-request (ARQ) retransmission protocols, such as hybrid ARQ in long term evolution (LTE) networks, trigger data retransmissions for lost packets once the end user’s signal-to-noise ratio (SNR) threshold is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The importance of data samples should be incorporated in transmission protocols to speed up the model training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' An importance-aware ARQ protocol is proposed for CNN-based classification model training in [192].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the protocol, both data importance levels and channel conditions are taken into account to determine the data retransmission threshold, which can enhance the model training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Model Training - Due to the distributed data and user privacy concerns, distributed training is suitable for training AI models in a network [206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' FL is one of the most promising distributed training paradigms, which can be applied in various fields such as smart healthcare and financial services [138], [207], [208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The FL operates as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Each end user iteratively trains a local model with its own data, and the local model is uploaded to an edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Then, the edge server aggregates the local models to obtain a global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The model training lasts multiple rounds until the global model achieves satisfactory accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Since the model is trained locally, FL is communication- efficient and can preserve data privacy of end users [148], [209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' with the increase of data sizes in state-of- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='TABLE VI: Summary of Related Works on Networking for AI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Topic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Work ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Contribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Highlight ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Collection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[190] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Scheduling data transmission based on users’ data importance levels and channel conditions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Data importance-aware spectrum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='allocation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[191] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Allocating users’ transmission power that can adjust the amount of collected data samples for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='multiple AI models to enhance the overall model accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Data amount-aware power allo- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='cation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[192] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Designing an importance-aware ARQ protocol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' in which users’ data importance levels and channel conditions are jointly considered to trigger data retransmission Data importance-aware retrans- mission protocol Model Training [193] Proposing an edge-cloud assisted FL framework,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' in which the edge and cloud servers alternatively ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='aggregate local models to reduce communication overhead ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Two-tier FL framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[194] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Proposing an over-the-air computation approach for model aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Over-the-air model aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[195] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Selecting users with more contribution to convergence for model aggregation based on users’ data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Data distribution-aware user se- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='lection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[196] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Selecting users with low training delay considering heterogeneity among users ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Training latency-aware user se- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='lection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[197] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Optimizing the number of local model updates given a resource budget ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Local ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='update ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='opti- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='mization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[198] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Scheduling model uploading based on end users’ model importance levels and channel conditions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Model importance-aware model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='uploading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Inference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[144] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Optimizing video frame rate and input image resolution to balance service latency and detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='accuracy for virtual reality users ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Data resolution optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[199] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Selecting the optimal DNN model for real-time video analytics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='DNN model selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[200] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Selecting the optimal DNN model’s cut layer to minimize inference latency for user-edge DNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='synergy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='User-edge DNN model partition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='[201] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='Partitioning a complicated DNN model across end users,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' the network edge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' and the cloud to reduce communication overhead User-edge-cloud DNN model partition [202] Designing a collaborative DNN model inference scheme with light-weight models at IoT devices and an uncompressed model at the network edge Collaborative DNN model infer- ence the-art AI models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='6 uploading local AI models still places a growing strain on spectrum-constrained wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, end users with powerful computing servers can conduct local model training with a low delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As such, the model uploading delay due to limited radio resources can be the dominant component in the entire FL delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Hence, it is necessary to maximize FL performance in resource- constrained wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Recent research works optimize FL performance from the following perspectives: FL Framework Design - A line of works focus on design- ing innovative FL frameworks to reduce communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A novel two-tier hierarchical FL framework is proposed in [193], which coordinates end users, edge servers, and the cloud server to perform FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Each edge server aggregates local models from end users in its cov- erage in every FL round, and the cloud server aggregates the models from edge servers in its coverage once in a few FL rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The proposed two-tier framework can accommodate a large number of end users for model training due to its broad coverage and, at the same time, reduce the backhaul data traffic between the cloud server and edge servers due to a low model aggregation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Such framework is applied to industrial IoT networks with geographically distributed data in [215];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Model Aggregation - Another line of works study ra- dio spectrum-efficient model aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Over-the-air computation based approaches are investigated in [194], [216], [217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The basic idea is to exploit the superposition property of wireless multiple-access channels to perform model aggregation, which can reduce radio resource consumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Resource Management - The FL performance can be 6The data sizes of ResNet32 [210], Inception-v3 [211], AlexNet [212] and VGG16 [213] models are 50 MB, 108 MB, 240 MB, and 552 MB, respectively [214].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' optimized via efficient resource management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As FL performance depends on multiple factors, such as end user selection, the number of local model updates, and local model importance levels, different resource man- agement schemes are developed as follows: (1) User selection - How to select participating end users in the FL process impacts model convergence and training delay and hence is crucial to FL performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A few end user selection algorithms are proposed based on princi- ples such as training data distribution [195] and local training latency [196];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' (2) FL parameters - To alleviate communication overhead, end users conduct a few local model updates before model uploading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Given a resource budget, the optimal number of local model updates is studied in [197], which provides a theoretical guideline for selecting the number of local model update;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' (3) Local model importance level, which is a concept extended from the idea of data importance7 - An importance-aware model uploading strategy is proposed in [198], in which end users with high model importance levels and good channel conditions are scheduled with high priority, to speed up the convergence of FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Model Inference - For many AI services in the network, AI models are usually deployed at end users and edge servers to achieve low service latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The model inference is computation-intensive, while end users and edge servers usually have limited computing capabilities and battery power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Executing model inference tasks usually results in long service latency and high energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Hence, performing model inference should satisfy service latency under node energy constraints, thereby calling for innovative model in- ference schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 7The model importance can be measured by layer-wise gradient norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Local models with larger gradient norm contribute more to global model convergence in FL [197].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 16 Existing studies on model inference can be categorized as follows: Data Resolution - Raw data are offloaded to edge or cloud servers for model inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The input data resolution influences the inference accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, the ac- curacy of object detection is related to the input image resolution [144], which in turn affects the offloaded data volume since the data size of high-resolution images is usually large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Taking into account the trade-off between the inference accuracy and the amount of offloaded data, the input image resolution should be optimized to satisfy the target AI service requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The optimal video frame rate and input image resolution are investigated in [144] to balance service latency and detection accuracy for virtual reality users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Model Selection - An appropriate AI model is selected to satisfy specific AI service requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition to the data resolution, the inference accuracy depends on the type of AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A DNN model with more hidden lay- ers can usually achieve a higher inference accuracy than a shallow DNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Considering multiple available DNN models deployed at the network edge, the optimal DNN model selection for real-time video analytics is investigated in [199];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Model Partition - With advanced model partition tech- niques, an AI model can be partitioned into multiple sub- models and then embedded into different network nodes to conduct model inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, leveraging the layered structure of DNNs, the entire DNN model can be partitioned into an end user-side model and a server-side model at a proper DNN layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', the cut layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As such, the end users and the edge servers can conduct model inference in a collaborative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' DNN models can be partitioned for achieving different goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, the optimal model partition for minimizing inference latency is studied in [200], in which an online learning algorithm can adaptively determine the optimal cut layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To reduce communication overhead among network nodes, compli- cated DNNs models can be partitioned into sub-models for end users, edge servers, and the cloud as in [201];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Model Compression - Light-weight models are used to facilitate prompt model inference at end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Computation-efficient compressed models can be ob- tained via various model compression techniques, such as weight pruning [218], knowledge distilling [219] and fast exiting [220].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, weight pruning tech- niques remove less important model weights to reduce the computational complexity of model inference, while achieving inference accuracy close to that of the un- compressed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To enhance service performance, a collaborative model inference scheme that deploys light-weight models at IoT devices and uncompressed models at the network edge is proposed in industrial IoT networks [202].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The IoT devices dynamically make AI task offloading decisions according to time-varying channel conditions to minimize the service delay while guaranteeing the accuracy requirements of DNN-based fault diagnosis services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 3) Research Challenges: Despite the aforementioned re- search efforts, facilitating AI services in a network faces vari- ous challenges, some of which are discussed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Complex Implementation Option Selection - An AI service can be implemented by various options with different model structures, training procedures, and inference processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, a service of object detection can be implemented via different neural networks, such as AlexNet [212] and SqueezeNet [221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Even if the model structure is the same, a model can be trained in different ways, such as centralized training, decentralized training (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', FL [207]), and semi- centralized training (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', split training [126]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, model inference can be conducted in various manners, such as end user-only inference, edge-only inference, and collaborative inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Different implementation options consume different amounts of computing, storage and communication resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Hence, it is necessary to select an implementation solution for AI services that suits the service characteristics and network dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Multi-Dimensional QoS Requirements - The QoS require- ments of AI services are multi-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI model ac- curacy is usually a key performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, AI services should be offered to end users with low latency in many use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, the service latency of object detection in autonomous driving should be less than 100 ms for safety considerations [143], whereas autonomous vehicles require an ultra-high accuracy in 3D object detection [222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover, these performance metrics are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' High- accuracy object detection usually requires high-resolution im- ages as input and advanced AI models to process the input images, which can result in long service latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' How to satisfy multi-dimensional QoS requirements of AI services requires further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 4) AI Slice: To better support AI services, we extend the network slice concept and propose an idea of AI slice with two subslices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The basic idea is to construct a training subslice for model training and an inference subslice for model inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The two subslices are logically isolated and use their own network resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The rationale behind training and inference separation is that the two stages can have different goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' An illustration of an AI slice is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the AI slice, the training and inference subslices share the same resource pool and are coordinated to jointly support the AI service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First, the multi-dimensional QoS requirement of the AI slice is decoupled into two separate QoS require- ments for the two subslices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For an object detection service in autonomous driving, both high detection accuracy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', 99%) and low service latency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', 100 ms) are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The training subslice should satisfy the detection accuracy requirement, while the inference subslice should satisfy the service latency requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Second, to satisfy the individual QoS requirements of the two subslices, the resources reserved for the AI slice are judiciously allocated between the two subslices, based on the performance of the two subslices and their QoS requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Then, given the allocated resources, the two subslices are configured to satisfy their individual QoS requirements, as described in the following: 17 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Physical Network Inference Subslice Training Subslice Slices AI Slice Pruned Model Uncompressed Model Partitioned Model Inference Subslice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Model Deployment BS 1 Core Network BS N Model Distribution Model Aggregation Local Model Training Local Model Training .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Training Subslice Local Model Uploading Local Model Uploading Time Well-Trained Model One Training Round Virtual Network Core Network BS Network Slicing Computing Resource Communication Resource Storage Resource .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Subslice Controller Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 5: Conceptual AI slice consisting of a training subslice and an inference subslice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the training subslice, based on the training data dis- tribution in the network, a subslice controller determines training configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', data collection schemes and model training methods) and schedules resources to net- work nodes to train a model given the target accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, since the training data vary over time in a dynamic network, the AI model may need to be retrained from time to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Note that allocating dedicated resources for the training subslice can effectively mitigate the straggler effect that plagues distributed learning in large-scale networks, thereby speeding up the model training process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the inference subslice, the subslice controller analyzes the service demand pattern at each BS and determines inference configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', model inference and input data compression schemes) to satisfy the inference la- tency requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, uncompressed models can be deployed at resource-abundant BSs, and parti- tioned and pruned models can be deployed at resource- limited BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This can achieve close inference service latency performance across different BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Overall, the two logically-isolated subslices focus on satisfying different QoS requirements and jointly support the AI service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To elaborate the idea of AI slices, we present the fol- lowing example on real-time video analytics in vehicular networks [199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Smart cameras are deployed in intersections to provide a video surveillance service such as vehicle plate recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In such service, a CNN model is trained using the video streams collected by smart cameras, and then the well-trained model is used to conduct video analytics tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Using the proposed AI slice framework, CNN model training is conducted in a training subslice, while real-time video analytics is conducted in an inference subslice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Specifically, in the training subslice, the CNN model can be trained via a FL framework for protecting data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The corresponding computing resources at smart cameras and spectrum resources in the network are allocated to satisfy model training require- ments, such as training accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the inference subslice, different user-edge orchestration schemes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', DNN model partition), input data compression schemes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', frame rate reduction), and network resource management policies can be configured to satisfy the inference delay requirement in video analytics services based on time-varying service demands and network conditions due to vehicle mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' With the AI slice for video analytics, both training accuracy and inference latency requirements can be satisfied in a dynamic network environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Summary In this section, we have reviewed some common AI tech- niques, explored the role of AI in 6G networks, and proposed a four-layer AI architecture for pervasive intelligence in 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Two perspectives of AI in wireless networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', AI for networking and networking for AI, have been discussed, which correspond to using AI as a powerful tool for network management and optimizing networks to support AI applications, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Recent advancements in ML algorithms have accelerated the deployment of AI in wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In 5G, AI techniques are used to address particular networking problems, whereas, in 6G, AI will penetrate every corner of wireless networks from network management to network services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Therefore, an architecture for AI is needed for identifying the role of AI and characterizing the functionalities of AI across a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 18 Appropriate AI techniques should be selected to tackle networking problems with different characteristics and on different decision time scales when it comes to AI for network- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Furthermore, the collaboration among intelligent modules is important to implement AI-driven networks efficiently and flexibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The idea of connected AI is to enable cooperative decision making among intelligent modules for network con- trol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In terms of networking for AI, a distributed architecture of AI algorithms connects AI models and network resources located at network edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The study of networking for AI is still in its infancy but essential to supporting an expanding group of AI services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Network slicing will remain to be an enabler for delivering AI services, but slicing policies should be customized according to the features of AI algorithms and the training and inference stages of AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A POTENTIAL NETWORK ARCHITECTURE FOR 6G In this section, we propose a conceptual network architec- ture for 6G, which integrates holistic network virtualization (including digital twins and network slicing) and pervasive network intelligence (including connected AI and AI slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Then, we illustrate three types of interplay enabled by the proposed architecture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', the interplay between digital twin paradigm and network slicing, model-driven methods and data- driven methods, and virtualization and AI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Related Studies on Architecture for 6G Several works have proposed architectures with various focuses for 6G networks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', space-air-ground integrated net- works for global coverage [8], cell-free massive multiple-input multiple-output (MIMO) architecture for inter-cell interference mitigation [223], and multi-tier computing architecture for ubiquitous computing service provisioning [224].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Pursuing the goal of advanced network management, most of the proposed architectures highlight AI techniques to optimize network architecture, control, and management [12], [183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For exam- ple, AI-based data analytics functions, which mine historical data for network operation troubleshooting, network resource optimization, and network traffic prediction, are incorporated in the network architecture in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The ITU specifies a high- level AI-based architectural framework for future networks, in which several novel components such as ML management and orchestration functionalities are incorporated for flexible AI- based function placement [183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition to AI techniques, some recent conceptual network architectures start to embrace digital twin techniques [75], [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For example, a digital twin-based network architecture constructs a digital twin for each end user to serve as its communication assistant and data asset manager [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Another digital twin-enabled network architecture adopts three categories of digital twins, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', edge- based, cloud-based, and hybrid digital twins, for supporting different types of services [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Different from the existing network architectures, our pro- posed network architecture features novel holistic network virtualization, which incorporates network slicing and digital twin paradigms, and pervasive network intelligence, which integrates AI for networking and networking for AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover, featuring the designs in Sections II and III, the proposed architecture enables various interplay among its key elements to empower 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the following subsections, we present the details of the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Architecture Overview The overall network architecture is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 6, which consists of the physical space and the cyber space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The physical space includes end users and network infrastructure at the edge and the core networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Data describing end users are collected from the physical network to create level-one digital twins as introduced in detail in Subsection II-C, and network slices are created for various services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The slices are further abstracted into level-two digital twins, which are supplemented with service-specific information aggregated from level-one digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The six-layer virtualization architecture in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 2 applies to the network slices and the digital twins, both of which reside in the cyber space in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI pervades the entire architecture, which supports both AI for networking and networking for AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First, AI is used to manage network slices and digital twins, as shown in the logic network control section in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For network management, a connected AI solution discussed in Subsection III-C is applied to enable intelligent modules, which in turn manage network slices and digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The connected AI solution corresponds to AL 3 and 4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Second, the architecture supports dedicated AI slices with training and inference separation for AI service provisioning, as mentioned in Subsection III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI slices provide services corresponding to AL 1 and 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 3, while the management of AI slices is conducted by intelligent modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' With the overall network architecture in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 6, we integrate holistic network virtualization and pervasive network intel- ligence for 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Virtualization is supported from the aspects of both the network and the end users, while intelligence is reflected through both AI for networking and networking for AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Taking advantage of digital twin paradigm and network slicing as well as those of virtualization and AI, the proposed architecture aims at exceeding flexibility, scalability, adaptiv- ity, and intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Components and Subsystems In the physical space, the proposed architecture includes both RANs and core networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Specifically, the following components are involved: Assorted APs: This component includes MBSs, SBSs, mobile APs (such as UAVs), satellites, and other non- cellular APs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Network controllers: This component includes local con- trollers located at APs or servers on network edge and the centralized controller located at servers in core networks or in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Each controller can consist of computing servers and affiliated network storage servers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' General computing servers: This component includes computing servers for implementing network functions, such as routing and firewall, and hosting the VNFs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 19 Level-One Digital Twin Cyber Space Physical Space Level-Two Digital Twin Aggregation Digital Twin Model Control .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Network Control Inference Subslice Training Subslice AI Slice Slices AI Slice Core Network Network Slicing Intelligent Modules for Network Management .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Gateway Local Controller Centralized Controller Data Flow (Physical) Control (Physical) Data Flow (Cyber) Control (Cyber) Generation/ Updating Generation/ Updating Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 6: The proposed network architecture for 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Application servers: This component includes computing and network storage servers for supporting general edge computing and AI services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' These servers are not used for network management or implementing network functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Other network devices: This component includes special- ized network hardware that are not general computing servers, such as baseband processing units and network switches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' End users: This component includes human mobile users, sensors, vehicles, and various IoT devices, such as meters, actuators, and robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the cyber space, the proposed architecture includes three subsystems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', network slices, digital twins, and connected AI, as follows: Network slices: This subsystem includes all virtual net- works created in network slicing, including AI slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A network slice can involve a RAN, a core network, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' General slices are inherited from existing networks, while AI slices are described in detail in Subsection III-D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Digital twins: This subsystem includes level-one and level-two digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The digital twin subsystem is described in detail in Subsection II-C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Connected AI: This subsystem includes intelligent mod- ules deployed across a network at both the local con- trollers and the centralized controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The connected AI subsystem is described in detail in Subsection III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Interconnections between different components and subsys- tems of the proposed architecture are elaborated in Subsec- tions IV-E to IV-G, which highlight the interplay between dig- ital twin paradigm and network slicing, between model-driven and data-driven methods, and between virtualization and AI, in the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Some open issues and challenges regarding the architecture are presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Note that the proposed conceptual architecture can apply to various types of physical networks, such as vehicular networks and integrated terrestrial-satellite networks, although Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 6 cannot illustrate every possible network scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In different physical networks, the implementation of holistic network virtualization and pervasive network intelligence can be different and require certain customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For example, the deployment of intelligent modules and the data flow among the modules in a satellite network segment can be different from those in a terrestrial network segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Furthermore, the f(x)f(x)f(x)20 migration of digital twins can be more important in a vehicular network than in a static IoT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Related discussions can be found in Section V, where we present challenges and open issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Nevertheless, the basic ideas in the proposed conceptual architecture, including the two-level digital twins, intelligent modules, and AI slices, are applicable in various physical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Implementation In this subsection, we provide a case study on a vehicular network to demonstrate the potential implementation of the proposed network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Roadside BSs co-located with edge computing and caching servers facilitate autonomous driving services for vehicles on the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To implement the proposed network architecture, the following steps are con- ducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Network Slice Establishment: Multiple network slices are established for autonomous driving services with different QoS requirements, achieving network virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Con- ventional network slices are established for non-AI based services, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', high-definition map downloading, while AI slices consisting of training and inference subslices are established for AI based services, such as deep learning based cooperative sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The network slices are stored and managed by a centralized controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Digital Twin Construction: By collecting extensive data from physical entities, digital twins are constructed for vehicle users, roadside BSs, and the established net- work slices, achieving the virtualization of end users and slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Digital twins of vehicle users and roadside BSs are located at edge servers, while digital twins of network slices are located at a cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Due to high vehicle mobility, digital twins of vehicle users should be migrated across edge servers to ensure service continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition to collected data, digital twins can include generated user and service specific data, such as predicted vehicle trajectory and spatial-temporal service demands, via mining historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The generated vehicle data will be used for network management and service provision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' AI Module Deployment: AI modules with different func- tionalities can be deployed at both the centralized and local network controllers, achieving intelligent network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The AI modules at the centralized network controller are in charge of network planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For guar- anteeing QoS requirements of different slices, these AI modules can make resource reservation decisions based on the predicted service demands from the digital twins of roadside BSs and collected slice performance data from the digital twins of network slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The AI modules at local network controllers are in charge of network operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For enhancing the perceived performance of the vehicle users, the AI modules schedule on-demand network resources based on the collected data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', vehicle users’ channel conditions) and the generated data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', predicted vehicle trajectory) from the digital twins of vehicle users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Interplay between Digital Twin Paradigm and Network Slicing As the two components of holistic network virtualization, digital twin paradigm and network slicing are connected in the following two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First, the digital twin paradigm for end user virtualization focuses on data management, while network slicing focuses on network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Data may be viewed as a new type of resources in future networks, in addition to communica- tion, computing, caching, and sensing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Meanwhile, as a resource, data has its unique features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First, data can be considered as an application-layer resource rather than a physical-layer resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Second, different from computing or communication resources, the amount of data resources available to a network is not fixed but progressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Last, the collection and processing of data, which is necessary for utilizing any data resource, consume other network resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' On one hand, effectual utilization of the data resource will benefit network management, and hence digital twin paradigm can enhance network slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' On the other hand, network management should take into account the need and cost of allocating other network resources for utilizing the data resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Hence, network slicing can facilitate digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Second, digital twins will enable user-centric networking in future networks, while network slicing enables service- centric networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Creating an isolated slice for each service and provisioning the service through managing the slice yield a service-centric focus in network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Meanwhile, creating a digital copy of each end user and administrating data that characterize the end user provide a user-centric perspective of network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Having a set of information, selected by the centralized controller through digital twin model con- trol, to describe various characteristics of the end users, such as their location, service request profile, resource utilization, and channel information, creates the possibility of user-specific scheduling within each slice in the network operation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, access control and resource allocation decisions for an end user may depend on the data profile from its digital twin, while different data profiles may lead to different scheduling policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Accordingly, future networks may feature service-centric network planning and user-centric network operations, which can improve the granularity of network management for handling highly diversified end users and dynamic network environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Interplay between Model-Driven and Data-Driven Methods The second interplay enabled by the proposed architecture is the interplay between model-driven and data-driven methods in network operation and service provision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This interplay applies to the intelligent modules for network management shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Network management mostly relied on model-driven or heuristic methods before 5G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Prior to the prevalence of AI, mathematical tools such as optimization methods and game theory have been widely used for network manage- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Optimization methods formulate the objective and con- straints in a closed form, and the corresponding network 21 management problems are solved using optimization algo- rithms [225]–[227].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Game-theoretic approaches analyze the interactions among network entities in either cooperative or non-cooperative scenarios to identify the optimal strategy of each entity [228]–[230].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Mechanism design, an analytical framework in game theory, has also been used to coordinate network entities with locally-held information to achieve desir- able network-wide solutions in network utility maximization problems [231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Through characterizing the relations among several key variables, model-driven methods can lead to either closed-form solutions or algorithms for network management problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Based on mathematical models, model-driven methods are usually explainable and generalize well for different specific problems [232].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='8 However, when networks become complex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', when there are a large number of variables and/or complicated correlation among them) or highly dynamic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', when the network environment changes too rapidly for an optimization algorithm to converge or for a game to achieve an equilibrium), model-driven methods may no longer be accurate or applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The investigation of data-driven methods for network man- agement has gained momentum since 5G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Through collecting and exploiting real-world data, data-driven methods implicitly characterize the relations among variables to generate and fine- tune policies for network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Given sufficient data and a stationary network environment, data-driven methods can provide close-to-optimal solutions to problems that are too complicated for model-driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' However, when the network environment is non-stationary so that new and unknown situations occur from time to time, the performance of data-driven methods can be questionable [233].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, data-driven methods may not generalize well due to their strong dependence on data collected from a specific network environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In 6G, data-driven and model-driven methods should work in synergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The proposed architecture enables the interplay between data-driven and model-driven methods for creating advanced hybrid data-model driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' There are differ- ent options of hybrid data-model driven methods, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 7 and elaborated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The first three options suit AI for networking, while the last option suits networking for AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Backup/Switching - Data-driven and model-driven meth- ods can be the backup for each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, models can be selected to back up data-driven methods, for the case when unknown situations occur in the network environment and degradation in the performance of data- driven methods appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Meanwhile, switching between data-driven and model-driven methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', based on the available resources, can potentially increase the adaptivity of network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Task Division - Date-driven and model-driven methods can target different steps and solve different subprob- lems of network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Specifically, data-driven 8For instance, the water-filling algorithm could be applied to various power allocation problems, and the Rayleigh fading model could characterize channels in various network environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Model Data Solution Solution Backup/ Switching Problem (a) Backup/switching Model Data Solution Part 1 Solution Part 2 Subproblem 1 Subproblem 2 (b) Task division Model Data Solution (Initial) Solution (Refined) Problem (c) Refinement Model Data Data Model Data (d) Mixing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 7: Options for hybrid data-model driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The “Data” and “Model” blocks represent “data-driven methods” and “model-driven methods”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' methods can solve the subproblems with a large number of variables or complicated coupling relations among variables, while model-driven methods can solve rela- tively isolated subproblems with a few key variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This would allow data-driven and model-driven methods to play to their respective strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Refinement - Model-driven methods can provide rough solutions based on general mathematical models, and then data-driven methods, taking the rough solutions as input and exploiting real-world data from the network, can refine the solutions for the specific network scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Having the initial solution generated from models may reduce either the amount of data or the amount of time needed by data-driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Mixing - In networking for AI, while deploying a service function chain for an AI service, some of the function modules can use data-driven methods, while other func- tion modules in the same service function chain can use model-driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For example, in an AI-based image processing service, a model-driven module can be used for image resolution adjustment prior to a data- 22 driven module for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The idea is similar to task division, except that the scenario here is networking for AI instead of AI for networking [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Interplay between Virtualization and AI The third interplay enabled by the proposed architecture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', the interplay between virtualization and AI, is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First and foremost, virtualization and AI are coupled through data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' With the introduction of digital twins, a vast amount of organized data regarding end users, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', level-one digital twins, and network services, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', level-two digital twins, become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The data included in the digital twins can be provided to the intelligent modules, the training or inference subslice of an AI slice, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, edge-hosted AI, possibly collaborating with end user-hosted AI, can perform user-specific data processing and prediction based on the data from digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The results, such as prediction results, resource scheduling schemes, or slicing policies, can be fed back to the digital twins to record certain predicted status, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', location and mobility, of the end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Correspondingly, data in the digital twins of end users, network infrastructure, and slices can be either the input or the output of AI modules, leading to a bidirectional interaction between virtualization and AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='9 The second connection between virtualization and AI is through control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Based on the data from digital twins, AI functions hosted at the edge and core networks can make the network management and service provisioning decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The decisions may include network slice control, which are fed back to the physical network and network slices for execution and, at the same time, to the level-two digital twins for data update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, the decisions may include digital twin model control for level-one and level-two digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Digital twin model control may include the determination of the type and the amount of data to be included in digital twins, the frequency and the method of data collection, the format and the precision of stored data, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The digital twin models affect the availability and quality of data available for network control, especially AI-driven network control, and thereby impact the network performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Therefore, from the perspective of network control, the interaction between virtualization and AI is also bi-directional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='10 The third and implicit connection between virtualization and AI is through resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Holistic network virtualization requires extensive resources, including computing resource for virtual network functions, caching resource for storing digital twins, and communication resource for the synchronization between end users and their digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Similarly, perva- sive network intelligence also requires extensive computing resource and possibly other resources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', communication resource for distributed training as mentioned in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Therefore, the network resources need to be shared and 9Interested readers are referred to [234] for the relation between AI and data life cycle, although the discussions therein do not involve virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 10The interaction between digital twin and AI for intelligent network control is discussed in [235].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Note that the definition of digital twins therein is different from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Physical Network Slices Data Control Digital Twins AI for Networking Networking for AI Level-Two Digital Twin Level-One Digital Twin Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 8: Interplay between virtualization and AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' coordinated between virtualization and AI functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' However, this does not mean that virtualization and AI functions simply compete for resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Instead, they can help each other improve resource utilization efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Digital twin paradigm may reduce the resource consumption of AI functions by providing high-importance data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This can be achieved by the aforementioned digital twin model control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Meanwhile, creating a digital twin for every end user may be too resource- demanding for networks in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Using AI to select representative end users for generating digital twins and optimizing digital twin models may reduce the resource consumption of maintaining and updating digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' One potential implementation is using AI to categorize end users and select a portion of users from each category for creating digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Alternatively, since it may be more challenging to provide QoE guarantee for some end users than others, using AI to select such end users for creating digital twins can potentially reduce the resource consumption on digital twins for 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Potential Network Architecture for 6G: A Summary This section has provided a potential network architecture for 6G, which integrates two key elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', pervasive network intelligence and holistic network virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the proposed network architecture, detailed network compo- nents, subsystems, and potential implementation have been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Moreover, three types of interplay in the archi- tecture are provided to characterize the proposed network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The proposed network architecture holds great potential for achieving advanced network management schemes and supporting AI services in 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Firstly, integrating digital twins and network slicing facilitates user-centric net- working and improves the granularity of network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 23 Secondly, integrating data-driven methods and model-driven methods enables novel hybrid data-model driven methods, which has the potential to outperform existing network man- agement methods in terms of adaptivity, granularity, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Thirdly, leveraging the network slicing concept in AI services facilitates AI services targeting QoS performance guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' CHALLENGES AND OPEN ISSUES Many challenges and open issues are yet to be addressed for holistic network virtualization and pervasive network intelli- gence in 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In the following, we present some key challenges and open issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Digital Twin The six-layer architecture in Subsection II-C provides a high-level design for integrating the digital twin paradigm into network virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Open issues to be investigated for practical implementation of this architecture include quantita- tive performance characterization of digital twins, the optimal digital twin model, digital twin migration, and data security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First, it is necessary to quantitatively characterize the network performance improvement from introducing digital twins, either from the perspective of QoS/QoE satisfaction or from the perspective of resource utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Second, level-one digital twin models configured by the centralized controller may be different for different edge networks to account for network heterogeneity, and how to determine effective digital twin models is a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Third, the mobility of end users such as vehicles creates a need for updating and migrating digital twins across different edge networks, which requires further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Last, ensuring the security of user data in the digital twin paradigm is yet another challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As the local and centralized network controllers have access to a vast amount of user data, developing proper security mechanisms for data col- lection, aggregation, and migration becomes essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Readers are referred to [32], [236]–[238] for discussions on some of the aforementioned challenges, such as the heterogeneity and migration of digital twins, and more open issues related to digital twins in 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Network Management Oriented Data Abstraction and Pro- cessing While digital twins provide data to enable AI for network- ing, including automated network slicing and AI-empowered network control, efficient data management can be challeng- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First, it is necessary to develop data abstraction methods to aggregate the data with different levels of granularity for making different network management decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, in network slicing, high-granularity data are required for determining the optimal network operation strategies and low- granularity data are sufficient for determining the optimal network planning strategies [11], [239].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' How to determine the appropriate data granularity for different network man- agement decisions is an open issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' A potential solution is to empirically adjust data granularity and the time scale for decision-making [240].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Meanwhile, as the number of variables and data types in network management can be huge, more scalable and efficient solutions are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Second, while applying the connected AI solution for network management, the settings of intelligent modules, such as the selection of algorithms, the input and output attributes, and the connections among intelligent modules, should be configured to maximize the utilization of data with low communication and processing overhead, yet finding the optimal settings is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The cooperation between model-driven and data-driven methods in intelligent modules can be a potential approach to address the challenge, yet how to support such cooperation among different types of intelligent modules requires further inves- tigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Third, as data can be generated, transmitted, and processed at different network stakeholders, configurable and regulation-compliant data management is also a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The integration of the blockchain and privacy-enhancing technolo- gies can be a potential solution, while the trade-offs between privacy preservation and processing efficiency need in-depth investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Readers are referred to [4], [131], [182], [234] for discussions on the aforementioned challenges, such as privacy preservation, AI model selection, intelligent modules, and more open issues about data abstraction and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Model and Resource Orchestration Networking for AI in Subsection III-D can facilitate AI services in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' One key issue is to optimize AI service performance, which requires judicious configuration of the network, including AI algorithm selection, data collection, and network resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The main challenge lies in modeling the relationship between AI performance and these network configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Establishing an accurate mathematical or empirical model requires extensive measurements in real- world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Even if establishing a model is viable, the model may be suitable only for a chosen AI algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, to adapt to network dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=', rapidly fluc- tuating service demands), an online network configuration scheme is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Since reinforcement learning algorithms are able to make online decisions in a dynamic environment, developing cost-effective reinforcement learning algorithms for high-dimensional network configuration problems can be a promising approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For example, a reinforcement learning al- gorithm is developed for joint AI model selection and resource allocation in industrial IoT [202].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For more discussions on the above challenges, interested readers are referred to [175], [241], [242].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Training and Inference Coordination The concept of AI slice is proposed to meet specific QoS requirements of AI services in Subsection III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The training and inference stages for an AI service consume multi- dimensional network resources [131], [243].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In an AI slice, two subslices share the virtualized network resource pool, and hence resource reservation decisions for the two subslices are closely correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' On the one hand, reserving abundant resources for the training subslice may help achieve a high training accuracy but potentially render resource insufficiency in the inference subslice, which can result in a long service 24 latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' On the other hand, insufficient resource provisioning for the training subslice may yield a model with low accuracy and consequently create a bottleneck for inference accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To optimize the performance of the AI service, resource reserva- tion for training and inference subslices should be coordinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Developing an accurate mathematical model to characterize the interplay between training and inference stages is difficult, since a large number of system factors should be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Hence, it is necessary to study efficient model-free approaches to characterize the interplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Energy Efficiency of AI With hundreds of neural network layers, thousands of neurons, and millions of parameters, state-of-the-art AI mod- els usually consume extensive energy and incur substantial environmental costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='11 Improving energy efficiency has be- come a major issue for wide deployment of AI services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, recent research shows that improving the accuracy of an AI model may come at an exponential increase in the computation, environmental and economic costs [245].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='12 Hence, deploying energy-efficient AI services in a network is necessary for reducing costs for the network operator and meeting environmental standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Several model compression techniques, such as weight pruning [218], parameter quanti- zation [246], and model compression [247], can be applied to alleviate the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' In addition, hybrid data-model driven methods can train AI models with a reduced amount of data, which can also decrease energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Hybrid Data-Model Driven Methods The four options listed in Subsection IV-F provide our initial ideas for hybrid data-model driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Related open issues to be investigated include the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' First, it is necessary to study how to determine which option to use and how to switch among options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Designing mechanisms for choosing and switching among options will allow networks to flexibly and adaptively integrate data-driven and model- driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Second, for a chosen option, it is important to understand how much the data-driven and model-driven com- ponents affect the overall performance and how much impact they have on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For instance, in the mixing option, the AI service performance may depend on the combined choices of data-driven and model-driven methods, and finding a proper combination can be a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Third, in addition to the four options as introduced, there should be other potential options for hybrid data-model driven methods, and identifying other promising options is an open issue of great importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Last, due to the lack of explainability in existing data-driven methods, careful investigations and analysis should be directed to the management of critical network operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' The role 11The estimated carbon footprint of training a state-of-the-art natural language processing model is about five times the life emissions of an average car [244].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' 12It is estimated that reducing the classification error probability from 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='5% to 5% over the ImageNet dataset needs to increase computation from 1014 to 1019 Gflops, carbon emissions from 106 to 1010 lbs, and economic costs from 106 to 1011 USD [245], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' of hybrid data-model driven methods in enhancing system robustness is an open issue that deserves further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' For more discussions on challenges in hybrid data-model driven methods for networks, interested readers are referred to [232], [248], [249].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content='13 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' CONCLUSION Designing an architecture for future networks is challenging, especially when the use cases and defining techniques are still beneath the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Nevertheless, the evolution of networks through the previous generations demonstrates a necessity to support increasingly heterogeneous networks, diverse services, and stringent QoS/QoE requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' This has been driving the trend of virtualization and generating significant interest in AI-driven networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Recognizing the insufficiency of the existing scope and level of virtualization and AI for future 6G networks, we have presented a conceptual architecture design that integrates holistic network virtualization and pervasive network intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' To complement and solidify our over- all network architecture, we have proposed several specific designs, including the six-layer holistic network virtualization based on digital twins, the connected AI solution for network management, as well as ideas, including AI slices and hybrid data-model driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' As a result, the proposed network architecture has the potential to achieve unprecedented scala- bility and flexibility due to the holistic network virtualization as well as exceeding adaptivity and intelligence due to the pervasive network intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' At last, we have identified some challenges and open issues related to the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' We hope this study will lead to further discussions and developments on the architecture of 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' ACKNOWLEDGEMENT The authors would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' Dongxiao Liu for helpful discussions on open issues related to data privacy and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' REFERENCES [1] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfo_jf/content/2301.00519v1.pdf'} +page_content=' You et al.' 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b/5NFAT4oBgHgl3EQfFRzt/content/tmp_files/2301.08427v1.pdf.txt @@ -0,0 +1,335 @@ +arXiv:2301.08427v1 [cs.CL] 20 Jan 2023 +Arxiv preprint +WHICH FEATURES ARE LEARNED BY CODEBERT: +AN EMPIRICAL STUDY OF THE BERT-BASED SOURCE +CODE REPRESENTATION LEARNING +Lan Zhang∗, Chen Cao∗, Zhilong Wang∗ and Peng Liu +The Pennsylvania State University +State College, PA 16801, USA +{lfz5092,cuc96,zzw169,pxl20}@psu.edu +ABSTRACT +The Bidirectional Encoder Representations from Transformers (BERT) were pro- +posed in the natural language process (NLP) and shows promising results. Re- +cently researchers applied the BERT to source-code representation learning and +reported some good news on several downstream tasks. However, in this pa- +per, we illustrated that current methods cannot effectively understand the logic of +source codes. The representation of source code heavily relies on the programmer- +defined variable and function names. We design and implement a set of experi- +ments to demonstrate our conjecture and provide some insights for future works. +1 +INTRODUCTION +Deep learning has demonstrated its great learning ability in natural language processing (NLP). +To deploy a natural language task, e.g. translation and text classification, researchers first pre- +train a model to embed words into vectors using ELMo +Sarzynska-Wawer et al. (2021), GPT +Radford et al. (2018) and BERT Devlin et al. (2018). These pre-trained models are first learned on a +large unsupervised text corpus and then fine-tuned on different downstream tasks. Those language- +based techniques have been deployed to the source code to learn a program representation. Simi- +lar to natural language, the program representation learned from the source code using pre-trained +models can be applied for several sub-tasks for example program analysis. In 2020, Feng et al. +proposed a pre-trained model called CodeBERT Feng et al. (2020) based on Bidirectional Encoder +Representations from Transformers (BERT) that learns general-purpose representations to support +downstream NL-PL applications such as natural language code search, code documentation genera- +tion, etc. In 2021, Guo et al. proposed a new pre-trained model called GraphCodeBERT Guo et al. +(2020), which improves the CodeBERT by enabling the model to capture more program semantic +information, such as data flow. +The difference between natural language and program language leads to an unintended consequence +if these methods are directly employed to program language. In natural language, the meaning of a +word is deterministic in a specific context, whereas in program language, a programmer can assign +any string to a variable, method, or function as their name. In such a case, most strings in the code +could be replaced by other words and may not have meaningful information. In this case, if a BERT +model still heavily relies on the literal meaning of a variable/methods/function name, it may leave +a pitfall when the assigned name does not literally contain any useful information or controversial +meaning. +Furthermore, limited words are used in natural language, while in the programming language, the +number of words can be unlimited because a programmer can casually create a string to name +a variable, no matter whether the created string is interpretable or not. Therefore, it is doubtful +whether the word embedding adopted in natural language is still efficient in solving the program +analysis tasks. If a model designer ignores the numerous difference between natural language and +programing language and naively adopt methods from NLP, the designed model may suffer from the +above limitations. +∗equal contribution +1 + +Arxiv preprint +In this paper, we aim to provide an explanation of these limitations of the BERT-based code rep- +resentation learning techniques. Specifically, we want to understand what kind of features can be +learned and cannot be learned by current pre-trained models. +1 +template:: +value_type>> +2 +inline void bubble_sort(It begin, It end, Pred pred=Pred()){ +3 +if ( std::distance( begin, end ) <= 1 ){ return; } +4 +auto it_end += end; +5 +bool finished += false; +6 +while ( !finished ){ +7 +finished = true; +8 +std::advance( it_end, -1 ); +9 +for (auto it = begin; it! = it_end; ++ it ){ +10 +auto next = detail::advance( it, 1 ); +11 +if (pred( * next, * it)){ +12 +std::swap( * it, * next); +13 +finished = false; +14 +} +15 +} +16 +} +17 +} +Code 1: A piece of code with meaningful variable/function names. +1 +template:: +value_type>> +2 +inline void fun1(It var1, It var2, Pred fun2=Fun2()){ +3 +if ( std::distance( var1, var2 ) <= 1 ){ return; } +4 +auto var3 += var2; +5 +bool var4 += false; +6 +while ( !var4 ){ +7 +var4 = true; +8 +std::advance( var3, -1 ); +9 +for (auto var5 = var1; var5! = var3; ++ var5 ){ +10 +auto var6 = detail::advance( var5, 1 ); +11 +if (fun2( * var6, * var5)){ +12 +std::swap( * var5, * var6); +13 +var4 = false; +14 +} +15 +} +16 +} +17 +} +Code 2: A piece of code without meaningful variable/function names. +Code 1 and Code 2 are two pieces of code that achieve the same logic – bubble sorting. The Code 1 +has well-named functions and variables whereas the Code 2 does not. If an analyst wants to know +their purpose, through a quick glance, even a beginner can easily conclude that Code 1 is a bubble- +sort function based on the literal meaning of the function name. However, it is much more chal- +lenging for an analyst to understand the purpose of Code 2. Therefore, despite the exactly the same +program logic that they have, Code 2 is much more difficult to analyze. We can draw the following +conclusions from the analysis of these two code examples: 1) a source code can be understood in +two ways: literal analysis, and logic analysis. 2) The literal analysis makes a conclusion based on +the name of variables and functions, which is easier to analyze but is not always reliable. 3) The +logic analysis requires a high-level understanding of the code, which is more reliable but hard to +analyze. +To understand whether the existing models learn the logic of the code, we identify two features in +the source code: 1) literal feature. 2) logic feature. For instance, a logical expression is the logic +feature, whereas the variable names in the expression are literal features. Then, we design a set of +experiments that mask out different kinds of features in the training set and observe corresponding +model performance. The result shows that the current models for source code representation learning +still have limited ability to learn logic features. +2 + +Arxiv preprint +2 +BACKGROUND +2.1 +DEEP LEARNING FOR PROGRAM ANALYSIS +Compared with traditional deep learning methods, researchers recognized several benefits of deep +learning for the program analysis: First, deep learning involves less domain knowledge. Second, the +representations learned by a DL model could be used for various downstream tasks. The applications +of deep learning in program analysis can be grouped into two categories: +Source code level deep learning. CodeBert and GraphCodeBERT Feng et al. (2020); Guo et al. +(2020) are pre-trained models based on Transformer which learns code representations through self- +supervised training tasks ( masked language modeling and structure-aware tasks) and a large-scale +unlabeled corpus. Specifically, CodeBERT, which is pre-trained over 6 programming languages, +is trained based on three tasks: masked language modeling, code structure edge predication, and +representation alignment. +Assembly code level deep learning. Previous research use DL to conduct various binary analysis +tasks Chua et al. (2017); Shin et al. (2015); Li et al. (2021). The main focus of these works is to +learn a good embedding from binary instructions or raw bytes, and then predict the label for a target +task through a classification output layer. +3 +INSIGHTS AND EXPERIMENTS +A source code file of a program consists of a sequence of tokens. The tokens can be grouped into +three categories: keywords, operators, and user-defined names. +Keywords are reserved words that have special meanings and purposes and can only be used for +specific purposes. For example, for, if, and break are widely known keywords used in many +programming languages. A programming language usually only contains a limited number of key- +words. For example, C programming language contains 32 keywords and Python3.7 contains 35 +keywords. +Besides the keywords, a programming language needs to define a set of operators. For example, +arithmetic operators (e.g., +, -, and *) and logical operators (e.g., and, or, and not) are two of +most important categories. The keywords and operators are defined by a programming language. A +programmer needs to define some tokens (i.e., names) to represent a variable, structure, function, +method, class, and package. When programmers write a code snippet, they can randomly choose +any string to name these elements. However, he/she has limited flexibility to choose the keywords +and operators. Only some keywords (such as for and while), operators (such as ++, +1) are +exchangeable. +Currently, GraphCodeBert takes code pieces of functions or class methods as data samples. It to- +kenizes keywords, operators, and user-defined names from the code pieces. Inside a function or a +method, we can group the user-defined names into three categories: 1) variable name. 2) method +name. 3) method invocation name. Program logic is not affected if we map these user-defined names +with other strings in the same namespace. To evaluate whether the model learns the code semantics, +we design 4 groups of experiments. For each group of experiments, we anonymize certain categories +of user-defined names. +1. In the first group of experiments, we anonymize the variable names. An example is the +change from it end to var3 and finished to var4 between Code 1 and Code 2. +2. In the second group of experiments, we anonymize the method names. An example is the +change from bubble sort to fun1 between Code 1 and Code 2. +3. In the third group of experiments, we anonymize the method/function invocation names. +An example is the change from swap to fun2 between Code 1 and Code 2. +4. The last group of experiments are a combination of the first three experiments, which +anonymize all three kinds of user-defined names. +Besides, we adopt two strategies to anonymize the name: The first strategy called “randomly- +generated” randomly generates strings (e.g., “oe4yqk4cit2maq7t”) with any literal meaning. The +3 + +Arxiv preprint +Table 1: Results on Code Search. +Language +Original +Anonymizing +w/o Variable +w/o Method Def. +w/o Method Inv. +All +Java +70.36% +Random +67.73% +60.89% +69.84% +17.42% +Meaningful +67.14% +58.36% +69.84% +17.03% +Python +68.17% +Random +59.8% +55.43% +65.61% +24.09% +Meaningful +59.78% +55.65% +65.61% +23.73% +Table 2: Results on Clone Detection. +Language +Original +Anonymizing +w/o Variable +w/o Method Def. +w/o Method Inv. +All +Java +94.87% +Random +92.64% +93.97% +94.72% +86.77% +Meaningful +92.52% +94.27% +93.67% +84.76% +second strategy called “meaningfully-generated” generates strings with a literal meaning. However +the literal meaning does not reflect the intention of the variable/function/invocation. For example, +this strategy could replace “bubble sort” with “aes encryption”. +Based on the four types of name-set to replace and two replacing strategies, we eventually generated +8 variants of the original dataset from Guo et al. (2020). Then, we retrain the existing models and +evaluated their performance on the existing 2 downstream tasks: natural language code search, and +clone detection. +3.1 +EXPERIMENT RESULTS +Figure 2 and Figure 1 show experiment results (accuracy) on the downstream task of code search +and code clone detection, respectively. The second column shows the module performance reported +by the original paper Guo et al. (2020). The fourth, fifth, and sixth columns show the module per- +formance when we anonymize the variable name, method definition name, and method invocation +name, respectively. The last column shows the model performance after we remove all three user- +defined names. +The results show that the anonymization of the variable names, method definition names, and method +invocation names will result in a huge downgrade in model performance not matter we replace user- +defined names with “randomly-generated” strings or a “meaningfully-generated” strings. Also, on +average the dateset with meaningfully-generated strings shows worse result then the dataset with +randomly-generatedstrings, which indicates that “meaningfully-generated”strings could misleading +the models. An adversarial machine learning could be trained to further exploit the weakness of the +CodeBert. +Overall, our experiments proves that current source-code level representation learning methods still +largely rely on the literal feature and ignore the logic feature. However, the literal feature is not +always reliable as mentioned in section 1. The current mode still cannot effectively learn the hidden +logic feature in the source code. +3.2 +DISCUSSION +Through a set of experiments and empirical analysis, this paper tries to explain the learning ability of +current BERT-based source code representation learning schemes. The results show that CodeBERT +and GraphCodeBERT are efficient to learn literal features but less efficient to learn logic features. +The insights provided by this paper can help future researchers or users in two aspects: Firstly, Code- +BERT and GraphCodeBERT, which open a new area for source analysis, are efficient methods for +“well-named” source code. However, the user and researcher should expect a lower model perfor- +mance if they want to apply them to analyze source code that does not provide enough information +in a variable, method, and function names, e.g., the code generated from decompilation Katz et al. +(2018) and code that does not follow standard code naming convention Butler et al. (2015). +4 + +Arxiv preprint +Secondly, this paper indicates that models borrowed from NLP are not very suitable for code anal- +ysis. The code analysis has some significant differences compared with NLP. Logical analysis is +more important in many sophisticated program analysis tasks, such as vulnerability analysis, and +patching generation. But it cannot be well performed by existing model designs. It is important to +investigate how to improve the model’s ability for logical analysis in future research. +REFERENCES +Simon Butler, Michel Wermelinger, and Yijun Yu. Investigating naming convention adherence in +java references. In 2015 IEEE International Conference on Software Maintenance and Evolution +(ICSME), pp. 41–50. IEEE, 2015. +Zheng Leong Chua, Shiqi Shen, Prateek Saxena, and Zhenkai Liang. Neural Nets Can Learn Func- +tion Type Signatures from Binaries. In 26th USENIX Security Symposium (USENIX Security 17), +pp. 99–116, 2017. +Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 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In ACM CCS, 2021. +Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language under- +standing by generative pre-training. 2018. +Justyna Sarzynska-Wawer, Aleksander Wawer, Aleksandra Pawlak, Julia Szymanowska, Izabela +Stefaniak, Michal Jarkiewicz, and Lukasz Okruszek. Detecting formal thought disorder by deep +contextualized word representations. Psychiatry Research, 304:114135, 2021. +Eui Chul Richard Shin, Dawn Song, and Reza Moazzezi. Recognizing functions in binaries with +neural networks. In 24th {USENIX} Security Symposium ({USENIX} Security 15), pp. 611–626, +2015. +5 + diff --git a/5NFAT4oBgHgl3EQfFRzt/content/tmp_files/load_file.txt b/5NFAT4oBgHgl3EQfFRzt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa5848a1ac859bf5cfef3c62d1be67cd84643916 --- /dev/null +++ b/5NFAT4oBgHgl3EQfFRzt/content/tmp_files/load_file.txt @@ -0,0 +1,240 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf,len=239 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='08427v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='CL] 20 Jan 2023 Arxiv preprint WHICH FEATURES ARE LEARNED BY CODEBERT: AN EMPIRICAL STUDY OF THE BERT-BASED SOURCE CODE REPRESENTATION LEARNING Lan Zhang∗, Chen Cao∗, Zhilong Wang∗ and Peng Liu The Pennsylvania State University State College, PA 16801, USA {lfz5092,cuc96,zzw169,pxl20}@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='edu ABSTRACT The Bidirectional Encoder Representations from Transformers (BERT) were pro- posed in the natural language process (NLP) and shows promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Re- cently researchers applied the BERT to source-code representation learning and reported some good news on several downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' However, in this pa- per, we illustrated that current methods cannot effectively understand the logic of source codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The representation of source code heavily relies on the programmer- defined variable and function names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' We design and implement a set of experi- ments to demonstrate our conjecture and provide some insights for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 1 INTRODUCTION Deep learning has demonstrated its great learning ability in natural language processing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' To deploy a natural language task, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' translation and text classification, researchers first pre- train a model to embed words into vectors using ELMo Sarzynska-Wawer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2021), GPT Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2018) and BERT Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' These pre-trained models are first learned on a large unsupervised text corpus and then fine-tuned on different downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Those language- based techniques have been deployed to the source code to learn a program representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Simi- lar to natural language, the program representation learned from the source code using pre-trained models can be applied for several sub-tasks for example program analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' In 2020, Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' proposed a pre-trained model called CodeBERT Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2020) based on Bidirectional Encoder Representations from Transformers (BERT) that learns general-purpose representations to support downstream NL-PL applications such as natural language code search, code documentation genera- tion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' In 2021, Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' proposed a new pre-trained model called GraphCodeBERT Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2020), which improves the CodeBERT by enabling the model to capture more program semantic information, such as data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The difference between natural language and program language leads to an unintended consequence if these methods are directly employed to program language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' In natural language, the meaning of a word is deterministic in a specific context, whereas in program language, a programmer can assign any string to a variable, method, or function as their name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' In such a case, most strings in the code could be replaced by other words and may not have meaningful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' In this case, if a BERT model still heavily relies on the literal meaning of a variable/methods/function name, it may leave a pitfall when the assigned name does not literally contain any useful information or controversial meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Furthermore, limited words are used in natural language, while in the programming language, the number of words can be unlimited because a programmer can casually create a string to name a variable, no matter whether the created string is interpretable or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Therefore, it is doubtful whether the word embedding adopted in natural language is still efficient in solving the program analysis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' If a model designer ignores the numerous difference between natural language and programing language and naively adopt methods from NLP, the designed model may suffer from the above limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' ∗equal contribution 1 Arxiv preprint In this paper, we aim to provide an explanation of these limitations of the BERT-based code rep- resentation learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Specifically, we want to understand what kind of features can be learned and cannot be learned by current pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 1 template:: value_type>> 2 inline void bubble_sort(It begin, It end, Pred pred=Pred()){ 3 if ( std::distance( begin, end ) <= 1 ){ return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' } 4 auto it_end = end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 5 bool finished = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 6 while ( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='finished ){ 7 finished = true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 8 std::advance( it_end, -1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 9 for (auto it = begin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' = it_end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' ++ it ){ 10 auto next = detail::advance( it, 1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 11 if (pred( * next, * it)){ 12 std::swap( * it, * next);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 13 finished = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 14 } 15 } 16 } 17 } Code 1: A piece of code with meaningful variable/function names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 1 template:: value_type>> 2 inline void fun1(It var1, It var2, Pred fun2=Fun2()){ 3 if ( std::distance( var1, var2 ) <= 1 ){ return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' } 4 auto var3 = var2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 5 bool var4 = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 6 while ( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='var4 ){ 7 var4 = true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 8 std::advance( var3, -1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 9 for (auto var5 = var1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' var5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' = var3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' ++ var5 ){ 10 auto var6 = detail::advance( var5, 1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 11 if (fun2( * var6, * var5)){ 12 std::swap( * var5, * var6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 13 var4 = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 14 } 15 } 16 } 17 } Code 2: A piece of code without meaningful variable/function names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Code 1 and Code 2 are two pieces of code that achieve the same logic – bubble sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The Code 1 has well-named functions and variables whereas the Code 2 does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' If an analyst wants to know their purpose, through a quick glance, even a beginner can easily conclude that Code 1 is a bubble- sort function based on the literal meaning of the function name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' However, it is much more chal- lenging for an analyst to understand the purpose of Code 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Therefore, despite the exactly the same program logic that they have, Code 2 is much more difficult to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' We can draw the following conclusions from the analysis of these two code examples: 1) a source code can be understood in two ways: literal analysis, and logic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 2) The literal analysis makes a conclusion based on the name of variables and functions, which is easier to analyze but is not always reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 3) The logic analysis requires a high-level understanding of the code, which is more reliable but hard to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' To understand whether the existing models learn the logic of the code, we identify two features in the source code: 1) literal feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 2) logic feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' For instance, a logical expression is the logic feature, whereas the variable names in the expression are literal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Then, we design a set of experiments that mask out different kinds of features in the training set and observe corresponding model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The result shows that the current models for source code representation learning still have limited ability to learn logic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 2 Arxiv preprint 2 BACKGROUND 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='1 DEEP LEARNING FOR PROGRAM ANALYSIS Compared with traditional deep learning methods, researchers recognized several benefits of deep learning for the program analysis: First, deep learning involves less domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Second, the representations learned by a DL model could be used for various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The applications of deep learning in program analysis can be grouped into two categories: Source code level deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' CodeBert and GraphCodeBERT Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2020) are pre-trained models based on Transformer which learns code representations through self- supervised training tasks ( masked language modeling and structure-aware tasks) and a large-scale unlabeled corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Specifically, CodeBERT, which is pre-trained over 6 programming languages, is trained based on three tasks: masked language modeling, code structure edge predication, and representation alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Assembly code level deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Previous research use DL to conduct various binary analysis tasks Chua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The main focus of these works is to learn a good embedding from binary instructions or raw bytes, and then predict the label for a target task through a classification output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 3 INSIGHTS AND EXPERIMENTS A source code file of a program consists of a sequence of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The tokens can be grouped into three categories: keywords, operators, and user-defined names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Keywords are reserved words that have special meanings and purposes and can only be used for specific purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' For example, for, if, and break are widely known keywords used in many programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' A programming language usually only contains a limited number of key- words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' For example, C programming language contains 32 keywords and Python3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='7 contains 35 keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Besides the keywords, a programming language needs to define a set of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' For example, arithmetic operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=', +, -, and *) and logical operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=', and, or, and not) are two of most important categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The keywords and operators are defined by a programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' A programmer needs to define some tokens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=', names) to represent a variable, structure, function, method, class, and package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' When programmers write a code snippet, they can randomly choose any string to name these elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' However, he/she has limited flexibility to choose the keywords and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Only some keywords (such as for and while), operators (such as ++, +1) are exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Currently, GraphCodeBert takes code pieces of functions or class methods as data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' It to- kenizes keywords, operators, and user-defined names from the code pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Inside a function or a method, we can group the user-defined names into three categories: 1) variable name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 2) method name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 3) method invocation name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Program logic is not affected if we map these user-defined names with other strings in the same namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' To evaluate whether the model learns the code semantics, we design 4 groups of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' For each group of experiments, we anonymize certain categories of user-defined names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' In the first group of experiments, we anonymize the variable names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' An example is the change from it end to var3 and finished to var4 between Code 1 and Code 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' In the second group of experiments, we anonymize the method names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' An example is the change from bubble sort to fun1 between Code 1 and Code 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' In the third group of experiments, we anonymize the method/function invocation names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' An example is the change from swap to fun2 between Code 1 and Code 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The last group of experiments are a combination of the first three experiments, which anonymize all three kinds of user-defined names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Besides, we adopt two strategies to anonymize the name: The first strategy called “randomly- generated” randomly generates strings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=', “oe4yqk4cit2maq7t”) with any literal meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The 3 Arxiv preprint Table 1: Results on Code Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Language Original Anonymizing w/o Variable w/o Method Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' w/o Method Inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' All Java 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='36% Random 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='73% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='89% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='84% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='42% Meaningful 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='14% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='36% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='84% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='03% Python 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='17% Random 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='8% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='43% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='61% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='09% Meaningful 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='78% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='65% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='61% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='73% Table 2: Results on Clone Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Language Original Anonymizing w/o Variable w/o Method Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' w/o Method Inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' All Java 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='87% Random 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='64% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='97% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='72% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='77% Meaningful 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='52% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='27% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='67% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='76% second strategy called “meaningfully-generated” generates strings with a literal meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' However the literal meaning does not reflect the intention of the variable/function/invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' For example, this strategy could replace “bubble sort” with “aes encryption”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Based on the four types of name-set to replace and two replacing strategies, we eventually generated 8 variants of the original dataset from Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Then, we retrain the existing models and evaluated their performance on the existing 2 downstream tasks: natural language code search, and clone detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='1 EXPERIMENT RESULTS Figure 2 and Figure 1 show experiment results (accuracy) on the downstream task of code search and code clone detection, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The second column shows the module performance reported by the original paper Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The fourth, fifth, and sixth columns show the module per- formance when we anonymize the variable name, method definition name, and method invocation name, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The last column shows the model performance after we remove all three user- defined names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The results show that the anonymization of the variable names, method definition names, and method invocation names will result in a huge downgrade in model performance not matter we replace user- defined names with “randomly-generated” strings or a “meaningfully-generated” strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Also, on average the dateset with meaningfully-generated strings shows worse result then the dataset with randomly-generatedstrings, which indicates that “meaningfully-generated”strings could misleading the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' An adversarial machine learning could be trained to further exploit the weakness of the CodeBert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Overall, our experiments proves that current source-code level representation learning methods still largely rely on the literal feature and ignore the logic feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' However, the literal feature is not always reliable as mentioned in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The current mode still cannot effectively learn the hidden logic feature in the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='2 DISCUSSION Through a set of experiments and empirical analysis, this paper tries to explain the learning ability of current BERT-based source code representation learning schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The results show that CodeBERT and GraphCodeBERT are efficient to learn literal features but less efficient to learn logic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The insights provided by this paper can help future researchers or users in two aspects: Firstly, Code- BERT and GraphCodeBERT, which open a new area for source analysis, are efficient methods for “well-named” source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' However, the user and researcher should expect a lower model perfor- mance if they want to apply them to analyze source code that does not provide enough information in a variable, method, and function names, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=', the code generated from decompilation Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2018) and code that does not follow standard code naming convention Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 4 Arxiv preprint Secondly, this paper indicates that models borrowed from NLP are not very suitable for code anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' The code analysis has some significant differences compared with NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Logical analysis is more important in many sophisticated program analysis tasks, such as vulnerability analysis, and patching generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' But it cannot be well performed by existing model designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' It is important to investigate how to improve the model’s ability for logical analysis in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' REFERENCES Simon Butler, Michel Wermelinger, and Yijun Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' Investigating naming convention adherence in java references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' In 2015 IEEE International Conference on Software Maintenance and Evolution (ICSME), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFAT4oBgHgl3EQfFRzt/content/2301.08427v1.pdf'} +page_content=' 41–50.' metadata={'source': 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Universit´e Toulouse I Capitole +3 Institut Universitaire de France +January 10, 2023 +Abstract +In this paper, we investigate a continuous time version of the Stochastic Langevin Monte Carlo +method, introduced in [39], that incorporates a stochastic sampling step inside the traditional over- +damped Langevin diffusion. This method is popular in machine learning for sampling posterior +distribution. We will pay specific attention in our work to the computational cost in terms of +n (the number of observations that produces the posterior distribution), and d (the dimension +of the ambient space where the parameter of interest is living). We derive our analysis in the +weakly convex framework, which is parameterized with the help of the Kurdyka-�Lojasiewicz (KL) +inequality, that permits to handle a vanishing curvature settings, which is far less restrictive when +compared to the simple strongly convex case. We establish that the final horizon of simulation +to obtain an ε approximation (in terms of entropy) is of the order (d log(n)2)(1+r)2[log2(ε−1) + +n2d2(1+r) log4(1+r)(n)] with a Poissonian subsampling of parameter +� +n(d log2(n))1+r�−1, where the +parameter r is involved in the KL inequality and varies between 0 (strongly convex case) and 1 +(limiting Laplace situation). +Keywords: Langevin Monte Carlo sampling; Log concave models; Weak convexity. +AMS classifications: Primary 6265C05; secondary ; 62C10; 65C30; 60H3520. +1 +1 +Markovian Stochastic Langevin Dynamics and main results +1.1 +Introduction +Motivations +In the recent past years, a huge amount of methods have been developed in machine +learning to handle large scale massive datasets with a large number n of observations (X1, . . . , Xn) +embedded in a high dimensional space Rd. These methods generally involve either optimization of a +data-dependent function (for frequentist learning) or sampling a data-dependent measure (for Bayesian +learning with posterior distributions). In both approaches, a bottleneck lies on the size of n and d +that usually generates numerical difficulties for the use of standard algorithms. We are interested +in this paper in the simulation of a posterior distribution following a Bayesian point of view with a +statistical model described by a collection of densities (pθ)θ∈Θ on X, where the parameter of interest +θ⋆ belongs to Θ = Rd and where the (Xi)1≤i≤n are assumed to be i.i.d. observations in X distributed +according to pθ⋆. A standard Bayesian approach consists in defining a prior distribution π0 on Θ and +then sample the posterior distribution denoted by µn (which will be denoted by exp(−Uνn) below) +using a numerical probabilistic approximation with the help of an over-damped Langevin diffusion: +dθt = −∇Uνn(t)dt + +√ +2dBt. +1We are grateful to Patrick Cattiaux and Arnaud Guillin for helpful discussions and references on functional inequal- +ities and especially on weak log Sobolev inequalities. +1 + +In this work, we manage to deal with an adaptation of the Langevin Monte Carlo (LMC) algorithm +proposed in [39], that exploits some old ideas of stochastic algorithms introduced in [36]: instead of +using the previous equation, the authors propose a modification of the diffusion that generates a noisy +drift in the LMC due to a sampling strategy among the set of observations (Xi)1≤i≤n. Before we +provide some details on the precise objects and algorithm necessary to properly define this method, +we first give some literature insights related to it. +State of the art +Ergodicity and quantitative mixing properties of over-damped LMC and many +other sampling algorithms is a popular subject of research initiated in the probabilistic works around, +roughly speaking, two strategies. The first one relies on pathwise considerations and dynamical proper- +ties of random dynamical system and is built with some coupling argument and Lyapunov controls. We +refer to the seminal contributions [32, 27], that exploits the approach of the Doeblin coupling and total +variation (TV) bounds. Many extensions may be derived from this Lyapunov approach and may lead +to Wasserstein or L2 upper bounds, we refer to [8] and the references therein of the same authors for a +description of the link between Lyapunov conditions and ergodicity. The second strategy derives from +spectral properties of Markov operators and is related to famous functional inequalities (Poincar´e and +Log-Sobolev among others). The general idea is to differentiate the distance along the time-evolution +and apply a Gronwall Lemma to obtain a quantitative estimate of the long-time evolution of the semi- +group. We refer to the seminal contributions of [26, 2], and to [3] for an almost exhaustive survey of +all possible inequalities and consequences on the ergodicity of the Markov semi-groups. Finally, let us +emphasize that some strong links exist between the spectral and the Lyapunov approaches, as pointed +out by [9]. If functional inequalities are then strongly related to mixing properties and especially from +a quantitative point of view, it is therefore necessary to develop a machinery that is able to assess these +inequalities carefully, especially with a specific attention to our statistical setting of large n and d in the +completely non-trivial situation where the target measure is log-concave but not strongly log-concave, +which is a common feature of Bayesian posterior distributions. +On the statistical side, the mixing properties of LMC has been largely investigated during the past +decade, strongly motivated by machine learning methods such as Exponentially Weighted Aggregation +introduced by [11], which involves sampling a non log-concave and heavy tailed posterior distribution. +A first paper of Dalalyan [12] establishes the cost of LMC to obtain an ε TV bound in terms of d +and ρ when the target measure is ρ strongly log-concave and proposes a penalized version of LMC to +circumvent the lack of strong log-concavity when the target distribution is only log-concave. Since this +pioneering paper, a huge impressive literature expanded. Among others, we refer to [16] that gives a +careful study of discretized LMC, [14] for a kinetic version of LMC and [15] where the penalized LMC in +non strongly-concave situation is studied in depth. Among all these papers, first, the lack of strong log- +concavity is dealt with a modification of the initial LMC using a surrogate and asymptotically vanishing +penalty. Second, these papers assume that a noiseless gradient of the log-posterior is available at each +iteration of the algorithm, which may not be realistic, especially with large n. +Stochastic LMC (SLMC below) has attracted the interest of several works: [39] introduced this +method and described its efficiency from a numerical point of view in the particular case of Bayesian +learning, which is exactly our framework. Some recent advances and related contributions may be also +cited: [13] studies a noisy version of LMC and derives some non-asymptotic upper bounds (in terms of +Wasserstein distance) of the sampling strategy in presence of a possibly biased noise for strongly log- +concave posterior distribution. The recent contribution of [40] is also related to our work: the authors +develop a machinery for the study of SLMC essentially based on the Poincar´e inequality but the way +the lower bound on the spectral gap involved in the LMC is dealt with appears to be inappropriate. In +particular, the diffusion involved in (Stochastic)-LMC is used at a very low-temperature, proportional +to 1/n, which generates some important troubles in the size of the spectral gap in non strongly log- +concave framework. In [35], the authors derives some close bounds to our framework for optimization +purpose, and the authors identify the important dependency of the spectral gap denoted by λ∗ in +their paper with the temperature level 1/β they introduced. They obtain some very highly pessimistic +bounds in some general situations (see their discussion in [35][Section 4]), they conclude their discussion +by the urgent need to find some non-trivial situations where some better lower bounds of λ∗ may be +derived. +2 + +Indeed, the final remark of [35][Section 4]) is related to the well known metastability phenomenon: +at a low temperature, the mixing rates of a lot of reversible and irreversible Markov semi-groups +are strongly deteriorated by the low temperature settings, which is implicitly induced by a Bayesian +posterior sampling problem with a large number n of observations. In a regime of variance noise +of the order O(β−1), the first study of large deviation principle of invariant measures traces back +to [18] where the authors establish the asymptotic of the spectral gap of the over-damped Langevin +diffusion as exp(−Iβ) ( [18][Chapter 6]) where I is an explicit constant that depends on the potential +of the Gibbs field. This result has been extended in depth by [26], which leads to the first precise +analyses of the so-called simulated annealing method (see e.g. [24, 33]). These works, and more recent +contributions with irreversible dynamical systems in a stochastic settings ([22, 19]) show that there +is almost nothing to expect in metastable situations in terms of asymptotic behaviour of the spectral +gap, and indirectly in terms of mixing rate. Hence, the only situation that may lead to reasonable +results is an intermediary situation between the (almost) trivial strongly log-concave case and the +metastable multi-welled case. This is the purpose of the weakly log-concave situation that is described +by the family of Kurdyka-�Lojasiewicz inequalities [28, 30] used in optimization theory [5] that have +shown to be efficient for stochastic optimization [20] or for sampling [21]. We also refer to the recent +contributions [6] that derives some functional inequalities within an intermediary framework in which +the curvature ρ is related to their keystone function α that controls the constants involved in the +functional inequalities they are studying. +Taking together the statistical considerations and limitations, we are motivated in this paper in +the study of the continuous time Stochastic Langevin Monte Carlo procedure. This process will be +described precisely in the next paragraph as well as the Kurdyka-�Lojasiewicz setup parametrized by a +real value r, which varies between 0 (strongly convex case) and 1 (limiting Laplace asymptotic tail). +We will show that the final horizon of simulation to obtain an ε approximation is of the order: +(d log(n)2)(1+r)2[log2(ε−1) + n2d2(1+r) log4(1+r)(n)] +with a Poissonian subsampling of parameter +1 +n(d log2(n))1+r . +The rest of the introduction consists in the definitions of the algorithm in Subsection 1.2, the way we +assess the quality of our result with an entropy criterion in Subsection 1.3, as well as the quantitative +weakly log-concave assumption in Subsection 1.4. We finally state our main result in Subsection 1.5. +1.2 +Continuous time evolution +Below, we briefly remind the continuous time SLMC algorithm for Bayesian learning, for which a +discretized form has been introduced in [39]. For this purpose, we consider a statistical model that +is built with the help of a function (x, θ) �−→ pθ(x) where θ ∈ Rd encodes the parameter of the +statistical model and x the observation in a space denoted by X. We then assume that we have n i.i.d. +observations denoted by (X1, . . . , Xn) distributed according to pθ. Given a prior distribution π0 on +Rd, the posterior distribution µn is then defined as: +µn(θ) ∝ π0(θ) × +n +� +i=1 +pθ(Xi). +We introduce the log-parametrization that leads to the Gibbs form: +Ux(θ) = −[log π0(θ) + n log pθ(x)], +and we then observe that: +µn(θ) ∝ exp +� +− 1 +n +n +� +i=1 +UXi(θ) +� += exp (−Uνn(θ)) , +where νn refers to the empirical distribution and Uνn the average value of UX(θ) when X ∼ νn: +νn(x) = 1 +n +n +� +i=1 +δXi(x) +and +Uνn(θ) = EX∼νn[UX(θ)]. +3 + +The standard Langevin Monte Carlo approach relies on the ergodic behaviour of the stochastic differ- +ential equation: +dθt = −∇Uνn(θt)dt + +√ +2dBt, +(1) +that possesses under some mild assumptions a unique invariant distribution µn. +The SLMC algorithm takes benefit of both sampling with a S.D.E. and homogenization of the drift +that may be written as an expectation on X that is sampled uniformly over the set of observations +according to νn. The leading idea is to replace the expectation in Uνn that depends on the overall set +of observations (X1, . . . , Xn) by a single unique observation that is randomized uniformly all along +the evolution of the stochastic differential equation, and modified according to a Markov exponential +clock. That being said, we can write an explicit formal definition of the algorithm as follows. We +define +� +ξ(n) +j +� +j≥1 an infinite sequence of exponential random variables of mean α−1 +n +that will be fixed +later on. +We also consider a sequence +� +V (n) +j +� +j≥0 of i.i.d. random variables uniformly distributed in {1, 2, . . ., n}. +We then define the process (Xt)t≥0 as a jump process that takes its values in {1, 2, . . ., n} such that: +Xt = + + + + + + + + + +XV (n) +1 +, +if +0 ≤ t < ξ(n) +1 +, +XV (n) +j +, +if +j−1 +� +k=1 +ξ(n) +k +≤ t < +j� +k=1 +ξ(n) +k , +j > 1. +(2) +Informally, (Xt)t≥0 should be understood as follows: the process takes the value of one observation +uniformly chosen from the n observations X1, . . . , Xn during exponential times with intensity αn. The +stochastic Langevin over-damped diffusion we consider is then given by the joint evolution (θt, Xt)t≥0 +and that is defined by: +dθt = −∇θUXt(θt)dt + +√ +2dBt, +t > 0, +(3) +where (Bt)t≥0 is a multivariate standard Brownian Motion. +Algorithm 1: Stochastic Langevin over-damped +Data: (X1, . . . , Xn) i.i.d. observations, n0 initial distribution, π0 prior distribution +1 t0 = 0 +2 Generate θ0 according to n0 +3 for k = 0, 1, . . . do +4 +Pick Xk uniformly in {X1, . . . , Xn} +5 +Generate ξk according to an Exponential distribution with mean α−1 +n +6 +tk+1 = tk + ξk +7 +θtk+1 = θtk − +� tk+1 +tk +∇θUXk(θs)ds + +√ +2Bξk +8 end +9 return lim +k→∞ θtk +1.3 +Entropic divergence +To assess the long-time behaviour of the SLMC, we introduce several notations related to the pair +(θt, Xt)t≥0. Below, we denote by λd the Lebesgue measure over Rd. The semi-group induced by L +being elliptic on the θ coordinate, trivially irreducible and finitely supported on the x coordinate, +makes the law of (θt, Xt) absolutely continuous with respect to the measure λd ⊗ νn as soon as t > 0. +We introduce the notation of mt to refer to the joint density of (θt, Xt) at time t with respect +to λd ⊗ νn. In the meantime, nt denotes the marginal distribution of θt and mt(·|θ) the conditional +distribution of Xt given θt = θ. That is: +Law(θt, Xt) = mt, +nt(θ) = +n +� +i=1 +mt(θ, Xi), +mt(x|θ) = mt(θ, x) +nt(θ) , +(4) +4 + +for θ ∈ Rd and x ∈ {X1, . . . , Xn}. +To show that the SLMC algorithm recovers the correct asymptotic behaviour, i.e. that nt(θ) −→ µn +when t −→ ∞, we consider the relative entropy (or Kullback-Leibler divergence) of nt with respect to +µn that is well defined thanks to the ellipticity, and given by: +Jt = Entµn +� nt +µn +� += +� +Rd +log +� nt(θ) +µn(θ) +� +dnt(θ). +(5) +Jt measures at any time t > 0 a divergence between the instantaneous law of the process at time t +and the (presumably) invariant distribution µn of the process (θt, Xt). It would also be possible to +measure this difference between the two distributions in terms of the L2 or the χ-square distance and +to produce a theoretical analysis with the help of functional analysis but it would rely on stronger +assumptions on the function Uνn. +In the meantime, we also introduce a weighted L2 distance between the conditional distribution of +Xt given θt = θ and the measure νn. This distance is denoted by It and is defined as: +It = +� +Rd +n +� +i=1 +�mt(Xi|θ) +νn(Xi) +− 1 +�2 +νn(Xi)dnt(θ). +(6) +This quantity measures the average closeness (w.r.t. θ) of the conditional law of x given θ at time t to +νn. +1.4 +Main assumptions +Weak convexity +We will study the SLMC into a weakly convex framework, i.e. when Uνn is assumed +to be convex but not necessarily strongly convex. SLMC has recently received an important interest in +the machine learning community and has been studied essentially in its explicit Euler discretized form +in various situations where functional inequalities are involved. We refer to [38] (uniform Log-Sobolev +inequality), to [35] (uniform Poincar´e inequality) where the authors develop a Wasserstein-2 analysis +of the algorithm, and to [40] (uniform Poincar´e inequality). In these works, the functional inequalities +play a crucial role to analyze the behaviour of SLMC and these inequalities are assumed, which is an +important hypothesis. Importantly, Poincar´e or Log-Sobolev inequalities are not so innocent since they +generally require convexity (see e.g. [4, 3]) to be reasonably dimension-dependent, and even strong +convexity to be dimension free. Otherwise, the constant involved in these functional inequalities are +exponentially degraded by the “temperature” (n−1(d log2β(n))−(1+r) in our case) and the dimension +(d for us) as indicated in [26]. +In our work, we have chosen to parameterize this lack of strong convexity with the help of the +Kurdyka-�Lojasiewicz inequality [28, 30], which is a standard tool in optimization to describe the tran- +sition between convexity and strong convexity and makes the bounds more explicit. This assumption +allows to observe how the entropy evolves according to the key exponent involved in the KL inequality. +In particular, it makes possible to understand the influence of the lack of strong convexity that is more +or less hidden in the uniform Poincar´e or Log-Sobolev inequalities that are assumed in the previous +works. We introduce a parametric form of the KL inequalities following [20]. +For this purpose, for any V twice differentiable function, we denote the spectrum of the Hessian +matrix of V as Sp(∇2V (θ)). Furthermore, if V is convex, we denote: +λ∇2V (θ) = inf Sp(∇2V (θ)). +Hypothesis Hr +KL(c, L) We say that a function V : Rd → R satisfies a Hr +KL(c, L)-condition if: +a) V is a C2-function. +b) V is a convex function and minθ∈RdV (θ) = V (θ∗) > 0. +c) ∇V is L-Lipschitz. +5 + +d) There exist some constants 0 ≤ r < 1 and c > 0 such that: +cV −r(θ) ≤ λ∇2V (θ) +∀θ ∈ Rd. +(7) +Let us briefly comment this assumption. +• In [21], a slightly different parametrization is used with the introduction of another exponent +q related to λ∇2V (θ) = sup Sp(∇2V (θ)). The authors also assume the upper bound λ∇2V (θ) ≤ +˜cV −q(θ). Here, we have chosen to simplify this assumption and use a rough upper bound on the +eigenvalues of the Hessian matrix given by the Lipschitz constant L, i.e. in the last inequality +we simply use ˜c = L and q = 0. +• We shall observe that if V (θ) = (1 + ∥θ∥2 +2)p with p ∈ [1/2, 1], then V satisfies Hr +KL(c, L) with +r = 1−p +p +and c = 2p(1 − 2(1 − p)), see Remark 7 of [21] for further details. In particular, the +larger p, the smaller r, which translates into a better curvature of the potential function V . +• When r = q, we recover a global standard KL inequality (see [20, 5]) and when r = 1 it +corresponds to the limiting Laplace case. +• The case r = 0 is of course associated to the strongly convex situation where the curvature of +the function is uniformly lower bounded by c. +Hence, it is expected that the complexity of SLMC increases with the lack of curvature, i.e. is an +increasing function of r. +In section 4 we recall some important consequences of the KL inequality obtained in Lemma 15 of +[21]. In particular, the growth of any function that satisfies Hr +KL(c, L) is lower and upper bounded by +a positive power of the distance to its minimizer. +If inequality (7) holds for a constant c, then it holds for all positive values less than c. For that +reason, in section 5 we assume c ≤ +� +8L +(1+r) +�1+r +. +Assumption on the prior π0 +We state below the important consequence of a “population” Hr +KL(c, L) +assumption, but before, let us state some mild assumptions on π0. +Hypothesis Hπ0(ℓ0) π0 is a log-concave C2-function such that minθ∈Rd − log π0(θ) > 0 and θ �→ +∇ log π0(θ) is ℓ0-Lipschitz. +Since the prior distribution is chosen by the user, our Hπ0(ℓ0) hypothesis is not restrictive and +some typical examples satisfy these conditions, such as Gaussian, Weibull and Gamma, both with +shape parameter larger than 1, Gumbel, among others. +Proposition 1.1. We assume Hπ0(ℓ0) and that there exist (c, r) such that for any x: θ �−→ − log pθ(x) +satisfies Hr +KL(c, L), then Uνn satisfies Hr +KL +� +cn1+r, nL + ℓ0 +� +, and in particular, for any Xi, UXi sat- +isfies Hr +KL +� +cn1+r, nL + ℓ0 +� +. +We introduce the notation a ≲uc b (a ≳uc b) which means a ≤ cb (a ≥ cb) where c is a universal +constant i.e. a positive constant independent of n and d. +We assume that the minimizers of the functions UXi are contained in a ball of radius which depends +of n and d. Additionally, we consider minθ∈RdUXi to be at most of order d. +Hypothesis Hmin There exists β ≥ 0 such that: +maxi∥ arg min UXi∥2 ≲uc +√ +d logβ(n) +and +maxi minθ∈Rd UXi(θ) ≲uc d. +Assumption Hmin is not restrictive. In dimension d = 1, it holds for many concentrated i.i.d. +samples (Xi)1≤i≤n with a suitable sub-Gaussian like behaviour for which the Laplace transform of +min UXi is upper bounded as: +E[exp(λmin UXi)] ≤ exp(σ2λk), +∀λ > 0. +6 + +The previous upper bound implies that, in this case, β involved in Hmin is given by β = k−1 +k . We +recover in particular the situation where β = 1/2 when k = 2. For larger dimensions, the result may +be extended using that ∥X∥2 +2 ≤ d max1≤j≤d(Xj)2, where Xj is the j-th component of X. We should +keep in mind from this last discussion that even if Hmin is stated (and makes sense) for any value of +β > 0, it holds in general for β ≤ 1. +This Hmin hypothesis together with Hπ0(ℓ0) lead to an almost similar behaviour of the minimizer +and the minimum of Uνn. Details appear in Proposition 4.4. +1.5 +Long-time entropy convergence +We introduce for any time t ≥ 0 the density of Law(θt) w.r.t. µn, which is given by: +ft(θ) = nt(θ) +µn(θ), +and n0 is chosen such that ∥f0∥∞ < +∞. The following hypothesis guarantees this result which will +be proved in Proposition 3.5. +Hypothesis Hn0(L, ℓ0) A positive constant σ2 exists such that n0 = N(0, σ2Id). Moreover, there +exist two universal constants c1 and c2 such that 0 < c1 ≤ c2 < 1 and +c1 +nL + ℓ0 +≤ σ2 ≤ +c2 +nL + ℓ0 +. +Futhermore, in Proposition 3.5, as an immediate consequence of the boundedness of ∥f0∥∞, we +obtain that J0 ≲uc nd1+r log2β(1+r)(n) + d log +� d +n +� +. +The next result assesses a mixing property in terms of decrease of the entropy and therefore states +the convergence of nt towards the correct measure µn. +Theorem 1.1. Assume Hπ0(ℓ0), Hmin, Hn0(L, ℓ0) and that each θ �→ − log pθ(Xi) satisfies Hr +KL(c, L), +then +• Uνn satisfies a Poincar´e inequality of constant CP (µn), indistinctly denoted as CP . +• Define cn,d := n4 � +d log2β(n) +�1+r +and On,d := +� C1d +n +� dr +2 exp +� +C2n +� +d log2β(n) +�1+r� +, where C1 +and C2 are universal constants, then for any t > 0: +Jt ≲uc +� +J0 + cn,d +αn +� +1 + +�CP +αn ++ +� +CP +� +e +√ +CP +√a + CP +3αn +� ++ On,d +� +(1 + t)1/4e− +√ +Cp +√a (√1+t−1). +(8) +• For any ε > 0, if αn = +1 +n(d log2β(n)) +1+r , then: +t ≳uc +� +d log2β(n) +�(1+r)2 � +log2(ε−1) + n2 � +d log2β(n) +�2(1+r) ++ d2 log2 d +� +=⇒ Jt ≤ ε. +If we denote tε the smallest value such that Jtε ≤ ε, then the choice of αn = +1 +n(d log2β(n)) +1+r +guarantees that the mean number of jumps αntε of the process (Xt)0≤t≤tε is the minimum possible. +In order to proof the main result, we first present in Section 2 the classical tools related to the +Markov semi-group, which could be skipped by the experienced reader in the subject. In Section 3 +we prove the main result. Sections 4 and 5 are reserved to the technical results of the Hr +KL(c, L) +hypothesis and Uνn, and the Markov Dynamics respectively. +7 + +2 +Markov tools +It is straightforward to verify that the joint evolution of (θt, Xt)t≥0 exists and is weakly unique (in +law) with the help of the Martingale Problem (MP below). For this purpose, we preliminary define +the operator L that acts on any function f ∈ C2(Rd × X) as: +Lf(θ, x) = −⟨∇θUx(θ), ∇θf(θ, x)⟩ + ∆θf(θ, x) +� +�� +� +:=L1f(θ,x) ++ αn +n +n +� +i=1 +[f(θ, Xi) − f(θ, x) +� +�� +� +L2f(θ,x) +], +(9) +for all (θ, x) ∈ Rd × X. +The operator L is divided into two terms, L1 acts on the component θ and is associated to the +diffusion part, while L2 is the jump operator that acts on the x component. Thanks to the finiteness +of the number of observations (X1, . . . , Xn), we can apply the results of Section 4 and 5 of chapter 4 +of [17] and deduce the following result: +Proposition 2.1. Assume that for any x ∈ X, Ux is C2(Rd) and ∇θUx is Lx-Lipschitz, then for any +initial distribution ν on Rd × X, the martingale problem (L, ν) is well-posed. +The associated (weakly) unique process (θt, Xt)t≥0 is a Feller Markov process associated to the +semi-group L. In particular, the θ component verifies the S.D.E. (3). +If we denote by L⋆ the adjoint operator of L in L2(Rd) × νn, the backward Kolmogorov Equation +yields: +∂tmt(θ, x) = L⋆mt(θ, x). +(10) +Using the ellipticity of the semi-group L on the θ coordinate, we can use the result of [25] and +deduce that for any t > 0, nt ∈ C∞(Rd, R) and the irreducibility yields ∀t ≥ 0, nt > 0. We will prove +in Proposition 3.5 some sufficient conditions that implies ∥f0∥∞ = ∥ n0(θ) +µn(θ)∥∞ < +∞ and an important +and standard consequence of the maximum principle, is as follows: if ∥f0∥∞ ≤ M, then +∀t ≥ 0, +∥ft∥∞ ≤ M. +We defer the details of this result to the Proposition 3.5 as they are not central to our analysis and +are rather technical. +Thanks to this master equation, it is possible to compute the derivative of the semi-group on some +time dependent function of θ. For this purpose, we introduce two keystone operators. The first one +describes the infinitesimal action on the θ coordinate under the average effect of Xt at time t that +applies ∀f ∈ C2(Rd, R) as: +Gtf(θ) = − +n +� +i=1 +⟨∇θf(θ), ∇θUXi(θ)⟩mt(Xi|θ) + ∆θf(θ). +(11) +The second one is very close to the first one except that the average effect of Xt is replaced by the +targeted ideal distribution νn. It leads to the definition ∀f ∈ C2(Rd, R): +Gf(θ) = − +n +� +i=1 +⟨∇θf(θ), ∇θUXi(θ)⟩νn(Xi) + ∆θf(θ) = −⟨∇θf(θ), ∇θUνn(θ)⟩ + ∆θf(θ). +(12) +This derivative is given in the next result, whose proof is deferred to the appendix. +Lemma 2.1. Let be ht a twice differentiable function with uniformly bounded first and second order +derivatives on Rd, then for t > 0: +∂t +�� +Rd ht(θ)dnt(θ) +� += +� +Rd ∂t{ht(θ)}dnt(θ) + +� +Rd Gtht(θ)dnt(θ), +(13) +where Gt is the diffusion operator under the average effect of Xt, defined in Equation (11). +8 + +3 +Proof of the main results +3.1 +Evolution of the entropy Jt +The entropy satisfies the following differential inequality. +Proposition 3.1. Assume Hmin, Hπ0(ℓ0) and for each Xi, θ → − log pθ(Xi) satisfies Hr +KL(c, L), then +a ”universal” constant C (independent from n and d) exists such that ∀t > 0: +∂t{Jt} ≤ − +� +Rd +�����∇θ +�� +nt(θ) +µn(θ) +������ +2 +2 +dµn(θ) + CI +1 +3 +t n +11 +3 +� +d log2β(n) +�1+r +. +Proof. We shall use the standard preliminary estimate that may be derived from Equation (3.14) of +[29] for elliptic diffusions to apply Lemma 2.1 to ft = log(ntµ−1 +n ). From Equation (13), we have: +∂t{Jt} = +� +Rd ∂t +� +log +� nt(θ) +µn(θ) +�� +dnt(θ) + +� +Rd Gt log +� nt(θ) +µn(θ) +� +dnt(θ), +The first term vanishes since: +� +Rd ∂t +� +log +� nt(θ) +µn(θ) +�� +dnt(θ) += +� +Rd +∂t{nt(θ)} +nt(θ) +dnt(θ) += +� +Rd ∂t {nt(θ)} dθ += +∂t +�� +Rd dnt(θ) +� += +0. +Then, the derivative is reduced to the second term, and we are led to: +∂t{Jt} += +� +Rd Gt log +� nt(θ) +µn(θ) +� +dnt(θ), += +� +Rd G log +� nt(θ) +µn(θ) +� +dnt(θ) +� +�� +� +J1,t ++ +� +Rd (Gt − G) log +� nt(θ) +µn(θ) +� +dnt(θ) +� +�� +� +J2,t +. +(14) +We study the two terms J1,t and J2,t separately. +• Study of J1,t. Since G is a diffusion operator and µn is the invariant measure associated to G, +then we can use the classical link between J1,t and the Dirichlet form (see [3]): +� +Rd G log +� nt(θ) +µn(θ) +� +dnt(θ) += +� +Rd +nt(θ) +µn(θ) G log +� nt(θ) +µn(θ) +� +dµn(θ) += +−4 +� +Rd +�����∇θ +�� +nt(θ) +µn(θ) +������ +2 +2 +dµn(θ). +(15) +• Study of J2,t. We use the difference between G and Gt, for any twice differentiable function f: +(Gt − G) f(θ) += +− +n +� +i=1 +⟨∇θf(θ), ∇θUXi(θ)⟩ [mt(Xi|θ) − νn(Xi)] += +− +n +� +i=1 +⟨∇θf(θ), ∇θUXi(θ)⟩ +�mt(Xi|θ) +νn(Xi) +− 1 +� +νn(Xi). +9 + +Then, the term J2,t may be computed as: +|J2,t| += +���� +� +Rd (Gt − G) log +� nt(θ) +µn(θ) +� +dnt(θ) +���� += +����� +� +Rd +n +� +i=1 +⟨∇θ log +� nt(θ) +µn(θ) +� +, ∇θUXi(θ)⟩ +�mt(Xi|θ) +νn(Xi) +− 1 +� +νn(Xi) dnt(θ) +����� . +Using the Cauchy-Schwartz inequality with respect to the measure νn(Xi) × dnt(θ) in the first +line, 2ab ≤ a2 + b2 in the second line and ∇ log f = 2∇ log √f = 2 ∇√f +√f +in the third line, we +obtain that: +|J2,t| +≤ +�� +Rd +����∇θ log +� nt(θ) +µn(θ) +����� +2 +2 +dnt(θ) +� 1 +2 �� +Rd +n +� +i=1 +��∇θUXi(θ) +��2 +2 +�mt(Xi|θ) +νn(Xi) +− 1 +�2 +νn(Xi) dnt(θ) +� 1 +2 +≤ +3 +4 +� +Rd +����∇θ log +� nt(θ) +µn(θ) +����� +2 +2 +dnt(θ) + 1 +3 +� +Rd +n +� +i=1 +��∇θUXi(θ) +��2 +2 +� mt(Xi|θ) +νn(Xi) +− 1 +�2 +νn(Xi) dnt(θ) +≤ +3 +� +Rd +�����∇θ +�� +nt(θ) +µn(θ) +������ +2 +2 +dµn(θ) + 1 +3 +� +Rd +n +� +i=1 +��∇θUXi(θ) +��2 +2 +�mt(Xi|θ) +νn(Xi) +− 1 +�2 +νn(Xi) dnt(θ). +Using Equation (15) and the previous line yields: +∂t{Jt} +≤ +− +� +Rd +�����∇θ +�� +nt(θ) +µn(θ) +������ +2 +2 +dµn(θ) + 1 +3 +� +Rd +n +� +i=1 +��∇θUXi(θ) +��2 +2 +�mt(Xi|θ) +νn(Xi) +− 1 +�2 +νn(Xi) dnt(θ) +� +�� +� +:=∆t +, +(16) +We then focus on the second term of the right hand side. For this purpose, we consider a non- +negative function g(t), which will be fixed later and we split ∆t into two terms as: +∆t += +� +Rd +n +� +i=1 +��∇θUXi(θ) +��2 +2 +� +1∥∇θUXi (θ)∥2≤g(t) + +1∥∇θUXi (θ)∥2>g(t) +� �mt(Xi|θ) +νn(Xi) +− 1 +�2 +νn(Xi) dnt(θ) +≤ +g2(t)It + +� +Rd +n +� +i=1 +��∇θUXi(θ) +��2 +2 +1∥∇θUXi (θ)∥2>g(t) +�mt(Xi|θ) +νn(Xi) +− 1 +�2 +νn(Xi) dnt(θ), +where It has been introduced in Equation (6) and measures the closeness of mt(Xi|θ) to νn. Finally, +for the last term we observe that 0 ≤ mt(Xi|θ) ≤ 1 and +��� mt(Xi|θ) +νn(Xi) − 1 +��� = n +��mt(Xi|θ) − 1 +n +�� ≤ n, which +implies that: +∆t ≤ g2(t)It + n2 1 +n +� +Rd +n +� +i=1 +∥∇θUXi(θ)∥2 +2 +1∥∇θUXi (θ)∥2>g(t)dnt(θ) +� +�� +� +:= ˜∆t +. +(17) +The Cauchy inequality leads to: +˜∆t +≤ +� +1 +n +� +Rd +n +� +i=1 +∥∇θUXi(θ)∥4 +2 dnt(θ) +� 1 +2 � +1 +n +� +Rd +n +� +i=1 +1∥∇θUXi (θ)∥2>g(t)dnt(θ) +� 1 +2 += +� +1 +n +n +� +i=1 +E +� +∥∇θUXi(θt)∥4 +2 +�� 1 +2 � +1 +n +n +� +i=1 +P (∥∇θUXi(θt)∥2 > g(t)) +� 1 +2 +. +(18) +We then use Proposition 4.1 and obtain that: +˜∆t +≤ +� +1 +n +n +� +i=1 +E +�� +2(nL + ℓ0)U 2 +Xi(θt) +�� +� 1 +2 � +1 +n +n +� +i=1 +P +� +2(nL + ℓ0)UXi(θt) > g2(t) +� +� 1 +2 +≤ +2(nL + ℓ0) +� +nE[U 2 +νn(θt)] +� 1 +2 +� +1 +n +n +� +i=1 +2(nL + ℓ0) +g2(t) +E [UXi(θt)] +� 1 +2 +≤ +[2(nL + ℓ0)] +3 +2 n +1 +2 E +� +U 2 +νn(θt) +� 1 +2 E [Uνn(θt)] +1 +2 +g(t) +, +10 + +where we used the Markov’s inequality and the relation ∥.∥2 ≤ ∥.∥1 in Rn. We apply Proposition 5.1 +with α = 2 and α = 1 and obtain that a constant C > 0 exists (whose value may change from line to +line) such that: +˜∆t +≤ +C +n +7 +2 +� +d log2β(n) +� 3(1+r) +2 +g(t) +. +We use this last bound in (17) and we deduce that: +∆t ≤ g2(t)It + C +n +11 +2 +� +d log2β(n) +� 3(1+r) +2 +g(t) +. +Optimizing this last bound with respect to g(t) leads to the upper bound: +∆t ≤ CI +1 +3 +t n +11 +3 +� +d log2β(n) +�1+r +, +∀t ≥ 0. +3.2 +Evolution of the weighted L2 distance It +The quantity It involved in Proposition 3.1 measures how close to νn the conditional distribution of +Xt|θt is. To study It, we first remark that it may be rewritten in a simpler way. +It += +� +Rd +n +� +i=1 +�mt(Xi|θ) +νn(Xi) +− 1 +�2 +νn(Xi) dnt(θ) += +� +Rd +n +� +i=1 +�m2 +t(Xi|θ) +ν2n(Xi) +− 2mt(Xi|θ) +νn(Xi) ++ 1 +� +νn(Xi) dnt(θ) += +� +Rd +n +� +i=1 +�m2 +t(Xi|θ) +νn(Xi) +− 2mt(Xi|θ) + νn(Xi) +� +dnt(θ) += +� +Rd +� n +� +i=1 +m2 +t(Xi|θ) +νn(Xi) +− 1 +� +dnt(θ) += +� +Rd +n +� +i=1 +m2 +t(Xi|θ) +νn(Xi) dnt(θ) − 1. +Using that mt(Xi|θ)nt(θ) = mt(θ, Xi) and νn(Xi) = 1 +n for i = 1, 2, . . ., n, we obtain that: +It = n +� +Rd +n +� +i=1 +m2 +t(θ, Xi) +nt(θ) +dθ − 1. +(19) +The next proposition then assesses how fast It decreases to 0 as t −→ +∞. +Proposition 3.2. For any t ≥ 0: +It ≤ I0e−2αnt ≤ (n − 1)e−2αnt. +(20) +Proof. Our starting point is Equation (19). We compute its derivative with respect to t: +∂t{It} += +2n +� +Rd +n +� +i=1 +mt(θ, Xi) +nt(θ) +∂tmt(θ, Xi)dθ − n +� +Rd +n +� +i=1 +m2 +t(θ, Xi) +n2 +t(θ) +∂tnt(θ)dθ += +2n +� +Rd +n +� +i=1 +mt(Xi|θ)∂tmt(θ, Xi)dθ − n +� +Rd +n +� +i=1 +m2 +t(Xi|θ)∂tnt(θ)dθ. +11 + +Using the Kolmogorov backward equation in the first line and L = L1 + L2 in the second one where +L1 and L2 are defined in Equation (9), we have: +∂t{It} += +2n +� +Rd +n +� +i=1 +Lmt(Xi|θ) mt(θ, Xi)dθ − n +� +Rd +n +� +i=1 +m2 +t(Xi|θ)∂tnt(θ)dθ += +2n +� +Rd +n +� +i=1 +L1mt(Xi|θ) mt(θ, Xi)dθ +� +�� +� +:=I3,t ++ 2n +� +Rd +n +� +i=1 +L2mt(Xi|θ) mt(θ, Xi)dθ +� +�� +� +:=I1,t +−n +� +Rd +n +� +i=1 +m2 +t(Xi|θ)∂tnt(θ)dθ +� +�� +� +:=I2,t +. +(21) +Then, ∂t{It} may be splitted into three terms that are studied separately. +• Study of I1,t. We observe that: +L2mt(Xi|θ) = αn +n +n +� +j=1 +[mt(Xj|θ) − mt(Xi|θ)] = αn +n − αn mt(Xi|θ). +(22) +We then use this last equation in the definition of I1(t) and obtain that: +I1,t += +2n +� +Rd +n +� +i=1 +L2mt(Xi|θ) mt(θ, Xi)dθ += +2αn +� +Rd +n +� +i=1 +mt(θ, Xi)dθ − 2αnn +� +Rd +n +� +i=1 +mt(Xi|θ)mt(θ, Xi)dθ += +2αn − 2αnn +� +Rd +n +� +i=1 +m2 +t(θ, Xi) +nt(θ) +dθ += +−2αnIt. +(23) +• Study of I2,t. Using the definition of nt, we obtain that: +I2,t += +−n +� +Rd +n +� +i=1 +m2 +t(Xi|θ)∂tnt(θ)dθ += +−n +� +Rd +n +� +i=1 +m2 +t(Xi|θ)∂t + + +n +� +j=1 +mt(θ, Xj) + + dθ += +−n +� +Rd +n +� +j=1 +n +� +i=1 +m2 +t (Xi|θ)∂tmt(θ, Xj)dθ += +−n +� +Rd +n +� +j=1 +� n +� +i=1 +Lm2 +t (Xi|θ) +� +mt(θ, Xj)dθ += +−n +� +Rd +n +� +i=1 +Lm2 +t(Xi|θ) dnt(θ). +where we used the Kolmogorov backward equation in the fourth line and again the definition of +nt in the last line. Again, the decomposition L = L1 + L2 yields: +I2,t += +−n +� +Rd +n +� +i=1 +L1m2 +t(Xi|θ) dnt(θ) − n +� +Rd +n +� +i=1 +L2m2 +t(Xi|θ) dnt(θ). +12 + +We repeat some similar computations as those developed in Equation (22) to study the action +of the jump component induced by L2 on m2 +t . We obtain that: +L2m2 +t(Xi|θ) = αn +n +n +� +k=1 +[m2 +t(Xk|θ) − m2 +t(Xi|θ)] = αn +n +n +� +k=1 +m2 +t (Xk|θ) − αn m2 +t(Xi|θ). +We use this last equation and obtain that: +I2,t += +−n +� +Rd +n +� +i=1 +L1m2 +t (Xi|θ) dnt(θ) − αn +� +Rd +n +� +i=1 +n +� +k=1 +m2 +t(Xk|θ) dnt(θ) ++αnn +� +Rd +n +� +i=1 +m2 +t(Xi|θ) dnt(θ) += +−n +� +Rd +n +� +i=1 +L1m2 +t (Xi|θ) dnt(θ) − αnn +� +Rd +n +� +k=1 +m2 +t(Xk|θ) dnt(θ) ++αnn +� +Rd +n +� +i=1 +m2 +t(Xi|θ) dnt(θ) += +−n +� +Rd +n +� +i=1 +L1m2 +t (Xi|θ) dnt(θ). +(24) +• Study of I2,t + I3,t. We observe that this sum involves only L1 (see Equation (9). We first +compute: +L1mt(Xi|θ) = −⟨∇θUXi(θ), ∇θmt(Xi|θ)⟩ + ∆θmt(Xi|θ), +and similarly: +L1m2 +t(Xi|θ) = −⟨∇θUXi(θ), ∇θm2 +t(Xi|θ), ⟩ + ∆θm2 +t(Xi|θ) += −2mt(Xi|θ)⟨∇θUXi(θ), ∇θmt(Xi|θ)⟩ + 2∥∇θmt(Xi|θ)∥2 +2 + 2mt(Xi|θ)∆θmt(Xi|θ). +Using these two equations into I2,t + I3,t and mt(Xi|θ)nt(θ) = mt(θ, Xi), we get: +I2,t + I3,t +n += 2 +� +Rd +n +� +i=1 +⟨∇θmt(Xi|θ), ∇θUXi(θ)⟩mt(θ, Xi)dθ +− 2 +� +Rd +n +� +i=1 +∥∇θmt(Xi|θ)∥2 +2 nt(θ)dθ − 2 +� +Rd +n +� +i=1 +∆θmt(Xi|θ) mt(θ, Xi)dθ +− 2 +� +Rd +n +� +i=1 +⟨∇θmt(Xi|θ), ∇θUXi(θ)⟩mt(θ, Xi)dθ + 2 +� +Rd +n +� +i=1 +∆θmt(Xi|θ) mt(θ, Xi)dθ += − +� +Rd +n +� +i=1 +∥∇θmt(Xi|θ)∥2 +2 dnt(θ) ≤ 0. +Gathering this last inequality with (23) into Equation (21) yields: +∂t{It} ≤ −2αnIt. +We conclude with a direct application of the Gronwall lemma while observing that I0 ≤ n − 1. +3.3 +Functional (weak) log-Sobolev inequalities +3.3.1 +Related works on functional inequalities +A straightforward consequence of Proposition 3.1 and Proposition 3.2 is the following differential +inequality on the relative entropy Jt: +∂t{Jt} ≤ − +� +Rd +�����∇θ +�� +nt(θ) +µn(θ) +������ +2 +2 +dµn(θ) + cn,de− 2αn +3 +t, +(25) +13 + +where cn,d is defined as: +cn,d ≲uc n4 � +d log2β(n) +�1+r +. +(26) +At this stage, we should observe that a standard approach consists in finding a functional inequality +that relates the key Dirichlet form E(f) defined by: +E(f) = +� +Rd ∥∇θf(θ)∥2 +2dµn(θ), +(27) +to Entµn(f 2), the entropy itself with respect to µn. These approaches rely on the initial works of [23] +where Logarithmic Sobolev Inequality (LSI for short) were introduced. The consequences of LSI to +exponential ergodicity has then been an extensive field of research and we refer to [3] for an overview +on this topic. A popular sufficient condition that ensures LSI is the log strong-convexity of the targeted +measure (see among other [2]) and an impressive amount of literature has been focused on the existing +links between these functional inequalities, ergodicity of the semi-group, transport inequalities and +Lyapunov conditions. We refer to [8, 1] (these two works are far from being exhaustive). The great +interest of LSI has then been observed in machine learning and statistics more recently as testified by +the recent works in Monte Carlo samplings of [31, 34]. A popular way to extend LSI from the strongly +convex situation to a more general case relies on the “strong convexity outside a ball” hypothesis using +the perturbation argument of the seminal contributions of [26]. If this method proves to be suitable +for the study of the simulated annealing process in [33], [26], it appears to be doubtful for the study +of sampling problems with convex potentials that satisfies Hr +KL(c, L) as this settings do not imply an +asymptotic strong convexity of θ �−→ U(θ) for large values of ∥θ∥2. That being said, and maybe an +even worst consequence of such approach, is the unavoidable dependency on the dimension for the LSI +constant when using a perturbation approach, which leads to a serious exponential degradation of the +convergence rates with the dimension of the ambient space. +To overcome these difficulties, we have chosen to use a slightly different functional inequality that +may be considered as an innocent modification of LSI, but that indeed appears to be well suited +to weakly log-concave setting described through an Hr +KL(c, L) assumption. +For this purpose, we +shall use weak log-Sobolev inequalities (WLSI for short below) that have been introduced in [37] +and whose interest has been extensively studied in many works to obtain exponentially sub-linear +rates of mixing, see among others for example [7]. +To derive such inequalities, our starting point +will be the contribution of [10] that makes the link between Lyapunov conditions and WLSI. Our +approach based on Hr +KL(c, L) certainly shares some similarities with the recent work of [6] where +some functional inequalities (Poincar´e and Transport inequalities) are obtained within a framework of +variable curvature bound. +3.3.2 +Weak log Sobolev inequalities +We briefly introduce the key theoretical ingredients, that are exhaustively described in [3]. We intro- +duce the following assumption, that will be suitable for the setting of bounded functions. +Definition 3.1 (Weak Log-Sobolev Inequality ). For any measurable space (Ω, F, µ) and for any nice +function f, let us define: +Entµ(f 2) := +� +Ω +f 2 log(f 2)dµ − +� +Ω +f 2dµ log +�� +Ω +f 2dµ +� +. +The measure µ satisfies a WLSI if a non-increasing function ϕWLS : (0, +∞) �→ R+ exists such that +for any f ∈ C +1 +b (Ω): +Entµ(f 2) ≤ ϕWLS(s)E(f) + s Osc2(f), +(28) +where Osc(f) := sup f − inf f. +Before establishing how to use this functional inequality, we first state the important relationship +between Poincar´e Inequality and WLSI. +14 + +Proposition 3.3. Assume that µ satisfies a Poincar´e Inequality of constant CP , i.e. for any smooth +integrable function f: +Cp(µ)V arµ(f) = Cp(µ) +� +Ω +(f − µ[f])2dµ ≤ +� +Ω +|∇f|2dµ, +then if log c = +3 +14e2 +� 1 +e + 1 +2 +� ++ 1 + log +� 14 +3 +� +, then µ satisfies a WLSI with: +ϕWLS(s) = +� +0, +s > 1 +e + 1 +2 +32 +CP log +� c +s +� +, +s ≤ 1 +e + 1 +2 +. +For the sake of readability, we introduce a universal a > 0 such that: +ϕWLS(s) = +� +0, +s > 1 +e + 1 +2 +a +1+log( 1 +s) +CP +, +s ≤ 1 +e + 1 +2 +. +(29) +Proof of Proposition 3.3. The proof of how the Poincar´e Inequality implies the WLSI in the bounded +setting described in Definition 28 is given for the sake of completeness. Technical details are skipped +and we refer to the references below. We use the measure-capacity inequality (see [3], Section 8.3). +We know that the Poincar´e Inequality implies a capacity inequality (Proposition 8.3.1 of [3]) with a +constant equal to 2CP . Then, we can apply Theorem 2.2 of [7] that induces a WLSI which is based +on the function ϕWLS given in the statement of the proposition. +3.3.3 +Weak log Sobolev inequalities under Hr +KL(c, L) +Of course, in the previous result, the only important dependency will be the one induced by CP , which +will deserve an ad-hoc study under Assumption Hr +KL(c, L). The numbers 32 and log(c) will be dealt +with as “universal constants” in what follows. +The next proposition states two lower bounds on the Poincar´e constant within the Hr +KL(c, L) +framework. The first one always holds, regardless the value of (X1, . . . , Xn) that may be been randomly +sampled. The second one has to be considered with high probability, with respect to the sampling +process (X1, . . . , Xn). +Proposition 3.4. Assume Hmin,Hn0(L, ℓ0), Hπ0(ℓ0) and for any x, θ �→ − log pθ(x) satisfies Hr +KL(c, L), +then: +i) For any sample (X1, . . . , Xn), it holds: +CP (µn) ≳uc +1 +� +d log2β(n) +�(1+r)2 +ii) Assume that θ �→ Pθ is injective and θ0 exists such that (X1, . . . , Xn) ∼ Pθ0. If locally around +θ0, θ �→ |θ − θ0|−αW1(Pθ, Pθ0) does not vanish, then: +E(X1,...,Xn)∼Pθ0[CP (µn)] ≳uc +� +n +Ld log n +�α +. +We are finally led to upper bound the oscillations of the function involved in the WLSI introduced +in (28), i.e. we are looking for an upper bound of Osc2 �� +nt +µn +� +for any time t > 0. For this purpose, +we observe that the Markov semi-group induces that ft = +nt +µn = Ptf0 where f0 = +n0 +µn . +The next +proposition implies the boundedness of ft over Rd when n0 is chosen as a Gaussian distribution with +a carefully tuned covariance matrix. +Proposition 3.5. Assume Hmin,Hn0(L, ℓ0), Hπ0(ℓ0) and that, for any x, θ �→ − log pθ(x) satisfies +Hr +KL(c, L), then: +15 + +i) Two positive constants C1 and C2 exist, which are independent from n and d and such that: +∥f0∥∞ ≲uc +�C1d +n +� dr +2 +exp +� +C2nd1+r log2β(1+r)(n) +� +. +ii) As a consequence: +Osc2( +� +ft) ≤ Osc2( +� +f0) ≲uc +�C1d +n +� dr +2 +exp +� +C2nd1+r log2β(1+r)(n) +� +. +iii) Moreover, a straightforward consequence of i) is: +J0 = +� +Rd log (f0(θ)) dn0(θ) ≲uc nd1+r log2β(1+r)(n) + d log +� d +n +� +. +3.4 +Entropic convergence of the SLMC +The purpose of this paragraph is to prove the main result of the paper, i.e. Theorem 1.1 that guarantees +the convergence of the SLMC algorithm. +Proof of Theorem 1.1. Our starting point is the semi-group inequality (25) associated with the func- +tional WLSI inequality (28). Using cn,d defined in (26), we obtain for any s > 0: +∂t{Jt} ≤ −E +�� nt +µn +� ++ cn,de− 2αn +3 +t +≤ − +Jt +ϕWLS(s) + +s +ϕWLS(s)Osc2 +�� nt +µn +� ++ cn,de− 2αn +3 +t +≤ − +Jt +ϕWLS(s) + +s On,d +ϕWLS(s) + cn,de− 2αn +3 +t, +where we applied Proposition 3.5 in the last line with On,d ≲uc +� C1d +n +� dr +2 exp +� +C2nd1+r log2β(1+r)(n) +� +and C1 and C2 two universal constants. We then choose s (that depends on t) such that: +st = e−A√t+1 +with +A > 1 +that will be chosen later on. +We observe that st < e−1 + 1/2, so that Equation (29) of Proposition 3.3 yields: +ϕWLS(st) = a +1 + log +� +1 +st +� +CP += a1 + A√1 + t +CP +. +We introduce ψ(t) = exp +� +CP +a +� t +0 +du +1+A√1+u +� +and deduce that +ψ(t) = exp + +CP +a +2A(√1 + t − 1) − 2 log +� +1+A√1+t +1+A +� +A2 + + ≤ exp +�2CP +aA ( +√ +1 + t − 1) +� +. +We now apply the Gronwall Lemma: +∂t {ψ(t)Jt} = +� +CP +a(1 + A√1 + t)Jt + J′ +t +� +ψ(t) +≤ +� +CP On,d +a +e−A√t+1 +1 + A√1 + t + cn,de− 2αn +3 +t +� +ψ(t) +≤ CP On,d +a +e−(A− 2CP +aA )√1+t + cn,de +2CP +aA (√1+t−1)− 2αn +3 +t. +16 + +We denote by t0 the positive real value that solves the equation 2CP +aA +√1 + t0 = αnt0 +3 . We then observe +that: +� t +0 +e +2CP +aA (√1+u−1)− 2αn +3 +udu ≤ +� t0 +0 +e +2CP +aA +√1+udu + +� +∞ +t0 +e− αn +3 udu +≤ t0e +2CP +aA +√1+t0 + 3 +αn += t0e +αnt0 +3 ++ 3 +αn +. +If A is chosen such that A > 2CP +aA , we then deduce that: +Jt ≤ +� +J0 + cn,dt0e +αnt0 +3 ++ 3cn,d +αn +� +ψ(t)−1 + CP On,d +a +ψ(t)−1 +� t +0 +e− +� +A− 2CP +aA +�√1+udu +≤ +� +J0 + cn,dt0e +αnt0 +3 ++ 3cn,d +αn +� +ψ(t)−1 + +2CP On,d +a +� +A − 2CP +aA +�2 ψ(t)−1, +where we used in the previous line the bound: +� t +0 +e−b√1+udu ≤ +� +∞ +0 +e−b√1+udu ≤ 2 +b2 . +To obtain the lowest upper bound, we are led to choose A such that 2CP +aA as large as possible and +below A, which naturally drives to the choice: +2CP +aA = A +2 =⇒ A = +2 +√a +� +CP . +Using this value of A in the previous bound, we observe that t0 ≤ 3√CP +αn +√a + CP +α2n , so that a constant C +exists such that: +Jt ≤ C +� +J0 + cn,d +αn +� +1 + +�CP +αn ++ +� +CP +� +e +√ +CP +√a + CP +3αn +� ++ On,d +� +(1 + t)1/4e− +√ +Cp +√a (√1+t−1). +(30) +In Proposition 3.4 we obtained CP ≥ +κ +(d log2β(n)) +(1+r)2 . If instead of using the constant CP , we use +directly +κ +(d log2β(n)) +(1+r)2 with κ < 1, then all the previous computations remain the same only replacing +CP by its lower bound and: +Jt ≤ C + + +J0 + cn,d +αn +e +√κ +� +1 +√a + +1 +3αn +� +(d log2β(n))(1+r)2/2 + On,d + + + (1 + t)1/4e +− +√κ(√1+t−1) +√a(d log2β(n))(1+r)2/2 . +(31) +Using the values of On,d, cn,d and the upper bound of J0, we finally observe that if αn = +1 +n(d log2β(n)) +1+r , then: +t ≥ ℵ +� +d log2β(n) +�(1+r)2 � +log2(ε−1) + n2 � +d log2β(n) +�2(1+r) ++ d2 log2 d +� +=⇒ Jt ≤ ε. +4 +Technical results on KL and Uνn +4.1 +Growth properties under the Kurdyka-�Lojasiewicz inequality +We remind here some important consequences of the KL inequality that implies several relationships +between the function and the norm of its gradient. The proof of these inequalities may be found in +Lemma 15 of [21] (a small mistake appears and we correct the statement with a factor 2 in our work). +17 + +Proposition 4.1. Assume that a function V satisfies Hr +KL(c, L), then: +2c +1 − r +� +V 1−r(θ) − min(V )1−r� +≤ ∥∇V (θ)∥2 +2 ≤ 2L [V (θ) − min(V )] , +∀θ ∈ Rd. +It is furthermore possible to assess a minimal and maximal growth property of any function that +satisfies Hr +KL(c, L), which is necessarily lower and upper bounded by a positive power of the distance +to its minimizer. +Proposition 4.2. Assume that a function V satisfies Hr +KL(c, L), then, ∀θ ∈ Rd: +V 1+r(θ) − min(V )1+r ≥ (1 + r)c +2 +∥θ − arg min V ∥2 +2, +and +V (θ) − min(V ) ≤ L +2 ∥θ − arg min V ∥2 +2. +A straightforward consequence of the first inequality is then +Proposition 4.3. Assume that a function V satisfies Hr +KL(c, L), then, ∀θ ∈ Rd: +V (θ) ≥ 2− +r +1+r +� +min(V ) + +�(1 + r)c +2 +� +1 +1+r +∥θ − arg min V ∥ +2 +1+r +2 +� +. +4.2 +Properties of Uνn +Proof of Proposition 1.1. First, we observe that if each θ �→ ∇ log pθ(Xi) is L-Lipschitz and θ �→ +∇ log π0 is ℓ0-Lipschitz, then the triangle inequality implies that +∥∇Uνn(θ1) − ∇Uνn(θ2)∥2 ≤ (nL + ℓ0)∥θ1 − θ2∥2. +Second, we consider the lower-bound property on the curvature and observe that: +λ∇2Uνn(θ) = +inf +e∈Rd:|e|=1 eT (∇2Uνn)(θ)e ≥ 1 +n +n +� +i=1 +inf +e∈Rd:|e|=1 eT (∇2UXi)(θ)e. +The log concavity of the prior yields +λ∇2Uνn(θ) ≥ 1 +n +n +� +i=1 +λ∇2(−n log pθ(Xi)) = +n +� +i=1 +λ∇2(− log pθ(Xi)). +Then, the Hr +KL(c, L) property applied to each term of the sum above and minθ∈Rd − log π0(θ) > 0 +yields +λ∇2Uνn(θ) ≥ c +n +� +i=1 +[− log pθ(Xi)]−r ≥ cnr +n +� +i=1 +U −r +Xi (θ) = cn1+r +� +1 +n +n +� +i=1 +U −r +Xi (θ) +� +. +From the Jensen inequality, we finally deduce that: +λ∇2Uνn(θ) ≥ cn1+r +� +1 +n +n +� +i=1 +U −r +Xi (θ) +� +≥ cn1+rU −r +νn (θ). +We conclude that Uνn satisfies Hr +KL +� +cn1+r, nL + ℓ0 +� +. For UXi, the proof is similar. +Proposition 4.4. We assume Hπ0(ℓ0), Hmin and that for any x: θ �−→ − log pθ(x) satisfies Hr +KL(c, L), +then: +∥ arg min Uνn∥2 ≲uc d +1+r +2 logβ(1+r)(n) +and +minθ∈Rd Uνn(θ) ≲uc nd log2β(n). +18 + +Proof. Proposition 1.1 shows that Uνn satisfies Hr +KL +� +cn1+r, nL + ℓ0 +� +. Therefore, we can apply Propo- +sition 4.2 with θ = 0 and deduce that: +∥ arg min Uνn∥2 +2 ≤ +2 +(1 + r)cn1+r +� +U 1+r +νn (0) − min U 1+r +νn +� +. +To obtain an upper bound of Uνn(0) we first bound UXi(0) using Proposition 4.2, for all i, as follows: +UXi(0) ≤ min UXi + nL + ℓ0 +2 +∥ arg min UXi∥2 +2 ≲uc d + nd log2β(n) ≲uc nd log2β(n), +then Uνn(0) ≲uc nd log2β(n). We deduce that: +∥ arg min Uνn∥2 +2 ≤ +2 +(1 + r)cn1+r U 1+r +νn (0) ≲uc d1+r log2β(1+r)(n). +The second part comes from min Uνn ≤ Uνn(0). +5 +Smoothness and boundedness of the semi-group +Proof of Proposition 3.4. i). The proof relies on an argument set up with a ”fixed” sample (X1, . . . , Xn). +Our starting point is Proposition 4.2 and the consequences of the Kurdyka-�Lojasiewicz inequality. +Since Hπ0(ℓ0) and θ �→ − log pθ(Xi) satisfies Hr +KL(c, L), then Proposition 1.1 shows that Uνn satisfies +Hr +KL +� +cn1+r, nL + ℓ0 +� +. Therefore, we can apply Proposition 4.2 and deduce that: +∥θ − arg min Uνn∥2 +2 ≤ +2 +(1 + r)cn1+r +� +U 1+r +νn (θ) − min U 1+r +νn +� +≤ +2 +(1 + r)cn1+r U 1+r +νn (θ). +If Id refers to the identity map, we use the fact that for any distribution µ, we have V ar[µ] ≤ µ[∥Id−a∥2 +2] +for any a ∈ Rd so that a straightforward consequence with a = arg min Uνn is then: +V ar(µn) ≤ +� +Rd ∥θ − arg min Uνn∥2 +2dµn(θ) ≤ +2 +(1 + r)cn1+r µn[U 1+r +νn ]. +We then use the ergodic behaviour of (θt)t≥0 and observe that there exists a constant C independent +from n and d such that: +V ar(µn) ≤ +2 +(1 + r)cn1+r lim sup +t≥0 +E[U 1+r +νn (θt)] +≤ C +� +d log2β(n) +�(1+r)2 +, +where the last inequality comes from Proposition 5.1. We now use the Bobkov bound on the Poincar´e +constant for log-concave distribution (see Theorem 1.2 of [4]) and deduce that a universal constant K +exists such that: +CP (µn) ≥ +1 +4K2V ar(µn). +Using the upper bound of the variance, we deduce that a universal κ > 0 exists such that: +CP (µn) ≥ +κ +� +d log2β(n) +�(1+r)2 . +ii). For the second point, we consider a situation on average over the samples and the result uses the +concentration of the posterior distribution around its mean. We know from Theorem 3 of [21] that a +constant c > 0 exists such that: +E(X1,...,Xn)∼Pθ0[Var(µn)] ≤ cǫ2 +n,d, +with ǫn,d = +� +Ld log n +n +�α−1 +. The result follows using the Jensen inequality and the Bobkov bound. +19 + +Proof of Proposition 3.5. i). We first establish the boundedness of f0. +From our assumptions, we +apply Proposition 1.1 and obtain that Uνn satisfies Hr +KL +� +cn1+r, nL + ℓ0 +� +. If θ⋆ +n = arg min Uνn, we +then deduce from Proposition 4.2 that: +f0(θ) = n0(θ) +µn(θ) = Zne− +∥θ∥2 +2 +2σ2 +Uνn(θ) +(2π)d/2σd +≤ Zne− +∥θ∥2 +2 +2σ2 +Uνn(θ⋆ +n)+ (nL+ℓ0) +2 +∥θ−θ⋆ +n∥2 +2 +(2π)d/2σd +. +(32) +We compute an upper bound of Zn and use the lower bound of Uνn induced by Proposition 4.3: +Zn = +� +Rd e−Uνn(θ)dθ +≤ +� +Rd e +−2 +− +r +1+r +� +Uνn(θ⋆ +n)+n( +(1+r)c +2 +) +1 +1+r ∥θ−θ⋆ +n∥ +2 +1+r +2 +� +dθ +≤ e−2 +− +r +1+r Uνn(θ⋆ +n) +� +Rd e−nar∥θ∥ +2 +1+r +2 +dθ, +with ar = ((1+r)c) +1 +1+r +2 +. Using the well known equality: +� +Rd e−a|θ|ℓdθ = +dπd/2Γ(d/ℓ) +ℓad/ℓΓ(d/2 + 1), +∀a > 0, +∀ℓ > 0. +we then deduce with a = nar and ℓ = +2 +1+r that: +Zn ≤ e−2 +− +r +1+r Uνn(θ⋆ +n) +� +Rd e−nar∥θ∥ +2 +1+r +2 +dθ ≤ d(1 + r) +2 +πd/2 +(nar) +d(1+r) +2 +Γ +� +d(1+r) +2 +� +Γ +� d +2 + 1 +� . +From standard relationships on the Gamma function: +Zn ≤ 2 +�21+rπ +cn1+r +� d +2 +d +dr +2 . +(33) +We gather Equations (32) and (33) and obtain that: +f0(θ) ≤ 2eUνn(θ⋆ +n) +� +2 +cσ2n1+r +� d +2 +d +dr +2 e− +∥θ∥2 +2 +2σ2 + (nL+ℓ0) +2 +∥θ−θ⋆ +n∥2 +2. +For all σ2 < +1 +nL+ℓ0 , a straightforward optimization on θ yields : +∥f0∥∞ ≤ 2eUνn(θ⋆ +n) +� +2 +cσ2n1+r +� d +2 +d +dr +2 exp +� +(nL + ℓ0) +2(1 − σ2(nL + ℓ0))∥θ⋆ +n∥2 +2 +� +. +Then, the choice +c1 +nL+ℓ0 ≤ σ2 ≤ +c2 +nL+ℓ0 , where 0 < c1 ≤ c2 < 1 in Hn0(L, ℓ0) and the bounds of ∥θ⋆ +n∥2 +2 +and Uνn(θ⋆ +n) in Proposition 4.4 lead to : +∥f0∥∞ ≤ 2 +�C1d +n +� dr +2 +exp +� +C2nd1+r log2β(1+r)(n) +� +, +where C1 and C2 are universal constants. +ii). This result is an almost standard consequence of the maximum principle for a Markov semi-group +property with a Brownian diffusion. For any bounded measurable h > 0, we observe that Pth > 0 +using the Markov property, and we are led to define gt as the following function gt := √Pth. We then +introduce θ(t) and θ(t) as: +θ(t) = arg max gt(θ) +and +θ(t) = arg min gt(θ). +20 + +The chain rule yields: +d +dtOsc(gt) += +d +dt +� +gt(θ(t)) − gt(θ(t)) +� += +dgt +dt (θ(t)) + +� +∇gt(θ(t)), dθ(t) +dt +� +− dgt +dt (θ(t)) − +� +∇gt(θ(t)), dθ(t) +dt +� +. +(34) +We compute: +dgt +dt (θ) += +1 +2√Pth +dPth +dt (θ) += +1 +2√PthGtPth(θ) += +1 +2 +� +Pth(θ) +� +− +n +� +i=1 +⟨∇θPth(θ), ∇θUXi(θ)⟩mt(Xi|θ) + ∆θPth(θ) +� +. +(35) +Now, we use that θ(t) = arg max gt = arg max Pth, (a similar argument holds for θ(t)): +∇θgt(θ(t)) = 0, +∇θPth(θ(t)) = 0 +and +∆θPth(θ(t)) ≤ 0. +then: +d +dtOsc(gt) += +dgt +dt (θ(t)) − dgt +dt (θ(t)) += +∆θPth +2√Pth(θ(t)) − ∆θPth +2√Pth(θ(t)) +(36) +≤ +0. +We have therefore shown that Osc(√Pth) is decreasing in t ≥ 0, which ends the proof. +Proof of Lemma 2.1. We proceed as in Proposition 3 of [33] to justify the use of the Lebesgue domi- +nated convergence theorem for the derivation of the integral involved in our statement. We can then +deduce that: +∂t +�� +Rd ft(θ)dnt(θ) +� += +� +Rd ∂t{ft(θ)}dnt(θ) + +� +Rd ft(θ)∂t{nt(θ)}dθ. +We leave the first term unchanged and now focus on the second term: +� +Rd ft(θ)∂t{nt(θ)}dθ += +� +Rd ft(θ)∂t +� n +� +i=1 +mt(θ, Xi) +� +dθ += +� +Rd +n +� +i=1 +ft(θ)∂t{mt(θ, Xi)}dθ += +� +Rd +n +� +i=1 +Lft(θ) mt(θ, Xi)dθ, +where we used the definition of nt in the first step and Kolmogorov backward equation (10) in the last +one. Since the function ft(θ) does not depend on x, we observe that L2ft(θ) = 0 and we only need to +compute the remaining term L1ft(θ): +� +Rd ft(θ)∂t{nt(θ)}dθ += +� +Rd +n +� +i=1 +L1ft(θ) mt(θ, Xi)dθ +(37) += +� +Rd +n +� +i=1 +[−⟨∇θft(θ), ∇θUXi(θ)⟩ + ∆θft(θ)] mt(θ, Xi)dθ += +− +� +Rd +n +� +i=1 +⟨∇θft(θ), ∇θUXi(θ)⟩mt(Xi|θ)dnt(θ) + +� +Rd ∆θft(θ)dnt(θ) += +� +Rd Gtft(θ)dnt(θ), +(38) +21 + +where we used the fact that mt(θ, Xi) = mt(Xi|θ)nt(θ). +5.1 +Moments upper bounds +Proposition 5.1. Assume Hn0(L, ℓ0), Hπ0(ℓ0), Hmin and that for each Xi, θ �→ − log pθ(Xi) satisfies +Hr +KL(c, L). Then: +i) Three positive constants C1, C2 and C3, independent from n and d, exist such that for any t > 0: +E +� +e +(1+r)nc +1 +1+r +16 +(∥θt∥2 +2+1) +1 +1+r +� +≤ C1 +� +d log2β(n) +� +r +1+r eC2nd log2β(n) + Cd +3e +(1+r)nc +1 +1+r +16 +. +ii) For any t > 0 and for any α ≥ 1: +E[U α +νn(θt)] ≲uc nα � +d log2β(n) +�α(1+r) +. +Proof of i). We consider the function f(θ) = exp +� a +2(∥θ∥2 +2 + 1)ρ� +where 0 < ρ < 1, which is twice +differentiable. The gradient of f is computed as: +∇f(θ) = aρ(∥θ∥2 +2 + 1)ρ−1f(θ)θ. +The Laplace operator is given as: +∆f(θ) = aρ(∥θ∥2 +2 + 1)ρ−2f(θ) +� +aρ(∥θ∥2 +2 + 1)ρ∥θ∥2 +2 + (d + 2ρ − 2)∥θ∥2 +2 + d +� +. +We then deduce that for any θ ∈ Rd: +Gtf(θ) += +− +n +� +i=1 +⟨∇UXi, ∇f(θ)⟩mt(Xi|θ) + ∆f(θ) += +aρ(∥θ∥2 +2 + 1)ρ−2f(θ) +� +− (∥θ∥2 +2 + 1) +n +� +i=1 +⟨θ, ∇θUXi(θ)⟩mt(Xi|θ) ++aρ(∥θ∥2 +2 + 1)ρ∥θ∥2 +2 + (d + 2ρ − 2) ∥θ∥2 +2 + d +� +≤ +aρ(∥θ∥2 +2 + 1)ρ−2f(θ) +� +− (∥θ∥2 +2 + 1) +n +� +i=1 +(UXi(θ) − UXi(0)) mt(Xi|θ) ++aρ(∥θ∥2 +2 + 1)ρ+1 + d +� +∥θ∥2 +2 + 1 +� � +≤ +aρ(∥θ∥2 +2 + 1)ρ−1f(θ) +� +− +n +� +i=1 +(UXi(θ) − UXi(0)) mt(Xi|θ) + aρ(∥θ∥2 +2 + 1)ρ + d +� +, +where we used the convexity of Ux for any position x. +Let us establish the bounds of UXi(θ) and UXi(0). +We denote by θi = arg min UXi and from +Hypothesis Hmin, there exist two positive constants K1 and K2 independent on n and d such that: +maxi ∥θi∥2 +2 ≤ K1d log2β(n) +and +maxi UXi(θi) ≤ K2d. +We apply Proposition 4.2 to each non-negative function UXi that satisfies Hr +KL +� +cn1+r, nL + ℓ0 +� +, then +we obtain that: +UXi(θ) ≥ n +�(1 + r)c +2 +� +1 +1+r +∥θ − θi∥ +2 +1+r +2 +. +Since +2 +1+r > 1, the Jensen inequality yields (u + v) +2 +1+r ≤ 2 +1−r +1+r +� +u +2 +1+r + v +2 +1+r +� +, for all (u, v) ∈ R2 ++ and +we deduce that: +∥θ − θi∥ +2 +1+r +2 +≥ 2 +r−1 +1+r ∥θ∥ +2 +1+r +2 +− ∥θi∥ +2 +1+r +2 +≥ 2 +r−1 +1+r ∥θ∥ +2 +1+r +2 +− +� +K1d log2β(n) +� +1 +1+r . +22 + +Then we use this inequality to obtain a lower bound of UXi: +UXi(θ) ≥ 2n +�(1 + r)c +8 +� +1 +1+r +∥θ∥ +2 +1+r +2 +− n +�(1 + r)c +2 +� +1 +1+r +(K1d log2β(n)) +1 +1+r . +Moreover an upper bound of max UXi(0) comes from Proposition 1.1 and 4.2 as follows: +UXi(0) ≤ UXi(θi) + nL + ℓ0 +2 +∥θi∥2 +2 ≤ K2d + K1(nL + ℓ0)d log2β(n) +2 +. +Using the previous bounds and the fact that �n +i=1 mt(Xi|θ) = 1, it yields: +n +� +i=1 +(UXi(θ) − UXi(0)) mt(Xi|θ) +≥ +2n +�(1 + r)c +8 +� +1 +1+r +∥θ∥ +2 +1+r +2 +− n +�(1 + r)c +2 +� +1 +1+r +(K1d log2β(n)) +1 +1+r − K2d − K1(nL + ℓ0)d log2β(n) +2 +≥ +nc +1 +1+r +4 +∥θ∥ +2 +1+r +2 +− nc +1 +1+r (K1d log2β(n)) +1 +1+r − K2d − K1(nL + ℓ0)d log2β(n) +2 +, +where we used some uniform upper bounds when r ∈ [0, 1). We then choose ρ = +1 +1+r and we deduce +that: +Gtf(θ) +≤ +a +1 + r (∥θ∥2 +2 + 1)− +r +1+r f(θ) +� +−nc +1 +1+r +4 +∥θ∥ +2 +1+r +2 ++ nc +1 +1+r (K1d log2β(n)) +1 +1+r + K2d ++K1(nL + ℓ0)d log2β(n) +2 ++ +a +(1 + r)(∥θ∥2 +2 + 1) +1 +1+r + d +� +≤ +a +1 + r (∥θ∥2 +2 + 1)− +r +1+r f(θ) +� +− +� +nc +1 +1+r +4 +− +a +(1 + r) +� +∥θ∥ +2 +1+r +2 ++ nc +1 +1+r (K1d log2β(n)) +1 +1+r ++(K2 + 1)d + K1(nL + ℓ0)d log2β(n) +2 ++ +a +(1 + r) +� +, +where we used (∥θ∥2 +2 + 1) +1 +1+r ≤ ∥θ∥ +2 +1+r +2 ++ 1 in the second line. +We now fix a = n(1+r)c +1 +1+r +8 +and deduce that: +Gtf(θ) +f(θ) +≤ +n2c +2 +1+r +64 +(∥θ∥2 +2 + 1)− +r +1+r +� +−∥θ∥ +2 +1+r +2 ++ 8(K1d log2β(n)) +1 +1+r + ++8(K2 + 1)d + 4K1(nL + ℓ0)d log2β(n) +nc +1 +1+r ++ 1 +� +. +(39) +We then study two complementary situations and below, we denote by Kn,d the radius of the key +compact set involved by the previous Lyapunov contraction: +K +2 +1+r +n,d = Cd log2β(n). +• When ∥θ∥2 is large enough (∥θ∥2 ≥ Kn,d), we observe that a large enough C > 0 independent from +n and d exists such that: +∥θ∥ +2 +1+r +2 +≥ Cd log2β(n) =⇒ Gtf(θ) +f(θ) +≤ − +n2 � +d log2β(n) +� +1 +1+r c +2 +1+r +128 += −an,d. +(40) +23 + +• When ∥θ∥2 is upper bounded (∥θ∥2 ≤ Kn,d), we use the upper bound stated in Equation (39) and +obtain that a universal C1 (whose value may change from line to line) exists such that : +∥θ∥ +2 +1+r +2 +≤ Cd log2β(n) =⇒ +Gtf(θ) ≤ C1n2f(θ) +� +8(K1d log2β(n)) +1 +1+r + 8(K2 + 1)d + 4K1(nL + ℓ0)d log2β(n) +nc +1 +1+r ++ 1 +� +≤ C1n2d log2β(n) exp +� +(C + 1)c +1 +1+r nd log2β(n) +8 +� +≤ bn,deδn,d. +(41) +We then use Equations (40) and (41) as follows. We define the function ψn,d as ψn,d(t) = E[f(θt)] and +use Lemma 2.1: +ψ′ +n,d(t) += +E[Gtf(θt)] += +E +� +Gtf(θt) +� +1∥θt∥2≥Kn,d + +1∥θt∥2≤Kn,d +�� +≤ +E +� +−an,df(θt)1∥θt∥2≥Kn,d + bn,deδn,d +1∥θt∥2≤Kn,d +� +≤ +−an,dψn,d(t) + an,d +sup +∥θ∥2≤Kn,d +f(θ) + bn,deδn,d +≤ +−an,dψn,d(t) + (an,d + bn,d)eδn,d. +We apply the Gronwall Lemma and obtain that: +∀t > 0 +ψn,d(t) ≤ +� +1 + bn,d +an,d +� +eδn,d + ψn,d(0)e−an,dt. +(42) +Using that n0 is a Gaussian distribution, which was fixed in Hn0(L, ℓ0) hypothesis, we find an +upper bound for ψn,d(0) = E[f(θ0)] = +� +Rd f(θ)dn0(θ) as follows : +ψn,d(0) += +� +2πσ2�− d +2 +� +Rd e +a +2(∥θ∥2 +2+1) +1 +1+r − +∥θ∥2 +2 +2σ2 dθ +≤ +� +2πσ2�− d +2 e +a +2 +� +Rd e− +∥θ∥2 +2 +2 ( 1 +σ2 −a)dθ, +if σ2 ≤ +1 +a = +8 +n(1+r)c +1 +1+r then the integral above is finite. Since c2 < 1 ≤ +8L +(1+r)c +1 +1+r , it guarantees +σ2 < 1 +a, then: +ψn,d(0) +≤ +� +1 − aσ2�− d +2 e +a +2 +≤ +Cd +3e +(1+r)nc +1 +1+r +16 +, +where C3 is a constant independent from n and d. +Finally, using the value of an,d and bn,d in (42), we deduce that: +E +� +e +(1+r)nc +1 +1+r +16 +(∥θt∥2 +2+1) +1 +1+r +� +≤ C1 +� +d log2β(n) +� +r +1+r eC2nd log2β(n) + Cd +3e +(1+r)nc +1 +1+r +16 +, +∀t > 0. +where C2 is another universal constant, which concludes the proof. +Proof of ii). We consider α > 1 and below, C > 0 refers to a “constant” independent from n and d, +whose value may change from line to line. Our starting point is the upper bound of the exponential +moments obtained in i). Proposition 1.1 shows that Uνn satisfies Hr +KL +� +cn1+r, nL + ℓ0 +� +, then thanks +to Proposition 4.2: +E[U α +νn(θt)] ≤ E +�� +min Uνn + Cn∥θt − θ∗ +n∥2 +2 +�α� +≤ E +�� +min Uνn + Cn∥θ∗ +n∥2 +2 + Cn∥θt∥2 +2 +�α� +, +24 + +where θ∗ +n = arg min Uνn. +By using Proposition 4.4 and the inequality derived from the Jensen inequality (a+b)β ≤ cβ(aβ+bβ) +for (a, b) ∈ R2 ++ and β ≥ 1, we obtain that: +(min Uνn+ Cn∥θ∗ +n∥2 +2 + Cn∥θt∥2 +2 +�α +≤ C +� +nd log2β(n) + nd1+r log2β(1+r)(n) + n∥θt∥2 +2 +�α +≤ Cnα +�� +d log2β(n) +�α(1+r) ++ ∥θt∥2α +2 +� +≤ Cnα +�� +d log2β(n) +�α(1+r) ++ k−α(1+r) logα(1+r) +� +ek∥θt∥ +2 +1+r +2 +�� +≤ Cnα +�� +d log2β(n) +�α(1+r) ++ k−α(1+r) logα(1+r) +� +eα(1+r)−1+k∥θt∥ +2 +1+r +2 +�� +. +The Jensen inequality and the concavity of x �→ logp(x) on [ep−1, +∞[ when p ≥ 1 yield +E[U α +νn(θt)] +≤ Cnα +�� +d log2β(n) +�α(1+r) ++ k−α(1+r)E +� +logα(1+r) +� +eα(1+r)−1+k∥θt∥ +2 +1+r +2 +��� +≤ Cnα +�� +d log2β(n) +�α(1+r) ++ k−α(1+r) logα(1+r) +� +E +� +eα(1+r)−1+k∥θt∥ +2 +1+r +2 +��� +≤ Cnα +�� +d log2β(n) +�α(1+r) ++ k−α(1+r) +� +α(1 + r) − 1 + log E +� +ek∥θt∥ +2 +1+r +2 +��α(1+r)� +≤ Cnα +�� +d log2β(n) +�α(1+r) ++ k−α(1+r) +� +α(1 + r) − 1 + log E +� +ek(∥θt∥2 +2+1) +1 +1+r +��α(1+r)� +, +where we used in the last inequality that ∥θ∥2 +2 ≤ ∥θ∥2 +2 + 1. +We then apply i) in Proposition 5.1, we choose k = (1+r)nc +1 +1+r +16 +and obtain that: +E[U α +νn(θt)] +≤ Cnα + + +� +d log2β(n) +�α(1+r) ++ +1 +nα(1+r) +� +1 + log E +� +e +(1+r)nc +1 +1+r +16 +(∥θt∥2 +2+1) +1 +1+r +��α(1+r) + +≤ C +� +nα � +d log2β(n) +�α(1+r) ++ 1 +nαr +� +1 + log +� +C1 +� +d log2β(n) +� +r +1+r eC2nd log2β(n) + Cd +3e +(1+r)nc +1 +1+r +16 +��α(1+r) + +≤ Cnα � +d log2β(n) +�α(1+r) +, +where we used in the previous lines simple algebra and log(a+b) ≤ log(2)+log(a)+log(b) when a ≥ 1 +and b ≥ 1. 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W. : Bayesian learning via stochastic gradient Langevin dynamics. +Proceedings of the 28th international conference on machine learning (ICML-11) 28, 3, (2011), +681–688. +[40] Xu, P. and Chen, J. and Zou, D. and Gu, Q. : Global Convergence of Langevin Dynamics Based +Algorithms for Nonconvex Optimization. Proceedings of the 32nd International Conference on +Neural Information Processing Systems. NIPS’18. Curran Associates Inc. (2018), 3126—3137. +27 + diff --git a/6tE1T4oBgHgl3EQfTQPr/content/tmp_files/load_file.txt b/6tE1T4oBgHgl3EQfTQPr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f056628e9752bc7e5c76283534384d05cb4c4efc --- /dev/null +++ b/6tE1T4oBgHgl3EQfTQPr/content/tmp_files/load_file.txt @@ -0,0 +1,774 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf,len=773 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='03077v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='ML] 8 Jan 2023 Stochastic Langevin Monte Carlo for (weakly) log-concave posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Marelys Crespo Navas1, S´ebastien Gadat2,3, Xavier Gendre1 1 ISAE-SUPAERO, Universit´e de Toulouse 2Toulouse School of Economics (CNRS UMR 5314), Universit´e Toulouse I Capitole 3 Institut Universitaire de France January 10, 2023 Abstract In this paper, we investigate a continuous time version of the Stochastic Langevin Monte Carlo method, introduced in [39], that incorporates a stochastic sampling step inside the traditional over- damped Langevin diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' This method is popular in machine learning for sampling posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We will pay specific attention in our work to the computational cost in terms of n (the number of observations that produces the posterior distribution), and d (the dimension of the ambient space where the parameter of interest is living).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We derive our analysis in the weakly convex framework, which is parameterized with the help of the Kurdyka-�Lojasiewicz (KL) inequality, that permits to handle a vanishing curvature settings, which is far less restrictive when compared to the simple strongly convex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We establish that the final horizon of simulation to obtain an ε approximation (in terms of entropy) is of the order (d log(n)2)(1+r)2[log2(ε−1) + n2d2(1+r) log4(1+r)(n)] with a Poissonian subsampling of parameter � n(d log2(n))1+r�−1, where the parameter r is involved in the KL inequality and varies between 0 (strongly convex case) and 1 (limiting Laplace situation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Keywords: Langevin Monte Carlo sampling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Log concave models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Weak convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' AMS classifications: Primary 6265C05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' secondary ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 62C10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 65C30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 60H3520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 1 1 Markovian Stochastic Langevin Dynamics and main results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 Introduction Motivations In the recent past years, a huge amount of methods have been developed in machine learning to handle large scale massive datasets with a large number n of observations (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn) embedded in a high dimensional space Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' These methods generally involve either optimization of a data-dependent function (for frequentist learning) or sampling a data-dependent measure (for Bayesian learning with posterior distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In both approaches, a bottleneck lies on the size of n and d that usually generates numerical difficulties for the use of standard algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We are interested in this paper in the simulation of a posterior distribution following a Bayesian point of view with a statistical model described by a collection of densities (pθ)θ∈Θ on X, where the parameter of interest θ⋆ belongs to Θ = Rd and where the (Xi)1≤i≤n are assumed to be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' observations in X distributed according to pθ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' A standard Bayesian approach consists in defining a prior distribution π0 on Θ and then sample the posterior distribution denoted by µn (which will be denoted by exp(−Uνn) below) using a numerical probabilistic approximation with the help of an over-damped Langevin diffusion: dθt = −∇Uνn(t)dt + √ 2dBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 1We are grateful to Patrick Cattiaux and Arnaud Guillin for helpful discussions and references on functional inequal- ities and especially on weak log Sobolev inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 1 In this work, we manage to deal with an adaptation of the Langevin Monte Carlo (LMC) algorithm proposed in [39], that exploits some old ideas of stochastic algorithms introduced in [36]: instead of using the previous equation, the authors propose a modification of the diffusion that generates a noisy drift in the LMC due to a sampling strategy among the set of observations (Xi)1≤i≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Before we provide some details on the precise objects and algorithm necessary to properly define this method, we first give some literature insights related to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' State of the art Ergodicity and quantitative mixing properties of over-damped LMC and many other sampling algorithms is a popular subject of research initiated in the probabilistic works around, roughly speaking, two strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The first one relies on pathwise considerations and dynamical proper- ties of random dynamical system and is built with some coupling argument and Lyapunov controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We refer to the seminal contributions [32, 27], that exploits the approach of the Doeblin coupling and total variation (TV) bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Many extensions may be derived from this Lyapunov approach and may lead to Wasserstein or L2 upper bounds, we refer to [8] and the references therein of the same authors for a description of the link between Lyapunov conditions and ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The second strategy derives from spectral properties of Markov operators and is related to famous functional inequalities (Poincar´e and Log-Sobolev among others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The general idea is to differentiate the distance along the time-evolution and apply a Gronwall Lemma to obtain a quantitative estimate of the long-time evolution of the semi- group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We refer to the seminal contributions of [26, 2], and to [3] for an almost exhaustive survey of all possible inequalities and consequences on the ergodicity of the Markov semi-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Finally, let us emphasize that some strong links exist between the spectral and the Lyapunov approaches, as pointed out by [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' If functional inequalities are then strongly related to mixing properties and especially from a quantitative point of view, it is therefore necessary to develop a machinery that is able to assess these inequalities carefully, especially with a specific attention to our statistical setting of large n and d in the completely non-trivial situation where the target measure is log-concave but not strongly log-concave, which is a common feature of Bayesian posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' On the statistical side, the mixing properties of LMC has been largely investigated during the past decade, strongly motivated by machine learning methods such as Exponentially Weighted Aggregation introduced by [11], which involves sampling a non log-concave and heavy tailed posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' A first paper of Dalalyan [12] establishes the cost of LMC to obtain an ε TV bound in terms of d and ρ when the target measure is ρ strongly log-concave and proposes a penalized version of LMC to circumvent the lack of strong log-concavity when the target distribution is only log-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Since this pioneering paper, a huge impressive literature expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Among others, we refer to [16] that gives a careful study of discretized LMC, [14] for a kinetic version of LMC and [15] where the penalized LMC in non strongly-concave situation is studied in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Among all these papers, first, the lack of strong log- concavity is dealt with a modification of the initial LMC using a surrogate and asymptotically vanishing penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Second, these papers assume that a noiseless gradient of the log-posterior is available at each iteration of the algorithm, which may not be realistic, especially with large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Stochastic LMC (SLMC below) has attracted the interest of several works: [39] introduced this method and described its efficiency from a numerical point of view in the particular case of Bayesian learning, which is exactly our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Some recent advances and related contributions may be also cited: [13] studies a noisy version of LMC and derives some non-asymptotic upper bounds (in terms of Wasserstein distance) of the sampling strategy in presence of a possibly biased noise for strongly log- concave posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The recent contribution of [40] is also related to our work: the authors develop a machinery for the study of SLMC essentially based on the Poincar´e inequality but the way the lower bound on the spectral gap involved in the LMC is dealt with appears to be inappropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In particular, the diffusion involved in (Stochastic)-LMC is used at a very low-temperature, proportional to 1/n, which generates some important troubles in the size of the spectral gap in non strongly log- concave framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In [35], the authors derives some close bounds to our framework for optimization purpose, and the authors identify the important dependency of the spectral gap denoted by λ∗ in their paper with the temperature level 1/β they introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' They obtain some very highly pessimistic bounds in some general situations (see their discussion in [35][Section 4]), they conclude their discussion by the urgent need to find some non-trivial situations where some better lower bounds of λ∗ may be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 2 Indeed, the final remark of [35][Section 4]) is related to the well known metastability phenomenon: at a low temperature, the mixing rates of a lot of reversible and irreversible Markov semi-groups are strongly deteriorated by the low temperature settings, which is implicitly induced by a Bayesian posterior sampling problem with a large number n of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In a regime of variance noise of the order O(β−1), the first study of large deviation principle of invariant measures traces back to [18] where the authors establish the asymptotic of the spectral gap of the over-damped Langevin diffusion as exp(−Iβ) ( [18][Chapter 6]) where I is an explicit constant that depends on the potential of the Gibbs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' This result has been extended in depth by [26], which leads to the first precise analyses of the so-called simulated annealing method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' [24, 33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' These works, and more recent contributions with irreversible dynamical systems in a stochastic settings ([22, 19]) show that there is almost nothing to expect in metastable situations in terms of asymptotic behaviour of the spectral gap, and indirectly in terms of mixing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Hence, the only situation that may lead to reasonable results is an intermediary situation between the (almost) trivial strongly log-concave case and the metastable multi-welled case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' This is the purpose of the weakly log-concave situation that is described by the family of Kurdyka-�Lojasiewicz inequalities [28, 30] used in optimization theory [5] that have shown to be efficient for stochastic optimization [20] or for sampling [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We also refer to the recent contributions [6] that derives some functional inequalities within an intermediary framework in which the curvature ρ is related to their keystone function α that controls the constants involved in the functional inequalities they are studying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Taking together the statistical considerations and limitations, we are motivated in this paper in the study of the continuous time Stochastic Langevin Monte Carlo procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' This process will be described precisely in the next paragraph as well as the Kurdyka-�Lojasiewicz setup parametrized by a real value r, which varies between 0 (strongly convex case) and 1 (limiting Laplace asymptotic tail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We will show that the final horizon of simulation to obtain an ε approximation is of the order: (d log(n)2)(1+r)2[log2(ε−1) + n2d2(1+r) log4(1+r)(n)] with a Poissonian subsampling of parameter 1 n(d log2(n))1+r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The rest of the introduction consists in the definitions of the algorithm in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2, the way we assess the quality of our result with an entropy criterion in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3, as well as the quantitative weakly log-concave assumption in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We finally state our main result in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 Continuous time evolution Below, we briefly remind the continuous time SLMC algorithm for Bayesian learning, for which a discretized form has been introduced in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For this purpose, we consider a statistical model that is built with the help of a function (x, θ) �−→ pθ(x) where θ ∈ Rd encodes the parameter of the statistical model and x the observation in a space denoted by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We then assume that we have n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' observations denoted by (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn) distributed according to pθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Given a prior distribution π0 on Rd, the posterior distribution µn is then defined as: µn(θ) ∝ π0(θ) × n � i=1 pθ(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We introduce the log-parametrization that leads to the Gibbs form: Ux(θ) = −[log π0(θ) + n log pθ(x)], and we then observe that: µn(θ) ∝ exp � − 1 n n � i=1 UXi(θ) � = exp (−Uνn(θ)) , where νn refers to the empirical distribution and Uνn the average value of UX(θ) when X ∼ νn: νn(x) = 1 n n � i=1 δXi(x) and Uνn(θ) = EX∼νn[UX(θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 3 The standard Langevin Monte Carlo approach relies on the ergodic behaviour of the stochastic differ- ential equation: dθt = −∇Uνn(θt)dt + √ 2dBt, (1) that possesses under some mild assumptions a unique invariant distribution µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The SLMC algorithm takes benefit of both sampling with a S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' and homogenization of the drift that may be written as an expectation on X that is sampled uniformly over the set of observations according to νn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The leading idea is to replace the expectation in Uνn that depends on the overall set of observations (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn) by a single unique observation that is randomized uniformly all along the evolution of the stochastic differential equation, and modified according to a Markov exponential clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' That being said, we can write an explicit formal definition of the algorithm as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We define � ξ(n) j � j≥1 an infinite sequence of exponential random variables of mean α−1 n that will be fixed later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We also consider a sequence � V (n) j � j≥0 of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' random variables uniformly distributed in {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We then define the process (Xt)t≥0 as a jump process that takes its values in {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=', n} such that: Xt = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 XV (n) 1 , if 0 ≤ t < ξ(n) 1 , XV (n) j , if j−1 � k=1 ξ(n) k ≤ t < j� k=1 ξ(n) k , j > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (2) Informally, (Xt)t≥0 should be understood as follows: the process takes the value of one observation uniformly chosen from the n observations X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn during exponential times with intensity αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The stochastic Langevin over-damped diffusion we consider is then given by the joint evolution (θt, Xt)t≥0 and that is defined by: dθt = −∇θUXt(θt)dt + √ 2dBt, t > 0, (3) where (Bt)t≥0 is a multivariate standard Brownian Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Algorithm 1: Stochastic Langevin over-damped Data: (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' observations, n0 initial distribution, π0 prior distribution 1 t0 = 0 2 Generate θ0 according to n0 3 for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' do 4 Pick Xk uniformly in {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn} 5 Generate ξk according to an Exponential distribution with mean α−1 n 6 tk+1 = tk + ξk 7 θtk+1 = θtk − � tk+1 tk ∇θUXk(θs)ds + √ 2Bξk 8 end 9 return lim k→∞ θtk 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3 Entropic divergence To assess the long-time behaviour of the SLMC, we introduce several notations related to the pair (θt, Xt)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Below, we denote by λd the Lebesgue measure over Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The semi-group induced by L being elliptic on the θ coordinate, trivially irreducible and finitely supported on the x coordinate, makes the law of (θt, Xt) absolutely continuous with respect to the measure λd ⊗ νn as soon as t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We introduce the notation of mt to refer to the joint density of (θt, Xt) at time t with respect to λd ⊗ νn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In the meantime, nt denotes the marginal distribution of θt and mt(·|θ) the conditional distribution of Xt given θt = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' That is: Law(θt, Xt) = mt, nt(θ) = n � i=1 mt(θ, Xi), mt(x|θ) = mt(θ, x) nt(θ) , (4) 4 for θ ∈ Rd and x ∈ {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' To show that the SLMC algorithm recovers the correct asymptotic behaviour, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' that nt(θ) −→ µn when t −→ ∞, we consider the relative entropy (or Kullback-Leibler divergence) of nt with respect to µn that is well defined thanks to the ellipticity, and given by: Jt = Entµn � nt µn � = � Rd log � nt(θ) µn(θ) � dnt(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (5) Jt measures at any time t > 0 a divergence between the instantaneous law of the process at time t and the (presumably) invariant distribution µn of the process (θt, Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' It would also be possible to measure this difference between the two distributions in terms of the L2 or the χ-square distance and to produce a theoretical analysis with the help of functional analysis but it would rely on stronger assumptions on the function Uνn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In the meantime, we also introduce a weighted L2 distance between the conditional distribution of Xt given θt = θ and the measure νn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' This distance is denoted by It and is defined as: It = � Rd n � i=1 �mt(Xi|θ) νn(Xi) − 1 �2 νn(Xi)dnt(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (6) This quantity measures the average closeness (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' θ) of the conditional law of x given θ at time t to νn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='4 Main assumptions Weak convexity We will study the SLMC into a weakly convex framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' when Uνn is assumed to be convex but not necessarily strongly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' SLMC has recently received an important interest in the machine learning community and has been studied essentially in its explicit Euler discretized form in various situations where functional inequalities are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We refer to [38] (uniform Log-Sobolev inequality), to [35] (uniform Poincar´e inequality) where the authors develop a Wasserstein-2 analysis of the algorithm, and to [40] (uniform Poincar´e inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In these works, the functional inequalities play a crucial role to analyze the behaviour of SLMC and these inequalities are assumed, which is an important hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Importantly, Poincar´e or Log-Sobolev inequalities are not so innocent since they generally require convexity (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' [4, 3]) to be reasonably dimension-dependent, and even strong convexity to be dimension free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Otherwise, the constant involved in these functional inequalities are exponentially degraded by the “temperature” (n−1(d log2β(n))−(1+r) in our case) and the dimension (d for us) as indicated in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In our work, we have chosen to parameterize this lack of strong convexity with the help of the Kurdyka-�Lojasiewicz inequality [28, 30], which is a standard tool in optimization to describe the tran- sition between convexity and strong convexity and makes the bounds more explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' This assumption allows to observe how the entropy evolves according to the key exponent involved in the KL inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In particular, it makes possible to understand the influence of the lack of strong convexity that is more or less hidden in the uniform Poincar´e or Log-Sobolev inequalities that are assumed in the previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We introduce a parametric form of the KL inequalities following [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For this purpose, for any V twice differentiable function, we denote the spectrum of the Hessian matrix of V as Sp(∇2V (θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Furthermore, if V is convex, we denote: λ∇2V (θ) = inf Sp(∇2V (θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Hypothesis Hr KL(c, L) We say that a function V : Rd → R satisfies a Hr KL(c, L)-condition if: a) V is a C2-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' b) V is a convex function and minθ∈RdV (θ) = V (θ∗) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' c) ∇V is L-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 5 d) There exist some constants 0 ≤ r < 1 and c > 0 such that: cV −r(θ) ≤ λ∇2V (θ) ∀θ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (7) Let us briefly comment this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In [21], a slightly different parametrization is used with the introduction of another exponent q related to λ∇2V (θ) = sup Sp(∇2V (θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The authors also assume the upper bound λ∇2V (θ) ≤ ˜cV −q(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Here, we have chosen to simplify this assumption and use a rough upper bound on the eigenvalues of the Hessian matrix given by the Lipschitz constant L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' in the last inequality we simply use ˜c = L and q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We shall observe that if V (θ) = (1 + ∥θ∥2 2)p with p ∈ [1/2, 1], then V satisfies Hr KL(c, L) with r = 1−p p and c = 2p(1 − 2(1 − p)), see Remark 7 of [21] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In particular, the larger p, the smaller r, which translates into a better curvature of the potential function V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' When r = q, we recover a global standard KL inequality (see [20, 5]) and when r = 1 it corresponds to the limiting Laplace case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The case r = 0 is of course associated to the strongly convex situation where the curvature of the function is uniformly lower bounded by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Hence, it is expected that the complexity of SLMC increases with the lack of curvature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' is an increasing function of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In section 4 we recall some important consequences of the KL inequality obtained in Lemma 15 of [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In particular, the growth of any function that satisfies Hr KL(c, L) is lower and upper bounded by a positive power of the distance to its minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' If inequality (7) holds for a constant c, then it holds for all positive values less than c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For that reason, in section 5 we assume c ≤ � 8L (1+r) �1+r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assumption on the prior π0 We state below the important consequence of a “population” Hr KL(c, L) assumption, but before, let us state some mild assumptions on π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Hypothesis Hπ0(ℓ0) π0 is a log-concave C2-function such that minθ∈Rd − log π0(θ) > 0 and θ �→ ∇ log π0(θ) is ℓ0-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Since the prior distribution is chosen by the user, our Hπ0(ℓ0) hypothesis is not restrictive and some typical examples satisfy these conditions, such as Gaussian, Weibull and Gamma, both with shape parameter larger than 1, Gumbel, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We assume Hπ0(ℓ0) and that there exist (c, r) such that for any x: θ �−→ − log pθ(x) satisfies Hr KL(c, L), then Uνn satisfies Hr KL � cn1+r, nL + ℓ0 � , and in particular, for any Xi, UXi sat- isfies Hr KL � cn1+r, nL + ℓ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We introduce the notation a ≲uc b (a ≳uc b) which means a ≤ cb (a ≥ cb) where c is a universal constant i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' a positive constant independent of n and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We assume that the minimizers of the functions UXi are contained in a ball of radius which depends of n and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Additionally, we consider minθ∈RdUXi to be at most of order d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Hypothesis Hmin There exists β ≥ 0 such that: maxi∥ arg min UXi∥2 ≲uc √ d logβ(n) and maxi minθ∈Rd UXi(θ) ≲uc d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assumption Hmin is not restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In dimension d = 1, it holds for many concentrated i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' samples (Xi)1≤i≤n with a suitable sub-Gaussian like behaviour for which the Laplace transform of min UXi is upper bounded as: E[exp(λmin UXi)] ≤ exp(σ2λk), ∀λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 6 The previous upper bound implies that, in this case, β involved in Hmin is given by β = k−1 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We recover in particular the situation where β = 1/2 when k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For larger dimensions, the result may be extended using that ∥X∥2 2 ≤ d max1≤j≤d(Xj)2, where Xj is the j-th component of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We should keep in mind from this last discussion that even if Hmin is stated (and makes sense) for any value of β > 0, it holds in general for β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' This Hmin hypothesis together with Hπ0(ℓ0) lead to an almost similar behaviour of the minimizer and the minimum of Uνn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Details appear in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='5 Long-time entropy convergence We introduce for any time t ≥ 0 the density of Law(θt) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' µn, which is given by: ft(θ) = nt(θ) µn(θ), and n0 is chosen such that ∥f0∥∞ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The following hypothesis guarantees this result which will be proved in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Hypothesis Hn0(L, ℓ0) A positive constant σ2 exists such that n0 = N(0, σ2Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Moreover, there exist two universal constants c1 and c2 such that 0 < c1 ≤ c2 < 1 and c1 nL + ℓ0 ≤ σ2 ≤ c2 nL + ℓ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Futhermore, in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='5, as an immediate consequence of the boundedness of ∥f0∥∞, we obtain that J0 ≲uc nd1+r log2β(1+r)(n) + d log � d n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The next result assesses a mixing property in terms of decrease of the entropy and therefore states the convergence of nt towards the correct measure µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assume Hπ0(ℓ0), Hmin, Hn0(L, ℓ0) and that each θ �→ − log pθ(Xi) satisfies Hr KL(c, L), then Uνn satisfies a Poincar´e inequality of constant CP (µn), indistinctly denoted as CP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Define cn,d := n4 � d log2β(n) �1+r and On,d := � C1d n � dr 2 exp � C2n � d log2β(n) �1+r� , where C1 and C2 are universal constants, then for any t > 0: Jt ≲uc � J0 + cn,d αn � 1 + �CP αn + � CP � e √ CP √a + CP 3αn � + On,d � (1 + t)1/4e− √ Cp √a (√1+t−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (8) For any ε > 0, if αn = 1 n(d log2β(n)) 1+r , then: t ≳uc � d log2β(n) �(1+r)2 � log2(ε−1) + n2 � d log2β(n) �2(1+r) + d2 log2 d � =⇒ Jt ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' If we denote tε the smallest value such that Jtε ≤ ε, then the choice of αn = 1 n(d log2β(n)) 1+r guarantees that the mean number of jumps αntε of the process (Xt)0≤t≤tε is the minimum possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In order to proof the main result, we first present in Section 2 the classical tools related to the Markov semi-group, which could be skipped by the experienced reader in the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In Section 3 we prove the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Sections 4 and 5 are reserved to the technical results of the Hr KL(c, L) hypothesis and Uνn, and the Markov Dynamics respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 7 2 Markov tools It is straightforward to verify that the joint evolution of (θt, Xt)t≥0 exists and is weakly unique (in law) with the help of the Martingale Problem (MP below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For this purpose, we preliminary define the operator L that acts on any function f ∈ C2(Rd × X) as: Lf(θ, x) = −⟨∇θUx(θ), ∇θf(θ, x)⟩ + ∆θf(θ, x) � �� � :=L1f(θ,x) + αn n n � i=1 [f(θ, Xi) − f(θ, x) � �� � L2f(θ,x) ], (9) for all (θ, x) ∈ Rd × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The operator L is divided into two terms, L1 acts on the component θ and is associated to the diffusion part, while L2 is the jump operator that acts on the x component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Thanks to the finiteness of the number of observations (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn), we can apply the results of Section 4 and 5 of chapter 4 of [17] and deduce the following result: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assume that for any x ∈ X, Ux is C2(Rd) and ∇θUx is Lx-Lipschitz, then for any initial distribution ν on Rd × X, the martingale problem (L, ν) is well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The associated (weakly) unique process (θt, Xt)t≥0 is a Feller Markov process associated to the semi-group L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' In particular, the θ component verifies the S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' If we denote by L⋆ the adjoint operator of L in L2(Rd) × νn, the backward Kolmogorov Equation yields: ∂tmt(θ, x) = L⋆mt(θ, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (10) Using the ellipticity of the semi-group L on the θ coordinate, we can use the result of [25] and deduce that for any t > 0, nt ∈ C∞(Rd, R) and the irreducibility yields ∀t ≥ 0, nt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We will prove in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='5 some sufficient conditions that implies ∥f0∥∞ = ∥ n0(θ) µn(θ)∥∞ < +∞ and an important and standard consequence of the maximum principle, is as follows: if ∥f0∥∞ ≤ M, then ∀t ≥ 0, ∥ft∥∞ ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We defer the details of this result to the Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='5 as they are not central to our analysis and are rather technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Thanks to this master equation, it is possible to compute the derivative of the semi-group on some time dependent function of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For this purpose, we introduce two keystone operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The first one describes the infinitesimal action on the θ coordinate under the average effect of Xt at time t that applies ∀f ∈ C2(Rd, R) as: Gtf(θ) = − n � i=1 ⟨∇θf(θ), ∇θUXi(θ)⟩mt(Xi|θ) + ∆θf(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (11) The second one is very close to the first one except that the average effect of Xt is replaced by the targeted ideal distribution νn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' It leads to the definition ∀f ∈ C2(Rd, R): Gf(θ) = − n � i=1 ⟨∇θf(θ), ∇θUXi(θ)⟩νn(Xi) + ∆θf(θ) = −⟨∇θf(θ), ∇θUνn(θ)⟩ + ∆θf(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (12) This derivative is given in the next result, whose proof is deferred to the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Let be ht a twice differentiable function with uniformly bounded first and second order derivatives on Rd, then for t > 0: ∂t �� Rd ht(θ)dnt(θ) � = � Rd ∂t{ht(θ)}dnt(θ) + � Rd Gtht(θ)dnt(θ), (13) where Gt is the diffusion operator under the average effect of Xt, defined in Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 8 3 Proof of the main results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 Evolution of the entropy Jt The entropy satisfies the following differential inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assume Hmin, Hπ0(ℓ0) and for each Xi, θ → − log pθ(Xi) satisfies Hr KL(c, L), then a ”universal” constant C (independent from n and d) exists such that ∀t > 0: ∂t{Jt} ≤ − � Rd �����∇θ �� nt(θ) µn(θ) ������ 2 2 dµn(θ) + CI 1 3 t n 11 3 � d log2β(n) �1+r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We shall use the standard preliminary estimate that may be derived from Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='14) of [29] for elliptic diffusions to apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 to ft = log(ntµ−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' From Equation (13), we have: ∂t{Jt} = � Rd ∂t � log � nt(θ) µn(θ) �� dnt(θ) + � Rd Gt log � nt(θ) µn(θ) � dnt(θ), The first term vanishes since: � Rd ∂t � log � nt(θ) µn(θ) �� dnt(θ) = � Rd ∂t{nt(θ)} nt(θ) dnt(θ) = � Rd ∂t {nt(θ)} dθ = ∂t �� Rd dnt(θ) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Then, the derivative is reduced to the second term, and we are led to: ∂t{Jt} = � Rd Gt log � nt(θ) µn(θ) � dnt(θ), = � Rd G log � nt(θ) µn(θ) � dnt(θ) � �� � J1,t + � Rd (Gt − G) log � nt(θ) µn(θ) � dnt(θ) � �� � J2,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (14) We study the two terms J1,t and J2,t separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Study of J1,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Since G is a diffusion operator and µn is the invariant measure associated to G, then we can use the classical link between J1,t and the Dirichlet form (see [3]): � Rd G log � nt(θ) µn(θ) � dnt(θ) = � Rd nt(θ) µn(θ) G log � nt(θ) µn(θ) � dµn(θ) = −4 � Rd �����∇θ �� nt(θ) µn(θ) ������ 2 2 dµn(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (15) Study of J2,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We use the difference between G and Gt, for any twice differentiable function f: (Gt − G) f(θ) = − n � i=1 ⟨∇θf(θ), ∇θUXi(θ)⟩ [mt(Xi|θ) − νn(Xi)] = − n � i=1 ⟨∇θf(θ), ∇θUXi(θ)⟩ �mt(Xi|θ) νn(Xi) − 1 � νn(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 9 Then, the term J2,t may be computed as: |J2,t| = ���� � Rd (Gt − G) log � nt(θ) µn(θ) � dnt(θ) ���� = ����� � Rd n � i=1 ⟨∇θ log � nt(θ) µn(θ) � , ∇θUXi(θ)⟩ �mt(Xi|θ) νn(Xi) − 1 � νn(Xi) dnt(θ) ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Using the Cauchy-Schwartz inequality with respect to the measure νn(Xi) × dnt(θ) in the first line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 2ab ≤ a2 + b2 in the second line and ∇ log f = 2∇ log √f = 2 ∇√f √f in the third line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' we obtain that: |J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='t| ≤ �� Rd ����∇θ log � nt(θ) µn(θ) ����� 2 2 dnt(θ) � 1 2 �� Rd n � i=1 ��∇θUXi(θ) ��2 2 �mt(Xi|θ) νn(Xi) − 1 �2 νn(Xi) dnt(θ) � 1 2 ≤ 3 4 � Rd ����∇θ log � nt(θ) µn(θ) ����� 2 2 dnt(θ) + 1 3 � Rd n � i=1 ��∇θUXi(θ) ��2 2 � mt(Xi|θ) νn(Xi) − 1 �2 νn(Xi) dnt(θ) ≤ 3 � Rd �����∇θ �� nt(θ) µn(θ) ������ 2 2 dµn(θ) + 1 3 � Rd n � i=1 ��∇θUXi(θ) ��2 2 �mt(Xi|θ) νn(Xi) − 1 �2 νn(Xi) dnt(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Using Equation (15) and the previous line yields: ∂t{Jt} ≤ − � Rd �����∇θ �� nt(θ) µn(θ) ������ 2 2 dµn(θ) + 1 3 � Rd n � i=1 ��∇θUXi(θ) ��2 2 �mt(Xi|θ) νn(Xi) − 1 �2 νn(Xi) dnt(θ) � �� � :=∆t , (16) We then focus on the second term of the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For this purpose, we consider a non- negative function g(t), which will be fixed later and we split ∆t into two terms as: ∆t = � Rd n � i=1 ��∇θUXi(θ) ��2 2 � 1∥∇θUXi (θ)∥2≤g(t) + 1∥∇θUXi (θ)∥2>g(t) � �mt(Xi|θ) νn(Xi) − 1 �2 νn(Xi) dnt(θ) ≤ g2(t)It + � Rd n � i=1 ��∇θUXi(θ) ��2 2 1∥∇θUXi (θ)∥2>g(t) �mt(Xi|θ) νn(Xi) − 1 �2 νn(Xi) dnt(θ), where It has been introduced in Equation (6) and measures the closeness of mt(Xi|θ) to νn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Finally, for the last term we observe that 0 ≤ mt(Xi|θ) ≤ 1 and ��� mt(Xi|θ) νn(Xi) − 1 ��� = n ��mt(Xi|θ) − 1 n �� ≤ n, which implies that: ∆t ≤ g2(t)It + n2 1 n � Rd n � i=1 ∥∇θUXi(θ)∥2 2 1∥∇θUXi (θ)∥2>g(t)dnt(θ) � �� � := ˜∆t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (17) The Cauchy inequality leads to: ˜∆t ≤ � 1 n � Rd n � i=1 ∥∇θUXi(θ)∥4 2 dnt(θ) � 1 2 � 1 n � Rd n � i=1 1∥∇θUXi (θ)∥2>g(t)dnt(θ) � 1 2 = � 1 n n � i=1 E � ∥∇θUXi(θt)∥4 2 �� 1 2 � 1 n n � i=1 P (∥∇θUXi(θt)∥2 > g(t)) � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (18) We then use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 and obtain that: ˜∆t ≤ � 1 n n � i=1 E �� 2(nL + ℓ0)U 2 Xi(θt) �� � 1 2 � 1 n n � i=1 P � 2(nL + ℓ0)UXi(θt) > g2(t) � � 1 2 ≤ 2(nL + ℓ0) � nE[U 2 νn(θt)] � 1 2 � 1 n n � i=1 2(nL + ℓ0) g2(t) E [UXi(θt)] � 1 2 ≤ [2(nL + ℓ0)] 3 2 n 1 2 E � U 2 νn(θt) � 1 2 E [Uνn(θt)] 1 2 g(t) , 10 where we used the Markov’s inequality and the relation ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='∥2 ≤ ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='∥1 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We apply Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 with α = 2 and α = 1 and obtain that a constant C > 0 exists (whose value may change from line to line) such that: ˜∆t ≤ C n 7 2 � d log2β(n) � 3(1+r) 2 g(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We use this last bound in (17) and we deduce that: ∆t ≤ g2(t)It + C n 11 2 � d log2β(n) � 3(1+r) 2 g(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Optimizing this last bound with respect to g(t) leads to the upper bound: ∆t ≤ CI 1 3 t n 11 3 � d log2β(n) �1+r , ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 Evolution of the weighted L2 distance It The quantity It involved in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 measures how close to νn the conditional distribution of Xt|θt is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' To study It, we first remark that it may be rewritten in a simpler way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' It = � Rd n � i=1 �mt(Xi|θ) νn(Xi) − 1 �2 νn(Xi) dnt(θ) = � Rd n � i=1 �m2 t(Xi|θ) ν2n(Xi) − 2mt(Xi|θ) νn(Xi) + 1 � νn(Xi) dnt(θ) = � Rd n � i=1 �m2 t(Xi|θ) νn(Xi) − 2mt(Xi|θ) + νn(Xi) � dnt(θ) = � Rd � n � i=1 m2 t(Xi|θ) νn(Xi) − 1 � dnt(θ) = � Rd n � i=1 m2 t(Xi|θ) νn(Xi) dnt(θ) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Using that mt(Xi|θ)nt(θ) = mt(θ, Xi) and νn(Xi) = 1 n for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=', n, we obtain that: It = n � Rd n � i=1 m2 t(θ, Xi) nt(θ) dθ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (19) The next proposition then assesses how fast It decreases to 0 as t −→ +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For any t ≥ 0: It ≤ I0e−2αnt ≤ (n − 1)e−2αnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (20) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Our starting point is Equation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We compute its derivative with respect to t: ∂t{It} = 2n � Rd n � i=1 mt(θ, Xi) nt(θ) ∂tmt(θ, Xi)dθ − n � Rd n � i=1 m2 t(θ, Xi) n2 t(θ) ∂tnt(θ)dθ = 2n � Rd n � i=1 mt(Xi|θ)∂tmt(θ, Xi)dθ − n � Rd n � i=1 m2 t(Xi|θ)∂tnt(θ)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 11 Using the Kolmogorov backward equation in the first line and L = L1 + L2 in the second one where L1 and L2 are defined in Equation (9), we have: ∂t{It} = 2n � Rd n � i=1 Lmt(Xi|θ) mt(θ, Xi)dθ − n � Rd n � i=1 m2 t(Xi|θ)∂tnt(θ)dθ = 2n � Rd n � i=1 L1mt(Xi|θ) mt(θ, Xi)dθ � �� � :=I3,t + 2n � Rd n � i=1 L2mt(Xi|θ) mt(θ, Xi)dθ � �� � :=I1,t −n � Rd n � i=1 m2 t(Xi|θ)∂tnt(θ)dθ � �� � :=I2,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (21) Then, ∂t{It} may be splitted into three terms that are studied separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Study of I1,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We observe that: L2mt(Xi|θ) = αn n n � j=1 [mt(Xj|θ) − mt(Xi|θ)] = αn n − αn mt(Xi|θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (22) We then use this last equation in the definition of I1(t) and obtain that: I1,t = 2n � Rd n � i=1 L2mt(Xi|θ) mt(θ, Xi)dθ = 2αn � Rd n � i=1 mt(θ, Xi)dθ − 2αnn � Rd n � i=1 mt(Xi|θ)mt(θ, Xi)dθ = 2αn − 2αnn � Rd n � i=1 m2 t(θ, Xi) nt(θ) dθ = −2αnIt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (23) Study of I2,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Using the definition of nt, we obtain that: I2,t = −n � Rd n � i=1 m2 t(Xi|θ)∂tnt(θ)dθ = −n � Rd n � i=1 m2 t(Xi|θ)∂t \uf8eb \uf8ed n � j=1 mt(θ, Xj) \uf8f6 \uf8f8 dθ = −n � Rd n � j=1 n � i=1 m2 t (Xi|θ)∂tmt(θ, Xj)dθ = −n � Rd n � j=1 � n � i=1 Lm2 t (Xi|θ) � mt(θ, Xj)dθ = −n � Rd n � i=1 Lm2 t(Xi|θ) dnt(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' where we used the Kolmogorov backward equation in the fourth line and again the definition of nt in the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Again, the decomposition L = L1 + L2 yields: I2,t = −n � Rd n � i=1 L1m2 t(Xi|θ) dnt(θ) − n � Rd n � i=1 L2m2 t(Xi|θ) dnt(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 12 We repeat some similar computations as those developed in Equation (22) to study the action of the jump component induced by L2 on m2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We obtain that: L2m2 t(Xi|θ) = αn n n � k=1 [m2 t(Xk|θ) − m2 t(Xi|θ)] = αn n n � k=1 m2 t (Xk|θ) − αn m2 t(Xi|θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We use this last equation and obtain that: I2,t = −n � Rd n � i=1 L1m2 t (Xi|θ) dnt(θ) − αn � Rd n � i=1 n � k=1 m2 t(Xk|θ) dnt(θ) +αnn � Rd n � i=1 m2 t(Xi|θ) dnt(θ) = −n � Rd n � i=1 L1m2 t (Xi|θ) dnt(θ) − αnn � Rd n � k=1 m2 t(Xk|θ) dnt(θ) +αnn � Rd n � i=1 m2 t(Xi|θ) dnt(θ) = −n � Rd n � i=1 L1m2 t (Xi|θ) dnt(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (24) Study of I2,t + I3,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We observe that this sum involves only L1 (see Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We first compute: L1mt(Xi|θ) = −⟨∇θUXi(θ), ∇θmt(Xi|θ)⟩ + ∆θmt(Xi|θ), and similarly: L1m2 t(Xi|θ) = −⟨∇θUXi(θ), ∇θm2 t(Xi|θ), ⟩ + ∆θm2 t(Xi|θ) = −2mt(Xi|θ)⟨∇θUXi(θ), ∇θmt(Xi|θ)⟩ + 2∥∇θmt(Xi|θ)∥2 2 + 2mt(Xi|θ)∆θmt(Xi|θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Using these two equations into I2,t + I3,t and mt(Xi|θ)nt(θ) = mt(θ, Xi), we get: I2,t + I3,t n = 2 � Rd n � i=1 ⟨∇θmt(Xi|θ), ∇θUXi(θ)⟩mt(θ, Xi)dθ − 2 � Rd n � i=1 ∥∇θmt(Xi|θ)∥2 2 nt(θ)dθ − 2 � Rd n � i=1 ∆θmt(Xi|θ) mt(θ, Xi)dθ − 2 � Rd n � i=1 ⟨∇θmt(Xi|θ), ∇θUXi(θ)⟩mt(θ, Xi)dθ + 2 � Rd n � i=1 ∆θmt(Xi|θ) mt(θ, Xi)dθ = − � Rd n � i=1 ∥∇θmt(Xi|θ)∥2 2 dnt(θ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Gathering this last inequality with (23) into Equation (21) yields: ∂t{It} ≤ −2αnIt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We conclude with a direct application of the Gronwall lemma while observing that I0 ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3 Functional (weak) log-Sobolev inequalities 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 Related works on functional inequalities A straightforward consequence of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 is the following differential inequality on the relative entropy Jt: ∂t{Jt} ≤ − � Rd �����∇θ �� nt(θ) µn(θ) ������ 2 2 dµn(θ) + cn,de− 2αn 3 t, (25) 13 where cn,d is defined as: cn,d ≲uc n4 � d log2β(n) �1+r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (26) At this stage, we should observe that a standard approach consists in finding a functional inequality that relates the key Dirichlet form E(f) defined by: E(f) = � Rd ∥∇θf(θ)∥2 2dµn(θ), (27) to Entµn(f 2), the entropy itself with respect to µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' These approaches rely on the initial works of [23] where Logarithmic Sobolev Inequality (LSI for short) were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The consequences of LSI to exponential ergodicity has then been an extensive field of research and we refer to [3] for an overview on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' A popular sufficient condition that ensures LSI is the log strong-convexity of the targeted measure (see among other [2]) and an impressive amount of literature has been focused on the existing links between these functional inequalities, ergodicity of the semi-group, transport inequalities and Lyapunov conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We refer to [8, 1] (these two works are far from being exhaustive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The great interest of LSI has then been observed in machine learning and statistics more recently as testified by the recent works in Monte Carlo samplings of [31, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' A popular way to extend LSI from the strongly convex situation to a more general case relies on the “strong convexity outside a ball” hypothesis using the perturbation argument of the seminal contributions of [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' If this method proves to be suitable for the study of the simulated annealing process in [33], [26], it appears to be doubtful for the study of sampling problems with convex potentials that satisfies Hr KL(c, L) as this settings do not imply an asymptotic strong convexity of θ �−→ U(θ) for large values of ∥θ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' That being said, and maybe an even worst consequence of such approach, is the unavoidable dependency on the dimension for the LSI constant when using a perturbation approach, which leads to a serious exponential degradation of the convergence rates with the dimension of the ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' To overcome these difficulties, we have chosen to use a slightly different functional inequality that may be considered as an innocent modification of LSI, but that indeed appears to be well suited to weakly log-concave setting described through an Hr KL(c, L) assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For this purpose, we shall use weak log-Sobolev inequalities (WLSI for short below) that have been introduced in [37] and whose interest has been extensively studied in many works to obtain exponentially sub-linear rates of mixing, see among others for example [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' To derive such inequalities, our starting point will be the contribution of [10] that makes the link between Lyapunov conditions and WLSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Our approach based on Hr KL(c, L) certainly shares some similarities with the recent work of [6] where some functional inequalities (Poincar´e and Transport inequalities) are obtained within a framework of variable curvature bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 Weak log Sobolev inequalities We briefly introduce the key theoretical ingredients, that are exhaustively described in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We intro- duce the following assumption, that will be suitable for the setting of bounded functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 (Weak Log-Sobolev Inequality ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For any measurable space (Ω, F, µ) and for any nice function f, let us define: Entµ(f 2) := � Ω f 2 log(f 2)dµ − � Ω f 2dµ log �� Ω f 2dµ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The measure µ satisfies a WLSI if a non-increasing function ϕWLS : (0, +∞) �→ R+ exists such that for any f ∈ C 1 b (Ω): Entµ(f 2) ≤ ϕWLS(s)E(f) + s Osc2(f), (28) where Osc(f) := sup f − inf f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Before establishing how to use this functional inequality, we first state the important relationship between Poincar´e Inequality and WLSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 14 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assume that µ satisfies a Poincar´e Inequality of constant CP , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' for any smooth integrable function f: Cp(µ)V arµ(f) = Cp(µ) � Ω (f − µ[f])2dµ ≤ � Ω |∇f|2dµ, then if log c = 3 14e2 � 1 e + 1 2 � + 1 + log � 14 3 � , then µ satisfies a WLSI with: ϕWLS(s) = � 0, s > 1 e + 1 2 32 CP log � c s � , s ≤ 1 e + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For the sake of readability, we introduce a universal a > 0 such that: ϕWLS(s) = � 0, s > 1 e + 1 2 a 1+log( 1 s) CP , s ≤ 1 e + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (29) Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The proof of how the Poincar´e Inequality implies the WLSI in the bounded setting described in Definition 28 is given for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Technical details are skipped and we refer to the references below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We use the measure-capacity inequality (see [3], Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We know that the Poincar´e Inequality implies a capacity inequality (Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 of [3]) with a constant equal to 2CP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Then, we can apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 of [7] that induces a WLSI which is based on the function ϕWLS given in the statement of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3 Weak log Sobolev inequalities under Hr KL(c, L) Of course, in the previous result, the only important dependency will be the one induced by CP , which will deserve an ad-hoc study under Assumption Hr KL(c, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The numbers 32 and log(c) will be dealt with as “universal constants” in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The next proposition states two lower bounds on the Poincar´e constant within the Hr KL(c, L) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The first one always holds, regardless the value of (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn) that may be been randomly sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The second one has to be considered with high probability, with respect to the sampling process (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assume Hmin,Hn0(L, ℓ0), Hπ0(ℓ0) and for any x, θ �→ − log pθ(x) satisfies Hr KL(c, L), then: i) For any sample (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn), it holds: CP (µn) ≳uc 1 � d log2β(n) �(1+r)2 ii) Assume that θ �→ Pθ is injective and θ0 exists such that (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn) ∼ Pθ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' If locally around θ0, θ �→ |θ − θ0|−αW1(Pθ, Pθ0) does not vanish, then: E(X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=',Xn)∼Pθ0[CP (µn)] ≳uc � n Ld log n �α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We are finally led to upper bound the oscillations of the function involved in the WLSI introduced in (28), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' we are looking for an upper bound of Osc2 �� nt µn � for any time t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For this purpose, we observe that the Markov semi-group induces that ft = nt µn = Ptf0 where f0 = n0 µn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The next proposition implies the boundedness of ft over Rd when n0 is chosen as a Gaussian distribution with a carefully tuned covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assume Hmin,Hn0(L, ℓ0), Hπ0(ℓ0) and that, for any x, θ �→ − log pθ(x) satisfies Hr KL(c, L), then: 15 i) Two positive constants C1 and C2 exist, which are independent from n and d and such that: ∥f0∥∞ ≲uc �C1d n � dr 2 exp � C2nd1+r log2β(1+r)(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' ii) As a consequence: Osc2( � ft) ≤ Osc2( � f0) ≲uc �C1d n � dr 2 exp � C2nd1+r log2β(1+r)(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' iii) Moreover, a straightforward consequence of i) is: J0 = � Rd log (f0(θ)) dn0(θ) ≲uc nd1+r log2β(1+r)(n) + d log � d n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='4 Entropic convergence of the SLMC The purpose of this paragraph is to prove the main result of the paper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 that guarantees the convergence of the SLMC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Our starting point is the semi-group inequality (25) associated with the func- tional WLSI inequality (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Using cn,d defined in (26), we obtain for any s > 0: ∂t{Jt} ≤ −E �� nt µn � + cn,de− 2αn 3 t ≤ − Jt ϕWLS(s) + s ϕWLS(s)Osc2 �� nt µn � + cn,de− 2αn 3 t ≤ − Jt ϕWLS(s) + s On,d ϕWLS(s) + cn,de− 2αn 3 t, where we applied Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='5 in the last line with On,d ≲uc � C1d n � dr 2 exp � C2nd1+r log2β(1+r)(n) � and C1 and C2 two universal constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We then choose s (that depends on t) such that: st = e−A√t+1 with A > 1 that will be chosen later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We observe that st < e−1 + 1/2, so that Equation (29) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3 yields: ϕWLS(st) = a 1 + log � 1 st � CP = a1 + A√1 + t CP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We introduce ψ(t) = exp � CP a � t 0 du 1+A√1+u � and deduce that ψ(t) = exp \uf8eb \uf8edCP a 2A(√1 + t − 1) − 2 log � 1+A√1+t 1+A � A2 \uf8f6 \uf8f8 ≤ exp �2CP aA ( √ 1 + t − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We now apply the Gronwall Lemma: ∂t {ψ(t)Jt} = � CP a(1 + A√1 + t)Jt + J′ t � ψ(t) ≤ � CP On,d a e−A√t+1 1 + A√1 + t + cn,de− 2αn 3 t � ψ(t) ≤ CP On,d a e−(A− 2CP aA )√1+t + cn,de 2CP aA (√1+t−1)− 2αn 3 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 16 We denote by t0 the positive real value that solves the equation 2CP aA √1 + t0 = αnt0 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We then observe that: � t 0 e 2CP aA (√1+u−1)− 2αn 3 udu ≤ � t0 0 e 2CP aA √1+udu + � +∞ t0 e− αn 3 udu ≤ t0e 2CP aA √1+t0 + 3 αn = t0e αnt0 3 + 3 αn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' If A is chosen such that A > 2CP aA , we then deduce that: Jt ≤ � J0 + cn,dt0e αnt0 3 + 3cn,d αn � ψ(t)−1 + CP On,d a ψ(t)−1 � t 0 e− � A− 2CP aA �√1+udu ≤ � J0 + cn,dt0e αnt0 3 + 3cn,d αn � ψ(t)−1 + 2CP On,d a � A − 2CP aA �2 ψ(t)−1, where we used in the previous line the bound: � t 0 e−b√1+udu ≤ � +∞ 0 e−b√1+udu ≤ 2 b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' To obtain the lowest upper bound, we are led to choose A such that 2CP aA as large as possible and below A, which naturally drives to the choice: 2CP aA = A 2 =⇒ A = 2 √a � CP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Using this value of A in the previous bound, we observe that t0 ≤ 3√CP αn √a + CP α2n , so that a constant C exists such that: Jt ≤ C � J0 + cn,d αn � 1 + �CP αn + � CP � e √ CP √a + CP 3αn � + On,d � (1 + t)1/4e− √ Cp √a (√1+t−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (30) In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='4 we obtained CP ≥ κ (d log2β(n)) (1+r)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' If instead of using the constant CP , we use directly κ (d log2β(n)) (1+r)2 with κ < 1, then all the previous computations remain the same only replacing CP by its lower bound and: Jt ≤ C \uf8eb \uf8ec \uf8edJ0 + cn,d αn e √κ � 1 √a + 1 3αn � (d log2β(n))(1+r)2/2 + On,d \uf8f6 \uf8f7 \uf8f8 (1 + t)1/4e − √κ(√1+t−1) √a(d log2β(n))(1+r)2/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (31) Using the values of On,d, cn,d and the upper bound of J0, we finally observe that if αn = 1 n(d log2β(n)) 1+r , then: t ≥ ℵ � d log2β(n) �(1+r)2 � log2(ε−1) + n2 � d log2β(n) �2(1+r) + d2 log2 d � =⇒ Jt ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 4 Technical results on KL and Uνn 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 Growth properties under the Kurdyka-�Lojasiewicz inequality We remind here some important consequences of the KL inequality that implies several relationships between the function and the norm of its gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The proof of these inequalities may be found in Lemma 15 of [21] (a small mistake appears and we correct the statement with a factor 2 in our work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 17 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assume that a function V satisfies Hr KL(c, L), then: 2c 1 − r � V 1−r(θ) − min(V )1−r� ≤ ∥∇V (θ)∥2 2 ≤ 2L [V (θ) − min(V )] , ∀θ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' It is furthermore possible to assess a minimal and maximal growth property of any function that satisfies Hr KL(c, L), which is necessarily lower and upper bounded by a positive power of the distance to its minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assume that a function V satisfies Hr KL(c, L), then, ∀θ ∈ Rd: V 1+r(θ) − min(V )1+r ≥ (1 + r)c 2 ∥θ − arg min V ∥2 2, and V (θ) − min(V ) ≤ L 2 ∥θ − arg min V ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' A straightforward consequence of the first inequality is then Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assume that a function V satisfies Hr KL(c, L), then, ∀θ ∈ Rd: V (θ) ≥ 2− r 1+r � min(V ) + �(1 + r)c 2 � 1 1+r ∥θ − arg min V ∥ 2 1+r 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 Properties of Uνn Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' First, we observe that if each θ �→ ∇ log pθ(Xi) is L-Lipschitz and θ �→ ∇ log π0 is ℓ0-Lipschitz, then the triangle inequality implies that ∥∇Uνn(θ1) − ∇Uνn(θ2)∥2 ≤ (nL + ℓ0)∥θ1 − θ2∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Second, we consider the lower-bound property on the curvature and observe that: λ∇2Uνn(θ) = inf e∈Rd:|e|=1 eT (∇2Uνn)(θ)e ≥ 1 n n � i=1 inf e∈Rd:|e|=1 eT (∇2UXi)(θ)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The log concavity of the prior yields λ∇2Uνn(θ) ≥ 1 n n � i=1 λ∇2(−n log pθ(Xi)) = n � i=1 λ∇2(− log pθ(Xi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Then, the Hr KL(c, L) property applied to each term of the sum above and minθ∈Rd − log π0(θ) > 0 yields λ∇2Uνn(θ) ≥ c n � i=1 [− log pθ(Xi)]−r ≥ cnr n � i=1 U −r Xi (θ) = cn1+r � 1 n n � i=1 U −r Xi (θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' From the Jensen inequality, we finally deduce that: λ∇2Uνn(θ) ≥ cn1+r � 1 n n � i=1 U −r Xi (θ) � ≥ cn1+rU −r νn (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We conclude that Uνn satisfies Hr KL � cn1+r, nL + ℓ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For UXi, the proof is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We assume Hπ0(ℓ0), Hmin and that for any x: θ �−→ − log pθ(x) satisfies Hr KL(c, L), then: ∥ arg min Uνn∥2 ≲uc d 1+r 2 logβ(1+r)(n) and minθ∈Rd Uνn(θ) ≲uc nd log2β(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 18 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 shows that Uνn satisfies Hr KL � cn1+r, nL + ℓ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Therefore, we can apply Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 with θ = 0 and deduce that: ∥ arg min Uνn∥2 2 ≤ 2 (1 + r)cn1+r � U 1+r νn (0) − min U 1+r νn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' To obtain an upper bound of Uνn(0) we first bound UXi(0) using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2, for all i, as follows: UXi(0) ≤ min UXi + nL + ℓ0 2 ∥ arg min UXi∥2 2 ≲uc d + nd log2β(n) ≲uc nd log2β(n), then Uνn(0) ≲uc nd log2β(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We deduce that: ∥ arg min Uνn∥2 2 ≤ 2 (1 + r)cn1+r U 1+r νn (0) ≲uc d1+r log2β(1+r)(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The second part comes from min Uνn ≤ Uνn(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 5 Smoothness and boundedness of the semi-group Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The proof relies on an argument set up with a ”fixed” sample (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Our starting point is Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 and the consequences of the Kurdyka-�Lojasiewicz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Since Hπ0(ℓ0) and θ �→ − log pθ(Xi) satisfies Hr KL(c, L), then Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 shows that Uνn satisfies Hr KL � cn1+r, nL + ℓ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Therefore, we can apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 and deduce that: ∥θ − arg min Uνn∥2 2 ≤ 2 (1 + r)cn1+r � U 1+r νn (θ) − min U 1+r νn � ≤ 2 (1 + r)cn1+r U 1+r νn (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' If Id refers to the identity map, we use the fact that for any distribution µ, we have V ar[µ] ≤ µ[∥Id−a∥2 2] for any a ∈ Rd so that a straightforward consequence with a = arg min Uνn is then: V ar(µn) ≤ � Rd ∥θ − arg min Uνn∥2 2dµn(θ) ≤ 2 (1 + r)cn1+r µn[U 1+r νn ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We then use the ergodic behaviour of (θt)t≥0 and observe that there exists a constant C independent from n and d such that: V ar(µn) ≤ 2 (1 + r)cn1+r lim sup t≥0 E[U 1+r νn (θt)] ≤ C � d log2β(n) �(1+r)2 , where the last inequality comes from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We now use the Bobkov bound on the Poincar´e constant for log-concave distribution (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 of [4]) and deduce that a universal constant K exists such that: CP (µn) ≥ 1 4K2V ar(µn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Using the upper bound of the variance, we deduce that a universal κ > 0 exists such that: CP (µn) ≥ κ � d log2β(n) �(1+r)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For the second point, we consider a situation on average over the samples and the result uses the concentration of the posterior distribution around its mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We know from Theorem 3 of [21] that a constant c > 0 exists such that: E(X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=',Xn)∼Pθ0[Var(µn)] ≤ cǫ2 n,d, with ǫn,d = � Ld log n n �α−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The result follows using the Jensen inequality and the Bobkov bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 19 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We first establish the boundedness of f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' From our assumptions, we apply Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 and obtain that Uνn satisfies Hr KL � cn1+r, nL + ℓ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' If θ⋆ n = arg min Uνn, we then deduce from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 that: f0(θ) = n0(θ) µn(θ) = Zne− ∥θ∥2 2 2σ2 +Uνn(θ) (2π)d/2σd ≤ Zne− ∥θ∥2 2 2σ2 +Uνn(θ⋆ n)+ (nL+ℓ0) 2 ∥θ−θ⋆ n∥2 2 (2π)d/2σd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (32) We compute an upper bound of Zn and use the lower bound of Uνn induced by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='3: Zn = � Rd e−Uνn(θ)dθ ≤ � Rd e −2 − r 1+r � Uνn(θ⋆ n)+n( (1+r)c 2 ) 1 1+r ∥θ−θ⋆ n∥ 2 1+r 2 � dθ ≤ e−2 − r 1+r Uνn(θ⋆ n) � Rd e−nar∥θ∥ 2 1+r 2 dθ, with ar = ((1+r)c) 1 1+r 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Using the well known equality: � Rd e−a|θ|ℓdθ = dπd/2Γ(d/ℓ) ℓad/ℓΓ(d/2 + 1), ∀a > 0, ∀ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' we then deduce with a = nar and ℓ = 2 1+r that: Zn ≤ e−2 − r 1+r Uνn(θ⋆ n) � Rd e−nar∥θ∥ 2 1+r 2 dθ ≤ d(1 + r) 2 πd/2 (nar) d(1+r) 2 Γ � d(1+r) 2 � Γ � d 2 + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' From standard relationships on the Gamma function: Zn ≤ 2 �21+rπ cn1+r � d 2 d dr 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (33) We gather Equations (32) and (33) and obtain that: f0(θ) ≤ 2eUνn(θ⋆ n) � 2 cσ2n1+r � d 2 d dr 2 e− ∥θ∥2 2 2σ2 + (nL+ℓ0) 2 ∥θ−θ⋆ n∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For all σ2 < 1 nL+ℓ0 , a straightforward optimization on θ yields : ∥f0∥∞ ≤ 2eUνn(θ⋆ n) � 2 cσ2n1+r � d 2 d dr 2 exp � (nL + ℓ0) 2(1 − σ2(nL + ℓ0))∥θ⋆ n∥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Then, the choice c1 nL+ℓ0 ≤ σ2 ≤ c2 nL+ℓ0 , where 0 < c1 ≤ c2 < 1 in Hn0(L, ℓ0) and the bounds of ∥θ⋆ n∥2 2 and Uνn(θ⋆ n) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='4 lead to : ∥f0∥∞ ≤ 2 �C1d n � dr 2 exp � C2nd1+r log2β(1+r)(n) � , where C1 and C2 are universal constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' This result is an almost standard consequence of the maximum principle for a Markov semi-group property with a Brownian diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' For any bounded measurable h > 0, we observe that Pth > 0 using the Markov property, and we are led to define gt as the following function gt := √Pth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We then introduce θ(t) and θ(t) as: θ(t) = arg max gt(θ) and θ(t) = arg min gt(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 20 The chain rule yields: d dtOsc(gt) = d dt � gt(θ(t)) − gt(θ(t)) � = dgt dt (θ(t)) + � ∇gt(θ(t)), dθ(t) dt � − dgt dt (θ(t)) − � ∇gt(θ(t)), dθ(t) dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (34) We compute: dgt dt (θ) = 1 2√Pth dPth dt (θ) = 1 2√PthGtPth(θ) = 1 2 � Pth(θ) � − n � i=1 ⟨∇θPth(θ), ∇θUXi(θ)⟩mt(Xi|θ) + ∆θPth(θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (35) Now, we use that θ(t) = arg max gt = arg max Pth, (a similar argument holds for θ(t)): ∇θgt(θ(t)) = 0, ∇θPth(θ(t)) = 0 and ∆θPth(θ(t)) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' then: d dtOsc(gt) = dgt dt (θ(t)) − dgt dt (θ(t)) = ∆θPth 2√Pth(θ(t)) − ∆θPth 2√Pth(θ(t)) (36) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We have therefore shown that Osc(√Pth) is decreasing in t ≥ 0, which ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We proceed as in Proposition 3 of [33] to justify the use of the Lebesgue domi- nated convergence theorem for the derivation of the integral involved in our statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We can then deduce that: ∂t �� Rd ft(θ)dnt(θ) � = � Rd ∂t{ft(θ)}dnt(θ) + � Rd ft(θ)∂t{nt(θ)}dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We leave the first term unchanged and now focus on the second term: � Rd ft(θ)∂t{nt(θ)}dθ = � Rd ft(θ)∂t � n � i=1 mt(θ, Xi) � dθ = � Rd n � i=1 ft(θ)∂t{mt(θ, Xi)}dθ = � Rd n � i=1 Lft(θ) mt(θ, Xi)dθ, where we used the definition of nt in the first step and Kolmogorov backward equation (10) in the last one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Since the function ft(θ) does not depend on x, we observe that L2ft(θ) = 0 and we only need to compute the remaining term L1ft(θ): � Rd ft(θ)∂t{nt(θ)}dθ = � Rd n � i=1 L1ft(θ) mt(θ, Xi)dθ (37) = � Rd n � i=1 [−⟨∇θft(θ), ∇θUXi(θ)⟩ + ∆θft(θ)] mt(θ, Xi)dθ = − � Rd n � i=1 ⟨∇θft(θ), ∇θUXi(θ)⟩mt(Xi|θ)dnt(θ) + � Rd ∆θft(θ)dnt(θ) = � Rd Gtft(θ)dnt(θ), (38) 21 where we used the fact that mt(θ, Xi) = mt(Xi|θ)nt(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 Moments upper bounds Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Assume Hn0(L, ℓ0), Hπ0(ℓ0), Hmin and that for each Xi, θ �→ − log pθ(Xi) satisfies Hr KL(c, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Then: i) Three positive constants C1, C2 and C3, independent from n and d, exist such that for any t > 0: E � e (1+r)nc 1 1+r 16 (∥θt∥2 2+1) 1 1+r � ≤ C1 � d log2β(n) � r 1+r eC2nd log2β(n) + Cd 3e (1+r)nc 1 1+r 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' ii) For any t > 0 and for any α ≥ 1: E[U α νn(θt)] ≲uc nα � d log2β(n) �α(1+r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proof of i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We consider the function f(θ) = exp � a 2(∥θ∥2 2 + 1)ρ� where 0 < ρ < 1, which is twice differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The gradient of f is computed as: ∇f(θ) = aρ(∥θ∥2 2 + 1)ρ−1f(θ)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The Laplace operator is given as: ∆f(θ) = aρ(∥θ∥2 2 + 1)ρ−2f(θ) � aρ(∥θ∥2 2 + 1)ρ∥θ∥2 2 + (d + 2ρ − 2)∥θ∥2 2 + d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We then deduce that for any θ ∈ Rd: Gtf(θ) = − n � i=1 ⟨∇UXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' ∇f(θ)⟩mt(Xi|θ) + ∆f(θ) = aρ(∥θ∥2 2 + 1)ρ−2f(θ) � − (∥θ∥2 2 + 1) n � i=1 ⟨θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' ∇θUXi(θ)⟩mt(Xi|θ) +aρ(∥θ∥2 2 + 1)ρ∥θ∥2 2 + (d + 2ρ − 2) ∥θ∥2 2 + d � ≤ aρ(∥θ∥2 2 + 1)ρ−2f(θ) � − (∥θ∥2 2 + 1) n � i=1 (UXi(θ) − UXi(0)) mt(Xi|θ) +aρ(∥θ∥2 2 + 1)ρ+1 + d � ∥θ∥2 2 + 1 � � ≤ aρ(∥θ∥2 2 + 1)ρ−1f(θ) � − n � i=1 (UXi(θ) − UXi(0)) mt(Xi|θ) + aρ(∥θ∥2 2 + 1)ρ + d � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' where we used the convexity of Ux for any position x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Let us establish the bounds of UXi(θ) and UXi(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We denote by θi = arg min UXi and from Hypothesis Hmin, there exist two positive constants K1 and K2 independent on n and d such that: maxi ∥θi∥2 2 ≤ K1d log2β(n) and maxi UXi(θi) ≤ K2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 to each non-negative function UXi that satisfies Hr KL � cn1+r, nL + ℓ0 � , then we obtain that: UXi(θ) ≥ n �(1 + r)c 2 � 1 1+r ∥θ − θi∥ 2 1+r 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Since 2 1+r > 1, the Jensen inequality yields (u + v) 2 1+r ≤ 2 1−r 1+r � u 2 1+r + v 2 1+r � , for all (u, v) ∈ R2 + and we deduce that: ∥θ − θi∥ 2 1+r 2 ≥ 2 r−1 1+r ∥θ∥ 2 1+r 2 − ∥θi∥ 2 1+r 2 ≥ 2 r−1 1+r ∥θ∥ 2 1+r 2 − � K1d log2β(n) � 1 1+r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 22 Then we use this inequality to obtain a lower bound of UXi: UXi(θ) ≥ 2n �(1 + r)c 8 � 1 1+r ∥θ∥ 2 1+r 2 − n �(1 + r)c 2 � 1 1+r (K1d log2β(n)) 1 1+r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Moreover an upper bound of max UXi(0) comes from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2 as follows: UXi(0) ≤ UXi(θi) + nL + ℓ0 2 ∥θi∥2 2 ≤ K2d + K1(nL + ℓ0)d log2β(n) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Using the previous bounds and the fact that �n i=1 mt(Xi|θ) = 1, it yields: n � i=1 (UXi(θ) − UXi(0)) mt(Xi|θ) ≥ 2n �(1 + r)c 8 � 1 1+r ∥θ∥ 2 1+r 2 − n �(1 + r)c 2 � 1 1+r (K1d log2β(n)) 1 1+r − K2d − K1(nL + ℓ0)d log2β(n) 2 ≥ nc 1 1+r 4 ∥θ∥ 2 1+r 2 − nc 1 1+r (K1d log2β(n)) 1 1+r − K2d − K1(nL + ℓ0)d log2β(n) 2 , where we used some uniform upper bounds when r ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We then choose ρ = 1 1+r and we deduce that: Gtf(θ) ≤ a 1 + r (∥θ∥2 2 + 1)− r 1+r f(θ) � −nc 1 1+r 4 ∥θ∥ 2 1+r 2 + nc 1 1+r (K1d log2β(n)) 1 1+r + K2d +K1(nL + ℓ0)d log2β(n) 2 + a (1 + r)(∥θ∥2 2 + 1) 1 1+r + d � ≤ a 1 + r (∥θ∥2 2 + 1)− r 1+r f(θ) � − � nc 1 1+r 4 − a (1 + r) � ∥θ∥ 2 1+r 2 + nc 1 1+r (K1d log2β(n)) 1 1+r +(K2 + 1)d + K1(nL + ℓ0)d log2β(n) 2 + a (1 + r) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' where we used (∥θ∥2 2 + 1) 1 1+r ≤ ∥θ∥ 2 1+r 2 + 1 in the second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We now fix a = n(1+r)c 1 1+r 8 and deduce that: Gtf(θ) f(θ) ≤ n2c 2 1+r 64 (∥θ∥2 2 + 1)− r 1+r � −∥θ∥ 2 1+r 2 + 8(K1d log2β(n)) 1 1+r + +8(K2 + 1)d + 4K1(nL + ℓ0)d log2β(n) nc 1 1+r + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (39) We then study two complementary situations and below, we denote by Kn,d the radius of the key compact set involved by the previous Lyapunov contraction: K 2 1+r n,d = Cd log2β(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' When ∥θ∥2 is large enough (∥θ∥2 ≥ Kn,d), we observe that a large enough C > 0 independent from n and d exists such that: ∥θ∥ 2 1+r 2 ≥ Cd log2β(n) =⇒ Gtf(θ) f(θ) ≤ − n2 � d log2β(n) � 1 1+r c 2 1+r 128 = −an,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (40) 23 When ∥θ∥2 is upper bounded (∥θ∥2 ≤ Kn,d), we use the upper bound stated in Equation (39) and obtain that a universal C1 (whose value may change from line to line) exists such that : ∥θ∥ 2 1+r 2 ≤ Cd log2β(n) =⇒ Gtf(θ) ≤ C1n2f(θ) � 8(K1d log2β(n)) 1 1+r + 8(K2 + 1)d + 4K1(nL + ℓ0)d log2β(n) nc 1 1+r + 1 � ≤ C1n2d log2β(n) exp � (C + 1)c 1 1+r nd log2β(n) 8 � ≤ bn,deδn,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (41) We then use Equations (40) and (41) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We define the function ψn,d as ψn,d(t) = E[f(θt)] and use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1: ψ′ n,d(t) = E[Gtf(θt)] = E � Gtf(θt) � 1∥θt∥2≥Kn,d + 1∥θt∥2≤Kn,d �� ≤ E � −an,df(θt)1∥θt∥2≥Kn,d + bn,deδn,d 1∥θt∥2≤Kn,d � ≤ −an,dψn,d(t) + an,d sup ∥θ∥2≤Kn,d f(θ) + bn,deδn,d ≤ −an,dψn,d(t) + (an,d + bn,d)eδn,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We apply the Gronwall Lemma and obtain that: ∀t > 0 ψn,d(t) ≤ � 1 + bn,d an,d � eδn,d + ψn,d(0)e−an,dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (42) Using that n0 is a Gaussian distribution, which was fixed in Hn0(L, ℓ0) hypothesis, we find an upper bound for ψn,d(0) = E[f(θ0)] = � Rd f(θ)dn0(θ) as follows : ψn,d(0) = � 2πσ2�− d 2 � Rd e a 2(∥θ∥2 2+1) 1 1+r − ∥θ∥2 2 2σ2 dθ ≤ � 2πσ2�− d 2 e a 2 � Rd e− ∥θ∥2 2 2 ( 1 σ2 −a)dθ, if σ2 ≤ 1 a = 8 n(1+r)c 1 1+r then the integral above is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Since c2 < 1 ≤ 8L (1+r)c 1 1+r , it guarantees σ2 < 1 a, then: ψn,d(0) ≤ � 1 − aσ2�− d 2 e a 2 ≤ Cd 3e (1+r)nc 1 1+r 16 , where C3 is a constant independent from n and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Finally, using the value of an,d and bn,d in (42), we deduce that: E � e (1+r)nc 1 1+r 16 (∥θt∥2 2+1) 1 1+r � ≤ C1 � d log2β(n) � r 1+r eC2nd log2β(n) + Cd 3e (1+r)nc 1 1+r 16 , ∀t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' where C2 is another universal constant, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proof of ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We consider α > 1 and below, C > 0 refers to a “constant” independent from n and d, whose value may change from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Our starting point is the upper bound of the exponential moments obtained in i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1 shows that Uνn satisfies Hr KL � cn1+r, nL + ℓ0 � , then thanks to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='2: E[U α νn(θt)] ≤ E �� min Uνn + Cn∥θt − θ∗ n∥2 2 �α� ≤ E �� min Uνn + Cn∥θ∗ n∥2 2 + Cn∥θt∥2 2 �α� , 24 where θ∗ n = arg min Uνn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' By using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='4 and the inequality derived from the Jensen inequality (a+b)β ≤ cβ(aβ+bβ) for (a, b) ∈ R2 + and β ≥ 1, we obtain that: (min Uνn+ Cn∥θ∗ n∥2 2 + Cn∥θt∥2 2 �α ≤ C � nd log2β(n) + nd1+r log2β(1+r)(n) + n∥θt∥2 2 �α ≤ Cnα �� d log2β(n) �α(1+r) + ∥θt∥2α 2 � ≤ Cnα �� d log2β(n) �α(1+r) + k−α(1+r) logα(1+r) � ek∥θt∥ 2 1+r 2 �� ≤ Cnα �� d log2β(n) �α(1+r) + k−α(1+r) logα(1+r) � eα(1+r)−1+k∥θt∥ 2 1+r 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' The Jensen inequality and the concavity of x �→ logp(x) on [ep−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' +∞[ when p ≥ 1 yield E[U α νn(θt)] ≤ Cnα �� d log2β(n) �α(1+r) + k−α(1+r)E � logα(1+r) � eα(1+r)−1+k∥θt∥ 2 1+r 2 ��� ≤ Cnα �� d log2β(n) �α(1+r) + k−α(1+r) logα(1+r) � E � eα(1+r)−1+k∥θt∥ 2 1+r 2 ��� ≤ Cnα �� d log2β(n) �α(1+r) + k−α(1+r) � α(1 + r) − 1 + log E � ek∥θt∥ 2 1+r 2 ��α(1+r)� ≤ Cnα �� d log2β(n) �α(1+r) + k−α(1+r) � α(1 + r) − 1 + log E � ek(∥θt∥2 2+1) 1 1+r ��α(1+r)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' where we used in the last inequality that ∥θ∥2 2 ≤ ∥θ∥2 2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' We then apply i) in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content='1, we choose k = (1+r)nc 1 1+r 16 and obtain that: E[U α νn(θt)] ≤ Cnα \uf8ee \uf8f0 � d log2β(n) �α(1+r) + 1 nα(1+r) � 1 + log E � e (1+r)nc 1 1+r 16 (∥θt∥2 2+1) 1 1+r ��α(1+r)\uf8f9 \uf8fb ≤ C � nα � d log2β(n) �α(1+r) + 1 nαr � 1 + log � C1 � d log2β(n) � r 1+r eC2nd log2β(n) + Cd 3e (1+r)nc 1 1+r 16 ��α(1+r)\uf8f9 \uf8fb ≤ Cnα � d log2β(n) �α(1+r) , where we used in the previous lines simple algebra and log(a+b) ≤ log(2)+log(a)+log(b) when a ≥ 1 and b ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' References [1] Bakry, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' and Cattiaux, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' and Guillin, A.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' and Zou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' and Gu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' : Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Proceedings of the 32nd International Conference on Neural Information Processing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' NIPS’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' Curran Associates Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' (2018), 3126—3137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} +page_content=' 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfTQPr/content/2301.03077v1.pdf'} diff --git a/7dAzT4oBgHgl3EQf-f5K/content/tmp_files/2301.01934v1.pdf.txt b/7dAzT4oBgHgl3EQf-f5K/content/tmp_files/2301.01934v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4468bf46169dd6d8e862d7a4e73ffa46d78da790 --- /dev/null +++ b/7dAzT4oBgHgl3EQf-f5K/content/tmp_files/2301.01934v1.pdf.txt @@ -0,0 +1,920 @@ +HYPERBOLIC AND SATELLITE LORENZ LINKS +OBTAINED BY TWISTING +THIAGO DE PAIVA AND JESSICA S. PURCELL +Abstract. A Lorenz link is equivalent to a T-link, which is a positive +braid built by concatenating torus braids of increasing size. When each +torus braid except the largest is obtained by full twists, then the T-link +can be described as the Dehn filling of a parent link. In this paper, we +completely classify when such parent links are hyperbolic. This gives +a classification of the geometry of T-links obtained by full twists when +the amount of twisting is large, although the bound on the number +of required twists is not effective. We also present effective results on +hyperbolicity for two families of T-links obtained by twisting. Finally, +we identify families of satellite T-links obtained by half-twists. +1. Introduction +Lorenz links are the closed periodic orbits of a system of equations in- +vestigated by Lorenz in the 1960s [18]. They exhibit interesting dynamics +that has led to significant further investigation over the years, in the fields of +dynamics, geometry, and topology; see for example [9]. These links can be +described as links on an embedded branched surface in R3, called the Lorenz +template, due to work of Guckenheimer and Williams [12], and Tucker [22]. +Birman and Williams were the first to investigate Lorenz links through the +lens of knot theory, in the 1980s [2], and the first to show such links are +closed positive braids. Birman and Kofman [1] showed that Lorenz links are +equivalent to T-links, which are positive braids with a particular form; see +Section 2 below. Thus techniques from braid theory can be brought to bear +upon Lorenz links via T-links. +We are interested in the complement of these links, and in particular their +geometrisation. Thurston showed in the 1980s that all knots in the 3-sphere +are either torus knots, satellite, or hyperbolic [20], and we refer to this as +the knot’s geometric type. The geometric type of Lorenz links has been +considered since work of Birman and Williams in the 1980s [2]. They showed +that all torus knots are Lorenz knots, and satellites obtained as certain cables +of Lorenz knots are Lorenz knots. Hyperbolic geometry has been considered +by Gomes, Franco, and Silva [10, 11], who proved hyperbolicity of Lorenz +links satisfying certain conditions based on the Lorenz template. Satellite +links have received additional attention, by El Rifai [7], de Paiva [4], and +de Paiva and Purcell [6]. +1 +arXiv:2301.01934v1 [math.GT] 5 Jan 2023 + +2 +THIAGO DE PAIVA AND JESSICA S. PURCELL +In spite of this work, there remains no systematic way of determining +whether a Lorenz link is hyperbolic, toroidal, or satellite using its description +either on the Lorenz template, or as a closed braid in the form of a T-link. +These descriptions uniquely determine a link, and hence uniquely determine +its geometric type, so it is natural to ask for a simple description of geometric +type based on the description. We focus on T-links in this paper. +The paper [6] begins a classification of the geometry of T-links, by finding +examples that are satellite and also by identifying certain “parent links”, +which give classes of T-links under Dehn filling. While the work in that +paper finds examples of satellite and hyperbolic links, it is incomplete for +two reasons: +(1) First, the hyperbolic geometry of the parent links is used to determine +geometry of T-links for many examples. But the classification of the +hyperbolic geometry of the parent links is incomplete. +(2) Second, because the results are obtained by Dehn filling, they apply +only to links that admit full twists as T-link parameters, which are +not required for general T-links. +In this paper, we extend the classification of geometry of T-links as +follows. First, we complete the classification of item (1) above: Theorem 3.8 +completely classifies when parent links of fully twisted T-links are hyperbolic. +This can be seen as an extension of work of Lee [16, Proposition 5.7], who +proved a similar result for twisted torus knots. Positive twisted torus knots +are T-links with only one additional torus braid besides the largest. Lee’s +result essentially proves Theorem 3.8 in the case of only one additional link +component in the parent. Our result applies to any number of additional +link components in the parent. +Theorem 3.8 leads to new infinite families of hyperbolic T-links, determined +only by parameters in a braid describing the link. +Theorem 1.1. Fix relatively prime integers q < p, and let a1, . . . , an be +integers less than p and increasing in value. +There exists B ≫ 0 with +the following property. Consider the T-link obtained from the (p, q)-torus +knot by full twisting at least B times in regions with a1, a2, . . . , an strands, +respectively. This Lorenz link is hyperbolic if and only if either all ai < q, or +there is ai > q that is not a multiple of q. +The T-links of Theorem 1.1 must be obtained by full twisting, and we +currently do not have a concrete, universal bound on the number of full +twists that are required in general; this is the constant B in the above result. +In Section 4 we improve this: We present two theorems that guarantee +hyperbolicity of T-links with full twists, given only their parameters, where +the bounds on numbers of full twists required are explicit and relatively +simple. The results are Theorem 4.3 and Theorem 4.5. +It seems much more difficult to address item (2), especially in the hyperbolic +case. There are some partial results known, for example by de Paiva for +torus knots [5]. In this paper, we give more results in the satellite case. We + +LORENZ LINKS OBTAINED BY TWISTING +3 +extend the results on satellite knots, requiring full twists in [6], to families of +T-links with both full twists and half twists, which gives many more families +in a very natural way. +Theorem 1.2. For q < p integers, let K be a T-link obtained from the +(p, q)-torus link by half-twisting in circles encircling less than q strands, or +encircling multiples of q strands. Then S3 − K is satellite. +The precise statement is Theorem 5.4. +1.1. Acknowledgements. This work was partially supported by the Aus- +tralian Research Council, grant DP210103136. +2. Results on braids +This section reviews results on braids that will be used throughout. As +usual, let σi be the standard generator of the braid group, giving a positive +crossing between the i-th and (i + 1)-th strands. +For 1 < p, q, define the (p, q)-torus braid as: +(σ1 . . . σp−1)q +Note that within the braid group on p strands, its closure is the torus link +T(p, q). When p, q are coprime, this is a torus knot, but we will not always +restrict to coprime p and q unless specifically stated. +We will also consider such braids within larger braid groups. When r < p, +the (r, s) braid within the braid group on p strands is still defined to be +(σ1 . . . σr−1)s, but now note this has p − r strands with no crossings lying to +the right of the braid, viewing the braid arranged from top to bottom. +Let r1, . . . , rk and si, . . . , sk be integers such that 2 ≤ r1 < · · · < rk, and +si > 0 for all i. The T-link T((r1, s1), . . . , (rk, sk)) is defined to be the closure +of the braid +(σ1σ2 . . . σr1−1)s1(σ1σ2 . . . σr2−1)s2 . . . (σ1σ2 . . . σrk−1)sk. +Thus T((r1, s1), . . . , (rk, sk)) is obtained by concatenating the braids (ri, si) +within the braid group on rk strands, and then taking the closure. +Taking closures of torus braids and related braids allows additional sym- +metries and restrictions on the braid. For example, we will use the following +standard result on torus knots and links. +Lemma 2.1. Let 1 < p, q be integers. Then the torus link T(p, q) is equiva- +lent to the torus link T(q, p) via a homeomorphism of S3 fixing the Heegaard +torus containing T(p, q) and switching the two solid tori bounded by F. +The proof of Lemma 2.1 is well known, and appears in many knot theory +texts. We visualise the proof in Figure 1. +The next result generalises [6, Lemma 2.7]. There the result only holds +when each si is a multiple of ri. Here we extend more generally. + +4 +THIAGO DE PAIVA AND JESSICA S. PURCELL +p +q +p +q +Figure 1. The equivalence of T(p, q) and T(q, p) is given by +rotating 180◦ in the diagonal axis shown for the Heegaard +torus for S3. This exchanges the solid tori in the standard +genus-1 Heegaard splitting for S3. +Proposition 2.2. Let 0 < r1 < · · · < ri−1 < q < ri+1 < · · · < rn < p be +integers. Then, for k > 0, the T-link +K = T((r1, s1), . . . , (ri−1, si−1), (q, qk), (ri+1, si+1), . . . , (rn, sn), (p, q)) +is equivalent to the T-link +K′ = T((r1, s1), . . . , (ri−1, si−1), (ri+1, si+1), . . . , (rn, sn), (p + qk, q)). +Note that Proposition 2.2 allows us to assume there are no full twists on +q strands in a T-link of the form T(· · · , (p, q)). +Proof. The braid (q, qk) is obtained by performing k full twists on q strands. +We know that these full twists commute in the braid group. Thus in the +braid representing K, we may isotope (q, qk) to the top of the braid, leaving +the rest of the braid unchanged. +Now perform the isotopy of K of Lemma 2.1, switching p and q in the +(p, q)-torus link. The rotation in the diagonal shown in Figure 1 takes the +(vertical) braids (r1, s1) ∗ · · · ∗ (rn, sn) to inverted braids, forming a tangle in +the horizontal direction on a quadrilateral representing the projection torus. +(The form of this tangle is not important for the argument here, but more +details can be found in [6, Lemma 2.3].) The result is a link of the form +T(q, p) with a tangle along the horizontal p-strands. The first such tangle +is the braid (q, qk), which is unchanged by this isotopy because it is a full +twist (see, for example, Birman and Kofman [1, Corollary 3]). Then the link +diagram is formed by the braid (q, p) followed by (q, qk). These two braids +can be combined to form the braid (q, p + qk). Now apply the inverse of the +isotopy of Figure 1. This changes the link from T(q, p + qk) with tangles +along the p horizontal strands to a link of the form T(p + qk, p) with these +tangles returned to their form as braids (r1, s1) ∗ · · · ∗ (rn, sn). The result is +the link K′. +□ +Proposition 2.3. Let p, q, and r be positive integers with 0 < q ≤ r < p. +Consider the (p, q) torus link, which is the closure of the braid on p strands +given by (σ1 . . . σp−1)q. There is an ambient isotopy of S3 taking this to the + +LORENZ LINKS OBTAINED BY TWISTING +5 +Figure 2. Illustration of Proposition 2.3 in the case that +q = 2, r = 4, p = 7, for an arbitrary tangle shown as a +gray box. The left-most picture shows the original link. The +(r + 1)-st strand, shown in blue, can be pulled tight beneath +the diagram, resulting in the middle picture. The right-most +picture shows the result after isotoping strands (r + 1) to p. +closure of the braid on r strands given by +(σr−1 . . . σr−q+1)p−r(σ1 . . . σr−1)q. +Moreover, an ambient isotopy realising the equivalence fixes the portion of +the braid (σ1 . . . σp−1)q corresponding to the r left-most strands at the top the +braid. Thus, we may replace a neighbourhood of these strands above the braid +(σ1 . . . σp−1)q with any tangle τ on r strands, and we find that the resulting +link is ambient isotopic to the closure of the link obtained by concatenating the +braid on r strands (σr−1 . . . σr−q+1)p−r, with τ, and then with (σ1 . . . σr−1)q. +See Figure 2. +Proof. Because r ≥ q, the (r + 1)-st strand at the top of the braid only runs +under the q overcrossing strands in the braid corresponding to the (p, q) +torus link. It then runs around the braid closure back to the top, returning +to the r − q + 1 position. Together with a horizontal line from the r − q + 1 +position to the r + 1 position, this strand bounds a disc in S3, lying under +the plane of projection. Use this disc to push the strand in S3 to become a +horizontal strand lying below the plane of projection, running from the r + 1 +position, then behind q strands, to the r + 1 − q position. Adjust slightly, +pulling the right side up, so that the result is a closed braid; see Figure 2, +middle. Note that the resulting braid consists of only p − 1 strands. This +isotopy generalises the isotopy given by Lee in [17, Figure 6], and by de Paiva +in [5, Figure 1]. + +6 +THIAGO DE PAIVA AND JESSICA S. PURCELL +This move can be repeated for all the p − r strands to the right of the +(r+1)-st strand. When finished, we obtain a link on r strands as claimed. +□ +2.1. Braid index. Recall that the braid index of a knot K, which we will +denote β(K), is the minimal number of strands required to form a braid +with closure isotopic to K. We will repeatedly use the following result of +Franks and Williams [8] on braid index of the closure of a positive braid. +Theorem 2.4 (Corollary 2.4 of [8]). Let B be a positive braid on p strands +that contains a full twist +∆2 = (σ1 . . . σp−1)p. +Then B has braid index p. +□ +Lemma 2.5. Let p, q, d and r be positive integers such that q ≤ r < p and +d + q ≥ r. Let Br be a positive braid on r strands, and let Bp denote the +braid on p strands obtained by adding p − r trivial strands to the right of the +braid Br. Then the closure of the braid on p strands +Bp(σ1 . . . σr−1)d(σ1 . . . σp−1)q +has braid index equal to r. +Proof. By Proposition 2.3, the closure of the given braid on p strands is +equivalent to the closure of the braid on r strands +B′ = (σr−1 . . . σr−q+1)p−rBr(σ1 . . . σr−1)d(σ1 . . . σr−1)q. +Because this is a positive braid, and because d + q ≥ r, the braid B′ has at +least one positive full twist on r strands. Thus Theorem 2.4 implies that the +closure of B (and B′) has braid index equal to r. +□ +Corollary 2.6. Suppose 0 < r1 < · · · < rn < p are integers, s1, . . . , sn and +q are positive integers, and suppose q ≤ rn ≤ sn + q. Then the T-link +K = T((r1, s1), . . . , (rn, sn), (p, q)) +has braid index equal to rn. +Proof. Let Brn be the braid on rn strands obtained as the concatenation of +torus braids (r1, s1) . . . (rn−1, sn−1), where we view each (ri, si) as a braid +on rn strands by adding rn − ri trivial strands to the right of the braid +(ri, si) = (σ1 . . . σri−1)si. Then the given T-link is the closure of the braid +Brn(σ1 . . . σrn−1)sn(σ1 . . . σp−1)q. +Since q ≤ rn ≤ sn + q, the result follows from Lemma 2.5. +□ +The next definition is from Williams [23]. +Definition 2.7. A generalized q-cabling of a link L is a link L′ contained in +the interior of a tubular neighbourhood L × D2 of L such that +(1) each fiber D2 intersects L′ transversely in q points; and +(2) all strands of L′ are oriented in the same direction as L itself. + +LORENZ LINKS OBTAINED BY TWISTING +7 +Williams showed the following result on generalised q-cablings for knotted +L in [23]. +Theorem 2.8 (Theorem 1 of Williams [23]). The braid index is multiplicative +under generalized cabling. That is, if L is a link with each component a +non-trivial knot and L′ is a generalized q-cabling of L then β(L′) = qβ(L), +where β(∗) is the braid index of ∗. +□ +This result was extended to unknotted L in the case of positive braids by +de Paiva in [3]. The following result is from that paper. +Lemma 2.9 (Lemma 2.3 of [3]). Let L′ be a generalized q-cabling of the +unknot L, with L given by a positive braid on n strands, where n > 1. Also, +assume the knot inside L is given by a positive braid. Then L′ has braid +index equal to q. +□ +3. Parents of T-links +In this section, we build the “parent links” mentioned in the introduction. +Dehn filling on such links produces T-links with full twists. By classifying +when such links are hyperbolic, and applying Thurston’s hyperbolic Dehn +filling theorem, we show that, in an appropriate sense, most T-links with +only full twists are hyperbolic. This is an extension of work by de Paiva and +Purcell [6]. There, the same links were constructed, and some conditions +were given to guarantee hyperbolicity. Here, we strengthen the result by +completely characterising when such links are hyperbolic. +Definition 3.1. Let p, q be relatively prime integers such that 1 < q < p. +Consider the (p, q)-torus braid on p strands, and its closure, the torus link +T(p, q). Let F denote the Heegaard torus on which T(p, q) lies. Let a be +an integer with 0 < a < p. Denote by Ja an unknot lying horizontally with +respect to the (p, q)-torus braid, positioned just above the crossings of the +braid, bounding a disc such that the interior of that disc meets F transversely +in a single arc intersecting the a leftmost strands of the braid. +More generally, given a1, . . . , an satisfying 1 < a1 < · · · < an < p, take +disjoint unknots Ja1, . . . , Jan as above, positioned so that the i-th is pushed +vertically above the (i + 1)-th with respect to the braid, so that all are +disjoint. Figure 3 shows an example. +Proposition 3.2. Let p, q be relatively prime integers with 1 < q < p. Let +an, . . . , a1 be integers such that 1 < a1 < · · · < an < p, with n > 1. Also, +assume that there is ai > q which is not a multiple of q. Then the link +K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 is atoroidal. +In [6], it is shown that K is hyperbolic if all the ai > q are not multiples +of q. Here, we show only one needs not be a multiple of q for hyperbolicity. +Proof. Suppose S3 − N(K) admits an essential torus T. Then T bounds a +solid torus V that must contain at least one component of K. + +8 +THIAGO DE PAIVA AND JESSICA S. PURCELL +Figure 3. Shows T(7, 2) augmented at the top right by J2, +J3, and J4. +First we show that we may choose V to contain T(p, q). For suppose V is +disjoint from T(p, q). Then it must contain at least one Jaj. The component +Jaj must have positive wrapping number in V , for otherwise T(p, q) and Jaj +would have zero linking number, which is a contradiction. Because there +is no essential torus in the exterior of the unknot in S3, it follows in this +case that T is unknotted in S3. Therefore, T bounds a second solid torus V ′ +containing T(p, q). Thus in all cases we may assume T bounds a solid torus +containing T(p, q). +As an ≥ q, by Proposition 2.3, the torus knot T(p, q) is isotopic to a +closed braid with an strands so that under the isotopy, the largest unknot +Jan becomes the braid axis. Because the isotopy moves only the right-most +p − an strands, all unknots Ja1, . . . , Jan are untouched by the isotopy. +The torus T is then contained in the solid torus S3 − N(Jan), and bounds +a solid torus V containing T(p, q). It follows that Jan is disjoint from V . +The torus T must intersect the disc Dan bounded by Jan in a series of +circles, with each circle bounding a meridian of V . Each meridian of V +can be isotoped to meet the same number of strands of T(p, q), as follows. +The boundary of a meridian defines an unknot in S3, and all such unknots +are isotopic in S3 − N(K), where the isotopy is obtained by pushing the +boundary of the meridian disc along the torus T. Because T(p, q) forms a +braid, it meets these discs monotonically. Let b denote the number of times +that a meridian of V intersects the strands of T(p, q) on the disc Dan. Note +b > 1, or else T would be boundary parallel. +Note also that V winds some number of times around the solid torus +S3 − N(Jan), and note that each meridian of this solid torus meets exactly +an strands of T(p, q), since this is the number of strands in the closed +braid isotopic to T(p, q) obtained from Proposition 2.3. Since V meets each + +LORENZ LINKS OBTAINED BY TWISTING +9 +meridian of S3 − N(Jan) a total of an times, and each meridian of V meets +T(p, q) a total of b times, b must divide an. +It follows that T(p, q) is a generalised b-cabling of L, where L is the core +of the solid torus V . +Observe that T is embedded in exterior of the torus knot S3 − N(T(p, q)). +By work of Tsau [21], there are no essential tori in a torus knot exterior. +Because b > 1, it follows that T must be compressible to its outside. That +is, V is unknotted in S3. Thus, Lemma 2.9 implies that T(p, q) has braid +index equal to b. +On the other hand, the torus knot T(p, q) with 1 < q < p has braid index +equal to q; for example this follows from Franks and Williams’ Theorem 2.4. +Then, b = q, and b divides an. Hence, q divides an. +By hypothesis, there is ai ∈ {a1, . . . , an} which is greater than q and not +a multiple of q. Since ai > q, it must be the case that Jai is disjoint from the +solid torus V . Since T(p, q) intersects the disc Dai bounded by Jai a total +of ai times, and T(p, q) is a generalised q-cabling of L, it must be the case +that L intersects the disc ai/q times. However, q does not divide ai. This is +a contradiction. +□ +Lemma 3.3. Let p, q be relatively prime integers with 1 < q < p. Let +an, . . . , a1 be integers such that 1 < a1 < · · · < an < p with n > 1. Then +the link K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 has no annuli with boundaries in two +different components. +Proof. Suppose that S3 − N(K) has an annulus A with boundaries ∂1A +and ∂2A that lie in two different components, C1 and C2, respectively, of +∂(S3 − N(K)). +Case 1: Consider first that C1 and C2 are Jaj and Jak, respectively, for +some j ̸= k ∈ {1, . . . , n}. +Note ∂1A and ∂2A are isotopic in S3−N(K). The linking number between +Cj and ∂jA is zero if and only if ∂jA is the longitude of Cj, in which case +Cj and ∂jA are isotopic, for j = 1, 2. +Suppose ∂1A is the longitude of C1, but ∂2A is not the longitude of C2. +Since ∂1A and ∂2A are isotopic, C1 and C2 would have nonzero linking +number in this case, but this is not possible. Similarly ∂2A cannot be the +longitude of C2 if ∂1A is not the longitude of C1. +Thus either ∂1A is the longitude of C1 and ∂2A is the longitude of C2, or +neither is a longitude. If both are longitudes, then C1 and C2 are isotopic, +which is not possible. Thus neither are longitudes. +Then the linking number between C2 and ∂2A is positive. However, C1 +and C2 have zero linking number, so ∂1A and C2 must have zero linking +number. But ∂2A is isotopic to ∂1A, and so ∂1A and C2 have nonzero linking +number equal to the linking number of C2 and ∂2A. This is a contradiction. +Case 2: Now suppose that C1 and C2 are Jaj and T(p, q), respectively, +for some j ∈ {1, . . . , n}. Again ∂1A and ∂2A are isotopic. + +10 +THIAGO DE PAIVA AND JESSICA S. PURCELL +Suppose first that ∂2A wraps at least one time along the longitude of +C2 = T(p, q). +Then ∂2A has positive linking number with each of the +components Jak, because T(p, q) has positive linking number with each. But +the linking number between ∂2A and Jak for Jak ̸= C1 is zero, because C1 +has linking number zero with each such component, and ∂2A has the same +linking number with C1 as ∂1A. This is a contradiction. +Thus ∂2A is a meridian of C2 = T(p, q). So ∂2A and T(p, q) have linking +number equal to one. The curve ∂1A is some torus knot T(a, b) on ∂N(C1). +If a is equal to zero, then ∂1A is a meridian of C1. Because a meridian +of C1 has linking number zero with C2 = T(p, q), it follows that ∂1A and +T(p, q) have linking number equal to zero. However, this is not possible as +∂1A and ∂2A are isotopic. So, a ̸= 0. The linking number between ∂1A and +C2 = T(p, q) is equal to a · aj, where C1 = Jaj. Because ∂2A and T(p, q) +have linking number 1, and ∂1A and T(p, q) have linking number identical +to ∂2A and T(p, q), it follows that a · aj = 1. This is impossible since aj > 1. +Therefore, no such annulus exists. +□ +Lemma 3.4. Let K be as in Proposition 3.2. Then K has no essential +annuli with both boundary components in ∂N(T(p, q)). +Proof. Suppose that S3 −N(K) has an essential annulus A with both bound- +ary components in ∂N(T(p, q)). +The exterior of a torus knot has just one essential annulus by work of +Tsau [21]. By work of Lee, [16, Lemma 5.1] that essential annulus would be +punctured by Jai, where ai > q is not a multiple of q. Thus A is not essential +in S3 − N(T(p, q)). Thus A is compressible, boundary compressible, or +boundary parallel in S3 − N(T(p, q)). Observe that a boundary compressible +annulus is in fact boundary parallel, using the fact that S3 − N(T(p, q)) is +irreducible and boundary irreducible. +Consider first that A is boundary parallel to an annulus B in ∂N(T(p, q)). +Then A ∪ B bounds a solid torus V in S3 − N(T(p, q)). Since A is not +boundary parallel in S3 − N(K), at least one Jaj must be inside V . In +addition, Jaj has wrapping number greater than zero in V , or else T(p, q) +and Jaj would have linking number equal to zero, which is a contradiction. +But Jaj is an unknot, whose complement admits no essential tori (e.g. [13, +page 15]). Thus V is also unknotted in S3. This implies that B is a meridional +annulus of ∂N(T(p, q)). If ∂V is boundary parallel to Jaj, then Jaj is the +core of ∂V . Hence, the linking number between T(p, q) and Jaj would be +one, which is not possible. Thus, as ∂V is not boundary parallel to Jaj, ∂V +is an essential torus for S3 − N(K). This contradicts Proposition 3.2. +Assume now that A is compressible in S3 − N(T(p, q)). Then there is a +compression disk D for A in S3 − N(T(p, q)). Surgering A along D yields +two discs, D1 and D2, such that ∂A = ∂D1 ∪ ∂D2. Since S3 − N(T(p, q)) +is boundary irreducible, ∂Di bounds a disk Ei on ∂N(T(p, q)). Thus, by +pushing Ei slightly off of ∂N(T(p, q)) in S3 −N(K), we obtain a compressing + +LORENZ LINKS OBTAINED BY TWISTING +11 +disc for A in S3−N(K), which contradicts our assumption that A is essential. +Therefore, A is not compressible. +Thus A cannot have both boundary components on ∂N(T(p, q)). +□ +Lemma 3.5. Let K be as in Proposition 3.2. Then K has no essential +annulus with both boundary components on one ∂N(Jaj). +Proof. Suppose that S3 −N(K) has an essential annulus A with both bound- +ary components on ∂N(Jaj). Since S3 −N(Jaj) is a solid torus, and the solid +torus admits no essential annuli, A is not essential in S3 − N(Jaj). Thus A +is either compressible or boundary parallel in S3 − N(Jaj). +Case A: Suppose A is boundary parallel, parallel to an annulus B in +∂N(Jaj). Then A ∪ B bounds a solid torus V in S3 − N(Jaj). Since A is not +boundary parallel in S3 − N(K), at least one component C of K must be +inside V . +Case A1: +Consider first that C = T(p, q). Then T(p, q) has wrapping +number greater than zero in V , for otherwise Jaj and T(p, q) would have zero +linking number, a contradiction. Note this implies that ∂V is incompressible +to its inside. +Suppose that some circle Jak with j ̸= k lies in S3 − V . Then we may +isotope Jak to lie outside of W = N(Jaj) ∪ V , which is a regular solid torus +neighbourhood of the unknot Jaj. Denote by ω the winding number of Jak in +S3 − W. If ω = 0, then the linking number between Jak and T(p, q) is zero. +Thus, ω ̸= 0. But then this implies that the linking number between Jaj and +Jak is nonzero, a contradiction. Thus all circles Ja1, . . . , Jai−1, Jai+1, . . . , Jan +are inside V in this case. Because at least two components of K lie inside V , +∂V is not boundary parallel to the inside. +The core of V forms a torus knot T(a, b) on N(Jaj). Note b > 0 or else +T(p, q) runs around a longitude of N(Jaj) and hence has linking number zero +with Jaj, a contradiction. +Suppose b = ±1, so the core of V has the form of the trivial knot T(a, ±1). +Then there exists a disc in S3 − N(K) that is a longitude for ∂N(Jaj) whose +boundary can be divided into two arcs, one of which meets A ⊂ ∂V in +a nontrivial arc, and the other meets ∂N(Jaj). See Figure 4. This is an +essential boundary compression disc for A, contradicting the fact that A is +essential. +Since |b| > 1, ∂V = ∂N(T(a, b)) is incompressible and not boundary +parallel to the outside, i.e. in the solid torus S3 − Jaj. +This implies that in all cases ∂V is essential in S3 − N(K) contradicting +Proposition 3.2. +Case A2: +The torus knot T(p, q) cannot lie inside V by the previous +case. So some C = Jak with j ̸= k lies inside V . The wrapping number of +Jak inside V must be different from zero as Jak and T(p, q) have positive +linking number. Since Jak and Jaj have zero linking number, V must be a +longitude of ∂N(Jaj). If Jak is the core of V , then Jaj and Jak are isotopic + +12 +THIAGO DE PAIVA AND JESSICA S. PURCELL +Figure 4. A disc with boundary an arc on each of A ⊂ ∂V +and ∂N(Jaj). +in S3 − N(T(p, q)), a contradiction. So Jak is not the core of V . But then +∂V is incompressible and not boundary parallel to the inside in S3 − K, +and incompressible and not boundary parallel to the outside in S3 − K, +contradicting Proposition 3.2. +Case B: +Suppose A is compressible in S3 − N(Jaj). Then there is a +compression disk D for A in S3 − N(Jaj). Surgering A along D yields two +discs, D1 and D2, such that ∂A = ∂D1 ∪ ∂D2. If one of ∂D1 or ∂D2 bounds +a disk E on ∂N(Jaj), then by considering a disc with boundary in A close +to E, we see that A is also compressible in S3 − N(K), a contradiction. So +suppose that neither ∂D1 nor ∂D2 bounds a disk on ∂N(Jaj). Then D1 and +D2 are discs in the solid torus S3 − N(Jaj) with nontrivial boundary on +∂N(Jaj) and hence both are meridians of S3 − N(Jaj), i.e. with ∂D1 and +∂D2 forming longitudes of ∂N(Jaj). Undoing the surgery along D, it follows +that A is boundary parallel in S3 − N(Jaj). Thus we have a contradiction +to Case A. +Therefore, S3 − N(K) has no essential annulus with both boundary com- +ponents in one ∂N(Jaj). +□ +Proposition 3.6. The link K as in Proposition 3.2 has no essential annuli. +Proof. By Lemma 3.3, any essential annulus has both boundary components +on the same component of K. By Lemma 3.4, the two boundary components +cannot lie on ∂N(T(p, q)). By Lemma 3.5 the two boundary components +cannot lie on one of the ∂N(Jaj). Thus no such annulus exists. +□ +Theorem 3.7. Let p, q be relatively prime integers with 1 < q < p. Let +an, . . . , a1 be integers such that 1 < a1 < · · · < an < p with n > 1. Also, +assume that there is ai > q which is not a multiple of q. Then, the link +K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 is hyperbolic. +Proof. By de Paiva and Purcell [6, Lemma 5.1], the link exterior is irre- +ducible and boundary irreducible. By Proposition 3.2, it is atoroidal. By +Proposition 3.6, it is anannular. Therefore it is hyperbolic by Thurston’s +hyperbolisation theorem for Haken manifolds [20]. +□ +Combining Theorem 3.7 and de Paiva and Purcell [6, Theorem 5.6], we +completely classify the geometric types of the links T(p, q) ∪ Ja1 ∪ . . . Jan. + +LORENZ LINKS OBTAINED BY TWISTING +13 +Theorem 3.8. Let p, q be relatively prime integers with 1 < q < p. Let +a1, . . . , an be integers such that 1 < a1 < · · · < an < p. Then the link +K = T(p, q) ∪ Ja1 ∪ . . . Jan is hyperbolic if and only if either all ai < q, or +there is ai > q which is not a multiple of q. +Proof. When n = 1, the link K = T(p, q) ∪ Ja1 is the Dehn-filling parent of +a twisted torus knot; this has been treated by Lee [15, 16]. If n = 1 and +a1 = q, then [15, Theorem 1] implies that infinitely many Dehn surgeries +along Ja1 yield non-hyperbolic knots. Therefore, Thurston’s hyperbolic Dehn +filling theorem [19] implies K is not hyperbolic. In fact, the proof of [15, +Theorem 1] implies K is annular. If n = 1 and a1 is not a multiple of q, then +K is hyperbolic by [16, Proposition 5.7]. +In the case n > 1, if there is ai > q that is not a multiple of q, then K is +hyperbolic by Theorem 3.7. +If n > 1 and all ai are less than q, then no ai is a multiple of q, and K is +hyperbolic by [6, Theorem 5.6]. +Finally, if n > 1, there is some ai > q and all ai > q are multiples of q, +then K is satellite by [6, Theorem 5.6]. +□ +Corollary 3.9. Let p, q be relatively prime integers with 1 < q < p, and let +a1, . . . , an and s1, . . . , sn be integers such that 1 < a1 < · · · < an < p and +si > 0 for all i. Then, there exists B ≫ 0 such that if each si > B, the +T-link +T((a1, a1s1), . . . , (an, ansn), (p, q)) +is hyperbolic if and only if either all ai < q, or there is ai > q which is not a +multiple of q. +Proof. By Theorem 3.8, the link K = T(p, q) ∪ Ja1 ∪ · · · ∪ Jan is hyperbolic +if and only if the ai satisfy the hypotheses of the corollary. Obtain the +given T-link by Dehn filling the link components Ja1, . . . , Jan along slopes +1/s1, . . . , 1/sn, respectively. When the link K is hyperbolic, the Dehn filling +remains hyperbolic by Thurston’s hyperbolic Dehn filling theorem [19] pro- +vided the si are sufficiently large. On the other hand, Dehn filling a satellite +K yields a satellite T-link, by de Paiva and Purcell [6, Theorem 5.6], and in +the case n = 1 and a1 = q, Dehn filling yields an annular link by Lee [15]. +□ +Note that Theorem 1.1 in the introduction follows immediately from +Corollary 3.9. +4. Hyperbolicity with effective full twist bounds +While Corollary Corollary 3.9 is quite broad, unfortunately the constant B +in that theorem is not explicit, and so it may be difficult to apply in practice. +In this section we find explicit parameters which produce hyperbolic T-knot +obtained by full twists. Because we are considering full twists exclusively in +this section, Proposition 2.2 implies that we may assume that none of the ai +are equal to q. + +14 +THIAGO DE PAIVA AND JESSICA S. PURCELL +Proposition 4.1. Let a1, . . . , an, s1, . . . , sn, and p, q, k be integers satisfying +the following hypotheses: +• p and q are relatively prime, +• 1 < a1 < · · · < an, and 0 < q < an < p, +• each si > 0, and sn ≥ 2, +• p and an are relatively prime, +• k ≥ 2. +Then the T-knot K = T((a1, a1s1), . . . , (an, ansn), (p, q + kp)) is atoroidal. +Proof. Suppose that the exterior of K in S3 admits an essential torus T. By +work of Ito [14, Theorem 1.2(3)], because K is the closure of a braid with at +least two positive full twists on p strands, the torus T does not intersect the +braid axis C. Moreover, the knot inside T is given by a braid. Thus there +exists some integer d > 1 such that K is a generalized d-cabling of a knot L, +where L is the core of the solid torus bounded by T. As a consequence, d +must divide p. +After (−1/k)-Dehn surgery along the braid axis C, the knot K becomes +the T-knot +K′ = T((a1, a1s1), . . . , (an, ansn), (p, q)) +and the torus T becomes a new torus T ′. This will bound a solid torus V ′ in +S3, with core L′. Because q < an < ansn + q, the knot K′ has braid index +equal to an by Corollary 2.6. +If L′ is trivial, then an is equal to d by Lemma 2.9. However, this is not +possible since gcd(p, an) = 1. +So L′ is knotted. Then by Theorem 2.8, an is equal to dβ(L′), where +β(L′) is the braid index of L′. But then d divides p and d divides an, again +contradicting gcd(p, an) = 1. +Therefore, the exterior of K admits no essential torus. +□ +We will combine the previous result with the following from [5], which +gives information on torus knots. +Theorem 4.2 (Theorem 1.2 of de Paiva [5]). Let p, q, a1, . . . , an, s1, . . . , sn +be positive integers such that 1 < q < p and 1 < a1 < · · · < an < p with +ai ̸= q. If gcd(p, q) = 1 and in addition one of the following hold: +• q < an, or +• q > an and p is not of the form bq + 1 for some b > 0, or +• q > an and p = bq + 1 for some b > 0, but s1 > 1, or +• q > an, p = bq + 1 for some b > 0, and s1 = 1, but a2 ̸= a1 + 1, +then T((a1, s1a1), (a2, s2a2), . . . , (an, snan), (p, q)) is not a torus knot. +□ +Theorem 4.3. Let a1, . . . , an, s1, . . . , sn, and p, q, k be integers satisfying +the following hypotheses: +• p and q are relatively prime, +• 1 < a1 < · · · < an and 1 < q < an < p, +• each si > 0 and sn ≥ 2, + +LORENZ LINKS OBTAINED BY TWISTING +15 +• p and an are relatively prime, +• k ≥ 2, n ≥ 2. +Then if in addition, one of the following hold: +• q ̸= 1, +• or s1 > 1, +• or a2 ̸= a1 + 1, +then the T-link K = T((a1, a1s1), . . . , (an, ansn), (p, q + kp)) is hyperbolic. +Proof. Because gcd(p, q) = 1, K is a knot. By Proposition 4.1, K is atoroidal, +so not a satellite knot. +The T-knot K is equivalent to the T-knot +T((a1, s1), . . . , (an, ansn), (q + kp, p)) +by [1, Corollary 3]. The integer q + kp does not have the form bp + 1 if and +only if q is different from 1. So under these conditions, K is not a torus knot +by Theorem 4.2. +Therefore, by Thurston’s hyperbolisation Theorem for knots [20], K is +hyperbolic. +□ +Proposition 4.4. Let a1, . . . , an, s1, . . . , sn, and p, q, k be integers satisfying +the following hypotheses: +• p and q are relatively prime, +• 1 < a1 < · · · < an, and 1 < q < an < p, +• each si > 0 and both sn and sn−1 are at least 2. +Suppose also that one of the following holds: +• q < an−1 and an and an−1 are relatively prime, or +• q > an−1 and an and q are relatively prime. +Then the knot K = T((a1, a1s1), . . . , (an, ansn), (p, q)) is atoroidal. +Proof. Suppose the exterior of K in S3 admits an essential torus T. +By Proposition 2.3, K is equivalent to the knot given by the closure of +the braid +B = (σan−1 . . . σan−q+1)p−an · τ · (σ1 . . . σan−1)snan+q, +where τ is the concatination of braids (a1, a1s1) . . . (an−1, an−1sn−1). +Since B has at least two positive full twists on an strands, it follows from +[14, Theorem 1.2(3)] that T does not intersect the braid axis C of B. Thus +there is an integer d > 0 such that K is a generalized d-cabling of the core L +of the solid torus bounded by T. Hence d divides an. +Perform (−1/sn)-Dehn surgery along the braid axis C to obtain the braid +B′ = (σan−1 . . . σan−q+1)p−an · τ · (σ1 . . . σan−1)q. +Its closure gives K′ = T((a1, a1s1), . . . , (an−1, an−1sn−1), (an, q)). The torus +T becomes a new essential torus T ′ in the exterior of K′. +The torus T ′ bounds a solid torus with core L′, which is either trivial or +knotted. + +16 +THIAGO DE PAIVA AND JESSICA S. PURCELL +Suppose first the case that q < an−1. Then q ≤ an−1 ≤ an−1sn−1 + q, so +Corollary 2.6 implies that K′ has braid index equal to an−1. If L′ is the +trivial knot, then an−1 is equal to d by Lemma 2.9. This implies that d +divides both an and an−1, contradicting the assumption in this case that +these are relatively prime. Similarly, if L′ is knotted, then Theorem 2.8 +implies that an−1 is a multiple of d, with the same contradiction. +Now suppose q > an−1. Then K′ has braid index q by Franks and Williams, +Theorem 2.4. If L′ is trivial, then as above, Lemma 2.9 implies q equals d, and +therefore d divides both an and q, contradicting the hypothesis. Similarly, if +L′ is knotted, Theorem 2.8 implies q is a multiple of d, and again d divides +both an and q, which is a contradiction. +□ +Theorem 4.5. Let a1, . . . , an, s1, . . . , sn, and p, q, k be integers satisfying +the following hypotheses: +• p and q are relatively prime, +• 1 < a1 < · · · < an and 1 < q < an < p, +• each si > 0 and both sn and sn−1 are at least 2. +Suppose also that one of the following holds: +• q < an−1 and an and an−1 are relatively prime, or +• q > an−1 and an and q are relatively prime. +Then K = T((a1, a1s1), . . . , (an, ansn), (p, q)) is hyperbolic. +Proof. By Proposition 4.4, the knot K is atoroidal. By Theorem 4.2, using +the fact that q < an, K is anannular. +Therefore, K is hyperbolic. +□ +5. Satellite T-links obtained by Half-twists +In this section we switch from discussions of hyperbolic links to satellite +links. We find families of Lorenz links that are satellites using half-twists, +rather than full-twists. Previous work by de Paiva and Purcell found con- +ditions that ensure a T-link is satellite, namely [6, Theorem 4.3]. Lee has +similar results for the case of twisted torus knots [15, Theorem 1]. We extend +these results. +Definition 5.1. Suppose B is a diagram given as a closed braid; we consider +the braid to have strands running vertically on the plane of projection. A +positive half-twist on the strands from a to b is the braid +∆a,b = (σa . . . σb)(σa . . . σb−1) . . . (σa). +This can be thought of as cutting the braid between the a-th and b-th strands, +rotating in the anticlockwise direction by 180◦, and gluing back. In braid +theory literature, the positive half-twist on all strands is well known as the +Garside fundamental braid. A negative half-twist is defined similarly, only +the rotation is in the clockwise direction. See Figure 5. + +LORENZ LINKS OBTAINED BY TWISTING +17 +Figure 5. An example of half twists when r = 2, q = 3, t = +1. Left: A positive half-twist ∆1,rq, a negative half-twist +∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,tq. The green +circle indicates the braid axis. Middle: The negative half- +twist cancels crossings above. Right: The additional positive +half-twist gives the braid (rq, tq). +Lemma 5.2. Let r, q, s be positive integers, and suppose s is not a multiple +of r. The (rq, sq)-torus braid is obtained by the following procedure. Start +with the trivial braid on rq strands; let J1,rq be an unknot encircling all rq +strands. Let t be an integer such that 0 < t < r and s = t + kr for some +integer k. Insert a positive half-twist ∆1,rq, followed by a negative half-twist +∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,rq. Finally, perform 1/k-Dehn +filling on J1,rq. The result is the (rq, sq)-torus braid. +Proof. The process is illustrated in Figure 5. The positive half-twist ∆1,rq +yields a braid +(σ1σ2 . . . σrq−1)(σ1 . . . σrq−2) . . . (σ1), +encircled by J1,rq. Perform the negative half-twist ∆1,(r−t)q. This concate- +nates the previous braid with +(σ−1 +(r−t)q−1 . . . σ−1 +2 σ−1 +1 )(σ−1 +(r−t)q−1 . . . σ−1 +2 ) . . . (σ−1 +(r−t)q−1). +This braid cancels with the positive half-twist along the first (r − t)q strands, +as shown in Figure 5, middle. Finally, the positive half-twist ∆(r−t)q+1,rq +concatenates a positive half-twist along the last tq strands, giving the braid +(σ1 . . . σrq−1)tq = (rq, tq), +still augmented by the unlink J1,rq. +To obtain the braid (rq, sq), perform 1/k Dehn filling on J1,rq, removing +that link component and inserting an additional krq overstrands into the +braid, for a total of tq+krq = sq overstrands, giving the desired (rq, sq)-torus +braid. +□ +Lemma 5.3. Let r, q, s be positive integers, with s not a multiple of r. +Consider the torus braid (rq, sq). At the top of the braid, consider r disjoint +discs arranged horizontally, each encircling q strands of the braid, and similar +discs at the bottom of the braid. The boundary of each disc at the top connects + +18 +THIAGO DE PAIVA AND JESSICA S. PURCELL +via a cylinder, embedded in the complement of the braid and enclosing q +strands, to the boundary of a disc at the bottom of the braid. +Moreover, the solid cylinders enclosed by these cylinders, containing q +strands each, forms the (r, s)-torus braid. +Proof. Let t be an integer such that 0 < t < r and s = t+kr for some integer +k. By Lemma 5.2, the (rq, sq) torus braid is formed from k full twists on +rq strands, followed by a positive half-twist ∆1,rq, then a negative half-twist +−∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,rq. Each half-twist is on a +multiple of q strands. +Observe that the cylinders described above can be arranged to completely +contain any half-twist on q strands. For a half-twist on a multiple of q strands, +say xq strands, x disjoint cylinders enter the top of the half-twist, and then +are half-twisted themselves, remaining disjoint, to exit the bottom of the +half-twist. Thus the cylinders remain embedded as claimed when passing +through half-twists. Finally, each full twist also preserves the cylinders, +sending each through a full twist. +To see that the braid formed by the solid cylinders is as claimed, observe +that the cylinders form k full twists, followed by one positive half-twist on all +strands. The (r−t) left-most cylinders then pass through a negative half-twist, +and the remaining t right-most cylinders pass through a positive half-twist. +As in Lemma 5.2, this creates braid on r strands, with rk overstrands at +the top coming from the full twists, followed by t overstrands coming from +the concatenation of half-twists. Thus this is an (r, rk + t) = (r, s)-torus +braid. +□ +Theorem 5.4. Let p, q be integers such that 1 < q < p, and let (a1, b1), . . . , +(an, bn) be pairs of integers such that 1 < a1 < · · · < an ≤ q and bi > 0 +for i = 1, . . . , n. Finally let r1, . . . , rm and s1, . . . , sm be integers such that +q < r1q < · · · < rmq < p, and si > 0 for i = 1, . . . , m. Then the T-link +K = T((a1, b1), . . . , (an, bn), (r1q, s1q), . . . , (rmq, smq), (p, q)) +is satellite with companion the T-link T((r1, s1), . . . , (rm, sm+1)) and pattern +given by the closure of the braid +(a1, b1) . . . (an, bn)(σq−1 . . . σ1)p−rm +� +q, q +� m +� +i=1 +risi +� ++ qrm +� +Proof. As before, we think of the T-link as the closure of a braid on p +strands arranged vertically, the concatenation of braids (a1, b1), . . . , (an, bn), +(r1q, s1q), . . . , (rmq, smq), (p, q) in that order. +First apply Proposition 2.3 to change the closed braid of the T-link to a +closed braid B′ on rmq strands. This isotopy fixes all of the rmq strands at +the top left of the original braid; thus it does not affect any of the braids +(aj, bj) or (riq, siq), for any i, j. In other words, B′ is the braid given by +concatenating (σrmq−1 . . . σrmq−q+1)p−rm with braids (a1, b1), . . . , (an, bn), +(r1q, s1q), . . . , (rmq, smq), and finally the braid (σ1 . . . σrmq−1)q. + +LORENZ LINKS OBTAINED BY TWISTING +19 +By Lemma 5.3, there are rm disjoint embedded cylinders in the complement +of the portion of the braid starting just above the braid (a1, b1), and ending +just below the braid (σ1 . . . σrmq−1)q at the bottom. These cylinders each +enclose q strands. They extend around the braid closure to give rm disjoint +embedded cylinders running to the top of the braid, each enclosing q strands, +arranged right to left across the top of the braid. +The only portion of the braid that is not already enclosed in one of these +cylinders is the braid (σrmq−1 . . . σrmq−q+1)p−rm lying at the top. This is +a braid whose left-most strand is the (rmq − q + 1)-th strand, and whose +right-most strand is the rmq-th strand. In other words, this is a braid on the +right-most q strands of the rmq-strand braid. Thus the right-most cylinder, +enclosing q strands, can be extended to enclose this braid. Then all cylinders +connect to form a closed embedded torus Σ, encircling q strands of the braid. +The torus Σ bounds a solid torus containing q strands, which we check +has the claimed form of the companion in the theorem statement. This solid +torus forms a braid on rm strands. By Lemma 5.3, each (riq, siq)-torus braid +from the original T-link causes the solid cylinder to form a braid (ri, si). +The braids (aj, bj) and (σrmq−1 . . . σrmq−q+1)p−rm lie completely inside the +solid cylinder, so they do not affect the braid it forms. Finally, consider +the braid (σ1 . . . σrmq−1)q at the bottom of B′. This is formed by q strands +running over all the rmq strands. When the collection of solid cylinders +encounter this braid, the left-most solid cylinder encircles exactly these q +strands, and runs over all others to lie on the right-most side. Thus it +forms a (rm, 1)-torus braid. So the solid torus enclosing q strands has the +form of the closure of a braid (r1, s1) . . . (rm, sm), (rm, 1). This is the T-link +T((r1, s1), . . . (rm, sm + 1)) as claimed. Since it forms a nontrivial knot in +S3, Σ is an incompressible torus. +Finally we check the form of the pattern. Starting at the top-left of +the braid B′, the torus Σ encloses the braid (a1, b1) . . . (an, bn), which will +form part of the braid describing the pattern. As Σ follows the companion +into each of the braids (ri, si), all the q strands will make one full twist +each time Σ runs completely through an overstrand. There are si of these, +i = 1, . . . m − 1, plus sm + 1 for the (rm, sm + 1) braid that the compan- +ion runs over. These will occur in some order, with Σ also enclosing the +braid (σq−1 . . . σ1)p−rm, coming from the top right of B′, at some point. +Because full twists commute in the braid group, we may write the braid +as (a1, b1) . . . (an, bn)(σq−1 . . . σ1)p−rm · τ where τ is an appropriate number +of full twists. To obtain the appropriate number of full twists, we need +to consider the homological longitude of the companion. The pattern is +the braid obtained when we apply a homeomorphism taking the solid torus +bounded by the companion to an unknotted solid torus, with homological +longitude mapped to a standard longitude of the unknot. The effect is to +add �m−1 +i=1 (ri − 1)si + (rm − 1)(sm + 1) additional full twists, for a total of + +20 +THIAGO DE PAIVA AND JESSICA S. PURCELL +�m +i=1 risi + rm full twists. Thus the pattern can be written as the braid +(a1, b1) . . . (an, bn)(σq−1 . . . σ1)p−rm(q, q( +� +risi) + qrm) +□ +References +1. Joan Birman and Ilya Kofman, A new twist on Lorenz links, J. Topol. 2 (2009), no. 2, +227–248. MR 2529294 [1, 4, 15] +2. Joan S. Birman and R. F. Williams, Knotted periodic orbits in dynamical systems. I. +Lorenz’s equations, Topology 22 (1983), no. 1, 47–82. MR 682059 [1] +3. Thiago de Paiva, Hyperbolic knots given by positive braids with at least two full twists, +Proc. Amer. Math. Soc. 150 (2022), no. 12, 5449–5458. MR 4494619 [7] +4. +, Satellite knots over lorenz knots which are not lorenz knots, arXiv:2211.12816, +2022. [1] +5. +, Torus Lorenz links obtained by full twists along torus links, Proc. Amer. Math. +Soc., to appear (2022), arXiv preprint arXiv:2203.10935. [2, 5, 14] +6. Thiago de Paiva and Jessica S. Purcell, Satellites and Lorenz knots, Int. Math. Res. +Not., to appear (2021), arXiv preprint arXiv:2103.09500. [1, 2, 3, 4, 7, 12, 13, 16] +7. E. A. El-Rifai, Necessary and sufficient condition for Lorenz knots to be closed under +satellite construction, Chaos Solitons Fractals 10 (1999), no. 1, 137–146. MR 1682295 +[1] +8. John Franks and R. F. Williams, Braids and the Jones polynomial, Trans. Amer. Math. +Soc. 303 (1987), no. 1, 97–108. MR 896009 [6] +9. E. +Ghys +and +J +Leys, +Lorenz +and +modular +flows: +a +visual +introduction, +www.ams.org/publicourtreach/feature-column/fcarc-lorenz, 2006. [1] +10. Paulo Gomes, Nuno Franco, and Lu´ıs Silva, Partial classification of Lorenz knots: +syllable permutations of torus knots words, Phys. D 306 (2015), 16–24. MR 3367570 +[1] +11. +, Farey neighbors and hyperbolic Lorenz knots, J. Knot Theory Ramifications +26 (2017), no. 9, 1743004, 14. MR 3687479 [1] +12. John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. +Hautes ´Etudes Sci. Publ. Math. (1979), no. 50, 59–72. MR 556582 [1] +13. Allen Hatcher, Notes on basic 3-manifold topology, 2007. [10] +14. 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Tsau, Incompressible surfaces in the knot manifolds of torus knots, Topology +33 (1994), no. 1, 197–201. MR 1259522 [9, 10] +22. Warwick Tucker, A rigorous ODE solver and Smale’s 14th problem, Found. Comput. +Math. 2 (2002), no. 1, 53–117. MR 1870856 [1] +23. R. F. Williams, The braid index of generalized cables, Pacific J. Math. 155 (1992), +no. 2, 369–375. MR 1178031 [6, 7] + diff --git a/7dAzT4oBgHgl3EQf-f5K/content/tmp_files/load_file.txt b/7dAzT4oBgHgl3EQf-f5K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d2e6c493b0f960c749239af788cb344c4de3d20 --- /dev/null +++ b/7dAzT4oBgHgl3EQf-f5K/content/tmp_files/load_file.txt @@ -0,0 +1,1136 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf,len=1135 +page_content='HYPERBOLIC AND SATELLITE LORENZ LINKS OBTAINED BY TWISTING THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' A Lorenz link is equivalent to a T-link, which is a positive braid built by concatenating torus braids of increasing size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When each torus braid except the largest is obtained by full twists, then the T-link can be described as the Dehn filling of a parent link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In this paper, we completely classify when such parent links are hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This gives a classification of the geometry of T-links obtained by full twists when the amount of twisting is large, although the bound on the number of required twists is not effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We also present effective results on hyperbolicity for two families of T-links obtained by twisting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, we identify families of satellite T-links obtained by half-twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Introduction Lorenz links are the closed periodic orbits of a system of equations in- vestigated by Lorenz in the 1960s [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' They exhibit interesting dynamics that has led to significant further investigation over the years, in the fields of dynamics, geometry, and topology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' see for example [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' These links can be described as links on an embedded branched surface in R3, called the Lorenz template, due to work of Guckenheimer and Williams [12], and Tucker [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Birman and Williams were the first to investigate Lorenz links through the lens of knot theory, in the 1980s [2], and the first to show such links are closed positive braids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Birman and Kofman [1] showed that Lorenz links are equivalent to T-links, which are positive braids with a particular form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' see Section 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus techniques from braid theory can be brought to bear upon Lorenz links via T-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We are interested in the complement of these links, and in particular their geometrisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thurston showed in the 1980s that all knots in the 3-sphere are either torus knots, satellite, or hyperbolic [20], and we refer to this as the knot’s geometric type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The geometric type of Lorenz links has been considered since work of Birman and Williams in the 1980s [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' They showed that all torus knots are Lorenz knots, and satellites obtained as certain cables of Lorenz knots are Lorenz knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Hyperbolic geometry has been considered by Gomes, Franco, and Silva [10, 11], who proved hyperbolicity of Lorenz links satisfying certain conditions based on the Lorenz template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Satellite links have received additional attention, by El Rifai [7], de Paiva [4], and de Paiva and Purcell [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='01934v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='GT] 5 Jan 2023 2 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL In spite of this work, there remains no systematic way of determining whether a Lorenz link is hyperbolic, toroidal, or satellite using its description either on the Lorenz template, or as a closed braid in the form of a T-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' These descriptions uniquely determine a link, and hence uniquely determine its geometric type, so it is natural to ask for a simple description of geometric type based on the description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We focus on T-links in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The paper [6] begins a classification of the geometry of T-links, by finding examples that are satellite and also by identifying certain “parent links”, which give classes of T-links under Dehn filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' While the work in that paper finds examples of satellite and hyperbolic links, it is incomplete for two reasons: (1) First, the hyperbolic geometry of the parent links is used to determine geometry of T-links for many examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But the classification of the hyperbolic geometry of the parent links is incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (2) Second, because the results are obtained by Dehn filling, they apply only to links that admit full twists as T-link parameters, which are not required for general T-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In this paper, we extend the classification of geometry of T-links as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' First, we complete the classification of item (1) above: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 completely classifies when parent links of fully twisted T-links are hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This can be seen as an extension of work of Lee [16, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7], who proved a similar result for twisted torus knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Positive twisted torus knots are T-links with only one additional torus braid besides the largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lee’s result essentially proves Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 in the case of only one additional link component in the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Our result applies to any number of additional link components in the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 leads to new infinite families of hyperbolic T-links, determined only by parameters in a braid describing the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Fix relatively prime integers q < p, and let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an be integers less than p and increasing in value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There exists B ≫ 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Consider the T-link obtained from the (p, q)-torus knot by full twisting at least B times in regions with a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an strands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This Lorenz link is hyperbolic if and only if either all ai < q, or there is ai > q that is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The T-links of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1 must be obtained by full twisting, and we currently do not have a concrete, universal bound on the number of full twists that are required in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' this is the constant B in the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In Section 4 we improve this: We present two theorems that guarantee hyperbolicity of T-links with full twists, given only their parameters, where the bounds on numbers of full twists required are explicit and relatively simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The results are Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' It seems much more difficult to address item (2), especially in the hyperbolic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There are some partial results known, for example by de Paiva for torus knots [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In this paper, we give more results in the satellite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We LORENZ LINKS OBTAINED BY TWISTING 3 extend the results on satellite knots, requiring full twists in [6], to families of T-links with both full twists and half twists, which gives many more families in a very natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' For q < p integers, let K be a T-link obtained from the (p, q)-torus link by half-twisting in circles encircling less than q strands, or encircling multiples of q strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then S3 − K is satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The precise statement is Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This work was partially supported by the Aus- tralian Research Council, grant DP210103136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Results on braids This section reviews results on braids that will be used throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As usual, let σi be the standard generator of the braid group, giving a positive crossing between the i-th and (i + 1)-th strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' For 1 < p, q, define the (p, q)-torus braid as: (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q Note that within the braid group on p strands, its closure is the torus link T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When p, q are coprime, this is a torus knot, but we will not always restrict to coprime p and q unless specifically stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We will also consider such braids within larger braid groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When r < p, the (r, s) braid within the braid group on p strands is still defined to be (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)s, but now note this has p − r strands with no crossings lying to the right of the braid, viewing the braid arranged from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , rk and si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sk be integers such that 2 ≤ r1 < · · · < rk, and si > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The T-link T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rk, sk)) is defined to be the closure of the braid (σ1σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr1−1)s1(σ1σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr2−1)s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (σ1σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrk−1)sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rk, sk)) is obtained by concatenating the braids (ri, si) within the braid group on rk strands, and then taking the closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Taking closures of torus braids and related braids allows additional sym- metries and restrictions on the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' For example, we will use the following standard result on torus knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let 1 < p, q be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the torus link T(p, q) is equiva- lent to the torus link T(q, p) via a homeomorphism of S3 fixing the Heegaard torus containing T(p, q) and switching the two solid tori bounded by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1 is well known, and appears in many knot theory texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We visualise the proof in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The next result generalises [6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There the result only holds when each si is a multiple of ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Here we extend more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 4 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL p q p q Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The equivalence of T(p, q) and T(q, p) is given by rotating 180◦ in the diagonal axis shown for the Heegaard torus for S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This exchanges the solid tori in the standard genus-1 Heegaard splitting for S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let 0 < r1 < · · · < ri−1 < q < ri+1 < · · · < rn < p be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then, for k > 0, the T-link K = T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (ri−1, si−1), (q, qk), (ri+1, si+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rn, sn), (p, q)) is equivalent to the T-link K′ = T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (ri−1, si−1), (ri+1, si+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rn, sn), (p + qk, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2 allows us to assume there are no full twists on q strands in a T-link of the form T(· · · , (p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The braid (q, qk) is obtained by performing k full twists on q strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We know that these full twists commute in the braid group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus in the braid representing K, we may isotope (q, qk) to the top of the braid, leaving the rest of the braid unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Now perform the isotopy of K of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1, switching p and q in the (p, q)-torus link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The rotation in the diagonal shown in Figure 1 takes the (vertical) braids (r1, s1) ∗ · · · ∗ (rn, sn) to inverted braids, forming a tangle in the horizontal direction on a quadrilateral representing the projection torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (The form of this tangle is not important for the argument here, but more details can be found in [6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=') The result is a link of the form T(q, p) with a tangle along the horizontal p-strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The first such tangle is the braid (q, qk), which is unchanged by this isotopy because it is a full twist (see, for example, Birman and Kofman [1, Corollary 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the link diagram is formed by the braid (q, p) followed by (q, qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' These two braids can be combined to form the braid (q, p + qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Now apply the inverse of the isotopy of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This changes the link from T(q, p + qk) with tangles along the p horizontal strands to a link of the form T(p + qk, p) with these tangles returned to their form as braids (r1, s1) ∗ · · · ∗ (rn, sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The result is the link K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q, and r be positive integers with 0 < q ≤ r < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Consider the (p, q) torus link, which is the closure of the braid on p strands given by (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There is an ambient isotopy of S3 taking this to the LORENZ LINKS OBTAINED BY TWISTING 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Illustration of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3 in the case that q = 2, r = 4, p = 7, for an arbitrary tangle shown as a gray box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The left-most picture shows the original link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The (r + 1)-st strand, shown in blue, can be pulled tight beneath the diagram, resulting in the middle picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The right-most picture shows the result after isotoping strands (r + 1) to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' closure of the braid on r strands given by (σr−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−q+1)p−r(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Moreover, an ambient isotopy realising the equivalence fixes the portion of the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q corresponding to the r left-most strands at the top the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus, we may replace a neighbourhood of these strands above the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q with any tangle τ on r strands, and we find that the resulting link is ambient isotopic to the closure of the link obtained by concatenating the braid on r strands (σr−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−q+1)p−r, with τ, and then with (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because r ≥ q, the (r + 1)-st strand at the top of the braid only runs under the q overcrossing strands in the braid corresponding to the (p, q) torus link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' It then runs around the braid closure back to the top, returning to the r − q + 1 position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Together with a horizontal line from the r − q + 1 position to the r + 1 position, this strand bounds a disc in S3, lying under the plane of projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Use this disc to push the strand in S3 to become a horizontal strand lying below the plane of projection, running from the r + 1 position, then behind q strands, to the r + 1 − q position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Adjust slightly, pulling the right side up, so that the result is a closed braid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' see Figure 2, middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note that the resulting braid consists of only p − 1 strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This isotopy generalises the isotopy given by Lee in [17, Figure 6], and by de Paiva in [5, Figure 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 6 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL This move can be repeated for all the p − r strands to the right of the (r+1)-st strand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When finished, we obtain a link on r strands as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Braid index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Recall that the braid index of a knot K, which we will denote β(K), is the minimal number of strands required to form a braid with closure isotopic to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We will repeatedly use the following result of Franks and Williams [8] on braid index of the closure of a positive braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4 (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4 of [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let B be a positive braid on p strands that contains a full twist ∆2 = (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then B has braid index p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q, d and r be positive integers such that q ≤ r < p and d + q ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let Br be a positive braid on r strands, and let Bp denote the braid on p strands obtained by adding p − r trivial strands to the right of the braid Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the closure of the braid on p strands Bp(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)d(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q has braid index equal to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, the closure of the given braid on p strands is equivalent to the closure of the braid on r strands B′ = (σr−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−q+1)p−rBr(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)d(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because this is a positive braid, and because d + q ≥ r, the braid B′ has at least one positive full twist on r strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4 implies that the closure of B (and B′) has braid index equal to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose 0 < r1 < · · · < rn < p are integers, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn and q are positive integers, and suppose q ≤ rn ≤ sn + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the T-link K = T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rn, sn), (p, q)) has braid index equal to rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let Brn be the braid on rn strands obtained as the concatenation of torus braids (r1, s1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (rn−1, sn−1), where we view each (ri, si) as a braid on rn strands by adding rn − ri trivial strands to the right of the braid (ri, si) = (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σri−1)si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the given T-link is the closure of the braid Brn(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrn−1)sn(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since q ≤ rn ≤ sn + q, the result follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ The next definition is from Williams [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' A generalized q-cabling of a link L is a link L′ contained in the interior of a tubular neighbourhood L × D2 of L such that (1) each fiber D2 intersects L′ transversely in q points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' and (2) all strands of L′ are oriented in the same direction as L itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' LORENZ LINKS OBTAINED BY TWISTING 7 Williams showed the following result on generalised q-cablings for knotted L in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 (Theorem 1 of Williams [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The braid index is multiplicative under generalized cabling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' That is, if L is a link with each component a non-trivial knot and L′ is a generalized q-cabling of L then β(L′) = qβ(L), where β(∗) is the braid index of ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ This result was extended to unknotted L in the case of positive braids by de Paiva in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The following result is from that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9 (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3 of [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let L′ be a generalized q-cabling of the unknot L, with L given by a positive braid on n strands, where n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Also, assume the knot inside L is given by a positive braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then L′ has braid index equal to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Parents of T-links In this section, we build the “parent links” mentioned in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Dehn filling on such links produces T-links with full twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By classifying when such links are hyperbolic, and applying Thurston’s hyperbolic Dehn filling theorem, we show that, in an appropriate sense, most T-links with only full twists are hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is an extension of work by de Paiva and Purcell [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There, the same links were constructed, and some conditions were given to guarantee hyperbolicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Here, we strengthen the result by completely characterising when such links are hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers such that 1 < q < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Consider the (p, q)-torus braid on p strands, and its closure, the torus link T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let F denote the Heegaard torus on which T(p, q) lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a be an integer with 0 < a < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Denote by Ja an unknot lying horizontally with respect to the (p, q)-torus braid, positioned just above the crossings of the braid, bounding a disc such that the interior of that disc meets F transversely in a single arc intersecting the a leftmost strands of the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' More generally, given a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an satisfying 1 < a1 < · · · < an < p, take disjoint unknots Ja1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Jan as above, positioned so that the i-th is pushed vertically above the (i + 1)-th with respect to the braid, so that all are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Figure 3 shows an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers with 1 < q < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let an, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , a1 be integers such that 1 < a1 < · · · < an < p, with n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Also, assume that there is ai > q which is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the link K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 is atoroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In [6], it is shown that K is hyperbolic if all the ai > q are not multiples of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Here, we show only one needs not be a multiple of q for hyperbolicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose S3 − N(K) admits an essential torus T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then T bounds a solid torus V that must contain at least one component of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 8 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Shows T(7, 2) augmented at the top right by J2, J3, and J4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' First we show that we may choose V to contain T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' For suppose V is disjoint from T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then it must contain at least one Jaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The component Jaj must have positive wrapping number in V , for otherwise T(p, q) and Jaj would have zero linking number, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because there is no essential torus in the exterior of the unknot in S3, it follows in this case that T is unknotted in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, T bounds a second solid torus V ′ containing T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus in all cases we may assume T bounds a solid torus containing T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As an ≥ q, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, the torus knot T(p, q) is isotopic to a closed braid with an strands so that under the isotopy, the largest unknot Jan becomes the braid axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because the isotopy moves only the right-most p − an strands, all unknots Ja1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Jan are untouched by the isotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The torus T is then contained in the solid torus S3 − N(Jan), and bounds a solid torus V containing T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' It follows that Jan is disjoint from V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The torus T must intersect the disc Dan bounded by Jan in a series of circles, with each circle bounding a meridian of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Each meridian of V can be isotoped to meet the same number of strands of T(p, q), as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The boundary of a meridian defines an unknot in S3, and all such unknots are isotopic in S3 − N(K), where the isotopy is obtained by pushing the boundary of the meridian disc along the torus T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because T(p, q) forms a braid, it meets these discs monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let b denote the number of times that a meridian of V intersects the strands of T(p, q) on the disc Dan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note b > 1, or else T would be boundary parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note also that V winds some number of times around the solid torus S3 − N(Jan), and note that each meridian of this solid torus meets exactly an strands of T(p, q), since this is the number of strands in the closed braid isotopic to T(p, q) obtained from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since V meets each LORENZ LINKS OBTAINED BY TWISTING 9 meridian of S3 − N(Jan) a total of an times, and each meridian of V meets T(p, q) a total of b times, b must divide an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' It follows that T(p, q) is a generalised b-cabling of L, where L is the core of the solid torus V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Observe that T is embedded in exterior of the torus knot S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By work of Tsau [21], there are no essential tori in a torus knot exterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because b > 1, it follows that T must be compressible to its outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' That is, V is unknotted in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9 implies that T(p, q) has braid index equal to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' On the other hand, the torus knot T(p, q) with 1 < q < p has braid index equal to q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' for example this follows from Franks and Williams’ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then, b = q, and b divides an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Hence, q divides an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By hypothesis, there is ai ∈ {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an} which is greater than q and not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since ai > q, it must be the case that Jai is disjoint from the solid torus V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since T(p, q) intersects the disc Dai bounded by Jai a total of ai times, and T(p, q) is a generalised q-cabling of L, it must be the case that L intersects the disc ai/q times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' However, q does not divide ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers with 1 < q < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let an, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , a1 be integers such that 1 < a1 < · · · < an < p with n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the link K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 has no annuli with boundaries in two different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose that S3 − N(K) has an annulus A with boundaries ∂1A and ∂2A that lie in two different components, C1 and C2, respectively, of ∂(S3 − N(K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case 1: Consider first that C1 and C2 are Jaj and Jak, respectively, for some j ̸= k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note ∂1A and ∂2A are isotopic in S3−N(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The linking number between Cj and ∂jA is zero if and only if ∂jA is the longitude of Cj, in which case Cj and ∂jA are isotopic, for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose ∂1A is the longitude of C1, but ∂2A is not the longitude of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since ∂1A and ∂2A are isotopic, C1 and C2 would have nonzero linking number in this case, but this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Similarly ∂2A cannot be the longitude of C2 if ∂1A is not the longitude of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus either ∂1A is the longitude of C1 and ∂2A is the longitude of C2, or neither is a longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If both are longitudes, then C1 and C2 are isotopic, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus neither are longitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the linking number between C2 and ∂2A is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' However, C1 and C2 have zero linking number, so ∂1A and C2 must have zero linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But ∂2A is isotopic to ∂1A, and so ∂1A and C2 have nonzero linking number equal to the linking number of C2 and ∂2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case 2: Now suppose that C1 and C2 are Jaj and T(p, q), respectively, for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Again ∂1A and ∂2A are isotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 10 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Suppose first that ∂2A wraps at least one time along the longitude of C2 = T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then ∂2A has positive linking number with each of the components Jak, because T(p, q) has positive linking number with each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But the linking number between ∂2A and Jak for Jak ̸= C1 is zero, because C1 has linking number zero with each such component, and ∂2A has the same linking number with C1 as ∂1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus ∂2A is a meridian of C2 = T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So ∂2A and T(p, q) have linking number equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The curve ∂1A is some torus knot T(a, b) on ∂N(C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If a is equal to zero, then ∂1A is a meridian of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because a meridian of C1 has linking number zero with C2 = T(p, q), it follows that ∂1A and T(p, q) have linking number equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' However, this is not possible as ∂1A and ∂2A are isotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So, a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The linking number between ∂1A and C2 = T(p, q) is equal to a · aj, where C1 = Jaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because ∂2A and T(p, q) have linking number 1, and ∂1A and T(p, q) have linking number identical to ∂2A and T(p, q), it follows that a · aj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is impossible since aj > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, no such annulus exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let K be as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then K has no essential annuli with both boundary components in ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose that S3 −N(K) has an essential annulus A with both bound- ary components in ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The exterior of a torus knot has just one essential annulus by work of Tsau [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By work of Lee, [16, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1] that essential annulus would be punctured by Jai, where ai > q is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus A is not essential in S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus A is compressible, boundary compressible, or boundary parallel in S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Observe that a boundary compressible annulus is in fact boundary parallel, using the fact that S3 − N(T(p, q)) is irreducible and boundary irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Consider first that A is boundary parallel to an annulus B in ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then A ∪ B bounds a solid torus V in S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since A is not boundary parallel in S3 − N(K), at least one Jaj must be inside V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In addition, Jaj has wrapping number greater than zero in V , or else T(p, q) and Jaj would have linking number equal to zero, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But Jaj is an unknot, whose complement admits no essential tori (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [13, page 15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus V is also unknotted in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This implies that B is a meridional annulus of ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If ∂V is boundary parallel to Jaj, then Jaj is the core of ∂V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Hence, the linking number between T(p, q) and Jaj would be one, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus, as ∂V is not boundary parallel to Jaj, ∂V is an essential torus for S3 − N(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This contradicts Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Assume now that A is compressible in S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then there is a compression disk D for A in S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Surgering A along D yields two discs, D1 and D2, such that ∂A = ∂D1 ∪ ∂D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since S3 − N(T(p, q)) is boundary irreducible, ∂Di bounds a disk Ei on ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus, by pushing Ei slightly off of ∂N(T(p, q)) in S3 −N(K), we obtain a compressing LORENZ LINKS OBTAINED BY TWISTING 11 disc for A in S3−N(K), which contradicts our assumption that A is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, A is not compressible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus A cannot have both boundary components on ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let K be as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then K has no essential annulus with both boundary components on one ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose that S3 −N(K) has an essential annulus A with both bound- ary components on ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since S3 −N(Jaj) is a solid torus, and the solid torus admits no essential annuli, A is not essential in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus A is either compressible or boundary parallel in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case A: Suppose A is boundary parallel, parallel to an annulus B in ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then A ∪ B bounds a solid torus V in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since A is not boundary parallel in S3 − N(K), at least one component C of K must be inside V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case A1: Consider first that C = T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then T(p, q) has wrapping number greater than zero in V , for otherwise Jaj and T(p, q) would have zero linking number, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note this implies that ∂V is incompressible to its inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose that some circle Jak with j ̸= k lies in S3 − V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then we may isotope Jak to lie outside of W = N(Jaj) ∪ V , which is a regular solid torus neighbourhood of the unknot Jaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Denote by ω the winding number of Jak in S3 − W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If ω = 0, then the linking number between Jak and T(p, q) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus, ω ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But then this implies that the linking number between Jaj and Jak is nonzero, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus all circles Ja1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Jai−1, Jai+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Jan are inside V in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because at least two components of K lie inside V , ∂V is not boundary parallel to the inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The core of V forms a torus knot T(a, b) on N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note b > 0 or else T(p, q) runs around a longitude of N(Jaj) and hence has linking number zero with Jaj, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose b = ±1, so the core of V has the form of the trivial knot T(a, ±1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then there exists a disc in S3 − N(K) that is a longitude for ∂N(Jaj) whose boundary can be divided into two arcs, one of which meets A ⊂ ∂V in a nontrivial arc, and the other meets ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is an essential boundary compression disc for A, contradicting the fact that A is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since |b| > 1, ∂V = ∂N(T(a, b)) is incompressible and not boundary parallel to the outside, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' in the solid torus S3 − Jaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This implies that in all cases ∂V is essential in S3 − N(K) contradicting Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case A2: The torus knot T(p, q) cannot lie inside V by the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So some C = Jak with j ̸= k lies inside V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The wrapping number of Jak inside V must be different from zero as Jak and T(p, q) have positive linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since Jak and Jaj have zero linking number, V must be a longitude of ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If Jak is the core of V , then Jaj and Jak are isotopic 12 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' A disc with boundary an arc on each of A ⊂ ∂V and ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' in S3 − N(T(p, q)), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So Jak is not the core of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But then ∂V is incompressible and not boundary parallel to the inside in S3 − K, and incompressible and not boundary parallel to the outside in S3 − K, contradicting Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case B: Suppose A is compressible in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then there is a compression disk D for A in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Surgering A along D yields two discs, D1 and D2, such that ∂A = ∂D1 ∪ ∂D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If one of ∂D1 or ∂D2 bounds a disk E on ∂N(Jaj), then by considering a disc with boundary in A close to E, we see that A is also compressible in S3 − N(K), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So suppose that neither ∂D1 nor ∂D2 bounds a disk on ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then D1 and D2 are discs in the solid torus S3 − N(Jaj) with nontrivial boundary on ∂N(Jaj) and hence both are meridians of S3 − N(Jaj), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' with ∂D1 and ∂D2 forming longitudes of ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Undoing the surgery along D, it follows that A is boundary parallel in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus we have a contradiction to Case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, S3 − N(K) has no essential annulus with both boundary com- ponents in one ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The link K as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2 has no essential annuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, any essential annulus has both boundary components on the same component of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4, the two boundary components cannot lie on ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5 the two boundary components cannot lie on one of the ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus no such annulus exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers with 1 < q < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let an, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , a1 be integers such that 1 < a1 < · · · < an < p with n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Also, assume that there is ai > q which is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then, the link K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By de Paiva and Purcell [6, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1], the link exterior is irre- ducible and boundary irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2, it is atoroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6, it is anannular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore it is hyperbolic by Thurston’s hyperbolisation theorem for Haken manifolds [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7 and de Paiva and Purcell [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6], we completely classify the geometric types of the links T(p, q) ∪ Ja1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' LORENZ LINKS OBTAINED BY TWISTING 13 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers with 1 < q < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an be integers such that 1 < a1 < · · · < an < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the link K = T(p, q) ∪ Ja1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Jan is hyperbolic if and only if either all ai < q, or there is ai > q which is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When n = 1, the link K = T(p, q) ∪ Ja1 is the Dehn-filling parent of a twisted torus knot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' this has been treated by Lee [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If n = 1 and a1 = q, then [15, Theorem 1] implies that infinitely many Dehn surgeries along Ja1 yield non-hyperbolic knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, Thurston’s hyperbolic Dehn filling theorem [19] implies K is not hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In fact, the proof of [15, Theorem 1] implies K is annular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If n = 1 and a1 is not a multiple of q, then K is hyperbolic by [16, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In the case n > 1, if there is ai > q that is not a multiple of q, then K is hyperbolic by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If n > 1 and all ai are less than q, then no ai is a multiple of q, and K is hyperbolic by [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, if n > 1, there is some ai > q and all ai > q are multiples of q, then K is satellite by [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers with 1 < q < p, and let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an and s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn be integers such that 1 < a1 < · · · < an < p and si > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then, there exists B ≫ 0 such that if each si > B, the T-link T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q)) is hyperbolic if and only if either all ai < q, or there is ai > q which is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8, the link K = T(p, q) ∪ Ja1 ∪ · · · ∪ Jan is hyperbolic if and only if the ai satisfy the hypotheses of the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Obtain the given T-link by Dehn filling the link components Ja1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Jan along slopes 1/s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , 1/sn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When the link K is hyperbolic, the Dehn filling remains hyperbolic by Thurston’s hyperbolic Dehn filling theorem [19] pro- vided the si are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' On the other hand, Dehn filling a satellite K yields a satellite T-link, by de Paiva and Purcell [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6], and in the case n = 1 and a1 = q, Dehn filling yields an annular link by Lee [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1 in the introduction follows immediately from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Hyperbolicity with effective full twist bounds While Corollary Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9 is quite broad, unfortunately the constant B in that theorem is not explicit, and so it may be difficult to apply in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In this section we find explicit parameters which produce hyperbolic T-knot obtained by full twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because we are considering full twists exclusively in this section, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2 implies that we may assume that none of the ai are equal to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 14 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn, and p, q, k be integers satisfying the following hypotheses: p and q are relatively prime, 1 < a1 < · · · < an, and 0 < q < an < p, each si > 0, and sn ≥ 2, p and an are relatively prime, k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the T-knot K = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q + kp)) is atoroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose that the exterior of K in S3 admits an essential torus T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By work of Ito [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2(3)], because K is the closure of a braid with at least two positive full twists on p strands, the torus T does not intersect the braid axis C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Moreover, the knot inside T is given by a braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus there exists some integer d > 1 such that K is a generalized d-cabling of a knot L, where L is the core of the solid torus bounded by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As a consequence, d must divide p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' After (−1/k)-Dehn surgery along the braid axis C, the knot K becomes the T-knot K′ = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q)) and the torus T becomes a new torus T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This will bound a solid torus V ′ in S3, with core L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because q < an < ansn + q, the knot K′ has braid index equal to an by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If L′ is trivial, then an is equal to d by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' However, this is not possible since gcd(p, an) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So L′ is knotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8, an is equal to dβ(L′), where β(L′) is the braid index of L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But then d divides p and d divides an, again contradicting gcd(p, an) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, the exterior of K admits no essential torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ We will combine the previous result with the following from [5], which gives information on torus knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2 (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2 of de Paiva [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn be positive integers such that 1 < q < p and 1 < a1 < · · · < an < p with ai ̸= q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If gcd(p, q) = 1 and in addition one of the following hold: q < an, or q > an and p is not of the form bq + 1 for some b > 0, or q > an and p = bq + 1 for some b > 0, but s1 > 1, or q > an, p = bq + 1 for some b > 0, and s1 = 1, but a2 ̸= a1 + 1, then T((a1, s1a1), (a2, s2a2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, snan), (p, q)) is not a torus knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn, and p, q, k be integers satisfying the following hypotheses: p and q are relatively prime, 1 < a1 < · · · < an and 1 < q < an < p, each si > 0 and sn ≥ 2, LORENZ LINKS OBTAINED BY TWISTING 15 p and an are relatively prime, k ≥ 2, n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then if in addition, one of the following hold: q ̸= 1, or s1 > 1, or a2 ̸= a1 + 1, then the T-link K = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q + kp)) is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because gcd(p, q) = 1, K is a knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1, K is atoroidal, so not a satellite knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The T-knot K is equivalent to the T-knot T((a1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (q + kp, p)) by [1, Corollary 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The integer q + kp does not have the form bp + 1 if and only if q is different from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So under these conditions, K is not a torus knot by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, by Thurston’s hyperbolisation Theorem for knots [20], K is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn, and p, q, k be integers satisfying the following hypotheses: p and q are relatively prime, 1 < a1 < · · · < an, and 1 < q < an < p, each si > 0 and both sn and sn−1 are at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose also that one of the following holds: q < an−1 and an and an−1 are relatively prime, or q > an−1 and an and q are relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the knot K = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q)) is atoroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose the exterior of K in S3 admits an essential torus T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, K is equivalent to the knot given by the closure of the braid B = (σan−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σan−q+1)p−an · τ · (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σan−1)snan+q, where τ is the concatination of braids (a1, a1s1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (an−1, an−1sn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since B has at least two positive full twists on an strands, it follows from [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2(3)] that T does not intersect the braid axis C of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus there is an integer d > 0 such that K is a generalized d-cabling of the core L of the solid torus bounded by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Hence d divides an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Perform (−1/sn)-Dehn surgery along the braid axis C to obtain the braid B′ = (σan−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σan−q+1)p−an · τ · (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σan−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Its closure gives K′ = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an−1, an−1sn−1), (an, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The torus T becomes a new essential torus T ′ in the exterior of K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The torus T ′ bounds a solid torus with core L′, which is either trivial or knotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 16 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Suppose first the case that q < an−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then q ≤ an−1 ≤ an−1sn−1 + q, so Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6 implies that K′ has braid index equal to an−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If L′ is the trivial knot, then an−1 is equal to d by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This implies that d divides both an and an−1, contradicting the assumption in this case that these are relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Similarly, if L′ is knotted, then Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 implies that an−1 is a multiple of d, with the same contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Now suppose q > an−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then K′ has braid index q by Franks and Williams, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If L′ is trivial, then as above, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9 implies q equals d, and therefore d divides both an and q, contradicting the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Similarly, if L′ is knotted, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 implies q is a multiple of d, and again d divides both an and q, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn, and p, q, k be integers satisfying the following hypotheses: p and q are relatively prime, 1 < a1 < · · · < an and 1 < q < an < p, each si > 0 and both sn and sn−1 are at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose also that one of the following holds: q < an−1 and an and an−1 are relatively prime, or q > an−1 and an and q are relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then K = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q)) is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4, the knot K is atoroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2, using the fact that q < an, K is anannular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, K is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Satellite T-links obtained by Half-twists In this section we switch from discussions of hyperbolic links to satellite links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We find families of Lorenz links that are satellites using half-twists, rather than full-twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Previous work by de Paiva and Purcell found con- ditions that ensure a T-link is satellite, namely [6, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lee has similar results for the case of twisted torus knots [15, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We extend these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose B is a diagram given as a closed braid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' we consider the braid to have strands running vertically on the plane of projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' A positive half-twist on the strands from a to b is the braid ∆a,b = (σa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σb)(σa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σb−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (σa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This can be thought of as cutting the braid between the a-th and b-th strands, rotating in the anticlockwise direction by 180◦, and gluing back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In braid theory literature, the positive half-twist on all strands is well known as the Garside fundamental braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' A negative half-twist is defined similarly, only the rotation is in the clockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' See Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' LORENZ LINKS OBTAINED BY TWISTING 17 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' An example of half twists when r = 2, q = 3, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Left: A positive half-twist ∆1,rq, a negative half-twist ∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The green circle indicates the braid axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Middle: The negative half- twist cancels crossings above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Right: The additional positive half-twist gives the braid (rq, tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let r, q, s be positive integers, and suppose s is not a multiple of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The (rq, sq)-torus braid is obtained by the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Start with the trivial braid on rq strands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' let J1,rq be an unknot encircling all rq strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let t be an integer such that 0 < t < r and s = t + kr for some integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Insert a positive half-twist ∆1,rq, followed by a negative half-twist ∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, perform 1/k-Dehn filling on J1,rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The result is the (rq, sq)-torus braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The process is illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The positive half-twist ∆1,rq yields a braid (σ1σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrq−1)(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrq−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (σ1), encircled by J1,rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Perform the negative half-twist ∆1,(r−t)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This concate- nates the previous braid with (σ−1 (r−t)q−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ−1 2 σ−1 1 )(σ−1 (r−t)q−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ−1 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (σ−1 (r−t)q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This braid cancels with the positive half-twist along the first (r − t)q strands, as shown in Figure 5, middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, the positive half-twist ∆(r−t)q+1,rq concatenates a positive half-twist along the last tq strands, giving the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrq−1)tq = (rq, tq), still augmented by the unlink J1,rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' To obtain the braid (rq, sq), perform 1/k Dehn filling on J1,rq, removing that link component and inserting an additional krq overstrands into the braid, for a total of tq+krq = sq overstrands, giving the desired (rq, sq)-torus braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let r, q, s be positive integers, with s not a multiple of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Consider the torus braid (rq, sq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' At the top of the braid, consider r disjoint discs arranged horizontally, each encircling q strands of the braid, and similar discs at the bottom of the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The boundary of each disc at the top connects 18 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL via a cylinder, embedded in the complement of the braid and enclosing q strands, to the boundary of a disc at the bottom of the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Moreover, the solid cylinders enclosed by these cylinders, containing q strands each, forms the (r, s)-torus braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let t be an integer such that 0 < t < r and s = t+kr for some integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2, the (rq, sq) torus braid is formed from k full twists on rq strands, followed by a positive half-twist ∆1,rq, then a negative half-twist −∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Each half-twist is on a multiple of q strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Observe that the cylinders described above can be arranged to completely contain any half-twist on q strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' For a half-twist on a multiple of q strands, say xq strands, x disjoint cylinders enter the top of the half-twist, and then are half-twisted themselves, remaining disjoint, to exit the bottom of the half-twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus the cylinders remain embedded as claimed when passing through half-twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, each full twist also preserves the cylinders, sending each through a full twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' To see that the braid formed by the solid cylinders is as claimed, observe that the cylinders form k full twists, followed by one positive half-twist on all strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The (r−t) left-most cylinders then pass through a negative half-twist, and the remaining t right-most cylinders pass through a positive half-twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2, this creates braid on r strands, with rk overstrands at the top coming from the full twists, followed by t overstrands coming from the concatenation of half-twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus this is an (r, rk + t) = (r, s)-torus braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be integers such that 1 < q < p, and let (a1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, bn) be pairs of integers such that 1 < a1 < · · · < an ≤ q and bi > 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally let r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , rm and s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sm be integers such that q < r1q < · · · < rmq < p, and si > 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the T-link K = T((a1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, bn), (r1q, s1q), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rmq, smq), (p, q)) is satellite with companion the T-link T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rm, sm+1)) and pattern given by the closure of the braid (a1, b1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (an, bn)(σq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ1)p−rm � q, q � m � i=1 risi � + qrm � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As before, we think of the T-link as the closure of a braid on p strands arranged vertically, the concatenation of braids (a1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, bn), (r1q, s1q), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rmq, smq), (p, q) in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' First apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3 to change the closed braid of the T-link to a closed braid B′ on rmq strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This isotopy fixes all of the rmq strands at the top left of the original braid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' thus it does not affect any of the braids (aj, bj) or (riq, siq), for any i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In other words, B′ is the braid given by concatenating (σrmq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−q+1)p−rm with braids (a1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, bn), (r1q, s1q), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rmq, smq), and finally the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' LORENZ LINKS OBTAINED BY TWISTING 19 By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, there are rm disjoint embedded cylinders in the complement of the portion of the braid starting just above the braid (a1, b1), and ending just below the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−1)q at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' These cylinders each enclose q strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' They extend around the braid closure to give rm disjoint embedded cylinders running to the top of the braid, each enclosing q strands, arranged right to left across the top of the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The only portion of the braid that is not already enclosed in one of these cylinders is the braid (σrmq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−q+1)p−rm lying at the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is a braid whose left-most strand is the (rmq − q + 1)-th strand, and whose right-most strand is the rmq-th strand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In other words, this is a braid on the right-most q strands of the rmq-strand braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus the right-most cylinder, enclosing q strands, can be extended to enclose this braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then all cylinders connect to form a closed embedded torus Σ, encircling q strands of the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The torus Σ bounds a solid torus containing q strands, which we check has the claimed form of the companion in the theorem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This solid torus forms a braid on rm strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, each (riq, siq)-torus braid from the original T-link causes the solid cylinder to form a braid (ri, si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The braids (aj, bj) and (σrmq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−q+1)p−rm lie completely inside the solid cylinder, so they do not affect the braid it forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, consider the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−1)q at the bottom of B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is formed by q strands running over all the rmq strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When the collection of solid cylinders encounter this braid, the left-most solid cylinder encircles exactly these q strands, and runs over all others to lie on the right-most side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus it forms a (rm, 1)-torus braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So the solid torus enclosing q strands has the form of the closure of a braid (r1, s1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (rm, sm), (rm, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is the T-link T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (rm, sm + 1)) as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since it forms a nontrivial knot in S3, Σ is an incompressible torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally we check the form of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Starting at the top-left of the braid B′, the torus Σ encloses the braid (a1, b1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (an, bn), which will form part of the braid describing the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As Σ follows the companion into each of the braids (ri, si), all the q strands will make one full twist each time Σ runs completely through an overstrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There are si of these, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' m − 1, plus sm + 1 for the (rm, sm + 1) braid that the compan- ion runs over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' These will occur in some order, with Σ also enclosing the braid (σq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ1)p−rm, coming from the top right of B′, at some point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because full twists commute in the braid group, we may write the braid as (a1, b1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (an, bn)(σq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ1)p−rm · τ where τ is an appropriate number of full twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' To obtain the appropriate number of full twists, we need to consider the homological longitude of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The pattern is the braid obtained when we apply a homeomorphism taking the solid torus bounded by the companion to an unknotted solid torus, with homological longitude mapped to a standard longitude of the unknot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The effect is to add �m−1 i=1 (ri − 1)si + (rm − 1)(sm + 1) additional full twists, for a total of 20 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL �m i=1 risi + rm full twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus the pattern can be written as the braid (a1, b1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (an, bn)(σq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ1)p−rm(q, q( � risi) + qrm) □ References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Joan Birman and Ilya Kofman, A new twist on Lorenz links, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 2 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 2, 227–248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 2529294 [1, 4, 15] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Joan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Birman and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Williams, Knotted periodic orbits in dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lorenz’s equations, Topology 22 (1983), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1, 47–82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 682059 [1] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thiago de Paiva, Hyperbolic knots given by positive braids with at least two full twists, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 150 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 12, 5449–5458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 4494619 [7] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Satellite knots over lorenz knots which are not lorenz knots, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='12816, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [1] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Torus Lorenz links obtained by full twists along torus links, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=', to appear (2022), arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='10935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [2, 5, 14] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thiago de Paiva and Jessica S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Purcell, Satellites and Lorenz knots, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Not.' metadata={'source': 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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 2 (2002), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1, 53–117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 1870856 [1] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Williams, The braid index of generalized cables, Pacific J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 155 (1992), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 2, 369–375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 1178031 [6, 7]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} diff --git a/7tFJT4oBgHgl3EQfmyyq/content/tmp_files/load_file.txt b/7tFJT4oBgHgl3EQfmyyq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1453be9202df2faea5c50c8cf3749634f4b025b1 --- /dev/null +++ b/7tFJT4oBgHgl3EQfmyyq/content/tmp_files/load_file.txt @@ -0,0 +1,476 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf,len=475 +page_content='Adversarial Learning for Implicit Semantic-Aware Communications Zhimin Lu∗, Yong Xiao∗†¶, Zijian Sun∗, Yingyu Li‡, Guangming Shi†§¶, Xianfu Chen∥, Mehdi Bennis∗∗, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Vincent Poor†† ∗School of Elect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' & Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=', Huazhong Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' of Science & Technology, Wuhan, China †Peng Cheng Laboratory, Shenzhen, China ‡School of Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' and Elect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=', China Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' of Geosciences, Wuhan, China §School of Artificial Intelligence, Xidian University, Xi’an, China ¶Pazhou Laboratory (Huangpu), Guangzhou, China ∥VTT Technical Research Centre of Finland, Finland ∗∗University of Oulu, Oulu, Finland ††Department of Elect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' & Computer Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=', Princeton University, Princeton, NJ, USA Abstract—Semantic communication is a novel communication paradigm that focuses on recognizing and delivering the desired meaning of messages to the destination users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Most existing works in this area focus on delivering explicit semantics, labels or signal features that can be directly identified from the source signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this paper, we consider the implicit semantic communication problem in which hidden relations and closely related semantic terms that cannot be recognized from the source signals need to also be delivered to the destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We develop a novel adversarial learning-based implicit semantic- aware communication (iSAC) architecture in which the source user, instead of maximizing the total amount of information transmitted to the channel, aims to help the recipient learn an inference rule that can automatically generate implicit semantics based on limited clue information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We prove that by applying iSAC, the destination user can always learn an inference rule that matches the true inference rule of the source messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Experimental results show that the proposed iSAC can offer up to a 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='69 dB improvement over existing non-inferential communication solutions, in terms of symbol error rate at the destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Index Terms—Semantic communication, implicit semantics, inference rule, semantic-aware, adversarial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' INTRODUCTION Semantic communication has recently been attracting sig- nificant interest, driven mostly by the recent surge in the demand of data-hungry and resource-consuming smart and human-oriented services, such as Tactile Internet [1], intel- ligent transportation systems [2], eXtended Reality (XR), and digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Different from the traditional content- agnostic communication solutions that focus on transmit- ting data packets from one user to another, while ignoring the semantics, semantic communication focuses on sensing, recognizing, utilizing, and delivering the key meaning of transported messages throughout the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Recent results This work has been accepted at the Proceedings of the IEEE International Conference on Communications (ICC) conference, Rome, Italy, May 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Copyright may be transferred without notice, after which this version may no longer be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' have demonstrated that semantic communication has the po- tential in significantly improving communication efficiency, quality-of-experience (QoE) of various services, and imbuing more high-level human-like capabilities into communication networks [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The concept of semantic communication was first in- troduced in 1949, where Weaver defined three levels of communication problems [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The Shannon theory has been considered as the solution for level 1 problem, also called the technical problem of communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The semantic and effective communication problems have been defined as level 2 and level 3 problems which address delivering “the desired meaning” and influencing the conduct of the destination user in the desired way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' These definitions have attracted significant interest in extending Shannon theory to investigate the seman- tic and effective communication problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Carnap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' [5] observed that there exists a fundamental paradox, called the Bar-Hillel-Carpnap (BHC) paradox, when applying Shannon theory to solve the semantic communication problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In the BHC paradox, semantically incorrect statements, especially those contradicting with common-sense knowledge can always maximize the Shannon information due to their rarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Recently, there has been a growing interest in applying deep learning (DL) algorithms to improve the performance of communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' This has sparked many works to convert the problem of semantic communication into pattern recognition and/or classification problems that can be solved by DL-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In these works, manually labeled objects, terms, and/or signal features that can be directly identified from various forms of signals are defined as se- mantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' For example, DL algorithms have been applied to identify semantics from text, image, and voice [6] for various downstream communication tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Recent observations suggest that information semantics can be much more than just the object labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In fact, according to the original definition of semantics first introduced by Breal in 1897, semantics are the “relationship between words arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='11589v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='LG] 27 Jan 2023 and the knowledge they signify”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Recent work in cognitive neuroscience also emphasizes that human users are able to express and infer complex implicit semantics by automatically inferring complex hidden relations among concepts and ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' This motivates the work of this paper to investigate the implicit semantic communication problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We define seman- tics that can be directly identified from the source messages as explicit semantics and focus on developing solutions to infer the implicit semantics, including hidden relations and relevant semantic concepts that cannot be identified from the source messages, by a destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' More specifically, we introduce a representation model of the implicit semantic information and develop an adversarial learning-based im- plicit semantic-aware communication (iSAC) architecture in which the source user, instead of maximizing the information transmitted across the channel, tries to assist the destination user to learn an inference rule that can automatically infer implicit semantics based on the received clue information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=', explicit semantics, sent by the source user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We prove that by applying iSAC, the destination user can always learn an inference rule that matches the true inference rule of the source messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Also, since the true inference rule of the source messages generated by human users can always be assumed to be semantically correct, the BHC paradox can be naturally solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We conduct extensive simulations based on real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Our results show that the proposed iSAC can achieve up to 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='69 dB improvement over existing non- inferential communication solutions, in terms of symbol error rate at the destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' RELATED WORKS Existing works in semantic communication can be roughly divided into three categorizes: information theory-based, machine learning-based, and cognitive neuroscience-inspired works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In particular, Carnap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' [5] elaborated semantic information measurements analogous to the binary-symbol- based information measurements in Shannon theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' More- over, Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' [7] further investigated the quantification of semantic information as well as semantic coding, and obtained initial results showing the feasibility of data compression and reliable communication from a semantic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Moti- vated by the fact that semantic information can be learned and evolved through interaction, in our recent work [8], we have proposed a strategic semantic communication framework by combining game theoretic models with rate-distortion theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Recently, the powerful capabilities of machine learning, especially the DNNs-based learning algorithms, in pattern recognition and data classification have been introduced to solve the semantic communication problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Most existing works have focused on the representation, identification, com- putation, and communication of explicit semantic, such as human-assigned labels and sample-related features or classes, that can be directly recognized from the transmitted mes- sages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' For example, the semantic communication systems for text transmission were developed in [9], while the se- mantic error was measured at the word level and sentence level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' A lite distributed semantic communication system, named L-DeepSC, was proposed in [10] also for text transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' For image transmission, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' [11] adopted a GAN-based image semantic coding method for sending and reconstructing images in extreme low bit rate using the proposed semantic communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' For transmitting speech signals, an attention mechanism based semantic communication system was developed in [12], which is capable of identifying the essential information of speech signals under dynamic channel environments in telephone systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Furthermore, Seo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' proposed a stochastic model of semantics-native communication (SNC) for generic tasks in [13] infused with contextual reasoning to cope with the situation where the semantics vary over time and in different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Inspired by the recent study in cognitive neuroscience sug- gesting that the human user is able to infer complex implicit semantics based on a limited clue/explicit information, we have investigated the implicit semantic communication in our recent works [8], [14]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We have derived an information theoretic bound of the implicit semantic communication chan- nel based on the rate distortion theory in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' By formulating the implicit semantic reasoning process of the source user as a reinforcement learning process, an imitation learning-based solution has been proposed in [15] for the destination user to estimate the reasoning policy of the source user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' One of the key issues of this work is that the proposed imitation learning- based solution always infers all the possible reasoning paths based on a maximum causal entropy framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Also, the state space in the formulated reasoning policy increases in an exponential scale with the path length which may re- sult in slow computation and reduce accuracy under certain scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this paper, we introduce a simple adversarial learning-based solution, iSAC, that allows the destination user to directly learn a much simplified approximated inference rule which will always output the most likely implicit term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Our proposed iSAC requires much less computational load and can deliver comparable performance to our previously proposed solution in many practical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' A GENERAL SEMANTIC COMMUNICATION MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Representation of Implicit Semantic Information As mentioned earlier, the semantics of a message should include the implicit relations that link the explicit semantic terms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=', the concepts and/or labels directly observed in the source messages, to the relevant implicit meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Therefore, in this paper, we define the semantic information of a given message as a tuple ω = ⟨v, uv⟩ where v is a set of explicit semantic terms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=', labels or features, that can be directly recognized from the message, uv consists of the implicit rela- tions and the connected semantic terms that are closely related to the explicit semantics v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' uv cannot be directly observed from the source message but will need to be inferred based on the previous communications and inference preference of the source user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Let V be the set of all possible explicit semantic terms that can be expressed by the source user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We assume relations are undirectional and each implicit semantic term in uv is linked to a term in v by a specific relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We can therefore abuse the notation and use uv to denote a set of implicit semantic terms that are related to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We use Rv to denote the set of all the possible implicit semantic terms that are relevant to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The implicit semantic meanings generally have the follow- ing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Randomness: Due to the human nature of the message generation users, the implicit semantic meaning that can be expressed by each user is generally not deterministic but exhibits a certain randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We use p(uv|v) to denote the probability of inferring implicit semantic terms uv when observing v, for v ⊆ V and uv ⊆ Rv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Polysemy: It is known that different users may infer different meanings when observing the same explicit term, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=', some users may infer the TV cartoon character when observing the term “Tweety”, while for others, the term “Tweety” may mean smartphone App of a social network website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Inference Rule: Recent study suggests that, for each indi- vidual user, its inference preference can be characterized by a function that maps the explicit semantics to the possible implicit semantic terms and/or concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this paper, we follow the same setting and, to characterize the randomness of implicit semantic inference process, we define the inference rule of the source user, denoted as π (v), as a function mapping each given explicit semantic term v to a probability distribution of all the possible implicit semantic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We assume π (v) is stationary and also � uv⊆Rv pπ(uv|v) = 1, where pπ(uv|v) is the probability of inferring uv based on inference rule π when observing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Note that we assume neither source user nor destination user can know the true inference rule of the source messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The source user can however observe a set of expert message samples, consisting of implicit semantics generated by the source user during the previous communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Semantic Communication Model A general semantic communication model consists of the following key components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' (1) Semantic Recognizer: extracts the key explicit semantic terms from the observed messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' For example, if the observed messages are in the form of images or voice and the semantic terms are object labels, it can directly apply existing object detecting algorithms, such as YOLO and wav2letter, to extract the explicit semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' (2) Semantic Encoder: converts the extracted semantic in- formation into a form that is suitable for physical channel transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In the traditional semantic communication model, the source user has two types of encoders: seman- tic (source) encoder for minimizing the redundancy in the semantic information and channel encoder for improving the robustness against noisy channel corruption by adding a certain amount of redundancy into the transmitted signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Recently, the joint source-and-channel encoding has also been considered to implement both types of encoders into a single encoding function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=', a DNN, that directly converts the input message into an output signal for physical channel transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this paper, we assume the semantic encoder directly converts the semantic information identified by semantic recognizer into the signal to be transmitted to the physical channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We will discuss in details later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' (3) Semantic Decoder: recovers the complete semantic in- formation at the destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In explicit semantic communication, the main objective of the destination user is to recover the explicit semantics identified by the semantic recognizer of the source user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this paper, we consider the implicit semantic communication, in which the destination user needs to recover both explicit and implicit semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Let ˆω = ⟨ˆv, ˆuv⟩ be the recovered semantic meaning of the destination user, where ˆv and ˆuv are the recovered explicit and implicit semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The main design objective of the semantic communication system is to minimize the semantic distance between the original semantic information and the recovered meaning at the destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Let Γ (ω, ˆω) be the semantic distance between ω and ˆω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this paper, we focus on the implicit semantic com- munication in which the destination user needs to recover both explicit and implicit semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Since implicit semantic cannot be directly identified by semantic recognizer, the main objective is then to learn an approximated inference rule πd that can minimize the semantic distance between the explicit semantics and possible implicit semantics inferred by the true inference rule of the source user as well as those recovered by the learned inference rule, we formulate the optimization problem of implicit semantic communication as: min πd E [Γ (⟨v, uv⟩, ⟨ˆv, ˆuv⟩)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' (1) A straightforward approach for implementing implicit se- mantic communication is to let the source user infer the implicit semantics from the explicit semantics identified by semantic recognizer, and then transmit both types of semantics to the destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' This approach however suffers from the following challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' First, as mentioned earlier, due to randomness and polysemy, implicit semantic information consists of the probability distributions of a set Rv of possible implicit semantic terms, characterized by a set of floating- point values, each requiring a large number of data packets for semantic information transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Second, allowing the source user to perform implicit semantic inference whenever it observes explicit semantics may result in extra delay for in- formation transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Moreover, the quality of the recovered implicit semantics may be closely related to the service need of the destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Always letting the source user send all the potential implicit information will result in reduced communication efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Finally, the implicit semantics are often generated by the personally preference-related inference rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Therefore, transmitting implicit semantics will also cause privacy information exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this paper, we propose a novel adversarial learning- based solution called iSAC for the destination user to learn an inference rule to directly infer implicit semantics based on the explicit semantics sent by the source user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this way, Source user Destination user Channel Semantic Evaluator Semantic Encoder Semantic Decoder messages Semantic Recognizer Feedback Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 1: Architecture of iSAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' during communication, the source user only needs to send explicit semantics observed from the source messages and the destination user can automatically recover the intended implicit semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' ISAC ARCHITECTURE In this section, we first define the semantic distance that is suitable to characterize the difference between any given pair of inference rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We then introduce the iSAC architecture and present the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Semantic Distance One of the key differences between the traditional data- oriented communication and the semantic communication solutions is that in the latter, instead of accurately reproducing the signal in its original form, the destination user tries to infer the semantic meaning that matches the real semantics associated with the source messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' This problem becomes more challenging for iSAC because in this system, the destina- tion user needs also to recover implicit semantics that cannot be directly observed in the message received by the source user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Due to the randomness and polysemy of the implicit semantics, an appropriate solution for evaluating the semantic distance is to adopt statistic-based distance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this paper, we focus on recovering the implicit semantics at the destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' More specifically, the destination user tries to learn the inference rule to infer the correct implicit meaning based on the received signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Let ˆv be the signal decoded by the destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The learned inference rule of the desti- nation user πd is a mapping function: πd : ˆv → Dπd(ˆuv | ˆv) where Dπd(ˆuv |ˆv) is the probability distribution of the possible implicit semantics ˆuv when observing ˆv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Similarly, we define the probability distribution of implicit semantics generated by the true inference rule of the message source when observing explicit semantics v as Sπ(uv | v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We use cross entropy, one of the most popular metrics for measuring statistical distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this case, the semantic distance between the true implicit semantic and the estimated implicit semantics at the destination user can be written as: Γ (⟨v, uv⟩, ⟨ˆv, ˆuv⟩) = � v⊆V � Euv∼Sπ(uv|v) [log p (uv|v)] +Eˆuv∼Dπd(ˆuv|ˆv) [log (1 − p (ˆuv|ˆv))] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' (2) Our proposed solution can be directly applied when other statistical distance measures are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' iSAC Architecture In this paper, we propose an adversarial learning-based iSAC architecture, in which the source user tries to assist in training an inference rule at the destination user, such that the source user only needs to send the explicit semantics observed directly from the original message and the destination user will be able to automatically infer implicit semantics based on the signal sent from the source user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We introduce a new component, the semantic evaluator, whose main objective is to compare the implicit semantics estimated by the destination user with a set of expert data samples generated by the true inference rule of the message source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' As will be proved later, by introducing the semantic evaluator at the source user, the destination user will be able to learn an inference rule without observing any expert samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In other words, the expert data samples can only be observed by the semantic evaluator and therefore, the implicit semantic message samples will not be exposed in both training and communication processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Let us describe the implementation details of different components of iSAC at the source and destination users as in the following: 1) Semantic encoder at the source user: The main ob- jective of the semantic encoder is to encode the recognized explicit semantics into a suitable form that can be transmitted through physical channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The source user, for instance, can use the joint source-channel encoder proposed in [17] to encode the explicit semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this paper, we use the bold font v to denote the encoded version of the explicit embedding sent by the source user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 2) Semantic decoder at the destination user: The main objective of the semantic decoder is to learn an inference rule that can generate implicit semantic terms based on the signal received from the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Suppose the encoded explicit semantic symbol sent by the source user is given by v, we can then write the received explicit semantic symbol received at the destination user as: vd = Hv + N, (3) where H is the channel gain and N is the additive noise received at the semantic decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Once received the noisy version of explicit semantics vd, the semantic decoder will output the most possible implicit semantics terms related to ˆv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We can therefore write the semantic decoder as a mapping function with parameters θd, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=', a graph neural network (GNN) with parameters θd, denoted as πd(ˆuv|ˆv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' θd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The output of the semantic decoder is the most possible ˆuv when observing vd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' For example, if we adopt a graph softmax based approach for the semantic decoder to estimate the inference probability, the probability for inferring ˆuv ⊆ Rv when observing ˆv can be calculated as πd (ˆuv | ˆv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' θd) = exp � θ⊤ d (ˆuv) · θd(ˆv) � � ˆuv⊆Rv exp � θ⊤ d (ˆuv) · θd(ˆv) �, (4) where θd(ˆuv) and θd(ˆv) are γ-dimension representation vector of ˆuv and ˆv, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The semantic decoder can be iteratively trained using SGD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=', we can calculate using θt d = θt−1 d − ξ∇θdΓd, (5) where θt d are the parameters of semantic decoder at t-th iteration, ξ is the learning rate and Γd is the semantic distance 3=3 =(oq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='a)b0()received from the semantic evaluator at the source user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' More specifically, at the beginning of the training phase, the semantic decoder randomly picks up implicit semantic terms in Rv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The selected implicit semantic terms will be feedback to the semantic evaluator at the source user for comparison with the expert message samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The calculated semantic distance will be sent to the semantic decoder for correcting its inference results and updating its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The decoder will then sample the possible implicit semantic terms in the next iteration according to the updated inference rule πd (ˆuv|ˆv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' θt d) calculated by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' As will be proved later, the finally trained inference rule at the semantic decoder will be able to approach the true inference rule of the message source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Note that, during the training phase, the semantic decoder only needs to feedback the indices of the inferred implicit semantic terms and therefore the communication overhead of the training phase is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Also, after the training phase, the semantic decoder will be able to directly generate the implicit semantics sent by the source user without incurring any feedback or communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 3) Semantic evaluator at the source user: The evaluator, with the objective of maximizing (2), evaluates the perfor- mance of the decoder in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In other words, the ability of differentiating the semantic distance between the implicit semantics generated by the source user based on π and that estimated by the destination user using πd will be enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The output p(uv | v) of the semantic evaluator is a probability of the connection existing between uv and v p (uv | v) = 1 1 + exp (−θe(uv)⊤θe(v)), (6) where θe is the parameters of the evaluator, and θe(uv), θe(v) are the γ-dimension representation vectors of semantic terms uv and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Any graph representation learning model, such as graph convolutional network (GCN) [18], can serve to facilitate the task of embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In this paper, we utilize a few graph convolutional layers, which are designed especially for data with graphical structure, to obtain the embeddings of knowledge entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The propagation process of the stacking layers can be written as: L(0) = X and L(l+1) = σ(ΦL(l)Wl), (7) where σ(·) is the Sigmoid function, L(l) is the output of layer l ∈ {0, 1, · · · , m} and X = [x1, · · · , xn] ∈ Rn×γ is the initial feature vector of n semantic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Φ = ˜D− 1 2 ˜A ˜D− 1 2 is a renormalized Laplacian matrix, where ˜Dii = � j ˜Aij, ˜A = A + I, I is an n × n identity matrix and A is the adjacent matrix of the semantic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Then the output of the final layer L(m) will be used as θe(v) to calculate the result of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Suppose p(uv|v) is differentiable with respect to θe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Then both the decoder and evaluator can be iteratively updated using SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' More specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' the semantic evaluator can be iteratively updated by θt e = θt−1 e − ξ∇θeΓe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' (8) Algorithm 1 iSAC Algorithm Input: The set of explicit semantic terms V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' the set of relevant implicit semantic terms Rv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' initial input feature vectors xv for semantic term v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' parameters of evaluator and decoder θe and θd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' learning rate ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' training iteration T Output: Inference rule π∗ d Initialization: Randomly initialize the parameters θe and θd For t = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' T do Encoder send the index of v to decoder and evaluator For decoder’s steps do Decode vd → ˆv Samples the most likely connected ˆuv by (4) Update θd: θt d = θt−1 d − ξ∇θdΓd end for For evaluator’s steps do Receive negative samples from decoder and derive some positive samples from original message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Calculate the gradient of Γ using (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Update θe: θt e = θt−1 e − ξ∇θeΓe end for end for Calculate π∗ d by (4) where the gradient of (2) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' θe is calculated as ∇θeΓe = � ∇θe log p (uv | v) , if uv ∼ Sπ(uv|v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' ∇θe (1 − log p (ˆuv | ˆv)) , if ˆuv ∼ Dπd(ˆuv|ˆv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' (9) Similarly, the semantic evaluator can calculate the gradient of (2) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' θd using the following equation: ∇θdΓd = ∇θd � v⊆V Eˆuv∼Dπd(ˆuv|ˆv) [log (1 − p (ˆuv | ˆv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' θe))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' (10) Note that, in each iteration, the semantic evaluator at the source user first receives the implicit semantic terms from the semantic decoder and updates its own parameters θe using (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The semantic evaluator will then send the calculated semantic distance value to the semantic decoder to update its parameters θd by (5) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The detailed algorithm is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Theoretical Analysis Let us now prove that the implicit semantics generated by the learned inference rule utilizing our proposed iSAC can approach that produced by the true inference rule of the source messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Suppose the inference rule that dominates the implicit semantics is a stationary process and, in each iter- ation, the semantic evaluator can always reach its optimal value given by θ∗ e(uv, v) = p(uv|v) p(uv|v)+p(ˆuv|ˆv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The probability distribution of implicit semantics generated by the learned inference rule Dπd(ˆuv|ˆv) approaches that generated by the true inference rule Sπ(uv|v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Proof: The proof proceeds similarly as the adversarial learning proposed in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We summarize the main idea due to 0 100 200 300 400 500 # of Iterations 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='3 Semantic Distance Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='Eva Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='Dec Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 2: Convergence rates of se- mantic evaluator and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 0 100 200 300 400 500 # of Iterations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='8 1 Inference Accuracy iSAC GAE VGAE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 3: Comparison of the infer- ence accuracy of implicit seman- tics at the destination user under iSAC, GAE, and VGAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' the limit of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' It can be directly observed that (2) is convex in Dπd(ˆuv|ˆv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' By substituting the optimal semantic evaluator into (2), the resulting function in (2) is also convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We can therefore apply the gradient descent update for Dπd(ˆuv|ˆv) and follow [19, Theorem 1], with sufficient small learning rate, Dπd(ˆuv|ˆv) can always converge to Sπ(uv|v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' EXPERIMENTAL RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Dataset and Experimental Setups To evaluate the performance of iSAC, we consider two real- world human knowledge datasets: arXiv-GrQc3 and Cora-ML, where arXiv-GrQc3 is a paper citation dataset based on arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' It consists of 5,242 vertices, corresponding to the authors, and 14,496 edges, specifying the collaborations between authors of papers in the general relativity and quantum cosmology categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Cora-ML is a dataset based on the paper topics in the field of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' It consists of 2,995 vertices, corresponding to feature vectors for each document, and 8,416 edges, specifying the citation links between documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' All experiments are performed on a Linux workstation (CPU: Intel(R) Xeon(R) CPU E5-2683 v3@ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='00GHz, GPU: four NVIDIA GeForce GTX 2080Ti (11GB)) and mainly using an open-source Python libraries, Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We set two-layer GCN at the semantic evaluator and set the learning rate of the SGD at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We use semantic decoder defined in (4) and randomly select the initialized parameters of the semantic decoder based on a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The dimensional size of both representation vectors of the semantic evaluator and decoder are set to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We randomly select 5% and 10% of the links to simulate the expert implicit data samples observed by the semantic evaluator that are connected with a selected number of explicit terms in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Experiment Results Let us now evaluate the performance of our proposed iSAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We compare iSAC with the following two solutions implemented to recover implicit semantics as the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' (1) Variational Graph Auto-encoder (VGAE)-based solution: The implicit semantics are first recovered by the semantic recognizer at the source user and then converted into low- dimensional embeddings using a GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The converted embeddings will then be sent to the physical channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 12 10 8 6 4 2 0 2 4 6 8 10 12 SNR (dB) 10-4 10-3 10-2 10-1 100 Symbol Error Rate No Inference GAE VGAE iSAC (a) 12 10 8 6 4 2 0 2 4 6 8 10 12 SNR (dB) 10-4 10-3 10-2 10-1 100 Symbol Error Rate No Inference GAE VGAE iSAC (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 4: Symbol error rate of semantic symbols (entities) when the inference rules learned by iSAC, GAE, and VGAE can be used in semantic symbol recovery, compared to the no inference solution, under datasets (a) arXiv-GrQc3 and (b) Cora-ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' The semantic decoder is also a GCN trained to recover the full implicit semantics based on the noisy version of the embeddings sent by the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' (2) Graph Auto-encoder (GAE)-based solution: This solution is almost the same as the VGAE-based solution with the only difference that the GCN in the semantic decoder has been replaced as a sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Let us first evaluate the convergence performance of iSAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 2, we consider dataset arXiv-GrQc3 and present the loss function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=', the semantic distance Γ, optimized by semantic evaluator and decoder using SGDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We can observe that the semantic evaluator and decoder of our proposed iSAC can always converge to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 3, we compare the accuracy of the inference rule learned by the semantic decoders of iSAC with that of VGAE and GAE-based solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We can observe that our proposed iSAC can always achieve the highest accuracy level for infer- ring the implicit semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In particular, with 100 iterations, VGAE and GAE-based solutions can achieve 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='61% and 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='79% inference accuracy levels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Our proposed iSAC is able to achieve the accuracy of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='01%, bringing over 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='40% and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='22% improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' It can be observed that the inference rule learned by the destination user can also be applied to recover semantic in- formation corrupted during the physical channel transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In particular, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 4, we compare the semantic symbol error rate under different inference rules learned by iSAC, VGAE and GAE-based solutions using two different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We can observe that for both datasets, our proposed iSAC always offers the lower symbol error rate, compared to VGAE and GAE-based solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In particular, when the SNR of the destination user is 2 dB, iSAC offers 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='69 dB improvements in terms of symbol error rate over the traditional data-oriented communication solutions without any semantic inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' When comparing to the VGAE and GAE-based solutions, the proposed iSAC can offer 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='73 dB and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='26 dB improvements, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We can also observe that the inference accuracy of all three solutions are better in arXiv-GrQc3, compared to Cora-ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' This is because the semantic terms are more closely linked (with higher degree) in arXiv-GrQc3 than that in Cora-ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In other words, our proposed inference-rule- based implicit semantic communication solutions are more suitable for message sources consisting of closely linked iSAC GAE VGAE No 0 200 400 600 800 # of Symbols 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='8 1 Ratio 50 318 249 360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='1389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='8833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='6917 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='0 Inference (a) iSAC GAE VGAE No 0 200 400 600 800 # of Symbols 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='8 1 Ratio 64 513 427 640 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='6672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='8016 Inference (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 5: Number of required transmitted symbols for recovering the same amount of implicit semantics at the destination users as well as the corresponding ratios compared to the no inference solution under iSAC, VGAE and GAE-based solutions, under datasets (a) arXiv-GrQc3 and (b) Cora-ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' semantic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' To evaluate the communication efficiency of iSAC, we compare the required number of fixed dimensional-sized trans- mitted symbols for recovering the same amount of implicit semantics at the destination user in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' 5 when implementing iSAC, VGAE and GAE-based solutions in arXiv-GrQc3 and Cora-ML datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We can observe that, compared to the non- inferential solution, iSAC, VGAE and GAE-based solutions can achieve over 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='89%, 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='33% and 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='17% improvements in terms of compression rate of transmitted symbols in dataset arXiv-GrQc3 and over 10%, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='16%, and 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='72% improvements in dataset Cora-ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Also, compared to VGAE and GAE-based solutions, iSAC achieves around 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='28% and 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='92% reductions in the total number of required transmitted symbols, respectively, in arXiv-GrQc3, and around 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='52% and 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='01% reductions, respectively, in Cora-ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' This is because, in our proposed iSAC, the source user only needs to send the explicit (clue) semantics to the destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' In the VGAE and GAE-based solutions, however, the source user needs to first recover all the implicit semantics and then convert all these semantics into the low-dimensional embed- ding representations for the physical channel transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' CONCLUSION This paper has considered the implicit semantic communi- cation problem in which, instead of sending explicit semantics, hidden relations and closely related semantic terms that cannot be recognized from the source signals need also be delivered to a destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We have developed a novel adversarial learning-based iSAC architecture in which the source user tries to assist the destination user to learn an inference rule that can automatically generate implicit semantics based on limited clue information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' We have proved that by applying iSAC, the destination user can always learn an inference rule that matches the true inference rule of the source messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Our experimental results have shown that the proposed iSAC can offer up to 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='69 dB improvement over existing non- inferential communication solutions, in terms of symbol error rate of the destination user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' ACKNOWLEDGMENT Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Xiao and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Shi were supported in part by the major key project of Peng Cheng Laboratory under grant PCL2021A12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Xiao was supported in part by the National Natural Science Foundation of China under grant 62071193 and the Key R & D Program of Hubei Province of China under grants 2021EHB015 and 2020BAA002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Shi was supported in part by the National Natural Science Foundation of China under grants 62293483, 61871304, and 61976169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Chen was supported in part by the Zhejiang Lab Open Program under Grant 2021LC0AB06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Bennis was supported in part by the SNC project under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' SCR6GE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Poor was supported in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' National Science Foundation under Grants CCF-1908308 and CNS-2128448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' REFERENCES [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFJT4oBgHgl3EQfmyyq/content/2301.11589v1.pdf'} +page_content=' Xiao and M.' metadata={'source': 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-0,0 +1,1553 @@ +1 +EMAHA-DB1: A New Upper Limb sEMG Dataset +for Classification of Activities of Daily Living +Naveen Kumar Karnam, Anish Chand Turlapaty, Member, IEEE, Shiv Ram Dubey, Senior Member, IEEE and +Balakrishna Gokaraju, Member, IEEE +Abstract—In this paper, we present electromyography analysis +of human activity - database 1 (EMAHA-DB1), a novel dataset +of multi-channel surface electromyography (sEMG) signals to +evaluate the activities of daily living (ADL). The dataset is +acquired from 25 able-bodied subjects while performing 22 activ- +ities categorised according to functional arm activity behavioral +system (FAABOS) (3 - full hand gestures, 6 - open/close office +draw, 8 - grasping and holding of small office objects, 2 - flexion +and extension of finger movements, 2 - writing and 1 - rest). The +sEMG data is measured by a set of five Noraxon Ultium wireless +sEMG sensors with Ag/Agcl electrodes placed on a human hand. +The dataset is analyzed for hand activity recognition classification +performance. The classification is performed using four state-of- +the-art machine learning classifiers, including Random Forest +(RF), Fine K-Nearest Neighbour (KNN), Ensemble KNN (sKNN) +and Support Vector Machine (SVM) with seven combinations of +time domain and frequency domain feature sets. The state-of-the- +art classification accuracy on five FAABOS categories is 83.21% +by using the SVM classifier with the third order polynomial +kernel using energy feature and auto regressive feature set +ensemble. The classification accuracy on 22 class hand activities +is 75.39% by the same SVM classifier with the log moments in +frequency domain (LMF) feature, modified LMF, time domain +statistical (TDS) feature, spectral band powers (SBP), channel +cross correlation and local binary patterns (LBP) set ensemble. +The analysis depicts the technical challenges addressed by the +dataset. The developed dataset can be used as a benchmark for +various classification methods as well as for sEMG signal analysis +corresponding to ADL and for the development of prosthetics and +other wearable robotics. +Index Terms—Machine learning, Classification Algorithms, +Surface Electromyography (sEMG), Activities of Daily Living +(ADL), Features, Dataset and Benchmark. +I. INTRODUCTION +P +ERFORMING hand movements during activities of daily +living (ADL) [1] without any difficulty provides func- +tional independence and a decent quality of life [2]. However, +it is quite difficult to perform simple hand movements for +individuals affected by the following disorders: upper limb +disabilities [3], [4], disorders related to aging [5], neuromus- +cular disorders [6], and stroke related disabilities [7], [8], [9], +[10]. Human computer interfaces and human robot interfaces +N.K. Karnam and A.C. Turlapaty are with the Biosignal Analysis Lab at +the Indian Institute of Information Technology, Sri City, A.P., India (email: +anish.turlapaty@iiits.in). +S.R. Dubey is with the Computer Vision and Biometrics Laboratory at +Indian Institute of Information Technology, Allahabad, Prayagraj-211015, +U.P., India (email: srdubey@iiita.ac.in). +B. Gokaraju is with the Visualizations and Computing Advanced Research +Center (ViCAR) and Department of Computational Data Science an Engineer- +ing, North Carolina A and T State University, Greensboro, North Carolina +(email: bgokaraju@ncat.edu). +can support the rehabilitation process to recover from the +above mentioned disorders. For instance, hand gesture-based +interfaces based on computer vision techniques for identifying +and classifying gestures are currently under development [11]. +Moreover, many researchers have explored robotic control +using visual gestures [12], [13], [14]. However, vision based +control methods are inadequate to determine the appropriate +control for actuation and the amount of force exerted by a +muscle during action. One approach to quantify the upper +limb activity is to use wearable sensors such as inertial +motion sensors (IMUs) including accelerometers, gyroscopes +and magnetometers. These sensors are utilised to measure +and monitor limb activities, quantify muscle motor deficits +[15], and classify the types of physical activity [16], [17]. +Although wearable sensors can recognize human activity, they +are deficient in precise identification of hand gestures, finer +finger movements and the amount of muscle strength used to +execute the movement [18]. +Alternatively, hand movement classification and the limb +control [19], [20] through surface electromyography (sEMG) +signals facilitates the design of prosthetic devices, exoskeleton +arms, advanced realistic bio-mechanical models, and rehabili- +tation therapies [21]. In these applications, utilization of multi- +modal signals is also very common. In the literature, fusion of +the IMU’s and sEMG signals [22], [23], [24] for hand activity +classification and estimation of the continuous orientation of +the forearm is analyzed. The electroencephalography signals +(EEG)) and sEMG signals are also fused to decode the +intention of the person. This fusion process can generate better +control signals compared to a standalone sEMG signal based +control [25], [26], [27]. In order to obtain better classification +accuracies the features from sEMG signals can be fused with +those from the vision based image classification network [28]. +In practice, the multi-modal methods increase the complexity +of the hardware as well as software systems, hence they pose +difficulty for different real-life applications. Hand movement +analysis and classification through standalone sEMG signals +is gaining attention [29], [30], [31], [32], [33] and is the focus +of our current work. +In this paper, we present electromyography analysis of +human activity - database 1 (EMAHA-DB1), a novel sEMG +dataset on ADL for the Indian population. Following are the +salient features of EMAHA-DB1: +• There are several sEMG datasets available that include +activities such as hand gestures, hand movements, wrist +movements, and grasping objects. These datasets are +mainly collected for western populations and there is no +arXiv:2301.03325v1 [eess.SP] 9 Jan 2023 + +2 +dataset for ADL from the Indian population. EMAHA- +DB1 fills this gap. +• There is a tradition of anthropometric data collection in +India [34], [35]. For any population, there is an influence +of anthropometrics on their kinematics and kinetics [36], +[37]. EMAHA-DB1 will compliment existing anthropo- +metrics, kinematics and kinetics datasets [30], [37] which +will be helpful in conducting upper limb rehabilitation +therapies, physiological studies and clinical studies for +Indian population. +• The ADL performance is analyzed by grouping the +actions according to the functional arm activity behav- +ioral observation system (FAABOS [38]). The functional +taxonomy provided by Uswatte et al. quantifies group of +hand actions based on the behavioral significance. +• There are publicly available ADL datasets such as the +NinaPro [39], the BioPatRec [40], the Ramikushaba [41] +and the UCI Gesture [42] that have not covered a few +important ADL categories. The hand activities are usually +performed in an experimental set up with a fixed duration +for each of the activities, however we have considered +different durations for distinct activities to approximate +corresponding durations of real time hand movements. +• The dataset can be used to benchmark classification +algorithms or perform statistical studies. The developed +dataset consists of a larger number of subjects and a +higher number of activity repetitions compared to any +other publicly available ADL datasets. +The main contributions of the paper are: +1) In this work, we have carried out muscle activity mea- +surements corresponding to activities of daily living and +collected a novel multichannel sEMG data from Indian +population. +2) The EMAHA-DB1 dataset is organized according to +custom FAABOS functional categories to perform anal- +ysis using state-of-the-art machine learning classifiers. +Specifically the sEMG signals are analyzed to classify +into the functional groups as well as individual activities. +3) We also perform extensive feature analysis with respect +to the FAABOS functional categories. +The rest of the paper is organised as follows: Section II +details about the proposed EMAHA-DB1 dataset; Section +III presents experiments in machine learning frameworks; +Section IV demonstrates the experimental results; and Section +V provides a conclusion along with the future scope. +II. EMAHA-DB1: PROPOSED SEMG DATASET +A. Data Collection +1) Study participants: The institutional ethics committee +of Indian Institute of Information Technology Sri City (No. +IIITS/EC/2022/01) approved the proposed data collection pro- +tocol developed in general accordance with the declaration of +Helsinki and specific accordance with the “National Ethical +Guidelines for Biomedical and Health Research involving hu- +man participants” of India. Twenty-five healthy subjects with +no history of upper limb pathology, including 22 males and 3 +females, participated in the sEMG data collection process. The +TABLE I: List of hand activities +Activity No. +Activity description +A0 +Hand at rest (sitting) +A1 +Tossing a coin (sitting) +A2 +Finger snapping (sitting) +A3 +Pulling an empty draw - Posterior view (sitting) +A4 +Pulling a draw with weight (2kg) - Posterior view (sitting) +A5 +Pulling an empty draw - Anterior view (sitting) +A6 +Pulling a draw with weight (2kg) - Anterior view (sitting) +A7 +Pushing an empty draw - Posterior view (sitting) +A8 +Pushing a draw with weight (2kg) - Posterior view (sitting) +A9 +Clasping both hands (sitting) +A10 +Hand clapping (sitting) +A11 +Grasping and holding 1L water bottle (sitting) +A12 +Grasping and holding small hammer (sitting) +A13 +Grasping and holding small saw (sitting) +A14 +Writing the phrase ”Bio signal lab” on paper with pen - +lateral grasp (sitting) +A15 +Writing the phrase ”Bio signal lab” on board with marker +- lateral grasp (standing) +A16 +Lifting a small bucket with 4L water (standing) +A17 +Typing the phrase ”Bio signal lab” on keyboard using +single finger (sitting) +A18 +Drinking tea/water from a cup - lateral grasp (sitting) +A19 +Picking up the phone, placing it to his/her ear and hanging +up the phone on table (sitting) +A20 +Grasping and holding a book (sitting) +A21 +Grasping and holding a tennis ball (sitting) +TABLE II: Sensor placement on hand muscle +Channel No. +Sensor No. +Hand muscle name +1 +21621 +Brachio Radialis (BR) muscle +2 +21623 +Flexor Carpi Radialis(FCR) muscle +3 +21624 +Flexor Carpi Ulnaris (FCU) muscle +4 +21625 +Biceps Brachii (BB) muscle +5 +21626 +Abductor Pollicis Brevis (APB) muscle +average age is 28±6 years. Before the first session of activities, +each of the participants gave written informed consent and the +data collection process is completely non-invasive. +2) Experimental setup and acquisition protocol: The 22 +activities performed by each subject are listed in Table I. +Each of the hand muscle activity is recorded with a 5-channel +Noraxon Ultium wireless sEMG sensor setup [43] as shown +in Fig. 1. Five self-adhesive Ag/AgCL dual electrodes were +placed at the centre of the five most representative muscle sites +of the right arm as shown in Fig. 1. Each subject is instructed +to sit comfortably with one elbow resting on a table and an arm +flexed 90◦ compared to the forearm. The muscle locations are +selected according to the atlas in chapter 17 [44] and is given +in Table II. At the beginning of each session, the participant’s +hands are cleaned with an alcohol based wet wipe. +Prior to each session, the subject is acquainted with the +experiment protocol including a video demonstration of the +proposed activities. The total duration of each session is up-to +one hour per subject depending on adaptability. Each activity +is performed for a maximum duration of 10s and repeated +10 times. There is a rest period of 5s between each of the +repetitions and a 30s gap between the sessions of different +activities. Each of the activities consists of two phases: (1) an +action and (2) rest. However, some of the activities included +an extra release phase. During the action phase, the subject +performs the corresponding activity; during the release phase, +the subject transitions from the action state to rest state; and +during the rest phase, the subject completely relaxes each of + +3 +Fig. 1: Learning steps from sEMG dataset collection to classification of hand activities +TABLE III: Phase-wise durations of each activity. +No. +TX TA TR TT +No. +TX TA TR TT +No. +TX TA TR TT +A1 +3 +5 +0 +8 +A8 +3 +5 +0 +8 +A15 3 +15 +2 +20 +A2 +3 +5 +0 +8 +A9 +3 +5 +0 +8 +A16 5 +5 +3 +13 +A3 +3 +5 +0 +8 +A10 3 +5 +0 +8 +A17 3 +10 +2 +15 +A4 +3 +5 +0 +8 +A11 3 +5 +3 +11 +A18 5 +5 +3 +13 +A5 +3 +5 +0 +8 +A12 3 +5 +3 +11 +A19 5 +5 +3 +13 +A6 +3 +5 +0 +8 +A13 3 +5 +3 +11 +A20 5 +5 +3 +13 +A7 +3 +5 +0 +8 +A14 3 +10 +2 +15 +A21 5 +5 +3 +13 +his/her muscles. The time duration for each activity is given +in Table III, where TX, TA, TR, and TT are the rest, action, +release, and total duration, respectively. +3) Comparisons with existing datasets : The characteristics +of EMAHA-DB1 data are compared against those of existing +sEMG hand activity datasets in TABLE IV. Apart from those +mentioned in salient features in Introduction, a few additional +and distinct characteristics of the EMAHA-DB1 are: 1) the +experiments are designed such that hand activities performed +consists of three phases of action (contraction/relaxation of +muscles), release (retreating of action), and rest (relaxing of +muscles), 2) the measurements are acquired with a minimal +number of sensors hence requires lower computational re- +sources compared to the existing datasets. +B. Data Preparation +1) Activity +segmentation: +For +the +sEMG +signals +in +EMAHA-DB1, the preliminary annotations for onset and +offset of the actions are performed based on the respective +durations of action phases shown in Table III. To improve the +quality of activity labels, based on the procedure developed in +[46], an improved signal segmentation process (listed below) +is implemented: +1) Initially, for each trial of each activity performed by each +subject, the multi-channel signal is rectified. +2) For each of these trials, the maximum and minimum +values are identified to determine the range R. +3) The signal strengths at R/ +√ +2 (3dB amplitude) are +considered the thresholds on either side. +4) The first signal strength, past the preliminary onset, +crossing the 3dB threshold is identified for each channel. +The earliest location among the threshold crossings from +the five channels is considered the onset of action. +5) The trial data is parsed backwards from the end of the +action. The first point from the end i.e., the final 3dB +crossing is identified for each channel. The right most +location among the crossings from these channels is +labelled the offset of action. +6) Finally, the signal samples between the onset and the +offsets are annotated as the action and assigned the +corresponding activity number, and the remaining signal +is considered to be rest state. +The above procedure is illustrated in Fig. 2 for a single trial +of ADL. It is observed that signal segmentation improves the +annotation process of activity vs. rest which further improves +veracity of the classification process. +2) FAABOS categories: The EMAHA-DB1 is mapped ac- +cording to function arm activity behavioral observation system +(FAABOS) [38], [29]. Specifically, actions in the EMAHA- +DB1 are reorganized into the following five major groups: +1) No object action, 2) object holding, 3) object grasping, 4) +Flexion and Extension of Fingers, and 5) writing. The action +categories that are mapped into these groups are listed in Table +V. + +Multi-channel sEMG signals +Output hand activity classification +A sample of hand +movements performed in +Learning +Activities of Daily Living +kinematic +(ADL) +characteristics of - +uu m d +the hand activities +ML classifier +Testing +training (KNN, RF, +SKNN, SVM3) +Grasping and holding +Grasping and holding +small hammer +a book +Preprocessing +O +Training +Notch filtering at 50Hz +Feature set visualisation by +sEMG Data acquired for +Low pass filtering with +fc = 500Hz +t-SNE plot +ADL by Noraxon Ultium +Wavelet denoising +Feature extraction with six +sEMG sensor setup +4 +feature sets of F0, F1, F2, F3, +Trial wise +segmentation +3 +F4, F5, and F6 +2 +LO +1 +0 +-1 +Data train and test split-up +3 +Train data with +Test data with +4 +trial no. +trial no. 2, 5 +1,3,4,6,8,9 and 10 +and 7 +6 +Datset is curated and +-4 +-2 +0 +Dimension1 +labelled using audio cue + Relabelled by an +algorithm4 +TABLE IV: Comparisons of basic data characteristics with benchmark datasets +Dataset +Name +Action categories +Sensor +No. +of +Subjects +(S) +No. +of +activities +(NA) +(including +rest) +No. +of +channels +(Nc) +Sampling +frequency +(Ns) +(samples +per sec) +Rest +dura- +tion +(TX)(s) +Action +dura- +tion +(TA)(s) +Release +dura- +tion +(TR)(s) +No. +of +repe- +titions +(NR) +Total +no. +of +pat- +terns +(N) +NinaPro +DB1 [39] +Gestures, Wrist move- +ments, +and +Grasping +Objects +Otto Bock +27 +53 +10 +100 +3 +5 +- +10 +14310 +NinaPro +DB2 [39] +Gestures, Wrist move- +ments, Grasping Ob- +jects, and Finger press- +ing movements +Delsys Trigno +wireless +40 +50 +12 +2000 +3 +5 +- +6 +12000 +NinaPro +DB4 [45] +Gestures, Wrist move- +ments, +and +Grasping +Objects +Cometa Mini- +Wave +10 +53 +12 +2000 +3 +5 +- +6 +3180 +BioPatRec +DB2 [40] +Gestures, +and +Wrist +and hand movements +Thalmic +myoarm band +17 +27 +8 +2000 +3 +3 +- +3 +1377 +UCI Ges- +ture [42] +Wrist and hand move- +ments +Myo Thalmic +bracelet +36 +7 +8 +1000 +3 +3 +- +4 +1008 +Rami- +kushaba +DB6 [41] +Hand movements +Delsys DE +2.x series +EMG sensors +11 +40 +7 +4000 +3-5 +5 +- +6 +2640 +EMAHA- +DB1 +(Our +dataset) +Daily activities - +Grasping and +holding, writing, and +draw open/close +Noraxon Ul- +tium +sEMG +sensor +25 +22 +5 +2000 +3-5 +5-15 +3-5 +10 +5500 +Fig. 2: Illustration of manual segmentation of sEMG signals for a trial of ADL +TABLE V: FAABOS groups of activities. +Group label +Group Name +Activity No. +0 +Rest +A0 +1 +No object action +A2, A9 and A10 +2 +Hold object +A3, A4, A5, A6, A7 and A8 +3 +Object grasping +A11, A12, A13, A16, A18, +A19, A20 and A21 +4 +Flexion +and +Exten- +sion of Fingers +A1 and A17 +5 +Writing +A14 and A15 +III. METHODOLOGY +A. Problem Statement +The total number of sEMG patterns in the EMAHA-DB1 +is N = S × NA × NR, where S is the total number of +subjects, NA is the number of different activities, and NR +corresponds to the number of repetitions per action per subject. +The proposed sEMG dataset can be represented as: +x = {xn}N +n=1 +(1) +where each observation array xn consists of multiple channels +as: +xn = {xn,m}NC +m=1, +n = 1, · · · , N +(2) +TABLE VI: Summary of extracted features +Feature +Set +Features +Feature Length +F0 [47] Mean Absolute Value (MAV), Temporal Spec- +tral Energies (TSE) and Spectral Band Ener- +gies (SBE) +1×NC, 4×NC, +and 4 × NC +F1 [48] MAV, Zero Crossings (ZC), Slope Changes +(SC), and Wavelength (WL) +1×NC, 1×NC, +1×NC, and 1× +NC +F2 [49] F1 and Auto Regression Coefficients (ARC) +9 × NC and 2 × +NC +F3 [50] F1, Myopulse Rate (MPR), Willison Ampli- +tude (WAMP), and Cardinality +9×NC, 1×NC, +1×NC, and 1× +NC +F4 [51] Log moments in frequency domain (LMF) +5 × NC +F5 [52] F4, modified LMF, Time domain statistics +(TDS), Spectral Band Powers (SBP), Max +channel cross correlations, and Local Binary +Patterns (LBP) +5 × NC, 10 × +NC, 4 × NC, +4×NC, 2×NC, +and 2 × NC +F6 [53] Root Mean Square (RMS), Time Dependent +Power spectrum Descriptors (TD-PSD) [51], +Difference Absolute Standard Deviation Value +(DASDV), and Difference Absolute Mean +Value (DAMV) +1×NC, 6×NC, +1×NC, and 1× +NC +where NC is the number of channels (from different elec- +trodes) and each of these channels consists of an array as: +xn,m = {xn,m(i)}NT +i=1 +(3) +where NT = Ns × TT is the number of values in one trial of +duration TT and Ns is the sampling rate (samples/sec). For a +given trial, for feature extraction purposes, the signal is divided +into Nseg segments. Each segment sg consists of an array as: +sj +g = {xn,m(i)}Ng +i=1 +j = 1, · · · Nseg +(4) +where Ng is the number of samples in one segment such that +NT = Nseg × Ng. +The objective of this study is to map the segmented sEMG +signals to the corresponding activity labels (i.e., tg - targets), + +Channel 3 +cue-ON +6 +Cue-OFF +OFF +40 +SEMG +20 +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +16000 +Channel 4 +10 +cue-OFFl +cue-ON +NO +OFF +sEMG Amp +５ +0 +8000 +2000 +4000 +6000 +10000 +12000 +14000 +16000 +0 +Rest vs. Action +Action-OFF +SEMG +Action- +norm +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +16000 +0 +Sample Index5 +(a) +(b) +(c) +(d) +(e) +Fig. 3: Performance comparison (a) with different Feature Ensembles with Cubic SVM (Polynomial SVM of order 3), (b) with different classifiers for the benchmark feature +ensemble F5, (c) against benchmark frameworks, (d) against benchmark frameworks in terms of various metrics, and (e) against FAABOS categories frameworks. +TABLE VII: Numerical setup for classifiers. +Classifier +Model Setup +Fine KNN +No.of neighbours = 5, Distance Metric = Cityblock, Dis- +tance weight = Squared Inverse +Ensemble +KNN +No.of learning cycles = 30, learners = KNN, Subspace +dimension = 25 +Cubic SVM +Polynomial kernel, Order = 3, Box constraint = 1, Multi- +class Method = one-vs-one +Random Forest No.of bags for bootstrapping = 300 +which can be formulated as: +f{sg} → tg +(5) +where tg denotes targets (group labels) as specified in TABLE +V or individual activity labels as specified in TABLE I. The +mapping function in (5) is implemented by a supervised +classifier. For the mapping function, appropriate features are +required that represent the underlying inverse kinematic rela- +tionships between the sEMG signals and the corresponding +activity performed. +B. Feature Extraction +In this work, the following feature sets are adapted from +[47]: F0, F1, F2, F3, F4, and F5 with an additional feature +set F6 consisting of root mean square (RMS), time dependent +power spectrum descriptors (TD-PSD), difference absolute +standard deviation value (DASDV), and difference absolute +mean value (DAMV). Note the features are computed for each +segment and concatenated to build the full feature vector. The +extracted feature sets are summarized in Table VI. +C. Supervised Classifiers +In this paper, four algorithms including random forest (RF), +fine K-nearest neighbour (FKNN), ensemble KNN (sKNN) +and cubic support vector machine (SVM3) are considered for +sEMG signal classification task. The classifiers are trained and +TABLE VIII: Feature ensemble vs benchmark classifier setup. +FE FL +BF +Classifier +FE FL +BF +Classifier +F0 +9 × NC +B0 [47] +Fine KNN +F4 +5 × NC +B4 [54] +SVM3 +F1 +4 × NC +B1 [48] +SVM3 +F5 +27×NC +B5 [52] +SVM3 +F2 +11×NC +B2 [49] +Fine KNN +F6 +9 × NC +B6 [39] +RF +F3 +12×NC +B3 [50] +SVM3 +tested with subject-wise data and the average performance is +reported. The hyperparameter settings for different machine +learning algorithms used in this work are summarized in Table +VII. The performance of classifiers is evaluated using the +standard metrics such as cross validation accuracy (α), testing +accuracy (β), Kappa coefficient (κ), precision (γ), recall (ρ) +and F-1 score (F1). +IV. CLASSIFICATION EXPERIMENTS, RESULTS & +ANALYSIS +The developed EMAHA-DB1 sEMG dataset is analyzed +using the state-of-the-art classification and feature extraction +methods as detailed below. +A. Pre-processing and Data Split-up +Based on the procedure described in [55], the recorded +sEMG data is pre-processed as follows. First, the sEMG data is +filtered to remove power line noise at 50Hz. Then a first order +Butterworth low pass filter is applied at a cut-off frequency of +500Hz. Finally, wavelet denoising of order 8 with the symlet +as the mother wavelet is applied. The data from each subject +is split trials-wise into 70% for training and 30% for testing as +per the splitting method in [55]. A non overlapping moving +window segment of Ng = 200 samples is considered with +duration Tseg = 100ms. The number of features obtained per +segment sg are summarized in Table VIII. + +80 +Cross Validation (α) +Testing (B) +75 +(%) +Accuracy +70 +65 +60 +F2 +F3 +F1 +F4 +F5 +F6 +F0 +Feature sets80 +Cross Validation (α) +Testing (β) +75 +% +70 +Accuracy +65 +60 +55 +RF +SKNN +SVM3 +FKNN +Classifiers80 +Cross Validation (α) +Testing (β) +75 +(%) +Accuracy +70 +65 +60 +B1 +B2 +B3 +B4 +B5 +B0 +B6 +Feature sets0.8 +-Precision ()kappa () +*F, score (F,) +Recall (p) +0.75 +Metric +0.7 +rmance +0.65 +0.6 +0.55 +B2 +B3 +B4 +B5 +B6 +B0 +B1 +Benchmarks85 +Feature set F0 +Feature set F5 +Feature set F2 +80 +75 +70 +65 +RF +SKNN +SVM3 +FKNN +Classifier6 +TABLE IX: Muscle vs action mapping. +Muscle +Major functionality of the mus- +cle +Biceps Brachii (BB) muscle +Flexes elbow joint, Supinates fore- +arm and hand at radioulnar joint +Brachio Radialis (BR) muscle +Flexes elbow joint +Flexor Carpi Radialis (FCR) muscle +Flexes and abducts hand at wrist +Flexor Carpi Ulnaris (FCU) muscle +Flexes and adducts wrist +Abductor Pollocis Brevis (APB) muscle Abducts joints of thumb +B. Experiments +In this paper, as mentioned earlier two case studies are +carried out as follows, 1) classification of individual action +categories listed in Table I, in this case study, the performance +is analyzed with respect to feature ensembles, classifiers, +benchmark classification frameworks and finally feature vi- +sualization; 2) classification of FAABOS categories listed in +Table V, in the second case study, the performance is analyzed +with respect to feature ensembles followed by an analysis of +the most relevant features with respect to the muscle sites. +C. Case Study 1: Results and Analysis +1) Comparison with feature ensembles: The feature sets +F0-F6 are analyzed in this comparison study. Each of the +feature set is utilised as input for SVM3 and their performance +metrics α and β are evaluated. As shown in Fig. 3a, the best +performance is produced by the feature set F5 (α = 77.42 and +β = 75.39). The next best feature ensemble F2 lags behind by +0.3% at α = 78.06 and β = 75.09. The feature ensemble F6 +has produced the least classification performance (α = 66.68 +and β = 66.79). +2) Comparison with classifiers: In this experiment, the +classification performance of the standard machine learning +algorithms such as the RF, FKNN, sKNN and SVM3 using +the F5 feature set is analyzed. As shown in Fig. 3b, the best +performance is produced by the SVM3 classifier (α = 77.42 +and β += 75.39) and then by FKNN (α = 74.83 and +β = 72.42). The least performance is obtained with SKNN +classifier (α = 58.4 and β = 58.3). Thus, it is observed from +this experiment that for the feature set F5 the SVM3 classifier +outperforms other benchmark classifiers. +3) Comparison with benchmark algorithms: The most suit- +able classification framework for the EMAHA-DB1 is deter- +mined by comparisons with the existing sEMG benchmark +classification methods consisting of respective combinations +of a feature ensemble and a classification framework as listed +in Table VIII. The benchmark Bi indicates the classification +framework with feature set Fi for i = 0, 1, · · · , 6. The param- +eter setups of the different classifiers used in the numerical +analysis are also shown in Table VII. The performance of these +classifiers is analyzed based on the cross validation accuracy +(α) and the test accuracy (β) with the corresponding results +shown in Fig. 3c. The benchmark B5 has achieved state-of- +the-art performance with α = 77.42 and β = 75.39. The lowest +performance among the compared benchmarks is for B6 with +α = 74.2 and β = 69.04. The other performance metrics (i.e., κ, +γ, ρ, and F1) of the benchmark frameworks are shown in Fig. +3d. The benchmark framework F5 has achieved highest values +for each of the performance metrics, i.e., κ = 0.73, γ=0.66, +ρ=0.71, and F1 = 0.68. The runner-up is B6 framework with +metric values κ = 0.66, γ= 0.60, ρ= 0.64, and F1 = 0.66. +4) Feature Visualization by t-SNE: The following analysis +is meant for the 22 individual action categories however +carried out FAABOS group wise. Among the feature sets F0 +to F6, it is observed that F5 is the best performing feature +set, hence used for sequential feature selection analysis (SFS). +From SFS, the most relevant features for each group of hand +activities are identified and further used for analysis with +t-distributed Stochastic Neighbourhood Embedding (t-SNE) +[56]. The top 6 feature columns of 84, 85, 96, 97, 101, and +105 are used in this study. The columns with higher ranking +are 84 and 85 that correspond to the features of mean and +variance respectively (from TDS feature set), and 96, 97, 101, +and 105 that correspond to the spectral bands [0 (Ns/8)] and +[(Ns/8) (Ns/4)] of SBP feature set [47]. The flexion and +extension of elbow and wrist flexion and extension are mainly +supported by the muscle groups BB, BR, FCR and FCU [57] +as given in TABLE IX. The action categories in group 2 and +group 3 involve the common muscle movements including +elbow flexion and extension, wrist flexion and extension and +pronation and supination as shown in TABLE X. Hence From +Fig. 4b and 4c, the clusters for some of the actions overlap +due to involvement of similar muscle groups across actions +with same basic muscle movements. The actions within group +1, group 4 and group 5 are clearly separable which can be +observed from Fig. 4a, Fig. 4d and Fig. 4e, respectively. +D. Case Study 2: Results and Analysis +1) Comparison of FAABOS categories with feature ensem- +bles: The sEMG signals from the EMAHA-DB1 are classified +based on FAABOS categories specified in Table V. The six +FAABOS categories of sEMG signals are trained and tested +with the top three feature sets such as F0, F2 and F5 and +the corresponding results are plotted in Fig. 3e. The best +performance is produced by the SVM3 classifier (α = 86.54 +and β = 83.21) with the feature set F2. The next best +performance is produced by the same SVM3 classifier (α = +85.85 and β = 83.14), but with the feature set F5 having a +slight variation of 0.07%. The least performance is observed +for feature set F0 with SKNN classifier (α = 85.66 and β = +82.39). +2) Feature Visualization by t-SNE for FAABOS groups: +This analysis is carried out for 6 FAABOS categories. Among +the feature sets F0, F2, and F5, it is observed that F2 is the +best performing feature set and used for SFS analysis. From +SFS, the top 6 feature columns 1, 3, 4, 5, 19, and 23 are +used in this study. The columns with higher ranking are 1, +3, 4, and 5 that correspond to mean absolute value (MAV), +19 corresponds to the waveform length, and 23 corresponds +to auto regressive coefficients. The t-SNE plot is generated +with high ranking column features as shown in Fig. 5. It is +observed that the action and rest clusters are clearly separable, +but clusters within action groups are overlapping due to similar +muscle group involvement. Based on a recent review of sEMG +studies of muscle groups and their functions [58], the muscles + +7 +(a) +(b) +(c) +(d) +(e) +Fig. 4: t-SNE plots of feature set for (a) group 1, (b) group 2, (c) group 3, (d) group 4, +and (e) group 5, respectively. +Fig. 5: t-SNE plot of feature set for six FAABOS groups +FCR, FCU, BR and BB are mapped to the major functions +involved in each of the FAABOS categories in our study and +detailed in Table X. +E. Discussion +The SVM3 method has the best classification performance +in case of the FAABOS categories (no. classes = 5). This can +be explained by relatively less number of classes and ability +of feature ensemble F5 to better capture the representation at +functional category level. The ML framework’s performance +TABLE X: FAABOS group vs actions vs muscle mapping. +Group +Major actions involved +Muscles +No object ac- +tion (1) +Wrist flexion & extension and hand digit ma- +nipulation +FCR, FCU, +BR +Hold +object +(2) +Elbow flexion & extension, Wrist flexion & ex- +tension, and Forearm Pronation & Supination +BB, +BR, +FCR, FCU +Object grasp- +ing (3) +Elbow flexion & extension, Wrist flexion & +extension, Forearm Pronation & Supination, +and hand digit manipulation +FCR, FCU, +BR, BB +Flexion +and +Extension +of +Fingers (4) +Wrist flexion & extension and hand digit ma- +nipulation +BB, +BR, +FCR, FCU +Writing (5) +Elbow flexion & extension, Wrist flexion & +extension, and hand digit manipulation +FCR, +BB, +FCU, APB +may need further improvement. This performance can be +explained by relatively higher number of activities and higher +intra-class correlations. The feature visualizations with t-SNE +has shown better separability of activities within FAABOS +groups. A clear separation between rest and action is also +observed in t-SNE plot across FAABOS groups. +V. CONCLUSION & FUTURE SCOPE +In this paper, we have collected a novel sEMG dataset +(EMAHA-DB1) of 22 activities of daily living from Indian +population. The EMAHA-DB1 includes a few activities that +are not considered in existing datasets. The sEMG EMAHA- +DB1 dataset is compared against the publicly available ex- +isting sEMG datasets. The dataset is analyzed from different +perspectives including feature set analysis in time domain and +frequency domain, individual action classification, FAABOS +category classification and feature visualization using t-SNE. +In the above mentioned analysis, the modified LMF, time +domain statistical (TDS) feature, spectral band powers (SBP), +channel cross correlation and local binary patterns (LBP) +ensemble feature set (F5) with Cubic SVM classifier has +obtained highest test accuracy of β = 75.39%. Additionally, +in the FAABOS groups classification, the best performance is +again produced by the cubic SVM classifier (β = 83.21) with +the feature set consisting of energy features and auto regressive +coefficients (F2). Finally, the visual analysis using t-SNE +plots showed that the extracted feature set is able to clearly +distinguish the ADL activities within a group. The obtained +results indicate that the EMAHA-DB1 can be successfully +used as a benchmark for the development of hand gesture +recognition system, physiological analysis and clinical studies +of sEMG for ADL. +In terms of future work, the framework may need further in- +novation in terms of features to improve the classification per- +formance; the EMAHA-DB1 is analysed using only machine +learning classifiers, there is a scope for improvement with deep +learning; the dataset can also be analysed by decomposing the +time series with wavelets or empirical mode decomposition +(EMD) techniques; finally, the EMAHA-DB1 dataset can also +be analysed for learning the statistical distributions. +ACKNOWLEDGMENT +This research is funded by SERB, Govt. of India under +Project Grant No. CRG/2019/003801. + +50 +40 +30 +20 +Dimension +10 +12 +10 +13 +16 +-20 +18 +-30 +19 +20 +-40 +21 +-50 +-60 +-40 +-20 +20 +40 +60 +Dimension. +717 +10 +5 +2 +Dimension +5 +-10 +-15 +-20 +-15 +-10 +5 +10 +-5 +15 +Dimension.40 +14 +15 +30 +20 +Dimension +10 +-10 +-20 +-20 +-10 +20 +30 +10 +0 +Dimension 180 +0 +60 +1 +2 +40 +3 +4 +20 +5 +Dimension +0 +-20 +-40 +-60 +-80 +-100 +-100 +-50 +50 +100 +0 +Dimension 12 +3 +9 +10 +2 +1 +Dimension +.2 +-3 +-5 +6 +2 +2 +4 +8 +0 +6 +Dimension20 +3 +4 +15 +5 +6 +10 +7 +8 +5 +Dimension +5 +-10 +-15 +-20 +-10 +10 +15 +-5 +5 +20 +08 +REFERENCES +[1] T.-H.-C. 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Vergara, “A systematic +review of EMG applications for the characterization of forearm and hand +muscle activity during activities of daily living: Results, challenges, and +open issues,” Sensors, vol. 21, no. 9, 2021. + diff --git a/ANE1T4oBgHgl3EQfowXh/content/tmp_files/load_file.txt b/ANE1T4oBgHgl3EQfowXh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e41e0198ebf12aca6ca3472be59476d2193a93a --- /dev/null +++ b/ANE1T4oBgHgl3EQfowXh/content/tmp_files/load_file.txt @@ -0,0 +1,997 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf,len=996 +page_content='1 EMAHA-DB1: A New Upper Limb sEMG Dataset for Classification of Activities of Daily Living Naveen Kumar Karnam, Anish Chand Turlapaty, Member, IEEE, Shiv Ram Dubey, Senior Member, IEEE and Balakrishna Gokaraju, Member, IEEE Abstract—In this paper, we present electromyography analysis of human activity - database 1 (EMAHA-DB1), a novel dataset of multi-channel surface electromyography (sEMG) signals to evaluate the activities of daily living (ADL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The dataset is acquired from 25 able-bodied subjects while performing 22 activ- ities categorised according to functional arm activity behavioral system (FAABOS) (3 - full hand gestures, 6 - open/close office draw, 8 - grasping and holding of small office objects, 2 - flexion and extension of finger movements, 2 - writing and 1 - rest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The sEMG data is measured by a set of five Noraxon Ultium wireless sEMG sensors with Ag/Agcl electrodes placed on a human hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The dataset is analyzed for hand activity recognition classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The classification is performed using four state-of- the-art machine learning classifiers, including Random Forest (RF), Fine K-Nearest Neighbour (KNN), Ensemble KNN (sKNN) and Support Vector Machine (SVM) with seven combinations of time domain and frequency domain feature sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The state-of-the- art classification accuracy on five FAABOS categories is 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='21% by using the SVM classifier with the third order polynomial kernel using energy feature and auto regressive feature set ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The classification accuracy on 22 class hand activities is 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='39% by the same SVM classifier with the log moments in frequency domain (LMF) feature, modified LMF, time domain statistical (TDS) feature, spectral band powers (SBP), channel cross correlation and local binary patterns (LBP) set ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The analysis depicts the technical challenges addressed by the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The developed dataset can be used as a benchmark for various classification methods as well as for sEMG signal analysis corresponding to ADL and for the development of prosthetics and other wearable robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Index Terms—Machine learning, Classification Algorithms, Surface Electromyography (sEMG), Activities of Daily Living (ADL), Features, Dataset and Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' INTRODUCTION P ERFORMING hand movements during activities of daily living (ADL) [1] without any difficulty provides func- tional independence and a decent quality of life [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' However, it is quite difficult to perform simple hand movements for individuals affected by the following disorders: upper limb disabilities [3], [4], disorders related to aging [5], neuromus- cular disorders [6], and stroke related disabilities [7], [8], [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Human computer interfaces and human robot interfaces N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Karnam and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Turlapaty are with the Biosignal Analysis Lab at the Indian Institute of Information Technology, Sri City, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=', India (email: anish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='turlapaty@iiits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Dubey is with the Computer Vision and Biometrics Laboratory at Indian Institute of Information Technology, Allahabad, Prayagraj-211015, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=', India (email: srdubey@iiita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Gokaraju is with the Visualizations and Computing Advanced Research Center (ViCAR) and Department of Computational Data Science an Engineer- ing, North Carolina A and T State University, Greensboro, North Carolina (email: bgokaraju@ncat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' can support the rehabilitation process to recover from the above mentioned disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' For instance, hand gesture-based interfaces based on computer vision techniques for identifying and classifying gestures are currently under development [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Moreover, many researchers have explored robotic control using visual gestures [12], [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' However, vision based control methods are inadequate to determine the appropriate control for actuation and the amount of force exerted by a muscle during action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' One approach to quantify the upper limb activity is to use wearable sensors such as inertial motion sensors (IMUs) including accelerometers, gyroscopes and magnetometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' These sensors are utilised to measure and monitor limb activities, quantify muscle motor deficits [15], and classify the types of physical activity [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Although wearable sensors can recognize human activity, they are deficient in precise identification of hand gestures, finer finger movements and the amount of muscle strength used to execute the movement [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Alternatively, hand movement classification and the limb control [19], [20] through surface electromyography (sEMG) signals facilitates the design of prosthetic devices, exoskeleton arms, advanced realistic bio-mechanical models, and rehabili- tation therapies [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' In these applications, utilization of multi- modal signals is also very common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' In the literature, fusion of the IMU’s and sEMG signals [22], [23], [24] for hand activity classification and estimation of the continuous orientation of the forearm is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The electroencephalography signals (EEG)) and sEMG signals are also fused to decode the intention of the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' This fusion process can generate better control signals compared to a standalone sEMG signal based control [25], [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' In order to obtain better classification accuracies the features from sEMG signals can be fused with those from the vision based image classification network [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' In practice, the multi-modal methods increase the complexity of the hardware as well as software systems, hence they pose difficulty for different real-life applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Hand movement analysis and classification through standalone sEMG signals is gaining attention [29], [30], [31], [32], [33] and is the focus of our current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' In this paper, we present electromyography analysis of human activity - database 1 (EMAHA-DB1), a novel sEMG dataset on ADL for the Indian population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Following are the salient features of EMAHA-DB1: There are several sEMG datasets available that include activities such as hand gestures, hand movements, wrist movements, and grasping objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' These datasets are mainly collected for western populations and there is no arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='03325v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='SP] 9 Jan 2023 2 dataset for ADL from the Indian population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' EMAHA- DB1 fills this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' There is a tradition of anthropometric data collection in India [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' For any population, there is an influence of anthropometrics on their kinematics and kinetics [36], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' EMAHA-DB1 will compliment existing anthropo- metrics, kinematics and kinetics datasets [30], [37] which will be helpful in conducting upper limb rehabilitation therapies, physiological studies and clinical studies for Indian population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The ADL performance is analyzed by grouping the actions according to the functional arm activity behav- ioral observation system (FAABOS [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The functional taxonomy provided by Uswatte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' quantifies group of hand actions based on the behavioral significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' There are publicly available ADL datasets such as the NinaPro [39], the BioPatRec [40], the Ramikushaba [41] and the UCI Gesture [42] that have not covered a few important ADL categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The hand activities are usually performed in an experimental set up with a fixed duration for each of the activities, however we have considered different durations for distinct activities to approximate corresponding durations of real time hand movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The dataset can be used to benchmark classification algorithms or perform statistical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The developed dataset consists of a larger number of subjects and a higher number of activity repetitions compared to any other publicly available ADL datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The main contributions of the paper are: 1) In this work, we have carried out muscle activity mea- surements corresponding to activities of daily living and collected a novel multichannel sEMG data from Indian population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 2) The EMAHA-DB1 dataset is organized according to custom FAABOS functional categories to perform anal- ysis using state-of-the-art machine learning classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Specifically the sEMG signals are analyzed to classify into the functional groups as well as individual activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 3) We also perform extensive feature analysis with respect to the FAABOS functional categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The rest of the paper is organised as follows: Section II details about the proposed EMAHA-DB1 dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Section III presents experiments in machine learning frameworks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Section IV demonstrates the experimental results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and Section V provides a conclusion along with the future scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' EMAHA-DB1: PROPOSED SEMG DATASET A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Data Collection 1) Study participants: The institutional ethics committee of Indian Institute of Information Technology Sri City (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' IIITS/EC/2022/01) approved the proposed data collection pro- tocol developed in general accordance with the declaration of Helsinki and specific accordance with the “National Ethical Guidelines for Biomedical and Health Research involving hu- man participants” of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Twenty-five healthy subjects with no history of upper limb pathology, including 22 males and 3 females, participated in the sEMG data collection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The TABLE I: List of hand activities Activity No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Activity description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Hand at rest (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Tossing a coin (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Finger snapping (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Pulling an empty draw - Posterior view (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Pulling a draw with weight (2kg) - Posterior view (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Pulling an empty draw - Anterior view (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Pulling a draw with weight (2kg) - Anterior view (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Pushing an empty draw - Posterior view (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Pushing a draw with weight (2kg) - Posterior view (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Clasping both hands (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Hand clapping (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Grasping and holding 1L water bottle (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Grasping and holding small hammer (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Grasping and holding small saw (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Writing the phrase ”Bio signal lab” on paper with pen - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='lateral grasp (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Writing the phrase ”Bio signal lab” on board with marker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='lateral grasp (standing) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Lifting a small bucket with 4L water (standing) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Typing the phrase ”Bio signal lab” on keyboard using ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='single finger (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Drinking tea/water from a cup - lateral grasp (sitting) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='A19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='Picking up the phone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' placing it to his/her ear and hanging up the phone on table (sitting) A20 Grasping and holding a book (sitting) A21 Grasping and holding a tennis ball (sitting) TABLE II: Sensor placement on hand muscle Channel No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Sensor No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Hand muscle name 1 21621 Brachio Radialis (BR) muscle 2 21623 Flexor Carpi Radialis(FCR) muscle 3 21624 Flexor Carpi Ulnaris (FCU) muscle 4 21625 Biceps Brachii (BB) muscle 5 21626 Abductor Pollicis Brevis (APB) muscle average age is 28±6 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Before the first session of activities, each of the participants gave written informed consent and the data collection process is completely non-invasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 2) Experimental setup and acquisition protocol: The 22 activities performed by each subject are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Each of the hand muscle activity is recorded with a 5-channel Noraxon Ultium wireless sEMG sensor setup [43] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Five self-adhesive Ag/AgCL dual electrodes were placed at the centre of the five most representative muscle sites of the right arm as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Each subject is instructed to sit comfortably with one elbow resting on a table and an arm flexed 90◦ compared to the forearm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The muscle locations are selected according to the atlas in chapter 17 [44] and is given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' At the beginning of each session, the participant’s hands are cleaned with an alcohol based wet wipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Prior to each session, the subject is acquainted with the experiment protocol including a video demonstration of the proposed activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The total duration of each session is up-to one hour per subject depending on adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Each activity is performed for a maximum duration of 10s and repeated 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' There is a rest period of 5s between each of the repetitions and a 30s gap between the sessions of different activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Each of the activities consists of two phases: (1) an action and (2) rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' However, some of the activities included an extra release phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' During the action phase, the subject performs the corresponding activity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' during the release phase, the subject transitions from the action state to rest state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and during the rest phase, the subject completely relaxes each of 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 1: Learning steps from sEMG dataset collection to classification of hand activities TABLE III: Phase-wise durations of each activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' TX TA TR TT No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' TX TA TR TT No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' TX TA TR TT A1 3 5 0 8 A8 3 5 0 8 A15 3 15 2 20 A2 3 5 0 8 A9 3 5 0 8 A16 5 5 3 13 A3 3 5 0 8 A10 3 5 0 8 A17 3 10 2 15 A4 3 5 0 8 A11 3 5 3 11 A18 5 5 3 13 A5 3 5 0 8 A12 3 5 3 11 A19 5 5 3 13 A6 3 5 0 8 A13 3 5 3 11 A20 5 5 3 13 A7 3 5 0 8 A14 3 10 2 15 A21 5 5 3 13 his/her muscles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The time duration for each activity is given in Table III, where TX, TA, TR, and TT are the rest, action, release, and total duration, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 3) Comparisons with existing datasets : The characteristics of EMAHA-DB1 data are compared against those of existing sEMG hand activity datasets in TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Apart from those mentioned in salient features in Introduction, a few additional and distinct characteristics of the EMAHA-DB1 are: 1) the experiments are designed such that hand activities performed consists of three phases of action (contraction/relaxation of muscles), release (retreating of action), and rest (relaxing of muscles), 2) the measurements are acquired with a minimal number of sensors hence requires lower computational re- sources compared to the existing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Data Preparation 1) Activity segmentation: For the sEMG signals in EMAHA-DB1, the preliminary annotations for onset and offset of the actions are performed based on the respective durations of action phases shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' To improve the quality of activity labels, based on the procedure developed in [46], an improved signal segmentation process (listed below) is implemented: 1) Initially, for each trial of each activity performed by each subject, the multi-channel signal is rectified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 2) For each of these trials, the maximum and minimum values are identified to determine the range R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 3) The signal strengths at R/ √ 2 (3dB amplitude) are considered the thresholds on either side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 4) The first signal strength, past the preliminary onset, crossing the 3dB threshold is identified for each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The earliest location among the threshold crossings from the five channels is considered the onset of action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 5) The trial data is parsed backwards from the end of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The first point from the end i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=', the final 3dB crossing is identified for each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The right most location among the crossings from these channels is labelled the offset of action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 6) Finally, the signal samples between the onset and the offsets are annotated as the action and assigned the corresponding activity number, and the remaining signal is considered to be rest state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The above procedure is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 2 for a single trial of ADL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' It is observed that signal segmentation improves the annotation process of activity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' rest which further improves veracity of the classification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 2) FAABOS categories: The EMAHA-DB1 is mapped ac- cording to function arm activity behavioral observation system (FAABOS) [38], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Specifically, actions in the EMAHA- DB1 are reorganized into the following five major groups: 1) No object action, 2) object holding, 3) object grasping, 4) Flexion and Extension of Fingers, and 5) writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The action categories that are mapped into these groups are listed in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Multi-channel sEMG signals Output hand activity classification A sample of hand movements performed in Learning Activities of Daily Living kinematic (ADL) characteristics of - uu m d the hand activities ML classifier Testing training (KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' RF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' SKNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' SVM3) Grasping and holding Grasping and holding small hammer a book Preprocessing O Training Notch filtering at 50Hz Feature set visualisation by sEMG Data acquired for Low pass filtering with fc = 500Hz t-SNE plot ADL by Noraxon Ultium Wavelet denoising Feature extraction with six sEMG sensor setup 4 feature sets of F0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' F2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' F3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Trial wise segmentation 3 F4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' F5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and F6 2 LO 1 0 1 Data train and test split-up 3 Train data with Test data with 4 trial no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' trial no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 2, 5 1,3,4,6,8,9 and 10 and 7 6 Datset is curated and 4 2 0 Dimension1 labelled using audio cue Relabelled by an algorithm4 TABLE IV: Comparisons of basic data characteristics with benchmark datasets Dataset Name Action categories Sensor No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' of Subjects (S) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' of activities (NA) (including rest) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' of channels (Nc) Sampling frequency (Ns) (samples per sec) Rest dura- tion (TX)(s) Action dura- tion (TA)(s) Release dura- tion (TR)(s) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' of repe- titions (NR) Total no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' of pat- terns (N) NinaPro DB1 [39] Gestures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Wrist move- ments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and Grasping Objects Otto Bock 27 53 10 100 3 5 10 14310 NinaPro DB2 [39] Gestures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Wrist move- ments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Grasping Ob- jects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and Finger press- ing movements Delsys Trigno wireless 40 50 12 2000 3 5 6 12000 NinaPro DB4 [45] Gestures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Wrist move- ments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and Grasping Objects Cometa Mini- Wave 10 53 12 2000 3 5 6 3180 BioPatRec DB2 [40] Gestures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and Wrist and hand movements Thalmic myoarm band 17 27 8 2000 3 3 3 1377 UCI Ges- ture [42] Wrist and hand move- ments Myo Thalmic bracelet 36 7 8 1000 3 3 4 1008 Rami- kushaba DB6 [41] Hand movements Delsys DE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='x series EMG sensors 11 40 7 4000 3-5 5 6 2640 EMAHA- DB1 (Our dataset) Daily activities - Grasping and holding, writing, and draw open/close Noraxon Ul- tium sEMG sensor 25 22 5 2000 3-5 5-15 3-5 10 5500 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 2: Illustration of manual segmentation of sEMG signals for a trial of ADL TABLE V: FAABOS groups of activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Group label Group Name Activity No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 0 Rest A0 1 No object action A2, A9 and A10 2 Hold object A3, A4, A5, A6, A7 and A8 3 Object grasping A11, A12, A13, A16, A18, A19, A20 and A21 4 Flexion and Exten- sion of Fingers A1 and A17 5 Writing A14 and A15 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Problem Statement The total number of sEMG patterns in the EMAHA-DB1 is N = S × NA × NR, where S is the total number of subjects, NA is the number of different activities, and NR corresponds to the number of repetitions per action per subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The proposed sEMG dataset can be represented as: x = {xn}N n=1 (1) where each observation array xn consists of multiple channels as: xn = {xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='m}NC m=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' N (2) TABLE VI: Summary of extracted features Feature Set Features Feature Length F0 [47] Mean Absolute Value (MAV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Temporal Spec- tral Energies (TSE) and Spectral Band Ener- gies (SBE) 1×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 4×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and 4 × NC F1 [48] MAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Zero Crossings (ZC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Slope Changes (SC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and Wavelength (WL) 1×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 1×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 1×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and 1× NC F2 [49] F1 and Auto Regression Coefficients (ARC) 9 × NC and 2 × NC F3 [50] F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Myopulse Rate (MPR),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Willison Ampli- tude (WAMP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and Cardinality 9×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 1×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 1×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and 1× NC F4 [51] Log moments in frequency domain (LMF) 5 × NC F5 [52] F4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' modified LMF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Time domain statistics (TDS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Spectral Band Powers (SBP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Max channel cross correlations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and Local Binary Patterns (LBP) 5 × NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 10 × NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 4 × NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 4×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 2×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and 2 × NC F6 [53] Root Mean Square (RMS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Time Dependent Power spectrum Descriptors (TD-PSD) [51],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Difference Absolute Standard Deviation Value (DASDV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and Difference Absolute Mean Value (DAMV) 1×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 6×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 1×NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and 1× NC where NC is the number of channels (from different elec- trodes) and each of these channels consists of an array as: xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='m = {xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='m(i)}NT i=1 (3) where NT = Ns × TT is the number of values in one trial of duration TT and Ns is the sampling rate (samples/sec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' For a given trial, for feature extraction purposes, the signal is divided into Nseg segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Each segment sg consists of an array as: sj g = {xn,m(i)}Ng i=1 j = 1, · · · Nseg (4) where Ng is the number of samples in one segment such that NT = Nseg × Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The objective of this study is to map the segmented sEMG signals to the corresponding activity labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=', tg - targets), Channel 3 cue-ON 6 Cue-OFF OFF 40 SEMG 20 0 2000 4000 6000 8000 10000 12000 14000 16000 Channel 4 10 cue-OFFl cue-ON NO OFF sEMG Amp ５ 0 8000 2000 4000 6000 10000 12000 14000 16000 0 Rest vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Action Action-OFF SEMG Action- norm 0 2000 4000 6000 8000 10000 12000 14000 16000 0 Sample Index5 (a) (b) (c) (d) (e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 3: Performance comparison (a) with different Feature Ensembles with Cubic SVM (Polynomial SVM of order 3), (b) with different classifiers for the benchmark feature ensemble F5, (c) against benchmark frameworks, (d) against benchmark frameworks in terms of various metrics, and (e) against FAABOS categories frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' TABLE VII: Numerical setup for classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Classifier Model Setup Fine KNN No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='of neighbours = 5, Distance Metric = Cityblock, Dis- tance weight = Squared Inverse Ensemble KNN No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='of learning cycles = 30, learners = KNN, Subspace dimension = 25 Cubic SVM Polynomial kernel, Order = 3, Box constraint = 1, Multi- class Method = one-vs-one Random Forest No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='of bags for bootstrapping = 300 which can be formulated as: f{sg} → tg (5) where tg denotes targets (group labels) as specified in TABLE V or individual activity labels as specified in TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The mapping function in (5) is implemented by a supervised classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' For the mapping function, appropriate features are required that represent the underlying inverse kinematic rela- tionships between the sEMG signals and the corresponding activity performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Feature Extraction In this work, the following feature sets are adapted from [47]: F0, F1, F2, F3, F4, and F5 with an additional feature set F6 consisting of root mean square (RMS), time dependent power spectrum descriptors (TD-PSD), difference absolute standard deviation value (DASDV), and difference absolute mean value (DAMV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Note the features are computed for each segment and concatenated to build the full feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The extracted feature sets are summarized in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Supervised Classifiers In this paper, four algorithms including random forest (RF), fine K-nearest neighbour (FKNN), ensemble KNN (sKNN) and cubic support vector machine (SVM3) are considered for sEMG signal classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The classifiers are trained and TABLE VIII: Feature ensemble vs benchmark classifier setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' FE FL BF Classifier FE FL BF Classifier F0 9 × NC B0 [47] Fine KNN F4 5 × NC B4 [54] SVM3 F1 4 × NC B1 [48] SVM3 F5 27×NC B5 [52] SVM3 F2 11×NC B2 [49] Fine KNN F6 9 × NC B6 [39] RF F3 12×NC B3 [50] SVM3 tested with subject-wise data and the average performance is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The hyperparameter settings for different machine learning algorithms used in this work are summarized in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The performance of classifiers is evaluated using the standard metrics such as cross validation accuracy (α), testing accuracy (β), Kappa coefficient (κ), precision (γ), recall (ρ) and F-1 score (F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' CLASSIFICATION EXPERIMENTS, RESULTS & ANALYSIS The developed EMAHA-DB1 sEMG dataset is analyzed using the state-of-the-art classification and feature extraction methods as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Pre-processing and Data Split-up Based on the procedure described in [55], the recorded sEMG data is pre-processed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' First, the sEMG data is filtered to remove power line noise at 50Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Then a first order Butterworth low pass filter is applied at a cut-off frequency of 500Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Finally, wavelet denoising of order 8 with the symlet as the mother wavelet is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The data from each subject is split trials-wise into 70% for training and 30% for testing as per the splitting method in [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' A non overlapping moving window segment of Ng = 200 samples is considered with duration Tseg = 100ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The number of features obtained per segment sg are summarized in Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 80 Cross Validation (α) Testing (B) 75 (%) Accuracy 70 65 60 F2 F3 F1 F4 F5 F6 F0 Feature sets80 Cross Validation (α) Testing (β) 75 % 70 Accuracy 65 60 55 RF SKNN SVM3 FKNN Classifiers80 Cross Validation (α) Testing (β) 75 (%) Accuracy 70 65 60 B1 B2 B3 B4 B5 B0 B6 Feature sets0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='8 Precision ()kappa () F, score (F,) Recall (p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='75 Metric 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='7 rmance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='55 B2 B3 B4 B5 B6 B0 B1 Benchmarks85 Feature set F0 Feature set F5 Feature set F2 80 75 70 65 RF SKNN SVM3 FKNN Classifier6 TABLE IX: Muscle vs action mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Muscle Major functionality of the mus- cle Biceps Brachii (BB) muscle Flexes elbow joint, Supinates fore- arm and hand at radioulnar joint Brachio Radialis (BR) muscle Flexes elbow joint Flexor Carpi Radialis (FCR) muscle Flexes and abducts hand at wrist Flexor Carpi Ulnaris (FCU) muscle Flexes and adducts wrist Abductor Pollocis Brevis (APB) muscle Abducts joints of thumb B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Experiments In this paper, as mentioned earlier two case studies are carried out as follows, 1) classification of individual action categories listed in Table I, in this case study, the performance is analyzed with respect to feature ensembles, classifiers, benchmark classification frameworks and finally feature vi- sualization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 2) classification of FAABOS categories listed in Table V, in the second case study, the performance is analyzed with respect to feature ensembles followed by an analysis of the most relevant features with respect to the muscle sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Case Study 1: Results and Analysis 1) Comparison with feature ensembles: The feature sets F0-F6 are analyzed in this comparison study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Each of the feature set is utilised as input for SVM3 and their performance metrics α and β are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 3a, the best performance is produced by the feature set F5 (α = 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='42 and β = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The next best feature ensemble F2 lags behind by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='3% at α = 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='06 and β = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The feature ensemble F6 has produced the least classification performance (α = 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='68 and β = 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 2) Comparison with classifiers: In this experiment, the classification performance of the standard machine learning algorithms such as the RF, FKNN, sKNN and SVM3 using the F5 feature set is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 3b, the best performance is produced by the SVM3 classifier (α = 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='42 and β = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='39) and then by FKNN (α = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='83 and β = 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The least performance is obtained with SKNN classifier (α = 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='4 and β = 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Thus, it is observed from this experiment that for the feature set F5 the SVM3 classifier outperforms other benchmark classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 3) Comparison with benchmark algorithms: The most suit- able classification framework for the EMAHA-DB1 is deter- mined by comparisons with the existing sEMG benchmark classification methods consisting of respective combinations of a feature ensemble and a classification framework as listed in Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The benchmark Bi indicates the classification framework with feature set Fi for i = 0, 1, · · · , 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The param- eter setups of the different classifiers used in the numerical analysis are also shown in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The performance of these classifiers is analyzed based on the cross validation accuracy (α) and the test accuracy (β) with the corresponding results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The benchmark B5 has achieved state-of- the-art performance with α = 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='42 and β = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The lowest performance among the compared benchmarks is for B6 with α = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='2 and β = 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The other performance metrics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=', κ, γ, ρ, and F1) of the benchmark frameworks are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The benchmark framework F5 has achieved highest values for each of the performance metrics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=', κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='73, γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='66, ρ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='71, and F1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The runner-up is B6 framework with metric values κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='66, γ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='60, ρ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='64, and F1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 4) Feature Visualization by t-SNE: The following analysis is meant for the 22 individual action categories however carried out FAABOS group wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Among the feature sets F0 to F6, it is observed that F5 is the best performing feature set, hence used for sequential feature selection analysis (SFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' From SFS, the most relevant features for each group of hand activities are identified and further used for analysis with t-distributed Stochastic Neighbourhood Embedding (t-SNE) [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The top 6 feature columns of 84, 85, 96, 97, 101, and 105 are used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The columns with higher ranking are 84 and 85 that correspond to the features of mean and variance respectively (from TDS feature set), and 96, 97, 101, and 105 that correspond to the spectral bands [0 (Ns/8)] and [(Ns/8) (Ns/4)] of SBP feature set [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The flexion and extension of elbow and wrist flexion and extension are mainly supported by the muscle groups BB, BR, FCR and FCU [57] as given in TABLE IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The action categories in group 2 and group 3 involve the common muscle movements including elbow flexion and extension, wrist flexion and extension and pronation and supination as shown in TABLE X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Hence From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 4b and 4c, the clusters for some of the actions overlap due to involvement of similar muscle groups across actions with same basic muscle movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The actions within group 1, group 4 and group 5 are clearly separable which can be observed from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 4a, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 4d and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 4e, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Case Study 2: Results and Analysis 1) Comparison of FAABOS categories with feature ensem- bles: The sEMG signals from the EMAHA-DB1 are classified based on FAABOS categories specified in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The six FAABOS categories of sEMG signals are trained and tested with the top three feature sets such as F0, F2 and F5 and the corresponding results are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The best performance is produced by the SVM3 classifier (α = 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='54 and β = 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='21) with the feature set F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The next best performance is produced by the same SVM3 classifier (α = 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='85 and β = 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='14), but with the feature set F5 having a slight variation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='07%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The least performance is observed for feature set F0 with SKNN classifier (α = 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='66 and β = 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 2) Feature Visualization by t-SNE for FAABOS groups: This analysis is carried out for 6 FAABOS categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Among the feature sets F0, F2, and F5, it is observed that F2 is the best performing feature set and used for SFS analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' From SFS, the top 6 feature columns 1, 3, 4, 5, 19, and 23 are used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The columns with higher ranking are 1, 3, 4, and 5 that correspond to mean absolute value (MAV), 19 corresponds to the waveform length, and 23 corresponds to auto regressive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The t-SNE plot is generated with high ranking column features as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' It is observed that the action and rest clusters are clearly separable, but clusters within action groups are overlapping due to similar muscle group involvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Based on a recent review of sEMG studies of muscle groups and their functions [58], the muscles 7 (a) (b) (c) (d) (e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 4: t-SNE plots of feature set for (a) group 1, (b) group 2, (c) group 3, (d) group 4, and (e) group 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' 5: t-SNE plot of feature set for six FAABOS groups FCR, FCU, BR and BB are mapped to the major functions involved in each of the FAABOS categories in our study and detailed in Table X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Discussion The SVM3 method has the best classification performance in case of the FAABOS categories (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' classes = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' This can be explained by relatively less number of classes and ability of feature ensemble F5 to better capture the representation at functional category level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The ML framework’s performance TABLE X: FAABOS group vs actions vs muscle mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Group Major actions involved Muscles No object ac- tion (1) Wrist flexion & extension and hand digit ma- nipulation FCR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' FCU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' BR Hold object (2) Elbow flexion & extension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Wrist flexion & ex- tension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and Forearm Pronation & Supination BB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' BR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' FCR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' FCU Object grasp- ing (3) Elbow flexion & extension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Wrist flexion & extension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Forearm Pronation & Supination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and hand digit manipulation FCR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' FCU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' BR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' BB Flexion and Extension of Fingers (4) Wrist flexion & extension and hand digit ma- nipulation BB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' BR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' FCR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' FCU Writing (5) Elbow flexion & extension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Wrist flexion & extension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' and hand digit manipulation FCR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' BB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' FCU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' APB may need further improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' This performance can be explained by relatively higher number of activities and higher intra-class correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The feature visualizations with t-SNE has shown better separability of activities within FAABOS groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' A clear separation between rest and action is also observed in t-SNE plot across FAABOS groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' CONCLUSION & FUTURE SCOPE In this paper, we have collected a novel sEMG dataset (EMAHA-DB1) of 22 activities of daily living from Indian population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The EMAHA-DB1 includes a few activities that are not considered in existing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The sEMG EMAHA- DB1 dataset is compared against the publicly available ex- isting sEMG datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The dataset is analyzed from different perspectives including feature set analysis in time domain and frequency domain, individual action classification, FAABOS category classification and feature visualization using t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' In the above mentioned analysis, the modified LMF, time domain statistical (TDS) feature, spectral band powers (SBP), channel cross correlation and local binary patterns (LBP) ensemble feature set (F5) with Cubic SVM classifier has obtained highest test accuracy of β = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='39%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Additionally, in the FAABOS groups classification, the best performance is again produced by the cubic SVM classifier (β = 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content='21) with the feature set consisting of energy features and auto regressive coefficients (F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' Finally, the visual analysis using t-SNE plots showed that the extracted feature set is able to clearly distinguish the ADL activities within a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' The obtained results indicate that the EMAHA-DB1 can be successfully used as a benchmark for the development of hand gesture recognition system, physiological analysis and clinical studies of sEMG for ADL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' In terms of future work, the framework may need further in- novation in terms of features to improve the classification per- formance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' the EMAHA-DB1 is analysed using only machine learning classifiers, there is a scope for improvement with deep learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' the dataset can also be analysed by decomposing the time series with wavelets or empirical mode decomposition (EMD) techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' finally, the EMAHA-DB1 dataset can also be analysed for learning the statistical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' ACKNOWLEDGMENT This research is funded by SERB, Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' of India under Project Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf'} +page_content=' CRG/2019/003801.' metadata={'source': 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b/CtAyT4oBgHgl3EQfefgG/content/tmp_files/2301.00320v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..de79b8baa6af19caf82844875e4a5e985e587179 --- /dev/null +++ b/CtAyT4oBgHgl3EQfefgG/content/tmp_files/2301.00320v1.pdf.txt @@ -0,0 +1,242 @@ +Relevance Classification of Flood-related Twitter Posts +via Multiple Transformers +Wisal Mukhtiar1,†, Waliiya Rizwan1,†, Aneela Habib1,†, Yasir Saleem Afridi1, +Laiq Hasan1 and Kashif Ahmad2 +1Department of Computer Systems Engineering, University of Engineering and Technology, Peshawar, Pakistan. +2Department of Computer Science, Munsters Technological University, Cork, Ireland. +Abstract +In recent years, social media has been widely explored as a potential source of communication and informa- +tion in disasters and emergency situations. Several interesting works and case studies of disaster analytics +exploring different aspects of natural disasters have been already conducted. Along with the great potential, +disaster analytics comes with several challenges mainly due to the nature of social media content. In this +paper, we explore one such challenge and propose a text classification framework to deal with Twitter noisy +data. More specifically, we employed several transformers both individually and in combination, so as to +differentiate between relevant and non-relevant Twitter posts, achieving the highest F1-score of 0.87. +1. Introduction +Natural disasters, which are hazardous events and occur frequently in different parts of the world, +can have devastating effects on society. Depending on the severity of the disaster, it may result in +significant damage to the infrastructure and human lives. Rapid response to natural disasters may +help in mitigating their adverse impact on society. In disasters and emergency situations, access +to relevant and timely information is key to a rapid and effective response. However, the literature +reports several situations where access to relevant and timely information may not be possible +due to several factors [1]. +In recent years, social media outlets, such as Twitter, Facebook, and Instagram, have been +explored as a source of communication and information dissemination in emergency situations +[2]. The literature already reports the feasibility and effectiveness of social media for a diversified +list of tasks in disaster analytics. For instance, Ahmad et al. [3] explored social media outlets as a +source of information collection and dissemination during natural disasters by proposing a system +that is able to collect and analyze disaster-related multimedia content from social media. Similarly, +social media content has also been utilized for disaster severity and damage assessment [4, 5]. +Despite being very effective in disaster analytics, social media data also come with several +limitations. For instance, social media content contains a lot of noise and irrelevant information. +This paper targets one of such challenges by proposing several solutions for the Relevance Classi- +fication of Twitter Posts (RCTP), sub-task introduced in DisasterMM challenge of MediaEval 2022 +MediaEval’22: Multimedia Evaluation Workshop, January 13–15, 2023, Bergen, Norwa,y and Online +*Corresponding author. +†These authors contributed equally. +� kashif.ahmad@mtu.ie (K. Ahmad) +© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). +CEUR +Workshop +Proceedings +http://ceur-ws.org +ISSN 1613-0073 +CEUR Workshop Proceedings (CEUR-WS.org) +arXiv:2301.00320v1 [cs.CL] 1 Jan 2023 + +[6]. The task aims at automatically analyzing and classifying flood-related tweets into relevant +and non-relevant tweets. +2. Related Work +Disaster analysis in social media content has been one of the active topics of research in the +domain over the last few years [2]. During this time, different aspects and applications of disaster +analytics in social media content have been explored [7]. Some key applications include com- +munication/information dissemination, damage assessment, response management, sentiment +analysis, and identification of the needs of affected individuals. The literature already reports +several interesting works on these applications. For instance, Nguyen et al. [8] utilized social +media content for damage assessment by analyzing disaster-related visual media posts. Ahmad +et al. [9] analyzed social media imagery for monitoring road conditions after floods. Moreover, +a vast majority of the literature demonstrates how social media outlets can be used as means of +communication in disasters and emergency situations [10, 1]. +In the literature, different types of disasters including natural disasters, such as earthquakes, +landslides, droughts, wildfires, and floods, as well as man-made disasters, such as accidents, have +been explored [1, 11]. However, the majority of the works have targeted floods, being one of +the most common natural disasters. The literature reports several interesting works on flood +analysis in social media content for different tasks. For instance, Ahmad et al. [9] proposed a +late fusion-based framework for the automatic detection of passable roads after a flood. For this +purpose, several deep learning models are trained on flood-related images from social media. Alam +et al. [4], on the other hand, employed social media imagery for post floods damage severity +assessment. +Flood detection and analysis in social content have also been a part of the MediaEval benchmark +initiative as a shared task for several years. Each time a separate aspect of flood analysis has +been explored. For instance, in MediaEval 2017 the task aimed at the retrieval of flood-related +images from social media. The task mainly involved analyzing the water level in different areas to +differentiate between floods and regular water reservoirs, such as lakes [12]. In MediaEval 2018, +the task was slightly modified by asking the participants to propose multi-modal classification +frameworks for flood-related multimedia content [13]. In MediaEval 2019 and 2020, the tasks +aimed at analyzing flood severity and flood events recognition in social media posts. +3. Approach +Figure 1 provides the block diagram of the proposed framework for the RCTP task. The framework +is composed of three main components namely (i) Pre-processing, (ii) Training and Classification, +and (iii) Fusion. In the first step, different pre-processing techniques are employed to clean the +dataset. Three different transformers are then trained on the data to obtain classification scores. +In the final step, the classification scores of the individual models are combined in a late fusion +scheme. The details of these steps are provided below. + +Figure 1: Block diagram of the proposed approach. +3.1. Pre-processing +In the pre-processing step, we employed different techniques for cleaning the dataset. More +specifically, we removed unnecessary information, such as user names, URLs, emojis, punctuation +marks, stop words, etc. Besides this, we also performed the necessary pre-possessing tasks that +are required to transform the raw text into a form that is suitable for the transformers. To achieve +this, we used the TF.text library1. +3.2. Classification via Transformers +After cleaning and pre-processing the data, we trained three different models, namely BERT [14], +RoBERTa [15], and XLNet [16]. The selection of these models for the task is motivated by their +proven performance on similar tasks [17]. A brief overview of these models is provided below. +• BERT: Bidirectional Encoder Representations from Transformers (BERT) is one of the state- +of-the-art NLP algorithms for text processing. The model is pre-trained on a large collection +of unlabeled text and can be fine-tuned for different text-analysis applications. The key +attributes of the model include its bi-directional nature, pre-training with Masked Language +Modeling (MLM), and Next Structure Prediction (NSP) objectives. In the experiments with +BERT, we used the Adam optimizer with a learning rate of 0.001 and a batch size of 8 for 3 +epochs. +• RoBERTa: Robustly Optimized BERT is a modified version of the BERT model with an +improved training mechanism. More specifically, in RoBERTa the NSP capabilities are +removed. Moreover, dynamic masking is introduced. In addition, a larger batch size and a +larger amount of training data were used in the training process. In this work, we used a +learning rate of 0.001, batch size of 20, and 10 epochs during the fine-tuning of the model +for the desired task. +• XLNet: XLNet is another state-of-the-art NLP algorithm. Similar to BERT, XLNet is also +a bidirectional transformer and uses an improved training approach. In contrast to BERT +and traditional NLP algorithms, XLNet relies on Permutation Language Modeling (PLM) by +predicting all the tokens in random order. This allows XLNet to handle dependencies and +bidirectional relationships in a better way. In this work, we used a learning rate of 0.002, a +batch size of 32, and 4 epochs during the fine-tuning of the model for the desired task. +1https://www.tensorflow.org/text/guide/bert_preprocessing_guide#text_preprocessing_with_tftext# + +Input Data +Data Pre-processing +Classification +Late Fusion +Model 1 +F = S1+S2....Sn +Score obtained with M2 +TextStreams +Pre-processing +Model 2 +Mn +Final Score +Score +Model NWe obtained the results in the form of posterior probabilities from these models, which are then +used in the fusion scheme to obtain the final predicted labels. The fusion method used in this work +is described in the next section. +3.3. Fusion +Our fusion method is based on late fusion, where we combined the classification scores obtained +with the individual models for the final classification decision as shown in Equ. 1. In the equation, +𝑆𝑓𝑖𝑛𝑎𝑙 represents the final classification score while 𝑠𝑛 is the score obtained with the nth model. +We note that in the current implementation, we used a simple fusion method by treating all the +models equally (i.e., simple aggregation of the individual scores). +𝑆𝑓𝑖𝑛𝑎𝑙 = 𝑆1 + 𝑆2 + 𝑠3 + .... + 𝑆𝑛 +(1) +4. Results and Analysis +Table 1 provides the experimental results of the proposed solutions on the development set. As +can be been in the table, overall better results are obtained with the BERT model, and surprisingly, +a lower F1-score is observed for RoBERTa. In the future, we will further investigate the potential +causes of the lower performance of RoBERTa by exploring different implementations and hyper- +parameter settings for it. As far as the performance of the fusion methods is concerned, overall +better results are obtained with the pair of XLNet and BERT. One of the potential reasons for the +lower performance of the fusion of all the models is the less accurate prediction of RoBERTa, as +also evident from the performance of the individual models. +Table 1 +Experimental results of the proposed solutions on the development set. +Method +F1-Score +BERT +0.94 +RoBERTa +0.78 +XLNet +0.93 +Fusion 1 (RoBERTa, BERT, XLNet) +0.75 +Fusion 2 (BERT, XLNet) +0.93 +Fusion 3 (RoBERTa, XLNet) +0.92 +Table 2 provides the official results of the proposed solutions on the test set. In total, three +different runs were submitted. The first run is based on the fusion of all three models used in this +work. The remaining two runs are based on the fusion of the models in pairs of two. In run 2, +BERT and XLNet are combined while in run 3 RoBERTa and XLNet are jointly used. As can be +seen in the table, better results are obtained for the fusion of the models in pairs of two where the +best performing pair of two models obtained an improvement of 20% over the fusion of all three +models. + +Table 2 +Experimental results of the proposed solutions on the test set. +Run +Precision +Recall +F1-Score +1 (Fusion of BERT, RoBERTa, XLNet) +0.6738 +0.5431 +0.6014 +2 (Fusion of BERT and XLNet) +0.8044 +0.6948 +0.7456 +3 (Fusion of RoBERTa and XLNet) +0.8977 +0.8598 +0.8784 +5. Conclusions +In this paper, we presented our solutions for the RCTP subtask of DisasterMM challenge posted +in MediaEval 2022. We proposed a late fusion framework incorporating several state-of-the-art +transformers for the task. In the current implementation, all the models are treated equally by +assigning them equal weights (i.e., 1). In the future, we aim to employ merit-based fusion methods +to further improve the final classification score. +References +[1] K. Ahmad, K. Pogorelov, M. Riegler, N. Conci, P. Halvorsen, Social media and satellites, Multimedia +Tools and Applications 78 (2019) 2837–2875. +[2] N. Said, K. Ahmad, M. Riegler, K. Pogorelov, L. Hassan, N. Ahmad, N. Conci, Natural disasters +detection in social media and satellite imagery: a survey, Multimedia Tools and Applications 78 (2019) +31267–31302. +[3] K. Ahmad, M. Riegler, A. Riaz, N. Conci, D.-T. Dang-Nguyen, P. Halvorsen, The jord system: Linking +sky and social multimedia data to natural disasters, in: Proceedings of the 2017 ACM on International +Conference on Multimedia Retrieval, 2017, pp. 461–465. +[4] F. Alam, M. Imran, F. Ofli, Image4act: Online social media image processing for disaster response, in: +Proceedings of the 2017 IEEE/ACM international conference on advances in social networks analysis +and mining 2017, 2017, pp. 601–604. +[5] F. Alam, F. Ofli, M. Imran, Crisismmd: Multimodal twitter datasets from natural disasters, in: Twelfth +international AAAI conference on web and social media, 2018. +[6] S. Andreadis, A. Bozas, I. Gialampoukidis, A. Moumtzidou, R. Fiorin, F. Lombardo, T. Mavropoulos, +D. Norbiato, S. Vrochidis, M. Ferri, I. Kompatsiaris, DisasterMM: Multimedia Analysis of Disaster- +Related Social Media Data Task at MediaEval 2022, in: Proceedings of the MediaEval 2022 Workshop, +Bergen, Norway and Online, 2023. +[7] F. Ofli, M. Imran, F. Alam, Using artificial intelligence and social media for disaster response and +management: an overview, AI and Robotics in Disaster Studies (2020) 63–81. +[8] D. T. Nguyen, F. Ofli, M. Imran, P. Mitra, Damage assessment from social media imagery data during +disasters, in: Proceedings of the 2017 IEEE/ACM international conference on advances in social +networks analysis and mining 2017, 2017, pp. 569–576. +[9] K. Ahmad, K. Pogorelov, M. Riegler, O. Ostroukhova, P. Halvorsen, N. Conci, R. Dahyot, Automatic +detection of passable roads after floods in remote sensed and social media data, Signal Processing: +Image Communication 74 (2019) 110–118. +[10] L. Palen, A. L. Hughes, Social media in disaster communication, Handbook of disaster research (2018) +497–518. +[11] K. Ahmad, A. Sohail, N. Conci, F. De Natale, A comparative study of global and deep features for +the analysis of user-generated natural disaster related images, in: 2018 IEEE 13th image, video, and +multidimensional signal processing workshop (IVMSP), IEEE, 2018, pp. 1–5. + +[12] B. Bischke, P. Helber, C. Schulze, V. Srinivasan, A. Dengel, D. Borth, The multimedia satellite task at +mediaeval 2017., in: MediaEval, 2017. +[13] B. Benjamin, H. Patrick, Z. Zhengyu, B. Damian, et al., The multimedia satellite task at mediaeval +2018: Emergency response for flooding events (2018). +[14] J. Devlin, M.-W. Chang, K. Lee, K. Toutanova, Bert: Pre-training of deep bidirectional transformers for +language understanding, arXiv preprint arXiv:1810.04805 (2018). +[15] Y. Liu, M. Ott, N. Goyal, J. Du, M. Joshi, D. Chen, O. Levy, M. Lewis, L. Zettlemoyer, V. Stoyanov, +Roberta: A robustly optimized bert pretraining approach, arXiv preprint arXiv:1907.11692 (2019). +[16] Z. Yang, Z. Dai, Y. Yang, J. Carbonell, R. R. Salakhutdinov, Q. V. Le, Xlnet: Generalized autoregressive +pretraining for language understanding, Advances in neural information processing systems 32 (2019). +[17] K. Ahmad, M. Ayub, J. Khan, N. Ahmad, A. Al-Fuqaha, Social media as an instant source of feedback +on water quality, IEEE Transactions on Technology and Society (2022). + diff --git a/CtAyT4oBgHgl3EQfefgG/content/tmp_files/load_file.txt b/CtAyT4oBgHgl3EQfefgG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6192a00f3dc4fabf0d2352b2da6581b1f27ee3af --- /dev/null +++ b/CtAyT4oBgHgl3EQfefgG/content/tmp_files/load_file.txt @@ -0,0 +1,271 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf,len=270 +page_content='Relevance Classification of Flood-related Twitter Posts via Multiple Transformers Wisal Mukhtiar1,†, Waliiya Rizwan1,†, Aneela Habib1,†, Yasir Saleem Afridi1, Laiq Hasan1 and Kashif Ahmad2 1Department of Computer Systems Engineering, University of Engineering and Technology, Peshawar, Pakistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' 2Department of Computer Science, Munsters Technological University, Cork, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Abstract In recent years, social media has been widely explored as a potential source of communication and informa- tion in disasters and emergency situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Several interesting works and case studies of disaster analytics exploring different aspects of natural disasters have been already conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Along with the great potential, disaster analytics comes with several challenges mainly due to the nature of social media content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In this paper, we explore one such challenge and propose a text classification framework to deal with Twitter noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' More specifically, we employed several transformers both individually and in combination, so as to differentiate between relevant and non-relevant Twitter posts, achieving the highest F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Introduction Natural disasters, which are hazardous events and occur frequently in different parts of the world, can have devastating effects on society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Depending on the severity of the disaster, it may result in significant damage to the infrastructure and human lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Rapid response to natural disasters may help in mitigating their adverse impact on society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In disasters and emergency situations, access to relevant and timely information is key to a rapid and effective response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' However, the literature reports several situations where access to relevant and timely information may not be possible due to several factors [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In recent years, social media outlets, such as Twitter, Facebook, and Instagram, have been explored as a source of communication and information dissemination in emergency situations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The literature already reports the feasibility and effectiveness of social media for a diversified list of tasks in disaster analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' For instance, Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' [3] explored social media outlets as a source of information collection and dissemination during natural disasters by proposing a system that is able to collect and analyze disaster-related multimedia content from social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Similarly, social media content has also been utilized for disaster severity and damage assessment [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Despite being very effective in disaster analytics, social media data also come with several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' For instance, social media content contains a lot of noise and irrelevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' This paper targets one of such challenges by proposing several solutions for the Relevance Classi- fication of Twitter Posts (RCTP), sub-task introduced in DisasterMM challenge of MediaEval 2022 MediaEval’22: Multimedia Evaluation Workshop, January 13–15, 2023, Bergen, Norwa,y and Online Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' †These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' � kashif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='ahmad@mtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='ie (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Ahmad) © 2022 Copyright for this paper by its authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Use permitted under Creative Commons License Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='0 International (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' CEUR Workshop Proceedings http://ceur-ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='org) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='00320v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='CL] 1 Jan 2023 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The task aims at automatically analyzing and classifying flood-related tweets into relevant and non-relevant tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Related Work Disaster analysis in social media content has been one of the active topics of research in the domain over the last few years [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' During this time, different aspects and applications of disaster analytics in social media content have been explored [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Some key applications include com- munication/information dissemination, damage assessment, response management, sentiment analysis, and identification of the needs of affected individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The literature already reports several interesting works on these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' For instance, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' [8] utilized social media content for damage assessment by analyzing disaster-related visual media posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' [9] analyzed social media imagery for monitoring road conditions after floods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Moreover, a vast majority of the literature demonstrates how social media outlets can be used as means of communication in disasters and emergency situations [10, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In the literature, different types of disasters including natural disasters, such as earthquakes, landslides, droughts, wildfires, and floods, as well as man-made disasters, such as accidents, have been explored [1, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' However, the majority of the works have targeted floods, being one of the most common natural disasters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The literature reports several interesting works on flood analysis in social media content for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' For instance, Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' [9] proposed a late fusion-based framework for the automatic detection of passable roads after a flood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' For this purpose, several deep learning models are trained on flood-related images from social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' [4], on the other hand, employed social media imagery for post floods damage severity assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Flood detection and analysis in social content have also been a part of the MediaEval benchmark initiative as a shared task for several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Each time a separate aspect of flood analysis has been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' For instance, in MediaEval 2017 the task aimed at the retrieval of flood-related images from social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The task mainly involved analyzing the water level in different areas to differentiate between floods and regular water reservoirs, such as lakes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In MediaEval 2018, the task was slightly modified by asking the participants to propose multi-modal classification frameworks for flood-related multimedia content [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In MediaEval 2019 and 2020, the tasks aimed at analyzing flood severity and flood events recognition in social media posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Approach Figure 1 provides the block diagram of the proposed framework for the RCTP task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The framework is composed of three main components namely (i) Pre-processing, (ii) Training and Classification, and (iii) Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In the first step, different pre-processing techniques are employed to clean the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Three different transformers are then trained on the data to obtain classification scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In the final step, the classification scores of the individual models are combined in a late fusion scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The details of these steps are provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Figure 1: Block diagram of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Pre-processing In the pre-processing step, we employed different techniques for cleaning the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' More specifically, we removed unnecessary information, such as user names, URLs, emojis, punctuation marks, stop words, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Besides this, we also performed the necessary pre-possessing tasks that are required to transform the raw text into a form that is suitable for the transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' To achieve this, we used the TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='text library1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Classification via Transformers After cleaning and pre-processing the data, we trained three different models, namely BERT [14], RoBERTa [15], and XLNet [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The selection of these models for the task is motivated by their proven performance on similar tasks [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' A brief overview of these models is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' BERT: Bidirectional Encoder Representations from Transformers (BERT) is one of the state- of-the-art NLP algorithms for text processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The model is pre-trained on a large collection of unlabeled text and can be fine-tuned for different text-analysis applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The key attributes of the model include its bi-directional nature, pre-training with Masked Language Modeling (MLM), and Next Structure Prediction (NSP) objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In the experiments with BERT, we used the Adam optimizer with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='001 and a batch size of 8 for 3 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' RoBERTa: Robustly Optimized BERT is a modified version of the BERT model with an improved training mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' More specifically, in RoBERTa the NSP capabilities are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Moreover, dynamic masking is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In addition, a larger batch size and a larger amount of training data were used in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In this work, we used a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='001, batch size of 20, and 10 epochs during the fine-tuning of the model for the desired task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' XLNet: XLNet is another state-of-the-art NLP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Similar to BERT, XLNet is also a bidirectional transformer and uses an improved training approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In contrast to BERT and traditional NLP algorithms, XLNet relies on Permutation Language Modeling (PLM) by predicting all the tokens in random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' This allows XLNet to handle dependencies and bidirectional relationships in a better way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In this work, we used a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='002, a batch size of 32, and 4 epochs during the fine-tuning of the model for the desired task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='org/text/guide/bert_preprocessing_guide#text_preprocessing_with_tftext# Input Data Data Pre-processing Classification Late Fusion Model 1 F = S1+S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='.Sn Score obtained with M2 TextStreams Pre-processing Model 2 Mn Final Score Score Model NWe obtained the results in the form of posterior probabilities from these models, which are then used in the fusion scheme to obtain the final predicted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The fusion method used in this work is described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Fusion Our fusion method is based on late fusion, where we combined the classification scores obtained with the individual models for the final classification decision as shown in Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In the equation, 𝑆𝑓𝑖𝑛𝑎𝑙 represents the final classification score while 𝑠𝑛 is the score obtained with the nth model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' We note that in the current implementation, we used a simple fusion method by treating all the models equally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=', simple aggregation of the individual scores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' 𝑆𝑓𝑖𝑛𝑎𝑙 = 𝑆1 + 𝑆2 + 𝑠3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='. + 𝑆𝑛 (1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Results and Analysis Table 1 provides the experimental results of the proposed solutions on the development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' As can be been in the table, overall better results are obtained with the BERT model, and surprisingly, a lower F1-score is observed for RoBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In the future, we will further investigate the potential causes of the lower performance of RoBERTa by exploring different implementations and hyper- parameter settings for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' As far as the performance of the fusion methods is concerned, overall better results are obtained with the pair of XLNet and BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' One of the potential reasons for the lower performance of the fusion of all the models is the less accurate prediction of RoBERTa, as also evident from the performance of the individual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Table 1 Experimental results of the proposed solutions on the development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Method F1-Score BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='94 RoBERTa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='78 XLNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='93 Fusion 1 (RoBERTa, BERT, XLNet) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='75 Fusion 2 (BERT, XLNet) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='93 Fusion 3 (RoBERTa, XLNet) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='92 Table 2 provides the official results of the proposed solutions on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In total, three different runs were submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The first run is based on the fusion of all three models used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' The remaining two runs are based on the fusion of the models in pairs of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In run 2, BERT and XLNet are combined while in run 3 RoBERTa and XLNet are jointly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' As can be seen in the table, better results are obtained for the fusion of the models in pairs of two where the best performing pair of two models obtained an improvement of 20% over the fusion of all three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Table 2 Experimental results of the proposed solutions on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Run Precision Recall F1-Score 1 (Fusion of BERT, RoBERTa, XLNet) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='6738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='5431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='6014 2 (Fusion of BERT and XLNet) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='8044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='6948 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='7456 3 (Fusion of RoBERTa and XLNet) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='8977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='8598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='8784 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Conclusions In this paper, we presented our solutions for the RCTP subtask of DisasterMM challenge posted in MediaEval 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' We proposed a late fusion framework incorporating several state-of-the-art transformers for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In the current implementation, all the models are treated equally by assigning them equal weights (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' In the future, we aim to employ merit-based fusion methods to further improve the final 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Ahmad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} +page_content=' Al-Fuqaha, Social media as an instant source of feedback on water quality, IEEE Transactions on Technology and Society (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfefgG/content/2301.00320v1.pdf'} diff --git a/DNE1T4oBgHgl3EQfqAUl/content/tmp_files/2301.03337v1.pdf.txt b/DNE1T4oBgHgl3EQfqAUl/content/tmp_files/2301.03337v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f7ead79502a38cd958110901d86f67edca7ff58b --- /dev/null +++ b/DNE1T4oBgHgl3EQfqAUl/content/tmp_files/2301.03337v1.pdf.txt @@ -0,0 +1,1067 @@ +Adiabatic theory of one-dimensional curved polariton waveguides +D. A. Zezyulin∗1 and I. A. Shelykh2, 1 +1Department of Physics, ITMO University, Saint Petersburg 197101, Russia +2Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland +(Dated: January 10, 2023) +We construct a general theory of adiabatic propagation of spinor exciton-polaritons in waveguides +of arbitrary shape, accounting for the effects of TE-TM splitting in linear polarizations and Zeeman +splitting in circular polarizations. The developed theory is applied for the description of waveguides +of periodically curved shape. We show that in this geometry the periodic rotation of the effective +in-plane magnetic field produced by TE-TM interaction results in a nontrivial band-gap structure, +which can be additionally tuned by application of an external magnetic field. It is also demonstrated, +that spin-dependent interactions between polaritons lead to the formation of stable gap solitons. +Introduction. +Exciton-polaritons are composite half- +light half-matter quasiparticles emerging in the regime +of the strong coupling between a photonic mode of a +planar semiconductor microcavity and an exciton in a +quantum well (QW) brought in resonance with it. They +possess a set of remarkable properties, which allow po- +laritonic systems to serve as a convenient playground for +study of collective nonlinear phenomena at elevated tem- +peratures [1]. From their photonic component polaritons +get extremely small effective mass (about 10−5 of the +mass of free electrons) and macroscopically large coher- +ence length [2], while the presence of an excitonic com- +ponent enables efficient polariton-polariton interactions +[3–5] and leads to the sensitivity of the polariton systems +to external electric [6–8] and magnetic [9–11] fields. +An important property of cavity polaritons is their spin +(or pseudo-spin) [12], inherited from the spins of QW ex- +citons and cavity photons. Similar to photons, polari- +tons have two possible spin projections on the structure +growth axis corresponding to the two opposite circular +polarizations which can be mixed by effective magnetic +fields of various origin. Real magnetic field applied along +the structure growth axis and acting on the excitonic +component splits in energy the polariton states with op- +posite circular polarizations, while TE-TM splitting of +the photonic modes of a planar resonator couples these +states to each other via a k-dependent term, thus playing +a role of an effective spin-orbit interaction [12]. Impor- +tantly, polariton-polariton interactions are also spin de- +pendent, as they stem from the interactions of excitonic +components which are dominated by the exchange term +[13]. This leads to the fact that polaritons of the same cir- +cular polarization interact orders of magnitude stronger +than polaritons with opposite circular polarizations [3]. +Remarkable tunability of cavity polaritons allows to +engineer their spatial confinement in a variety of ex- +perimental geometries, ranging from individual micropil- +lars [14–17] to systems of several coupled pillars form- +∗email: d.zezyulin@gmail.com +ing so-called polariton molecules [18, 19] or periodically +arranged arrays of the pillars forming polariton super- +lattices [20–24]. +Realization of quasi one-dimensional +(1D) geometries, where the motion of the polaritons is +restricted to individual waveguides [7, 25], rings [26–28] +or systems of coupled waveguides [29, 30], represents par- +ticular interest from the point of view of the applications +of polaritonics, as they can form basis for classical [31–33] +and quantum [34, 35] polaritonic circuits. +Current state of technology allows routine production +of quasi 1D polariton waveguides of arbitrary shape, in- +cluding ones with periodically modulated curvature. Cre- +ation of the general theory of the polariton propagation +in these structures, which includes polarization dynam- +ics and polariton-polariton interactions, is the goal of the +present Letter. +The model. The presence of the in-plane spatial con- +finement results in the strong nonequivalency of the +states polarized normally and tangentially to a waveg- +uide, which leads to the appearance of a local effective +magnetic field, acting on a polariton pseudospin and di- +rected tangentially to the waveguide. Although one can +safely assume that in the case of a narrow waveguide of +a constant width the absolute value of this field remains +constant (see Supplementary material [36] for further de- +tails), its direction changes along the curved waveguide, +and, as we demonstrate below, this has crucial effect on +polariton dynamics. +Let us suppose that the shape of a waveguide in (x, y)- +plane is given parametrically as x = x(ξ), y = y(ξ). The +components of the effective magnetic field Ωx,y produced +by TE-TM interaction are proportional to the compo- +nents of the unit vector tangential to a waveguide τx,y +and thus read +Ωx = Ω0τx = +Ω0x′(ξ) +� +x′(ξ)2 + y′(ξ)2 , +(1) +Ωy = Ω0τy = +Ω0y′(ξ) +� +x′(ξ)2 + y′(ξ)2 , +(2) +arXiv:2301.03337v1 [cond-mat.mes-hall] 9 Jan 2023 + +2 +FIG. 1: (a) Schematic representation of the considered geom- +etry of a 1D polariton waveguide etched in planar semicon- +ductor microcavity. The arc length ℓ measures the distance +along the waveguide. Direction of the in-plane tangential unit +vector ⃗τ = (τx, τy) changes along the waveguide and leads to +emergence of an effective space-dependent field for the spinor +polariton wavefunction. +(b,c) Real and imaginary parts of +the L-periodic effective potentials Ω(ℓ) for a waveguide com- +posed of a chain of touching halfcircles (b) and a sine-shaped +waveguide (c). +where primes correspond to derivatives, and +Ω0 ≈ ℏ2 +4d2 +� 1 +ml +− 1 +mt +� +. +(3) +In the above equation, ml and mt stand for the effective +longitudinal and transverse masses of 2D polaritons, and +d is an effective width of a polariton channel [37]. As it +was already mentioned, the presence of the field Ω splits +in energy the modes polarized normally and tangentially +to a waveguide. +Additional splitting in circular polar- +izations, denoted by ∆z, can be induced by application +of an external magnetic field perpendicular to a cavity +interface. +Let us introduce the coordinate ℓ along the waveguide, +ℓ = +� ξ +0 +� +x′(η)2 + y′(η)2dη. +In the adiabatic approxi- +mation, the effective 1D Hamiltonian governing the dy- +namics of the spinor wavefunction of polaritons can be +then represented in the following form (see Supplemen- +tary material [36] for corresponding derivation): +ˆH = +� +� +� +� +− +ℏ2 +2meff +d2 +dℓ2 + ∆z +2 +Ω− +Ω+ +− +ℏ2 +2meff +d2 +dℓ2 − ∆z +2 +� +� +� +� , (4) +where +Ω± = Ω(ℓ) = Ω0(τx ± iτy)2, +(5) +and meff is the effective mass. +The physical meaning of the above Hamiltonian is +pretty clear: it describes a motion of a one-dimensional +spinor particle affected by a constant z-directed magnetic +field and in-plane magnetic field whose direction changes +along the way, being always tangential to the waveguide. +In what follows, we will work with the effective Hamil- +tonian rewritten in the dimensionless form. To this end, +we introduce the unit length λ0 and the unit energy +ε0 ≡ ℏ2/(2meffλ2 +0), and then rescale the variables of +(22) as ℓ → λ0ℓ and ∆z → ε0∆z. +Additionally, we +rescale time as t → (ℏ/ε0)t. +Assuming, for instance, +that the unit length λ0 corresponds to 5 µm and meff +is about 10−5 of the free electron mass, we obtain that +the unit energy ε0 is about 0.2 meV, and the time unit +ℏ/ε0 is equivalent to few picoseconds. Supplementing the +obtained dimensionless Hamiltonian with the interaction +terms [38], we obtain the following nonlinear evolution +problem that governs the dynamics of the spinor wave- +function (Ψ1, Ψ2): +i∂Ψ1 +∂t += −∂2Ψ1 +∂ℓ2 + ∆z +2 Ψ1 + Ω−(ℓ)Ψ2 ++(|Ψ1|2 + σ|Ψ2|2)Ψ1, +(6) +i∂Ψ2 +∂t += −∂2Ψ2 +∂ℓ2 − ∆z +2 Ψ2 + Ω+(ℓ)Ψ1 ++(|Ψ2|2 + σ|Ψ1|2)Ψ2. +(7) +Small negative coefficient σ takes into account weak at- +traction between polaritons of opposite polarizations (in +our numerical calculations the value σ = −0.05 was +used). +Examples: The chain of halfcircles and the sine-shaped +waveguide. +In what follows, we focus on the situation +when the shape of the curved waveguide can be de- +scribed by function y(x), see Fig. 1(a) for a schemat- +ics of the assumed geometry. +Then the effective field, +as a function of the arc length ℓ, can be computed as +Ω±(ℓ) = Ω0 exp{±2i arctan(dy/dx)}, where the deriva- +tive dy/dx should be expressed as a function of ℓ. In our +further consideration we focus on the case of periodically +curved waveguides. +As a first analytically tractable example we consider +the situation when the waveguide is composed of a peri- +odic chain of touching halfcircles of a radius R. In terms + +a +yRe +[m +Re, Im (2/20) +(°/) I +0 +Re, +0.25 +0.5 +0 +0.75 +1 +0 +0.25 +0.5 +0.75 +L3 +FIG. 2: Transformation of the band-gap structure for the sine-shaped waveguide under the fixed TE-TM splitting coefficient +Ω0 = 0.45 and increasing strength of the external magnetic field ∆z. Here the Bloch quasimomentum k varies within the reduced +Brillouin zone [−π/L, π/L), where L is the spatial period of the structure. The periodic curvature results in a nontrivial band- +gap structure. Finite bandgaps are present even in the absence of the external magnetic field (∆z = 0). The increase of ∆z +leads to the anticrossings of the bands touching at k = 0 and related shift of the band minima and maxima to k ̸= 0. +of coordinates x and y, the unit cell of the resulting +periodic structure is given as y(x) = +� +R2 − (x − R)2 +for x +∈ +[0, 2R] (the upper halfcircle) and y(x) += +− +� +R2 − (x − 3R)2 for x ∈ [2R, 4R] (the lower halfcir- +cle). +In terms of the arc length ℓ, the unit cell cor- +responds to the interval ℓ ∈ [0, L] where L = 2πR +is the period of the structure. +The first halfperiod +ℓ ∈ [0, πR] corresponds to the first halfcircle, where +x(ℓ) = R[1 − cos(ℓ/R)] and y(ℓ) = R sin(ℓ/R), and the +second halfperiod ℓ ∈ [πR, 2πR] corresponds to the sec- +ond halfcircle, where we have parametrization x(ℓ) = +R[3 + cos(ℓ/R)] and y(ℓ) = R sin(ℓ/R), and the rest of +waveguide is obtained by the periodic repetition of the +unit cell. Performing straightforward calculations, we ob- +tain that within the unit cell the resulting potential reads +Ω±(ℓ) = −Ω0 exp{∓2iℓ sign (πR − ℓ)/R}. +The shape +of the resulting dependency is illustrated in Fig. 1(b). +While the obtained dependence is rather simple, its imag- +inary part is not a smooth function: it has a cusp exactly +at the center of the unit cell ℓ = πR, where the two half- +circles touch. +As a second example, which results in a smooth peri- +odic potential (which is therefore better suited for the +numerical analysis), we consider a sine-shaped waveg- +uide y(x) = V0 sin x. +Then the arc length along the +waveguide is given by the incomplete elliptic integral of +the second kind [39]: ℓ(x) = +� +1 + V 2 +0 E(sin x, m), where +m = V 2 +0 /(1 + V 2 +0 ). To the best of our knowledge, there +is neither a commonly used special function nor a closed- +form expression that allows to invert the incomplete el- +liptic integral of the second kind, i.e., to express x and +y through ℓ in our case. In the meantime, there exists a +simple iterative numerical procedure for inversion of the +incomplete elliptic integral of the second kind [40]. Us- +ing this procedure, one can easily obtain the dependence +Ω(ℓ), see Fig. 1(c) for a representative example. +The +resulting 1D Hamiltonian ˆH defined by (22) becomes ef- +fectively periodic with the spatial period in ℓ given as +L = 4E(m), where E(m) is the complete elliptic integral +of the second kind. +Band structure. Periodic nature of the resulting sys- +tem suggests to look at the band structure which can +be presented in the form of the dependencies of the en- +ergy E versus Bloch quasimomentum k, which, without +loss of generality, can be assumed to belong to the Bril- +louin zone [−π/L, π/L), where L is the period. For sinu- +soidal waveguide the result computed for system (6)–(7) +with omitted nonlinear terms (|Ψ1,2|2 + σ|Ψ2,1|2)Ψ1,2 is +shown in Fig. 2. +We have focused on the transforma- +tion of the spectral structure subject the the increase +of the external magnetic field, which is characterized by +the Zeeman splitting coefficient ∆z. As one can see, the +periodic curvature of a waveguide results in a nontriv- +ial band-gap structure as the effective periodic potential +Ω(ℓ) opens finite gaps even in the absence of the external +magnetic field (∆z = 0). The increase of ∆z leads to +a transformation of the band-gap structure. In particu- +lar, it leads to the anticrossing of the bands touching at +k = 0 and related shift of the band minima and max- +ima to k ̸= 0. Dispersion curves having two degenerate +extrema at k = ±k0 ̸= 0 can be, in particular, relevant +for the observation of the so-called stripe phase charac- +terized by spinor wavefunctions carrying a more complex +internal structure, see e.g. [41–45] and [46] for discussion +of stripe phase and stripe solitons in spin-orbit coupled +atomic and polariton condensates, respectively. +Gap solitons. The presence of finite gaps in the band- +gap structure suggests that when the repulsive interac- +tions between the polaritons of the same circular po- +larization are taken into account, the waveguide can +support formation of polariton gap solitons [22, 46–51]. +These localized states can be found using the substitu- +tion Ψ1,2(t, ℓ) = e−iµtψ1,2(ℓ), where stationary wavefunc- +tions ψ1,2(ℓ) satisfy zero boundary conditions at ℓ → ∞ +and ℓ → −∞, and µ characterizes the chemical poten- +tial of the polariton condensate. +The numerical study + +=0 +△= 0.4 +△= 1.0 +△z = 1.4 +△= 2.0 +8 +8 +8 +6 +6 +6 +6 +E +2 +2 +0 +0 +0 +0 +kL/π +kL/π +kL/π +kL/π +kL/π4 +indicates that the system supports a variety of solitons +which form continuous families, i.e., can be parameter- +ized by the continuous change of the chemical potential +µ within the energy spectrum bandgap. To describe the +found solitons, we introduce the polariton density inte- +gral N = +� ∞ +−∞(|ψ1|2 + |ψ2|2)dℓ which characterizes the +squared norm of the solution. In Fig. 3(a) we illustrate +the family of fundamental (simplest) gap solitons as a de- +pendence N on µ. The soliton family detaches from the +left edge of the bandgap, where the soliton norm van- +ishes: N → 0. +In this limit, small-amplitude solitons +transform to a linear Bloch wave. As the chemical po- +tential increases towards the right gap edge, the total +norm N grows monotonously. To quantify the degree of +the soliton localization, we introduce an additional char- +acteristics n99 which amounts to the number of spatial +periods where 99% of quasiparticles are confined. The de- +pendence n99 on µ is also plotted in Fig. 3(a). It demon- +strates nonmonotonic behavior approaching its minimal +values in the center of the gap. In this regime the soli- +tons are most localized, and almost all energy can be +trapped in the segment of waveguide composed of ap- +proximately from five to ten unit cells. At the same time, +the quantity n99 becomes extremely large near the edges +of the gap, which means that the corresponding solitons +are very broad and relatively poorly localized. Examples +of spatial profiles of solitons having different amplitudes +and degrees of localization are shown in Fig. 3(b). +It is known that gap solitons and, in particular, those +in systems dominated by repulsive nonlinearities, can be +be prone to dynamical instabilities [52–55]. In the mean- +time, using the dynamical simulations, we found that the +family of fundamental gap solitons presented in Fig. 3(a) +contains stable solutions which can robustly preserve the +steady shape for the indefinite simulation time (much +larger than typical polariton lifetimes), even if the ini- +tial profiles are perturbed by a small-amplitude random +noise. Example of such stable dynamics is presented in +Fig. 3(c,d). At the same time, more complex solitons can +develop dynamical instabilities which eventually lead to +their delocalization. The corresponding example is shown +in Fig. 3(e,f). +Conclusion. In conclusion, we constructed a theory of +the propagation of cavity polaritons in narrow quasi-1D +waveguides of arbitrary shape and applied it to the case of +periodically curved waveguides. We demonstrated that +the periodic rotation of an effective in-plane magnetic +field produced by TE-TM splitting in linear polarizations +leads to the formation of nontrivial band structure. The +shape of the bands, the bandgaps and the positions of +the band extrema can be tuned by application of an ex- +ternal magnetic field. In the nonlinear regime the system +supports formation of dynamically stable gap solitons. +Acknowledgements. +The research was supported by +Priority 2030 Federal Academic Leadership Program. +IAS acknowledges support from Icelandic Research Fund +FIG. 3: (a) Gap solitons norm N and the localization measure +n99 as functions of chemical potential µ for a family of funda- +mental gap solitons in the first finite gap. Here the coefficient +of TE-TM splitting Ω0 = 0.4 and amplitude of the Zeeman +splitting ∆z = 0.3. Shaded regions correspond to the values of +µ that belong to spectral bands. 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Alfimov, Phys- +ica D: Nonlinear Phenomena 337, 58 (2016), ISSN 0167- +2789, URL https://www.sciencedirect.com/science/ +article/pii/S0167278916301440. + +7 +SUPPLEMENTAL MATERIAL: DERIVATION OF +THE 1D ADIABATIC HAMILTONIAN +The two-dimensional Hamiltonian of a polariton mov- +ing inside a waveguide defined by a confining potential +U(x, y) is [38]: +ˆH2D = +� +� +� +� +− +ℏ2 +2meff +� ∂2 +∂x2 + ∂2 +∂y2 +� ++ ∆z +2 + U(x, y) +β +� +∂ +∂y + i ∂ +∂x +�2 +β +� +∂ +∂y − i ∂ +∂x +�2 +− +ℏ2 +2meff +� ∂2 +∂x2 + ∂2 +∂y2 +� +− ∆z +2 + U(x, y) +� +� +� +� , +(8) +where +β = ℏ2 +4 +� 1 +ml +− 1 +mt +� +. +(9) +Let us introduce in each point of a waveguide local +coordinate system with axis ℓ directed tangential to it +and n normal to it. The elementary lengths dℓ and dn +read: +dℓ = τx(ℓ)dx + τy(ℓ)dy, +(10) +dn = −τy(ℓ)dx + τx(ℓ)dy +(11) +where τx,y are components of the unit vector tangential +to the waveguide at a given point characterized by coor- +dinate ℓ along the waveguide. +We can now right down: +∂ +∂x = ∂ℓ +∂x +∂ +∂ℓ + ∂n +∂x +∂ +∂n = τx +∂ +∂ℓ − τy +∂ +∂n, +(12) +∂ +∂y = ∂ℓ +∂y +∂ +∂ℓ + ∂n +∂y +∂ +∂n = τy +∂ +∂ℓ + τx +∂ +∂n, +(13) +∂ +∂y ± i ∂ +∂x = ±iτ∓ +∂ +∂ℓ + τ∓ +∂ +∂n, +(14) +where +τ± = τx ± iτy. +(15) +We thus have: +∂2 +∂x2 + ∂2 +∂y2 = ∂2 +∂ℓ2 + ∂2 +∂n2 + +� +τy +∂τx +∂ℓ − τx +∂τy +∂ℓ +� ∂ +∂n,(16) +where we used that +τ 2 +x + τ 2 +y = 1. +(17) +Similarly +� ∂ +∂y ± i ∂ +∂x +�2 += (18) += τ 2 +∓ +∂2 +∂n2 − τ∓ +∂ +∂ℓτ∓ +∂ +∂ℓ ± iτ∓ +� +τ∓ +∂ +∂ℓ + ∂ +∂ℓτ∓ +� ∂ +∂n. +Let us now suggest that the confining potential locally +depends on the transverse coordinate n only, and use adi- +abatic approximation for the spinor wavefunction Ψ(x, y) +representing it as: +Ψ(x, y) = ψ(ℓ)φ(n), +(19) +where the part ψ(ℓ) describes the propagation of the po- +laritons along the waveguide, and φ(n) corresponds to +their 1D lateral confinement and can be taken real. This +approximation holds if an effective thickness of a waveg- +uide d is much less then its local curvature R, which for +a parametrically given curve is given by +R = +� +x′(ξ)2 + y′(ξ)2�3/2 +|x′(ξ)y′′(ξ) − y′(ξ)x′′(ξ)|. +(20) +Multiplying the Schr¨odinger equation ˆH2DΨ = EΨ by +φ(n) and integrating by n from −∞ to +∞, one gets for +the dynamics of the propagation along the channel the +following 1D Schr¨odinger equation: +ˆHψ(ℓ) = Eψ(ℓ), +(21) +where + +8 +ˆH = +� +� +� +� +� +� +E0 − +ℏ2 +2meff +d2 +dℓ2 + ∆z +2 +Ω− − βτ− +d +dℓτ− +d +dℓ +Ω+ − βτ+ +d +dℓτ+ +d +dℓ +E0 − +ℏ2 +2meff +d2 +dℓ2 − ∆z +2 +� +� +� +� +� +� +, +(22) +and we have used that +� +∞ +−∞ +φ(n)dφ +dn dn = 0, +(23) +and +E0 = +� +∞ +−∞ +φ(n) +� +− +ℏ2 +2meff +d2 +dn2 + U(n) +� +φ(n)dn +(24) +is the energy of the confinement, and +Ω± = βτ 2 +± +� +∞ +−∞ +φ(n) d2φ +∂n2 dn ≈ β +d2 τ 2 +± = Ω0τ 2 +±, +(25) +where d is an effective width of the confining channel, and +we used Gaussion approximation, φ(n) = d√πe−n2/(2d2) +Note, that E0 is just a constant, which can be safely +dropped. As for the off-diagonal terms βτ± d +dℓτ± d +dℓ, one +can note, that by the order of magnitude d/dℓ ∼ k, where +k is a wavenumber, describing the propagation of the +polaritons along the waveguide. Therefore, for narrow +waveguides and small k, when k ≪ d−1, these terms +can be neglected as compared to Ω±, and one gets the +Hamiltonian (4) of the main text. + diff --git a/DNE1T4oBgHgl3EQfqAUl/content/tmp_files/load_file.txt b/DNE1T4oBgHgl3EQfqAUl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..26c40b42fa49f43b6352cc1d82113efbe7ff9a41 --- /dev/null +++ b/DNE1T4oBgHgl3EQfqAUl/content/tmp_files/load_file.txt @@ -0,0 +1,1096 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf,len=1095 +page_content='Adiabatic theory of one-dimensional curved polariton waveguides D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Zezyulin∗1 and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Shelykh2, 1 1Department of Physics, ITMO University, Saint Petersburg 197101, Russia 2Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland (Dated: January 10, 2023) We construct a general theory of adiabatic propagation of spinor exciton-polaritons in waveguides of arbitrary shape, accounting for the effects of TE-TM splitting in linear polarizations and Zeeman splitting in circular polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The developed theory is applied for the description of waveguides of periodically curved shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' We show that in this geometry the periodic rotation of the effective in-plane magnetic field produced by TE-TM interaction results in a nontrivial band-gap structure, which can be additionally tuned by application of an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' It is also demonstrated, that spin-dependent interactions between polaritons lead to the formation of stable gap solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Exciton-polaritons are composite half- light half-matter quasiparticles emerging in the regime of the strong coupling between a photonic mode of a planar semiconductor microcavity and an exciton in a quantum well (QW) brought in resonance with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' They possess a set of remarkable properties, which allow po- laritonic systems to serve as a convenient playground for study of collective nonlinear phenomena at elevated tem- peratures [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' From their photonic component polaritons get extremely small effective mass (about 10−5 of the mass of free electrons) and macroscopically large coher- ence length [2], while the presence of an excitonic com- ponent enables efficient polariton-polariton interactions [3–5] and leads to the sensitivity of the polariton systems to external electric [6–8] and magnetic [9–11] fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' An important property of cavity polaritons is their spin (or pseudo-spin) [12], inherited from the spins of QW ex- citons and cavity photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Similar to photons, polari- tons have two possible spin projections on the structure growth axis corresponding to the two opposite circular polarizations which can be mixed by effective magnetic fields of various origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Real magnetic field applied along the structure growth axis and acting on the excitonic component splits in energy the polariton states with op- posite circular polarizations, while TE-TM splitting of the photonic modes of a planar resonator couples these states to each other via a k-dependent term, thus playing a role of an effective spin-orbit interaction [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Impor- tantly, polariton-polariton interactions are also spin de- pendent, as they stem from the interactions of excitonic components which are dominated by the exchange term [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' This leads to the fact that polaritons of the same cir- cular polarization interact orders of magnitude stronger than polaritons with opposite circular polarizations [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Remarkable tunability of cavity polaritons allows to engineer their spatial confinement in a variety of ex- perimental geometries, ranging from individual micropil- lars [14–17] to systems of several coupled pillars form- ∗email: d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='zezyulin@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='com ing so-called polariton molecules [18, 19] or periodically arranged arrays of the pillars forming polariton super- lattices [20–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Realization of quasi one-dimensional (1D) geometries, where the motion of the polaritons is restricted to individual waveguides [7, 25], rings [26–28] or systems of coupled waveguides [29, 30], represents par- ticular interest from the point of view of the applications of polaritonics, as they can form basis for classical [31–33] and quantum [34, 35] polaritonic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Current state of technology allows routine production of quasi 1D polariton waveguides of arbitrary shape, in- cluding ones with periodically modulated curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Cre- ation of the general theory of the polariton propagation in these structures, which includes polarization dynam- ics and polariton-polariton interactions, is the goal of the present Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The presence of the in-plane spatial con- finement results in the strong nonequivalency of the states polarized normally and tangentially to a waveg- uide, which leads to the appearance of a local effective magnetic field, acting on a polariton pseudospin and di- rected tangentially to the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Although one can safely assume that in the case of a narrow waveguide of a constant width the absolute value of this field remains constant (see Supplementary material [36] for further de- tails), its direction changes along the curved waveguide, and, as we demonstrate below, this has crucial effect on polariton dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Let us suppose that the shape of a waveguide in (x, y)- plane is given parametrically as x = x(ξ), y = y(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The components of the effective magnetic field Ωx,y produced by TE-TM interaction are proportional to the compo- nents of the unit vector tangential to a waveguide τx,y and thus read Ωx = Ω0τx = Ω0x′(ξ) � x′(ξ)2 + y′(ξ)2 , (1) Ωy = Ω0τy = Ω0y′(ξ) � x′(ξ)2 + y′(ξ)2 , (2) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='03337v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='mes-hall] 9 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 1: (a) Schematic representation of the considered geom- etry of a 1D polariton waveguide etched in planar semicon- ductor microcavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The arc length ℓ measures the distance along the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Direction of the in-plane tangential unit vector ⃗τ = (τx, τy) changes along the waveguide and leads to emergence of an effective space-dependent field for the spinor polariton wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (b,c) Real and imaginary parts of the L-periodic effective potentials Ω(ℓ) for a waveguide com- posed of a chain of touching halfcircles (b) and a sine-shaped waveguide (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' where primes correspond to derivatives, and Ω0 ≈ ℏ2 4d2 � 1 ml − 1 mt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (3) In the above equation, ml and mt stand for the effective longitudinal and transverse masses of 2D polaritons, and d is an effective width of a polariton channel [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' As it was already mentioned, the presence of the field Ω splits in energy the modes polarized normally and tangentially to a waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Additional splitting in circular polar- izations, denoted by ∆z, can be induced by application of an external magnetic field perpendicular to a cavity interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Let us introduce the coordinate ℓ along the waveguide, ℓ = � ξ 0 � x′(η)2 + y′(η)2dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In the adiabatic approxi- mation, the effective 1D Hamiltonian governing the dy- namics of the spinor wavefunction of polaritons can be then represented in the following form (see Supplemen- tary material [36] for corresponding derivation): ˆH = � � � � − ℏ2 2meff d2 dℓ2 + ∆z 2 Ω− Ω+ − ℏ2 2meff d2 dℓ2 − ∆z 2 � � � � , (4) where Ω± = Ω(ℓ) = Ω0(τx ± iτy)2, (5) and meff is the effective mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The physical meaning of the above Hamiltonian is pretty clear: it describes a motion of a one-dimensional spinor particle affected by a constant z-directed magnetic field and in-plane magnetic field whose direction changes along the way, being always tangential to the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In what follows, we will work with the effective Hamil- tonian rewritten in the dimensionless form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' To this end, we introduce the unit length λ0 and the unit energy ε0 ≡ ℏ2/(2meffλ2 0), and then rescale the variables of (22) as ℓ → λ0ℓ and ∆z → ε0∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Additionally, we rescale time as t → (ℏ/ε0)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Assuming, for instance, that the unit length λ0 corresponds to 5 µm and meff is about 10−5 of the free electron mass, we obtain that the unit energy ε0 is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='2 meV, and the time unit ℏ/ε0 is equivalent to few picoseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Supplementing the obtained dimensionless Hamiltonian with the interaction terms [38], we obtain the following nonlinear evolution problem that governs the dynamics of the spinor wave- function (Ψ1, Ψ2): i∂Ψ1 ∂t = −∂2Ψ1 ∂ℓ2 + ∆z 2 Ψ1 + Ω−(ℓ)Ψ2 +(|Ψ1|2 + σ|Ψ2|2)Ψ1, (6) i∂Ψ2 ∂t = −∂2Ψ2 ∂ℓ2 − ∆z 2 Ψ2 + Ω+(ℓ)Ψ1 +(|Ψ2|2 + σ|Ψ1|2)Ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (7) Small negative coefficient σ takes into account weak at- traction between polaritons of opposite polarizations (in our numerical calculations the value σ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='05 was used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Examples: The chain of halfcircles and the sine-shaped waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In what follows, we focus on the situation when the shape of the curved waveguide can be de- scribed by function y(x), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 1(a) for a schemat- ics of the assumed geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Then the effective field, as a function of the arc length ℓ, can be computed as Ω±(ℓ) = Ω0 exp{±2i arctan(dy/dx)}, where the deriva- tive dy/dx should be expressed as a function of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In our further consideration we focus on the case of periodically curved waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' As a first analytically tractable example we consider the situation when the waveguide is composed of a peri- odic chain of touching halfcircles of a radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In terms a yRe [m Re, Im (2/20) (°/) I 0 Re, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='75 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='75 L3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 2: Transformation of the band-gap structure for the sine-shaped waveguide under the fixed TE-TM splitting coefficient Ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='45 and increasing strength of the external magnetic field ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Here the Bloch quasimomentum k varies within the reduced Brillouin zone [−π/L, π/L), where L is the spatial period of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The periodic curvature results in a nontrivial band- gap structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Finite bandgaps are present even in the absence of the external magnetic field (∆z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The increase of ∆z leads to the anticrossings of the bands touching at k = 0 and related shift of the band minima and maxima to k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' of coordinates x and y, the unit cell of the resulting periodic structure is given as y(x) = � R2 − (x − R)2 for x ∈ [0, 2R] (the upper halfcircle) and y(x) = − � R2 − (x − 3R)2 for x ∈ [2R, 4R] (the lower halfcir- cle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In terms of the arc length ℓ, the unit cell cor- responds to the interval ℓ ∈ [0, L] where L = 2πR is the period of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The first halfperiod ℓ ∈ [0, πR] corresponds to the first halfcircle, where x(ℓ) = R[1 − cos(ℓ/R)] and y(ℓ) = R sin(ℓ/R), and the second halfperiod ℓ ∈ [πR, 2πR] corresponds to the sec- ond halfcircle, where we have parametrization x(ℓ) = R[3 + cos(ℓ/R)] and y(ℓ) = R sin(ℓ/R), and the rest of waveguide is obtained by the periodic repetition of the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Performing straightforward calculations, we ob- tain that within the unit cell the resulting potential reads Ω±(ℓ) = −Ω0 exp{∓2iℓ sign (πR − ℓ)/R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The shape of the resulting dependency is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' While the obtained dependence is rather simple, its imag- inary part is not a smooth function: it has a cusp exactly at the center of the unit cell ℓ = πR, where the two half- circles touch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' As a second example, which results in a smooth peri- odic potential (which is therefore better suited for the numerical analysis), we consider a sine-shaped waveg- uide y(x) = V0 sin x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Then the arc length along the waveguide is given by the incomplete elliptic integral of the second kind [39]: ℓ(x) = � 1 + V 2 0 E(sin x, m), where m = V 2 0 /(1 + V 2 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' To the best of our knowledge, there is neither a commonly used special function nor a closed- form expression that allows to invert the incomplete el- liptic integral of the second kind, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=', to express x and y through ℓ in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In the meantime, there exists a simple iterative numerical procedure for inversion of the incomplete elliptic integral of the second kind [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Us- ing this procedure, one can easily obtain the dependence Ω(ℓ), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 1(c) for a representative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The resulting 1D Hamiltonian ˆH defined by (22) becomes ef- fectively periodic with the spatial period in ℓ given as L = 4E(m), where E(m) is the complete elliptic integral of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Periodic nature of the resulting sys- tem suggests to look at the band structure which can be presented in the form of the dependencies of the en- ergy E versus Bloch quasimomentum k, which, without loss of generality, can be assumed to belong to the Bril- louin zone [−π/L, π/L), where L is the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' For sinu- soidal waveguide the result computed for system (6)–(7) with omitted nonlinear terms (|Ψ1,2|2 + σ|Ψ2,1|2)Ψ1,2 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' We have focused on the transforma- tion of the spectral structure subject the the increase of the external magnetic field, which is characterized by the Zeeman splitting coefficient ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' As one can see, the periodic curvature of a waveguide results in a nontriv- ial band-gap structure as the effective periodic potential Ω(ℓ) opens finite gaps even in the absence of the external magnetic field (∆z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The increase of ∆z leads to a transformation of the band-gap structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In particu- lar, it leads to the anticrossing of the bands touching at k = 0 and related shift of the band minima and max- ima to k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Dispersion curves having two degenerate extrema at k = ±k0 ̸= 0 can be, in particular, relevant for the observation of the so-called stripe phase charac- terized by spinor wavefunctions carrying a more complex internal structure, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' [41–45] and [46] for discussion of stripe phase and stripe solitons in spin-orbit coupled atomic and polariton condensates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Gap solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The presence of finite gaps in the band- gap structure suggests that when the repulsive interac- tions between the polaritons of the same circular po- larization are taken into account, the waveguide can support formation of polariton gap solitons [22, 46–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' These localized states can be found using the substitu- tion Ψ1,2(t, ℓ) = e−iµtψ1,2(ℓ), where stationary wavefunc- tions ψ1,2(ℓ) satisfy zero boundary conditions at ℓ → ∞ and ℓ → −∞, and µ characterizes the chemical poten- tial of the polariton condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The numerical study =0 △= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='4 △= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='0 △z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='4 △= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='0 8 8 8 6 6 6 6 E 2 2 0 0 0 0 kL/π kL/π kL/π kL/π kL/π4 indicates that the system supports a variety of solitons which form continuous families, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=', can be parameter- ized by the continuous change of the chemical potential µ within the energy spectrum bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' To describe the found solitons, we introduce the polariton density inte- gral N = � ∞ −∞(|ψ1|2 + |ψ2|2)dℓ which characterizes the squared norm of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 3(a) we illustrate the family of fundamental (simplest) gap solitons as a de- pendence N on µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The soliton family detaches from the left edge of the bandgap, where the soliton norm van- ishes: N → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In this limit, small-amplitude solitons transform to a linear Bloch wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' As the chemical po- tential increases towards the right gap edge, the total norm N grows monotonously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' To quantify the degree of the soliton localization, we introduce an additional char- acteristics n99 which amounts to the number of spatial periods where 99% of quasiparticles are confined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The de- pendence n99 on µ is also plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' It demon- strates nonmonotonic behavior approaching its minimal values in the center of the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In this regime the soli- tons are most localized, and almost all energy can be trapped in the segment of waveguide composed of ap- proximately from five to ten unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' At the same time, the quantity n99 becomes extremely large near the edges of the gap, which means that the corresponding solitons are very broad and relatively poorly localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Examples of spatial profiles of solitons having different amplitudes and degrees of localization are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' It is known that gap solitons and, in particular, those in systems dominated by repulsive nonlinearities, can be be prone to dynamical instabilities [52–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In the mean- time, using the dynamical simulations, we found that the family of fundamental gap solitons presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 3(a) contains stable solutions which can robustly preserve the steady shape for the indefinite simulation time (much larger than typical polariton lifetimes), even if the ini- tial profiles are perturbed by a small-amplitude random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Example of such stable dynamics is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 3(c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' At the same time, more complex solitons can develop dynamical instabilities which eventually lead to their delocalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The corresponding example is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 3(e,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In conclusion, we constructed a theory of the propagation of cavity polaritons in narrow quasi-1D waveguides of arbitrary shape and applied it to the case of periodically curved waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' We demonstrated that the periodic rotation of an effective in-plane magnetic field produced by TE-TM splitting in linear polarizations leads to the formation of nontrivial band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The shape of the bands, the bandgaps and the positions of the band extrema can be tuned by application of an ex- ternal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' In the nonlinear regime the system supports formation of dynamically stable gap solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The research was supported by Priority 2030 Federal Academic Leadership Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' IAS acknowledges support from Icelandic Research Fund FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 3: (a) Gap solitons norm N and the localization measure n99 as functions of chemical potential µ for a family of funda- mental gap solitons in the first finite gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Here the coefficient of TE-TM splitting Ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='4 and amplitude of the Zeeman splitting ∆z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Shaded regions correspond to the values of µ that belong to spectral bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (b) Example of a broad soli- ton near the left edge of the gap (specifically, at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='24) and a strongly localized soliton in the center of the gap at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (c,d) Stable dynamics of the gap soliton with chemical poten- tial µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Initial conditions correspond to the stationary wavefunctions perturbed with a random noise whose ampli- tude is about 2% of the soliton’s amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (e,f) Example of unstable evolution of a gap soliton of more complex shape corresponding to Ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='4, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='4, and ∆z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (Rannis), project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Caputo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Mu˜noz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' De Giorgi, 10 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='1 (b) 20 99 N5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='5 2 10 a 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='6 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='4 μ l/L 亚1 亚2 (c) (d) 0 0 0 [亚1 [亚2] (e) (f) 1> 05 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Dominici, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' H.' metadata={'source': 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Bloch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 106, 126401 (2011), URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='org/doi/10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Sagnes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Lemaˆıtre, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Le Gratiet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Harouri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Ravets, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Bloch, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} 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Galopin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Sagnes, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' X 5, 011034 (2015), URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='org/doi/10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Carusotto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Galopin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Lemaˆıtre, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Le Gratiet, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Sagnes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Bloch, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Schneider, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 121, 257402 (2018), URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='org/doi/10.' metadata={'source': 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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Zezyulin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Alfimov, Phys- ica D: Nonlinear Phenomena 337, 58 (2016), ISSN 0167- 2789, URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content='com/science/ article/pii/S0167278916301440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' 7 SUPPLEMENTAL MATERIAL: DERIVATION OF THE 1D ADIABATIC HAMILTONIAN The two-dimensional Hamiltonian of a polariton mov- ing inside a waveguide defined by a confining potential U(x, y) is [38]: ˆH2D = � � � � − ℏ2 2meff � ∂2 ∂x2 + ∂2 ∂y2 � + ∆z 2 + U(x, y) β � ∂ ∂y + i ∂ ∂x �2 β � ∂ ∂y − i ∂ ∂x �2 − ℏ2 2meff � ∂2 ∂x2 + ∂2 ∂y2 � − ∆z 2 + U(x, y) � � � � , (8) where β = ℏ2 4 � 1 ml − 1 mt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (9) Let us introduce in each point of a waveguide local coordinate system with axis ℓ directed tangential to it and n normal to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' The elementary lengths dℓ and dn read: dℓ = τx(ℓ)dx + τy(ℓ)dy, (10) dn = −τy(ℓ)dx + τx(ℓ)dy (11) where τx,y are components of the unit vector tangential to the waveguide at a given point characterized by coor- dinate ℓ along the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' We can now right down: ∂ ∂x = ∂ℓ ∂x ∂ ∂ℓ + ∂n ∂x ∂ ∂n = τx ∂ ∂ℓ − τy ∂ ∂n, (12) ∂ ∂y = ∂ℓ ∂y ∂ ∂ℓ + ∂n ∂y ∂ ∂n = τy ∂ ∂ℓ + τx ∂ ∂n, (13) ∂ ∂y ± i ∂ ∂x = ±iτ∓ ∂ ∂ℓ + τ∓ ∂ ∂n, (14) where τ± = τx ± iτy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (15) We thus have: ∂2 ∂x2 + ∂2 ∂y2 = ∂2 ∂ℓ2 + ∂2 ∂n2 + � τy ∂τx ∂ℓ − τx ∂τy ∂ℓ � ∂ ∂n,(16) where we used that τ 2 x + τ 2 y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (17) Similarly � ∂ ∂y ± i ∂ ∂x �2 = (18) = τ 2 ∓ ∂2 ∂n2 − τ∓ ∂ ∂ℓτ∓ ∂ ∂ℓ ± iτ∓ � τ∓ ∂ ∂ℓ + ∂ ∂ℓτ∓ � ∂ ∂n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Let us now suggest that the confining potential locally depends on the transverse coordinate n only, and use adi- abatic approximation for the spinor wavefunction Ψ(x, y) representing it as: Ψ(x, y) = ψ(ℓ)φ(n), (19) where the part ψ(ℓ) describes the propagation of the po- laritons along the waveguide, and φ(n) corresponds to their 1D lateral confinement and can be taken real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' This approximation holds if an effective thickness of a waveg- uide d is much less then its local curvature R, which for a parametrically given curve is given by R = � x′(ξ)2 + y′(ξ)2�3/2 |x′(ξ)y′′(ξ) − y′(ξ)x′′(ξ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (20) Multiplying the Schr¨odinger equation ˆH2DΨ = EΨ by φ(n) and integrating by n from −∞ to +∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' one gets for the dynamics of the propagation along the channel the following 1D Schr¨odinger equation: ˆHψ(ℓ) = Eψ(ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (21) where 8 ˆH = � � � � � � E0 − ℏ2 2meff d2 dℓ2 + ∆z 2 Ω− − βτ− d dℓτ− d dℓ Ω+ − βτ+ d dℓτ+ d dℓ E0 − ℏ2 2meff d2 dℓ2 − ∆z 2 � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (22) and we have used that � +∞ −∞ φ(n)dφ dn dn = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (23) and E0 = � +∞ −∞ φ(n) � − ℏ2 2meff d2 dn2 + U(n) � φ(n)dn (24) is the energy of the confinement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' and Ω± = βτ 2 ± � +∞ −∞ φ(n) d2φ ∂n2 dn ≈ β d2 τ 2 ± = Ω0τ 2 ±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' (25) where d is an effective width of the confining channel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' and we used Gaussion approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' φ(n) = d√πe−n2/(2d2) Note,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' that E0 is just a constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' which can be safely dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' As for the off-diagonal terms βτ± d dℓτ± d dℓ, one can note, that by the order of magnitude d/dℓ ∼ k, where k is a wavenumber, describing the propagation of the polaritons along the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} +page_content=' Therefore, for narrow waveguides and small k, when k ≪ d−1, these terms can be neglected as compared to Ω±, and one gets the Hamiltonian (4) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf'} diff --git a/DdAyT4oBgHgl3EQf4vob/vector_store/index.pkl b/DdAyT4oBgHgl3EQf4vob/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..6ec49ef5f6d78d20999c9fd9bbf49f8cfc94d3b4 --- /dev/null +++ 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Shamimabad Road, Sylhet +3100, Bangladesh + +Corresponding author: a) 19h0901@ubd.edu.bn or rajarshiry@gmail.com + +ABSTRACT. The usage of technologically advanced devices has seen a boom in many domains, including education, +automation, and healthcare; with most of the services requiring Internet-connectivity. To secure a network, device +identification plays key role. In this paper, a device fingerprinting (DFP) model, which is able to distinguish between +Internet of Things (IoT) and non-IoT devices, as well as uniquely identify individual devices, has been proposed. Four +statistical features have been extracted from the consecutive five device-originated packets, to generate individual device +fingerprints. The method has been evaluated using the Random Forest (RF) classifier and different datasets. Experimental +results have shown that the proposed method achieves up to 99.8% accuracy in distinguishing between IoT and non-IoT +devices and over 97.6% in classifying individual devices. These signify that the proposed method is useful in assisting +operators in making their networks more secure and robust to security breaches and unauthorised access. +Keywords : digital footprint; network traffic traces; machine learning algorithm; internet of things; device +fingerprinting + +INTRODUCTION + +It has been predicted that the number of network-connected Internet of Things (IoT) and non-IoT devices +worldwide will reach approximately 30.9 billion and 10.3 billion, respectively, by the year 2025 [1]⁠. Proliferated +growth of these devices with their heterogeneous functionalities, has imposed new challenges to network +administrators and operators, in providing, managing, and controlling the operations and security of the network +services [2]⁠. Accurate device identification is one key aspect that needs to be seriously considered in securing +network-connected devices. Conventionally, internet protocol (IP) enabled devices have been using user-defined +identifiers, such as IP and media access control (MAC) addresses, as a form of identifications. However, these +identifiers have been proven to be vulnerable [3]⁠ to various attacks, such as spoofing [4]⁠ and device mobility, due to +the availability of malicious software [5]⁠, for performing such attacks. Device fingerprinting (DFP) [3]⁠ represents +one technique that may be used to identify devices based on their communication traffic traces (or digital footprints) +without using explicit identifiers, and it can be performed, either actively or passively, from different layers of the +communication model [6]⁠. + +Due to the prominent characteristics of network traffic features, many researchers [2, 7]⁠ have used packet-level +features for different purposes [8]⁠, including for device identification [9]⁠. Sivanathan et al. [10]⁠ have described a +DFP scheme based on the analysis of passively observed network traffic traces. A total of 11 statistical features are +used as device fingerprints, from packet traffic-flows over a period of one day, by looking at the devices’ sleeping +time, average packet size and traffic rate, active time, number of servers and protocols used in a flow, number of + +8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei +unique domain name system (DNS) request, and intervals of DNS and network time protocol (NTP) requests. +Subsequently, these features are used to train an ML model for classification. It has been shown that the DFP +scheme is able to distinguish between IoT and non-IoT devices with high accuracy and achieve over 95% accuracy +in identifying individual IoT devices. The same researchers [9]⁠ have also presented another device fingerprinting +scheme, by utilizing statistical characteristics of hourly network traffic traces, to generate 8 device-specific +fingerprints. Experimental result has shown over 99% accuracy using the UNSW dataset. Charyyev et al. [11]⁠ have +utilized Nilsimsa hash value of packet flows (n packets) for device-specific fingerprints, to classify individual IoT +devices, to achieve 93% precision. + +Researchers in [2, 12]⁠ have used 12 packets information, to generate device signatures for classifying IoT +devices, with 81.5% global accuracy and 76.15% accuracy using an aggregated model, whilst Aksoy and Gunes [13]⁠ +have presented a DFP approach, known as SysID, which utilizes features from a single packet, for identifying smart +home IoT devices with 82% average classification accuracy. Bezawada et al. [14]⁠ have utilized 5 consecutive +packets information, including protocols headers and payload (20 features), for classifying IoT devices uniquely +with mean identification accuracy of 93% to 100% using a laboratory dataset of 14 IoT devices. In [15]⁠, the authors +have used a one second window to group packets, for generating statistical fingerprinting features. These features +are then used to train a binary classifier for categorizing IoT and non-IoT devices with high accuracy of 99%, whilst +a multi-class classifier has been used to uniquely identify IoT devices with about 96% accuracy. All these existing +DFP models, however, require either a large number of features set from different layers of the communication +model, or a large number of network packets information for generating fingerprints. Consequently, these models +consume a long period of time, and require complex computation. As such, a more efficient DFP model is required +for classifying devices with high accuracy, but with less computation cost. + +In this paper, a supervised machine learning (ML) based DFP model, which generates device-specific signatures +by computing four statistical features from consecutive five packets of the network traffic, has been proposed. An +intuition that these features carry device-specific characteristics in terms of device memory and processing speed. +Experimental results have shown that over 97.0% accuracy is achievable in classifying individual non-IoT devices +from traffic collected in a laboratory environment, and 97.3% accuracy on the non-IoT traffic traces from the +UNSW dataset. The proposed DFP model is also capable of distinguishing between IoT and non-IoT devices with +up to 99.8% accuracy on the UNSW dataset. The key contributions of this research work are: + +• +Identifying device-specific features from the device-originated communication traffic traces, to generate +device signatures for classification. +• +Instrument an experimental testbed of non-IoT devices in a laboratory environment for data collection. +• +Evaluate the proposed DFP scheme performance based on a supervised ML algorithm, to distinguish between +IoT and non-IoT devices and identify individual devices. + +The rest of the paper is organized as follows. The proposed ML-based device fingerprinting method, as well as +the datasets, data collection procedure, and an ML classifier are described in Section II. Section III describes +experimental results on various datasets, and finally, conclusion is given in Section IV. + +METHODOLOGY + +The proposed DFP method is used to extract unique device features from network traffic traces. These features +are used to train an ML classifier, and subsequently, used to test the performance of the proposed DFP method on +different datasets. This section describes the proposed DFP method, the datasets used for training and testing, as +well as the classification method used to test the model. + +Datasets: IoT and Non-IoT + +The proposed device fingerprinting model performance has been evaluated by utilizing a publicly available +dataset: UNSW [9]⁠, and a testbed dataset of non-IoT devices, which has been collected from a laboratory +environment. Summary of the datasets are listed in Table 1. The UNSW dataset comprises network traffic traces +from both IoT and non-IoT devices, including TP-Link camera, smart bulb, Belkin camera, smart doorbell, printer, + +8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei +smart photo frame, laptop, smartphone, and tablet devices, with these heterogeneous devices coming from different +manufacturers: Belkin, Philips Hue, Netatmo, TP-Link, Withings, HP, Apple. On the other hand, the laboratory +dataset comprises 7 non-IoT devices, including laptops, smartphones, and desktops, from different manufacturers. +The data collection procedure from the 7 non-IoT devices is described in the following section. + +TABLE 1. List of IoT and non-IoT Datasets. +Dataset +Devices +Total Packets +Source +IoT +Non-IoT +UNSW +22 +-- +6,844,740 +[9]⁠ +-- +7 +3,515,705 +Lab Dataset +-- +7 +442,970 +-- + +TABLE 2. List of non-IoT devices for experimental set up. +No. +Device Category +Device Name/Model +Operating System +Connectivity +MAC Address +1 +Laptop +Aspire-S7 +Windows +WiFi +34:23:87:b7:56:17 +2 +ProBook-4410s +WiFi/Ethernet +00:25:b3:47:da:6f +3 +Desktop +Asus +Ethernet +08:60:6e:c1:79:c2 +4 +HP-EliteDesk +Ethernet +80:e8:2c:d6:9e:49 +5 +Smart Phone +MYA-U29 +Android +WiFi +d0:ff:98:95:57:af +6 +MLXP2ZA-A +iOS +WiFi +e0:c7:67:45:a3:62 +7 +MWC22KH-A +WiFi +06:44:b7:aa:20:98 + +Dataset Collection Methodology + +An experimental design, consisting of local area network (LAN) and wireless local area network (WLAN) with +non-IoT devices, was set up in a laboratory environment at Universiti Brunei Darussalam (UBD). Design of the +testbed is depicted in Figure 1, with the seven non-IoT devices from different manufacturers and of different types, as +listed in Table 2. These devices were configured, to connect with an access point (AP) either using ethernet or wireless +fidelity (WiFi) interfaces. + + + + + + + + + + + + + + + + + + +FIGURE 1. An experimental testbed of non-IoT devices network (LAN/WLAN). + +DNS +NTP +Server +Connectivity: +Server +Server +N +Ethernet +WiFi +Other +a +Internet +WiFi +Hotspot +Gateway +ubuntu? +(UBD Network) +Hub +Ethernet +USB Ethernet +Port +Port +Monitoring Station +(Capture Network Traffic)8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei +A laptop was used to configure an access point (AP), which was used to provide network services to the non-IoT +devices, as well as to monitor and capture communication footprints from the devices. The Dell Inspiron 15 5000 +Series laptop runs Ubuntu 18.04 as an operating system (OS), and was connected to the UBD network via its built-in +Ethernet interface, to provide the Internet connections. The built-in WiFi interface was configured as a WiFi Hotspot, +providing wireless connectivity to the WiFi-enabled (IEEE 802.11 standard) devices. Additionally, a TU3-ETG USB +Ethernet adapter was connected to the laptop, and used to set up a LAN network using the D-Link Switch Hub DES- +1005A hub for providing network services to the connected non-IoT devices. On the Ubuntu OS, the network +connection editor tool, i.e. nm-connection-editor, was been utilised for connection establishment. + +Devices generally generate two types of traffic [9]⁠: autonomous traffic, including traffic generated for +connection establishment, application and system synchronizations, and activity traffic, which is generated due to +human or object interactions. These inbound and outbound communication traffic traces, flowing over both +interfaces (external Ethernet and built-in WiFi interfaces) were captured using tcpdump 4.9.3 utility, and stored into +.pcap (packet capture) files format, similar to [16]⁠. Device-originated traffic traces were then extracted using TShark +utility and stored in .csv (comma-separated values) files format, along with labelling of individual devices names. +Finally, the recorded dataset was cleaned for further processing, by eliminating inconsistent instances, including +empty rows, and duplicate values. + +Device Fingerprinting Model + +The proposed DFP scheme architecture is depicted in Figure 2, which uses device-originated communication +traffic traces to generate device fingerprints for classification. Device-originated traffic traces are filtered according +to individual device MAC addresses, with tcp.window_size and ip.len values extracted from each packet from the +available captured data. These two values of a network packet carry significant device-specific information. +tcp.window_size value depends on a device buffer size and computation speed [14]⁠ whilst ip.len value specifies the +total length of a packet to represent unique characteristics of a devices communication pattern [15]⁠. tcp.window_size +and ip.len values from five consecutive packets (as one instance) are utilized, to compute mean (µ) and standard +deviation (σ), for constructing device-specific fingerprints, i.e. iplen_µ, iplen_σ, tcpwinsiz_µ, and tcpwinsiz_σ. +These 4 statistical fingerprints have been used for training a machine learning (ML) model, and subsequently, to +evaluate the performance, of the model in classifying devices using datasets, which have been randomly split into +training (80% instances) and testing (20% instances) datasets. + +FIGURE 2. The proposed device fingerprinting scheme. + +Random Forest Classifier + +Random Forest (RF) classifier is a supervised machine learning (ML) algorithm, that can be used for both +classification [9]⁠ and regression [17]⁠ problems. The algorithm randomly generates a group of trees, with majority +voting used to make a decision from the ensemble of decision trees [18, 19]⁠, for the classification task, as presented +in Figure 3. This assists in avoiding over-fitting problem. Researchers in different domains have utilized RF +classifier for different classification tasks. In [9]⁠, the RF algorithm has been used for classifying IoT devices with +high accuracy. Primartha et al. [20] have performed anomaly detection using the algorithm, and it has also been used + +Testing Dataset +(20%) +Training Dataset +(80%) +Capture +Filter and Extract +Fingerprint Generation +Training Model +Test Model +Classification +Network Traffic +Traffie Traces +(Mean, Standard Deviation) +ML Algorithm +ML Algorithm +IoT and Non-IoT +# Inbound and outbound +# Outbound traffic traces +# Statistical analysis +# Train a machine +# Test model performance +# Category: IoT and non-loT +traffic traces +# Packet header features +# Device fingerprint/Signature +Learning (ML) model +# Device identification +(Store in pcap files fomat) +# Label instances +# Tune hyperparameter +# Train and test datasets (csv files)8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei +for disease identification in medical science [21]⁠. In this paper, RF classifier is used to appraise the performance of +the proposed DFP method, by using the extracted features for training the RF classifier, and subsequently, using the +trained RF classifier to determine classification performance. Some of the significant tunable hyper-parameters are +set experimentally, including the number of iterations (or number of trees) = 100, seed = 1, and batch size (number +of instances) = 100, to improve classification accuracy and reduce the root mean squared error (RMSE) [22]⁠. + + +FIGURE 3. An abstract representation of a RF classifier. + +RESULTS AND DISCUSSION + +The proposed DFP method has been evaluated using waikato environment for knowledge analysis (Weka) tool +[23]⁠. An online dataset: UNSW [9]⁠ dataset, and an experimental dataset, as presented in Table 3, have been utilized +to evaluate the classification performance based on the RF classifier. The UNSW dataset consists of network traffic +traces from IoT and non-IoT devices, which are referred to as the U-IoT and U-NonIoT datasets, respectively. On +the other hand, the experimental dataset contains only network traffic traces from non-IoT devices, and it is referred +to as the L-NonIoT dataset. + +TABLE 3. Total number of instances used for evaluating the proposed DFP model. +Dataset +Devices +Training Dataset +(80%) +Test Dataset +(20%) +Total Instances +(100%) +IoT +Non-IoT +UNSW (U-IoT) +* +--- +1,095,158 +273,790 +1,368,948 +UNSW (U-NonIoT) +--- +* +562,513 +140,628 +703,141 +Lab +(L-NonIoT) +--- +* +70,875 +17,719 +88,594 + + +The proposed DFP method utilises 5 network traffic packets as one instance to generate fingerprint. As such, a +total of 1,368,948 (6,844,740 / 5) and 703,141 (3,515,705 / 5) instances have been used from the U-IoT and U- +NonIoT datasets, respectively, whilst a total of 88,594 (442,970 / 5) instances have been used from the L-NonIoT +dataset. 80% of the datasets have been used for training and the remainder for testing. The performance of the +trained RF classifier has been measured with respect to its ability to a) distinguish between IoT and non-IoT devices, +and b) classify individual devices. + +Device Category: IoT and Non-IoT Devices + +Classification performances of the proposed DFP model in distinguishing between IoT and non-IoT devices are +presented in Figure 4, on combined U-IoT and U-NonIoT datasets (i.e. UNSW dataset), and combined U-IoT and L- + +Dataset +Data +Subset of Data +Subset of Data +Subset of Data +Random Samples +1 +2 +n +Decision Trees +Selected Class +Selected Class +Class +Selected Class +(Vote) +(Vote) +(Vote) +Majority Voting +Final Decision +(Class)8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei +NonIoT datasets. The figure shows that device categorization accuracy reaches up to 99.9% using the RF classifier +on the combined U-IoT and L-NonIoT datasets. On the UNSW dataset [9]⁠, which consists of instances from 22 IoT +and 7 non-IoT devices, the proposed DFP method achieves 99.8% accuracy. + + +FIGURE 4. Categorize IoT and non-IoT devices: UNSW and Lab datasets. + + +FIGURE 5. Classification performance of the non-IoT devices: UNSW and Lab datasets. + +Individual Device Classification + +The performances of the proposed DFP method in classifying individual IoT and non-IoT devices on different +datasets, are depicted in Figure 5 and Figure 6. In Figure 5, the proposed DFP model achieves over 97.0% accuracy +in classifying non-IoT devices from the L-NonIoT and U-NonIoT datasets, with accuracy a little bit higher on the U- +NonIoT dataset. Individual IoT devices classification performance of the proposed DFP model, on the U-IoT dataset +with 22 IoT devices, is given in Figure 6. Most of the IoT devices in the dataset can be classified with over 97.6% +accuracy, with the exception of the BlipcareBPmeter, the BelkinWemoSensor and BelkinWemoSwitch devices, +which give classification accuracies of about 75.0%, 96.5% and 91.4%, respectively. The lowest accuracy for the +BlipcareBPmeter device is due to the limited number of instances available from this device for training and testing. + +CONCLUSION + +A large number of heterogeneous IoT and non-IoT devices from different manufacturers are being connected to +the Internet, to obtain network-based services. In terms of network security, it is challenging for network +administrators and operators to identify the connected devices using conventional identifiers, as they are prone to +security breaches. In this paper, a DFP model based on the analysis of network traffic traces has been proposed, +which is capable of distinguishing between IoT and non-IoT devices as well as classifying individual IoT and non- +IoT devices. As opposed to other methods in the literature, which require relatively large number of features and + +loTvs NonloT +U-loT: UNSW-loT, U-NonloT: UNSW-NonloT, L-NonloT: Lab-NonloT Datasets +U-loT vs +U-NonloT +0.998 +Datasets +U-loT vs +L-NonloT +0.999 +0.00 +0.25 +0.50 +0.75 +1.00 +AccuracyNonloTDevices +L-NonloT: Lab-NonloT, U-NonloT: UNSW-NonloT Datasets +L-NonloT +0.970 +Datasets +U-NonloT +0.973 +0.00 +0.25 +0.50 +0.75 +1.00 +Accuracy8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei +requiring longer sequence of packet network traffics to construct their DFP features, only 4 statistical features from +5 consecutive packet network traffics are required to construct the DFP features. These are used for training and +testing an ML classifier. Evaluations on the UNSW dataset have shown that the proposed DFP method is able to +distinguish between IoT and non-IoT devices with up to 99.8% accuracy, and individually classify most of the IoT +and non-IoT devices with over 97.6% accuracy. On the laboratory collected network traces, the proposed DFP +model is able to classify individual devices with 97.0% accuracy. The research outcomes signify that the proposed +DFP model is useful for device identification and may assist network administrators in providing a more secure +network. + + + + + + + + + + + + + + + + + + + +FIGURE 6. Individual IoT device classification performance: U-IoT dataset. + +ACKNOWLEDGEMENTS + +The authors are profoundly grateful to the Faculty of Integrated Technologies (FIT), Universiti Brunei +Darussalam (UBD), for supporting this research work, as well as to UBD for awarding the UBD Graduate +Scholarship (UGS) to the first author. + +REFERENCES + +1. +L. +S. +Vailshery, +IoT +and +non-IoT +connections +worldwide +2010-2025, +(2020). +https://www.statista.com/statistics/1101442/iot-number-of-connected-devices-worldwide/ (accessed May 12, +2021). +2. +M. Miettinen, S. Marchal, I. Hafeez, N. Asokan, A.-R. Sadeghi, and S. Tarkoma, IoT Sentinel: Automated +device-type identification for security enforcement in IoT, in 2017 IEEE 37th International Conference on +Distributed Computing Systems (2017) 2177–2184. +3. +Q. Xu, R. Zheng, W. Saad, and Z. Han, Device fingerprinting in wireless networks: Challenges and +opportunities, IEEE Commun. Surv. 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Morgan Kaufmann (2016). + + + + + + + + + + + + + + + + + + + diff --git a/EtAyT4oBgHgl3EQfevi1/content/tmp_files/load_file.txt b/EtAyT4oBgHgl3EQfevi1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b413a0b6768d47d83ffa7d415cff639f6f140af --- /dev/null +++ b/EtAyT4oBgHgl3EQfevi1/content/tmp_files/load_file.txt @@ -0,0 +1,523 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf,len=522 +page_content='8th Brunei International Conference on Engineering and Technology (BICET 2021),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Universiti Teknologi Brunei Internet of Things: Digital Footprints Carry A Device Identity Rajarshi Roy Chowdhury1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Azam Che Idris1 and Pg Emeroylariffion Abas1 1Faculty of Integrated Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Universiti Brunei Darussalam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Jalan Tungku Link,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Gadong BE1410,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Brunei Darussalam 2Department of Computer Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Sylhet International University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Shamimabad Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Sylhet 3100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Bangladesh Corresponding author: a) 19h0901@ubd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='bn or rajarshiry@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='com ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The usage of technologically advanced devices has seen a boom in many domains, including education, automation, and healthcare;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' with most of the services requiring Internet-connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' To secure a network, device identification plays key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' In this paper, a device fingerprinting (DFP) model, which is able to distinguish between Internet of Things (IoT) and non-IoT devices, as well as uniquely identify individual devices, has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Four statistical features have been extracted from the consecutive five device-originated packets, to generate individual device fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The method has been evaluated using the Random Forest (RF) classifier and different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Experimental results have shown that the proposed method achieves up to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='8% accuracy in distinguishing between IoT and non-IoT devices and over 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='6% in classifying individual devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' These signify that the proposed method is useful in assisting operators in making their networks more secure and robust to security breaches and unauthorised access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Keywords : digital footprint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' network traffic traces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' machine learning algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' internet of things;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' device fingerprinting INTRODUCTION It has been predicted that the number of network-connected Internet of Things (IoT) and non-IoT devices worldwide will reach approximately 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='9 billion and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='3 billion, respectively, by the year 2025 [1]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Proliferated growth of these devices with their heterogeneous functionalities, has imposed new challenges to network administrators and operators, in providing, managing, and controlling the operations and security of the network services [2]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Accurate device identification is one key aspect that needs to be seriously considered in securing network-connected devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Conventionally, internet protocol (IP) enabled devices have been using user-defined identifiers, such as IP and media access control (MAC) addresses, as a form of identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' However, these identifiers have been proven to be vulnerable [3]\u2060 to various attacks, such as spoofing [4]\u2060 and device mobility, due to the availability of malicious software [5]\u2060, for performing such attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Device fingerprinting (DFP) [3]\u2060 represents one technique that may be used to identify devices based on their communication traffic traces (or digital footprints) without using explicit identifiers, and it can be performed, either actively or passively, from different layers of the communication model [6]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Due to the prominent characteristics of network traffic features, many researchers [2, 7]\u2060 have used packet-level features for different purposes [8]\u2060, including for device identification [9]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Sivanathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' [10]\u2060 have described a DFP scheme based on the analysis of passively observed network traffic traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' A total of 11 statistical features are used as device fingerprints, from packet traffic-flows over a period of one day, by looking at the devices’ sleeping time, average packet size and traffic rate, active time, number of servers and protocols used in a flow, number of 8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei unique domain name system (DNS) request, and intervals of DNS and network time protocol (NTP) requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Subsequently, these features are used to train an ML model for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' It has been shown that the DFP scheme is able to distinguish between IoT and non-IoT devices with high accuracy and achieve over 95% accuracy in identifying individual IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The same researchers [9]\u2060 have also presented another device fingerprinting scheme, by utilizing statistical characteristics of hourly network traffic traces, to generate 8 device-specific fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Experimental result has shown over 99% accuracy using the UNSW dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Charyyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' [11]\u2060 have utilized Nilsimsa hash value of packet flows (n packets) for device-specific fingerprints, to classify individual IoT devices, to achieve 93% precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Researchers in [2, 12]\u2060 have used 12 packets information, to generate device signatures for classifying IoT devices, with 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='5% global accuracy and 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='15% accuracy using an aggregated model, whilst Aksoy and Gunes [13]\u2060 have presented a DFP approach, known as SysID, which utilizes features from a single packet, for identifying smart home IoT devices with 82% average classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Bezawada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' [14]\u2060 have utilized 5 consecutive packets information, including protocols headers and payload (20 features), for classifying IoT devices uniquely with mean identification accuracy of 93% to 100% using a laboratory dataset of 14 IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' In [15]\u2060, the authors have used a one second window to group packets, for generating statistical fingerprinting features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' These features are then used to train a binary classifier for categorizing IoT and non-IoT devices with high accuracy of 99%, whilst a multi-class classifier has been used to uniquely identify IoT devices with about 96% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' All these existing DFP models, however, require either a large number of features set from different layers of the communication model, or a large number of network packets information for generating fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Consequently, these models consume a long period of time, and require complex computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' As such, a more efficient DFP model is required for classifying devices with high accuracy, but with less computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' In this paper, a supervised machine learning (ML) based DFP model, which generates device-specific signatures by computing four statistical features from consecutive five packets of the network traffic, has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' An intuition that these features carry device-specific characteristics in terms of device memory and processing speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Experimental results have shown that over 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='0% accuracy is achievable in classifying individual non-IoT devices from traffic collected in a laboratory environment, and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='3% accuracy on the non-IoT traffic traces from the UNSW dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The proposed DFP model is also capable of distinguishing between IoT and non-IoT devices with up to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='8% accuracy on the UNSW dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The key contributions of this research work are: Identifying device specific features from the device originated communication traffic traces, to generate device signatures for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Instrument an experimental testbed of non IoT devices in a laboratory environment for data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Evaluate the proposed DFP scheme performance based on a supervised ML algorithm, to distinguish between IoT and non IoT devices and identify individual devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The proposed ML-based device fingerprinting method, as well as the datasets, data collection procedure, and an ML classifier are described in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Section III describes experimental results on various datasets, and finally, conclusion is given in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' METHODOLOGY The proposed DFP method is used to extract unique device features from network traffic traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' These features are used to train an ML classifier, and subsequently, used to test the performance of the proposed DFP method on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' This section describes the proposed DFP method, the datasets used for training and testing, as well as the classification method used to test the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Datasets: IoT and Non IoT The proposed device fingerprinting model performance has been evaluated by utilizing a publicly available dataset: UNSW [9]\u2060, and a testbed dataset of non-IoT devices, which has been collected from a laboratory environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Summary of the datasets are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The UNSW dataset comprises network traffic traces from both IoT and non-IoT devices, including TP-Link camera, smart bulb, Belkin camera, smart doorbell, printer, 8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei smart photo frame, laptop, smartphone, and tablet devices, with these heterogeneous devices coming from different manufacturers: Belkin, Philips Hue, Netatmo, TP-Link, Withings, HP, Apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' On the other hand, the laboratory dataset comprises 7 non-IoT devices, including laptops, smartphones, and desktops, from different manufacturers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The data collection procedure from the 7 non-IoT devices is described in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' List of IoT and non-IoT Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Dataset Devices Total Packets Source IoT Non-IoT UNSW 22 -- 6,844,740 [9]\u2060 -- 7 3,515,705 Lab Dataset -- 7 442,970 -- TABLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' List of non-IoT devices for experimental set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Device Category Device Name/Model Operating System Connectivity MAC Address 1 Laptop Aspire-S7 Windows WiFi 34:23:87:b7:56:17 2 ProBook-4410s WiFi/Ethernet 00:25:b3:47:da:6f 3 Desktop Asus Ethernet 08:60:6e:c1:79:c2 4 HP-EliteDesk Ethernet 80:e8:2c:d6:9e:49 5 Smart Phone MYA-U29 Android WiFi d0:ff:98:95:57:af 6 MLXP2ZA-A iOS WiFi e0:c7:67:45:a3:62 7 MWC22KH-A WiFi 06:44:b7:aa:20:98 Dataset Collection Methodology An experimental design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' consisting of local area network (LAN) and wireless local area network (WLAN) with non-IoT devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' was set up in a laboratory environment at Universiti Brunei Darussalam (UBD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Design of the testbed is depicted in Figure 1, with the seven non-IoT devices from different manufacturers and of different types, as listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' These devices were configured, to connect with an access point (AP) either using ethernet or wireless fidelity (WiFi) interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' An experimental testbed of non-IoT devices network (LAN/WLAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' DNS NTP Server Connectivity: Server Server N Ethernet WiFi Other a Internet WiFi Hotspot Gateway ubuntu?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' (UBD Network) Hub Ethernet USB Ethernet Port Port Monitoring Station (Capture Network Traffic)8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei A laptop was used to configure an access point (AP), which was used to provide network services to the non-IoT devices, as well as to monitor and capture communication footprints from the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The Dell Inspiron 15 5000 Series laptop runs Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='04 as an operating system (OS), and was connected to the UBD network via its built-in Ethernet interface, to provide the Internet connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The built-in WiFi interface was configured as a WiFi Hotspot, providing wireless connectivity to the WiFi-enabled (IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='11 standard) devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Additionally, a TU3-ETG USB Ethernet adapter was connected to the laptop, and used to set up a LAN network using the D-Link Switch Hub DES- 1005A hub for providing network services to the connected non-IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' On the Ubuntu OS, the network connection editor tool, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' nm-connection-editor, was been utilised for connection establishment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Devices generally generate two types of traffic [9]\u2060: autonomous traffic, including traffic generated for connection establishment, application and system synchronizations, and activity traffic, which is generated due to human or object interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' These inbound and outbound communication traffic traces, flowing over both interfaces (external Ethernet and built-in WiFi interfaces) were captured using tcpdump 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='3 utility, and stored into .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='pcap (packet capture) files format, similar to [16]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Device-originated traffic traces were then extracted using TShark utility and stored in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='csv (comma-separated values) files format, along with labelling of individual devices names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Finally, the recorded dataset was cleaned for further processing, by eliminating inconsistent instances, including empty rows, and duplicate values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Device Fingerprinting Model The proposed DFP scheme architecture is depicted in Figure 2, which uses device-originated communication traffic traces to generate device fingerprints for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Device-originated traffic traces are filtered according to individual device MAC addresses, with tcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='window_size and ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='len values extracted from each packet from the available captured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' These two values of a network packet carry significant device-specific information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' tcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='window_size value depends on a device buffer size and computation speed [14]\u2060 whilst ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='len value specifies the total length of a packet to represent unique characteristics of a devices communication pattern [15]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' tcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='window_size and ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='len values from five consecutive packets (as one instance) are utilized, to compute mean (µ) and standard deviation (σ), for constructing device-specific fingerprints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' iplen_µ, iplen_σ, tcpwinsiz_µ, and tcpwinsiz_σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' These 4 statistical fingerprints have been used for training a machine learning (ML) model, and subsequently, to evaluate the performance, of the model in classifying devices using datasets, which have been randomly split into training (80% instances) and testing (20% instances) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The proposed device fingerprinting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Random Forest Classifier Random Forest (RF) classifier is a supervised machine learning (ML) algorithm, that can be used for both classification [9]\u2060 and regression [17]\u2060 problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The algorithm randomly generates a group of trees, with majority voting used to make a decision from the ensemble of decision trees [18, 19]\u2060, for the classification task, as presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' This assists in avoiding over-fitting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Researchers in different domains have utilized RF classifier for different classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' In [9]\u2060, the RF algorithm has been used for classifying IoT devices with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Primartha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' [20] have performed anomaly detection using the algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' and it has also been used Testing Dataset (20%) Training Dataset (80%) Capture Filter and Extract Fingerprint Generation Training Model Test Model Classification Network Traffic Traffie Traces (Mean,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Standard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Deviation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='IoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Non-IoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Inbound ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='outbound ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Outbound ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='traffic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='traces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Statistical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Train ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Category: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='IoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='non-loT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='traffic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='traces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Packet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='header ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Device ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='fingerprint/Signature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='(ML) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Device ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Conference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Engineering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='Technology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='(BICET ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='2021),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Universiti Teknologi Brunei for disease identification in medical science [21]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' In this paper, RF classifier is used to appraise the performance of the proposed DFP method, by using the extracted features for training the RF classifier, and subsequently, using the trained RF classifier to determine classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Some of the significant tunable hyper-parameters are set experimentally, including the number of iterations (or number of trees) = 100, seed = 1, and batch size (number of instances) = 100, to improve classification accuracy and reduce the root mean squared error (RMSE) [22]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' An abstract representation of a RF classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' RESULTS AND DISCUSSION The proposed DFP method has been evaluated using waikato environment for knowledge analysis (Weka) tool [23]\u2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' An online dataset: UNSW [9]\u2060 dataset, and an experimental dataset, as presented in Table 3, have been utilized to evaluate the classification performance based on the RF classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The UNSW dataset consists of network traffic traces from IoT and non-IoT devices, which are referred to as the U-IoT and U-NonIoT datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' On the other hand, the experimental dataset contains only network traffic traces from non-IoT devices, and it is referred to as the L-NonIoT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' TABLE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Total number of instances used for evaluating the proposed DFP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Dataset Devices Training Dataset (80%) Test Dataset (20%) Total Instances (100%) IoT Non-IoT UNSW (U-IoT) * --- 1,095,158 273,790 1,368,948 UNSW (U-NonIoT) --- * 562,513 140,628 703,141 Lab (L-NonIoT) --- * 70,875 17,719 88,594 The proposed DFP method utilises 5 network traffic packets as one instance to generate fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' As such, a total of 1,368,948 (6,844,740 / 5) and 703,141 (3,515,705 / 5) instances have been used from the U-IoT and U- NonIoT datasets, respectively, whilst a total of 88,594 (442,970 / 5) instances have been used from the L-NonIoT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' 80% of the datasets have been used for training and the remainder for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The performance of the trained RF classifier has been measured with respect to its ability to a) distinguish between IoT and non-IoT devices, and b) classify individual devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Device Category: IoT and Non-IoT Devices Classification performances of the proposed DFP model in distinguishing between IoT and non-IoT devices are presented in Figure 4, on combined U-IoT and U-NonIoT datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' UNSW dataset), and combined U-IoT and L- Dataset Data Subset of Data Subset of Data Subset of Data Random Samples 1 2 n Decision Trees Selected Class Selected Class Class Selected Class (Vote) (Vote) (Vote) Majority Voting Final Decision (Class)8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei NonIoT datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The figure shows that device categorization accuracy reaches up to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='9% using the RF classifier on the combined U-IoT and L-NonIoT datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' On the UNSW dataset [9]\u2060, which consists of instances from 22 IoT and 7 non-IoT devices, the proposed DFP method achieves 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='8% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' FIGURE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Categorize IoT and non-IoT devices: UNSW and Lab datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' FIGURE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Classification performance of the non-IoT devices: UNSW and Lab datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Individual Device Classification The performances of the proposed DFP method in classifying individual IoT and non-IoT devices on different datasets, are depicted in Figure 5 and Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' In Figure 5, the proposed DFP model achieves over 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='0% accuracy in classifying non-IoT devices from the L-NonIoT and U-NonIoT datasets, with accuracy a little bit higher on the U- NonIoT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Individual IoT devices classification performance of the proposed DFP model, on the U-IoT dataset with 22 IoT devices, is given in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Most of the IoT devices in the dataset can be classified with over 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='6% accuracy, with the exception of the BlipcareBPmeter, the BelkinWemoSensor and BelkinWemoSwitch devices, which give classification accuracies of about 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='0%, 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='5% and 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='4%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The lowest accuracy for the BlipcareBPmeter device is due to the limited number of instances available from this device for training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' CONCLUSION A large number of heterogeneous IoT and non-IoT devices from different manufacturers are being connected to the Internet, to obtain network-based services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' In terms of network security, it is challenging for network administrators and operators to identify the connected devices using conventional identifiers, as they are prone to security breaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' In this paper, a DFP model based on the analysis of network traffic traces has been proposed, which is capable of distinguishing between IoT and non-IoT devices as well as classifying individual IoT and non- IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' As opposed to other methods in the literature, which require relatively large number of features and loTvs NonloT U-loT: UNSW-loT, U-NonloT: UNSW-NonloT, L-NonloT: Lab-NonloT Datasets U-loT vs U-NonloT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='998 Datasets U-loT vs L-NonloT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='00 AccuracyNonloTDevices L-NonloT: Lab-NonloT, U-NonloT: UNSW-NonloT Datasets L-NonloT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='970 Datasets U-NonloT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='00 Accuracy8th Brunei International Conference on Engineering and Technology (BICET 2021), Universiti Teknologi Brunei requiring longer sequence of packet network traffics to construct their DFP features, only 4 statistical features from 5 consecutive packet network traffics are required to construct the DFP features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' These are used for training and testing an ML classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Evaluations on the UNSW dataset have shown that the proposed DFP method is able to distinguish between IoT and non-IoT devices with up to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='8% accuracy, and individually classify most of the IoT and non-IoT devices with over 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='6% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' On the laboratory collected network traces, the proposed DFP model is able to classify individual devices with 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content='0% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' The research outcomes signify that the proposed DFP model is useful for device identification and may assist network administrators in providing a more secure network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' FIGURE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Individual IoT device classification performance: U-IoT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors are profoundly grateful to the Faculty of Integrated Technologies (FIT), Universiti Brunei Darussalam (UBD), for supporting this research work, as well as to UBD for awarding the UBD Graduate Scholarship (UGS) to the first author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Vailshery, IoT and 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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Frank, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Hall, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Witten, The WEKA Workbench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Online Appendix for Data Mining: Practical Machine Learning Tools and Techniques, 4th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} +page_content=' Morgan Kaufmann (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf'} diff --git a/EtFLT4oBgHgl3EQfFS_K/content/tmp_files/2301.11987v1.pdf.txt b/EtFLT4oBgHgl3EQfFS_K/content/tmp_files/2301.11987v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..90e0af0cbc3333be6b36274f400de27e4d01fcd6 --- /dev/null +++ b/EtFLT4oBgHgl3EQfFS_K/content/tmp_files/2301.11987v1.pdf.txt @@ -0,0 +1,1063 @@ +J/ψ polarization in large-PT semi-inclusive deep-inelastic scattering at the EIC +Umberto D’Alesio,1, 2, ∗ Luca Maxia,1, 2, † Francesco Murgia,2, ‡ Cristian Pisano,1, 2, § and Sangem Rajesh3, 4, ¶ +1Dipartimento di Fisica, Universit`a di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy +2INFN, Sezione di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy +3Department of Physics, School of Advanced Sciences, +Vellore Institute of Technology, Vellore, Tamil Nadu 632014, India +4INFN, Sezione di Perugia, via A. Pascoli snc, 06123, Perugia, Italy +(Dated: January 31, 2023) +We present a detailed phenomenological study of J/ψ polarization in semi-inclusive deep inelastic +scattering processes, focusing on the kinematics accessible at the future Electron-Ion Collider. We +show theoretical estimates for the standard polarization parameters for different frames usually +adopted in the literature, in the large PT region, namely PT ≫ ΛQCD, where collinear factorization +is expected to hold. We adopt both the Color Singlet Model and the Nonrelativistic QCD approach, +paying special attention to the role of different sets of Long Distance Matrix Elements. Finally we +present a preliminary analysis of some frame independent polarization invariants. +I. +INTRODUCTION +Our understanding of the J/ψ production mechanism at high energies has improved significantly since its discovery +almost 50 years ago [1, 2], thanks to the combined efforts from both the theoretical and experimental communities. +However, there are still major problems in the theoretical analyses of the available data, such as the long-standing +J/ψ polarization puzzle. Namely, J/ψ polarization measurements cannot yet be explained in a way entirely consistent +with the world experimental results for the unpolarized J/ψ yields. +The present theoretical frameworks all agree in providing a perturbative description of the creation of the charm +quark-antiquark (c¯c) pair. The charm mass mc plays the role of the hard scale, since it is much larger than the +asymptotic scale parameter of QCD, ΛQCD. These approaches nonetheless differ in the treatment of the subsequent +nonperturbative transition to the hadronic bound state. For instance, in the traditional Color-Singlet Model (CSM) [3] +the c¯c pair is produced at short distances directly with the quantum numbers of the J/ψ meson, i.e. in a color-singlet +(CS) state with spin one and no orbital angular momentum. This is possible by the emission of an additional hard +gluon, which implies the suppression of the cross section by one power of the strong coupling constant αs. However, +the CSM cannot be considered as a complete theory, since at the next-to-leading order (NLO) P-wave quarkonia are +affected by uncanceled infrared singularities. +These singularities are properly removed in the effective field theory approach of nonrelativistic QCD (NRQCD), +based on a rigorous factorization theorem, which was assumed in the original paper by Bodwin, Braaten, and Lep- +age [4], and later explicitly proven to next-to-next-to-leading order (NNLO) [5]. NRQCD therefore implies a sep- +aration of process-dependent short-distance coefficients, to be calculated perturbatively as expansions in αs, from +long-distance matrix elements (LDMEs), which are expected to be universal and have to be extracted from experi- +ments. Scaling rules [6] predict each of the LDMEs to scale with a definite power of the relative velocity v of the heavy +quark-antiquark pair in the quarkonium rest frame in the limit v ≪ 1. Observables are hence evaluated by means of +a double expansion in αs and in v, with αs ≃ 0.2 and v2 ≃ 0.3 for charmonium states. An essential feature of this +approach is that the c¯c pair at short distance can be produced in any Fock state n = 2S+1L[c] +J with definite orbital +angular momentum L, spin S, total angular momentum J and color configuration c = 1, 8. NRQCD hence predicts +the existence of intermediate color-octet (CO) states, which subsequently evolve into physical, CS quarkonia by the +emission of soft gluons. For S-wave quarkonia, the CSM is recovered in the limit v → 0. In the specific case of J/ψ +production, the CSM prediction is based only on the 3S[1] +1 +CS state, while NRQCD includes the leading relativistic +corrections as well, which at the relative order O(v4) are given by the CO states 1S[8] +0 , 3S[8] +1 , and 3P [8] +J +with J = 0, 1, 2. +The values of the CO LDMEs extracted from different fits to data on J/ψ and Υ yields [7–11] are not compatible +with each other, even within the large uncertainties [12–14]. Therefore, any new method to determine them with +better precision is worth exploring [15–17]. In this paper we propose to look at the J/ψ polarization parameters in +∗ umberto.dalesio@ca.infn.it +† luca.maxia@ca.infn.it +‡ francesco.murgia@ca.infn.it +§ cristian.pisano@unica.it +¶ sangem.rajesh@vit.ac.in +arXiv:2301.11987v1 [hep-ph] 27 Jan 2023 + +2 +semi-inclusive deep-inelastic scattering (SIDIS), e p → e′ J/ψ X, in a kinematic region where the transverse momentum +of the J/ψ meson PT is large, namely PT ≫ ΛQCD, and collinear factorization is expected to hold. Analysing SIDIS at +finite values of the exchanged photon virtuality Q2 has certain experimental and theoretical advantages as compared to +photoproduction. Namely, as Q2 increases theoretical uncertainties in the different frameworks decrease and resolved +photon contributions are expected to be negligible. Moreover, background from diffractive J/ψ production is expected +to decrease with Q2 faster than the SIDIS cross section. The distinct signature of the scattered lepton makes the +process particularly easy to detect. Clearly, cross sections are smaller than those expected in the photoproduction +case, however, considering the achievable high luminosities, this study should be feasible at the future Electron-Ion +Collider (EIC) planned in the United States [18–20]. +So far, only a single experimental study of J/ψ polarization in SIDIS has been performed, by the H1 Collaboration +at HERA [21]. Such a measurement is limited to the polarization parameter λ in the helicity frame. This result turns +out to be compatible with the predictions provided in Refs. [22, 23], but it can hardly discriminate among the different +models. In analogy with Refs. [22, 23], our phenomenological analysis has been carried out at the perturbative order +α2 +s, which has to be considered as the state of the art for these observables. Higher-order effects have been calculated +very recently only for the unpolarized cross section within the CSM [24]. Anyway, we expect these effects (at least +in the large Q2 region) to be small for the observables we are investigating, because they are ratios of cross sections. +We point out that our estimates include also the polarization parameters µ and ν, not addressed in Refs. [22, 23], +which are studied in different reference frames. Furthermore, we perform a preliminary study of rotational invariant +combinations of these parameters. +The remainder of the paper is organized as follows. In section II we recall the standard SIDIS variables and collect +the expressions of the differential cross section for quarkonium production and its leptonic decay in terms of the helicity +structure functions and the polarization parameters. In section III we discuss the three polarization parameters λ, µ, +ν, showing their estimates in two reference frames and paying special attention to their energy, z and PT dependences +as well as to the impact of the LDME set adopted. To overcome the intrinsic frame dependence of the polarization +parameters, in section IV we present two classes of the so-called rotational invariant quantities, and show, as a case +of study, some results for one of them. Finally in section V we gather our conclusions. +II. +KINEMATICS AND FORMALISM +In this section we provide the main analytic expressions needed to carry out the phenomenological analysis. For +more details and the complete formalism we refer the reader to Ref. [25]. We consider the SIDIS process +e(k) + p(P) → e′(k′) + J/ψ(Pψ) + X(PX) , +(1) +with the subsequent J/ψ decay into a lepton pair +J/ψ(Pψ) → l+(l) + l−(l′) , +(2) +where, in brackets, we have shown the four-momenta of each particle. The J/ψ meson is produced via the partonic +subprocess +γ∗(q) + a(pa) → c¯c[n](Pψ) + a(p′ +a) , +(3) +with q2 = −Q2 and P 2 +ψ = M 2 +ψ = (2mc)2. The initial parton momentum, pa, is related to the parent proton one, P, as +pa = ξP . +(4) +We adopt the following three standard invariant quantities, defined in terms of the photon and hadron momenta +xB = +Q2 +2P · q , +y = P · q +P · k , +z = P · Pψ +P · q , +(5) +where xB is the Bjorken variable, y is the inelasticity and z is the energy fraction carried out by the J/ψ (in the +proton rest frame). All these variables are constrained in the region 0 ≤ xB, y, z ≤ 1 and they are connected to other +kinematical quantities of the system, like the total center-of-mass (cm) energy √s and the virtual photon-proton cm +energy, W. +The cross section that describes the J/ψ formation and its decay into a lepton pair can be written as +1 +Bll +dσ +dxB dy dz d2PT dΩ = +α +8 y z Q2 +3 +8π +� +WT (1 + cos2 θ) + WL(1 − cos2 θ) + W∆ sin 2θ cos φ + W∆∆ sin2 θ cos 2φ +� +, +(6) + +3 +where PT is the J/ψ transverse momentum in the cm frame of the virtual photon and the proton, Bll is the branching +ratio for the decay process J/ψ → ℓ+ℓ− and Ω(θ, φ) refers to the solid angle spanned by the lepton ℓ+ in a reference +frame where the system formed by ℓ+ and ℓ− is at rest. Moreover, we have introduced the following helicity structure +functions +WT ≡ W11 = W−1,−1 , +WL ≡ W00 , +W∆ ≡ +1 +√ +2 (W10 + W01) = +√ +2 Re [W10] , +W∆∆ ≡ W1,−1 = W−1,1 , +(7) +where the subscripts refer to the J/ψ polarization states. More specifically, WT and WL are respectively the structure +functions for transversely and longitudinally polarized J/ψ mesons, W∆ is the single-helicity flip structure function, +and W∆∆ is the double-helicity flip one. Notice that in Eq. (6) we have introduced a proper overall constant factor +w.r.t. Eq. (2.35) of Ref. [25] to ensure the normalization when integrated over the solid angle, see Eq. (8) below. +This does not affect any conclusion of Ref. [25], where all relevant quantities are defined as ratios of helicity structure +functions. +As shown in Ref. [25], the structure functions in Eq. (7) can be further decomposed in terms of the contributions +coming from the longitudinal ( ) and transverse (⊥) polarizations of the virtual photon. Moreover, within a collinear +factorization scheme, they are given as convolutions of collinear parton distribution functions (PDFs) with partonic +helicity structure functions (weighted by proper LDMEs). These, in turn, can be expressed as functions of the partonic +Mandelstam invariants. +The unpolarized cross section is obtained by integrating Eq. (6) over the solid angle Ω, +1 +Bll +dσ +dxB dy dz d2PT += +α +8 y z Q2 (2WT + WL) . +(8) +It is then useful to introduce the ratio of polarized and unpolarized cross sections +dN +dΩ ≡ +dσ +dxB dy dz d2PT dΩ +� +dσ +dxB dy dz d2PT +�−1 +, +(9) +which can be expressed as follows +dN +dΩ = 3 +4π +1 +λ + 3 +� +1 + λ cos2 θ + µ sin 2θ cos ϕ + 1 +2 ν sin2 θ cos 2ϕ +� +, +(10) +where we have defined the polarization parameters +λ = W11 − W00 +W11 + W00 +, +µ = +√ +2 Re [W10] +W11 + W00 +, +ν = +W1, −1 +W11 + W00 +, +(11) +or alternatively adopting Eq. (7), +λ = WT − WL +WT + WL +, +µ = +W∆ +WT + WL +, +ν = +2 W∆∆ +WT + WL +. +(12) +The parameterizations shown in Eqs. (6) and (10) are standard for the study of the angular distribution of a spin-one +particle decay into a lepton pair and, indeed, they are commonly adopted in Drell-Yan processes [26] and in J/ψ +photoproduction [27]. +Among the polarization coefficients, λ, µ and ν, the most investigated experimentally is λ. +Moreover, from +the phenomenological point of view it has a very intuitive interpretation, with λ = +1(−1) describing a trans- +verse(longitudinal) polarization state for the J/ψ (i.e. a J/ψ helicity equal to ±1 or 0), while λ = 0 for an unpolarized +one. +The main goal of our study is to present estimates for these polarization quantities, within both the CSM and the +NRQCD frameworks, focusing on the kinematic region accessible at the future EIC. As we will show in the following, +such a detailed phenomenological study could help in disentangling among the production mechanisms. + +4 +LDME Set +⟨O1[ 3S1]⟩ +� +GeV3� ⟨O8[ 1S0]⟩ +� +GeV3� ⟨O8[ 3S1]⟩ +� +GeV3� ⟨O8[ 3P0]⟩ +� +GeV5� +C12 +1.16 +0.089 +0.003 +0.0126 +G13 +1.16 +0.097 +−0.0046 +−0.0214 +BK11 +1.32 +0.0304 +0.00168 +−0.00908 +Table I. LDME set (central) values for the J/ψ state: C12 [8], G13 [28] and BK11 [29]. For the other 3PJ states we use the +standard spin-symmetry relation ⟨O8[ 3PJ]⟩ = (2J + 1) ⟨O8[ 3P0]⟩. +III. +ANGULAR DISTRIBUTIONS +In this section we analyze the polarization parameters defined in Eq. (11) showing both their z and PT distributions. +The explicit analytic expressions of the underlying partonic structure functions, calculated at the perturbative order +α2 +s, are presented in Ref. [25] for the so-called Gottfried-Jackson frame, together with all prescriptions needed to +transform them in the other relevant frames. For the predictions based on the NRQCD approach, in addition to the +CS contribution, given by a pure gluon fusion channel, we consider the CO channels up to the order v4, which involve +both gluon and quark final states. The CTEQ6L1 set [30] is used for the unpolarized parton distribution functions. +Moreover, in order to assess the stability of our results against higher order corrections, we produce uncertainty bands +by varying the factorization scale µF in the range µ0/2 < µF < 2µ0, around the central value µ0 = +� +Q2 + M 2 +ψ. +Concerning the CO LDME values, three different sets are adopted, see Table I. Here we only recall their main +features: the C12 set [8] has been extracted simultaneously from both polarized and unpolarized J/ψ production +data in pp collision at PT > 7 GeV, measured by the CDF (Run II) Collaboration; the G13 set [28] is obtained +including only PT > 7 GeV unpolarized data from the CDF and LHCb Collaborations and then used to predict +the J/ψ polarization in pp collisions; it is in agreement with the C12 set if feed-down contribution is negligible; the +BK11 set [29] is based on a fit without any polarization data, but starting from a lower PT value, around 3 GeV, and +including both photoproduction and hadroproduction data. +The high cm energy kinematical set-ups expected at the EIC are an ideal environment to study J/ψ polarization in +electroproduction. Moreover, they will allow to better explore high photon virtualities (Q), avoiding the competing +contributions from photoproduction. Furthermore, since we are interested in the region where collinear factorization +holds, our results will be shown only for PT values above PT min = 1 GeV. Notice that around this value we actually +enter the region where the transverse momentum dependent (TMD) factorization could be applied and therefore our +estimates are pushed down to the overlapping region of validity of the two factorization schemes. +A. +The λ parameter +In Fig. 1 we present our predictions for λ at √s = 140 GeV, as a function of both the J/ψ energy fraction z +(left panels) and its transverse momentum PT (right panels). Two quarkonium rest frames are explicitly considered: +the Gottfried-Jackson (upper panels) and the Helicity (lower panels) ones. In this and in the following figures, the +kinematical ranges explored are indicated in the legend boxes. For completeness we report here the corresponding +regions explored in xB and y at √s = 140 GeV, 10−3 ≲ xB ≲ 0.2 and y ≲ 0.5 respectively, even if the effectively +probed maximum value in xB is around 0.07. +Concerning other typical frames, like the Target and Collins-Soper ones, we only notice that the first one give +estimates very close to those in the Helicity frame, while predictions obtained in the second one, at least for the +kinematics considered, are in general much smaller than those in the Gottfried-Jackson frame or even close to zero. +Notice that for such observable, defined as a ratio of cross sections, the dependence on the scale µF in the range +[µ0/2, 2µ0] is barely appreciable and therefore is not shown. +The study of the λ parameter as a function of z presents very interesting features from the phenomenological +point of view. The reasons are manifold: first of all its expected relative large size as compared to the µ and ν +parameters. Moreover, it is experimentally under more active investigation. On the other hand, theoretical estimates +for λ as a function of z (for small and moderate values) do not vary significantly adopting different frameworks +(Fig. 1, left panels), which implies that, in order to get information on the quarkonium formation mechanism, one +would need highly precise measurements. The same problem was found in different analyses performed by the HERA +Collaborations, Refs. [21, 23]. +The situation changes considerably at z > 0.6, which represents a very interesting region from the phenomenological +point of view. As is well known, NRQCD estimates for the unpolarized cross section manifest a divergent behavior as + +5 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Gottfried-Jackson +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.2 +0.4 +0.6 +0.8 +z +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Helicity +CSM +NRQCD (C12) +NRQCD (BK11) +NRQCD (G13) +2 +4 +6 +8 +10 +PT [GeV] +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +s = 140 GeV +9 GeV2 < Q2 < 100 GeV2 +20 GeV < W < 100 GeV +0.2 < z < 0.9 or PT > 1 GeV +Figure 1. Estimates for λ at √s = 140 GeV as a function of z (left panels) and PT (right panels) for different models and +LDME sets and two reference frames: Gottfried-Jackson (upper panels) and Helicity (lower panels) frames. Integration ranges +are given in the light-blue legend box. +z → 1, due to the corresponding ˆt → 0 singularities. This can potentially spoil the validity of NRQCD factorization. +As shown in Ref. [31], in order to extend the region of applicability of NRQCD up to 1 − z ∼ v2, one can introduce +a new set of functions, the so-called shape functions [32], that allow to improve noticeably the convergence for +photoproduction. We expect such quantities to be relevant also for the SIDIS process, together with their TMD +extensions, which have been adopted in the study of pp collisions in Refs. [33, 34] and whose perturbative tails have +been derived in Ref. [35] for unpolarized and in Ref. [25] for polarized J/ψ SIDIS. On the other hand, the impact of +the shape functions on λ is expected to be strongly reduced since λ is a ratio of cross sections. This can be tested +with future available data. +A much more powerful tool to assess the relevance of the CO contributions is the study of the PT distribution +(Fig. 1, right panels). In the Gottfried-Jackson frame (upper panel) we see a clear separation as well as a different +behavior between the CSM and NRQCD curves, in particular in the region 4 < PT < 7 GeV; similarly in the Helicity +frame there is a wide separation between the CSM and the NRQCD curves, while different LDME sets give predictions +much closer to each other and closer to λ = 0. It is worth noticing that, even if the unpolarized cross section decreases +as PT increases, a good separation can be found already around PT ≃ 5 GeV, which is also far away from the TMD +region. +Before concluding the analysis of λ at large cm energies, a comment on the contributions from different partonic +channels and/or different NRQCD waves can be useful. Concerning the z distribution, we find that the main con- +tribution to the numerator of λ comes from the (gluon) CS wave, while the differences among NRQCD predictions, +especially around z → 0.9, are due to the gluon P-wave, modulated by the corresponding LDME parameter. For the +PT distribution we find, similarly, that the CS term is on the whole the most relevant contribution, followed again by +the gluon P-wave one. In particular at PT → 1 GeV the size of the gluon P-wave contribution becomes comparable +to (or even larger than) the CS one; moreover, since the low-PT region dominates the integration over PT , one can +also understand why the gluon P-wave is so relevant in our estimates vs. z, with the most visible effects for z → 0.9. +At medium PT values the quark P-wave starts becoming important and at even higher PT values it is similar in +size to the gluon one; this means that in this region, the full P-wave contribution (gluon+quark) dominates over the +CS one. +Another interesting possibility given by the future EIC facility is the corresponding analysis at smaller energies: +in the following we will adopt √s = 45 GeV. In this case, different integration ranges have been considered for W +and Q2, as reported in the legend box of Fig. 2. These, in turn, correspond to 10−3 ≲ xB ≲ 0.5 (with an effective + +6 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Gottfried-Jackson +0.2 +0.0 +0.2 +0.4 +0.2 +0.4 +0.6 +0.8 +z +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Helicity +s = 45 GeV +2.5 GeV2 < Q2 < 100 GeV2 +10 GeV < W < 40 GeV +0.2 < z < 0.9 or PT > 1 GeV +2 +4 +6 +8 +10 +PT [GeV] +0.2 +0.0 +0.2 +0.4 +CSM +NRQCD (C12) +NRQCD (BK11) +NRQCD (G13) +Figure 2. +Estimates for λ at cm energy √s = 45 GeV. The integration region, different with respect to the higher-energy case, +is given in the red legend box, while curves and panels have the same meaning as in Fig. 1. The scale error bands are sizable +and explicitly shown only for the CSM prediction as a function of PT . +upper limit around xB ≃ 0.2) and y ≲ 0.8, a more valence-like region w.r.t. the previous case. Moreover, since at +lower energies it is more difficult to reach high photon virtualities, we get contributions mostly from moderately low +Q2. Consistently we adopt a lower limit, Qmin ≃ 1.6 GeV, in the integration. Notice that in this kinematic region, at +least for the high PT dependence of λ within the CSM, the scale error bands are once again sizeable enough. +From Fig. 2 (left panels) we can see that the z distribution does not depend significantly on the energy for z ≤ 0.6, +while at higher z values the estimates are closer to zero, at variance with those at higher cm energy. As said, a +polarization study pushed up to this regime can suffer from factorization breaking effects in NRQCD even if data in +this region could be relevant from the phenomenological point of view. We also observe a rapid variation of all curves +in the Helicity frame at z ∼ 0.1. This is due to geometrical factors which are energy dependent (see also Eq. (A16) +of Ref. [25]). The same variation is also present at higher cm energy, but for z < 0.1 (outside the range shown in the +lower-left panel of Fig. 1). +Concerning the PT dependence, Fig. 2 (right panels), we notice that the CSM results are very different with respect +to the corresponding ones in Fig. 1, while the same is not true for the NRQCD cases. This is related to the different +virtualities explored, on which the CSM estimates depend heavily. This difference can be considered as an extra tool +in the quest of discerning among different frameworks. +Finally, we briefly comment on how the parton and/or wave contributions vary with the energy. +While the z +distribution manifests almost no energy dependence, the PT spectrum presents interesting features in the two frames +considered. For the Gottfried-Jackson one the relative contribution from the quark P-wave is widely increased at this +lower energy, making it the leading term in the numerator at medium/high PT . Regarding the Helicity frame the +situation is, potentially, even more interesting, since the CSM and P-wave (both gluon and quark) contributions are +highly suppressed at this energy, especially at large PT . The main role is then played by the 3S(8) +1 +quark wave, which +is responsible for the difference among the predictions based on the LDME sets considered. Even if in this region it +is quite hard to expect precise enough data to discriminate between models, it is nevertheless worth stressing that it +could be very useful in constraining the nonperturbative physics. + +7 +0.8 +0.6 +0.4 +0.2 +0.0 +0.2 +Gottfried-Jackson +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.2 +0.4 +0.6 +0.8 +z +0.8 +0.6 +0.4 +0.2 +0.0 +0.2 +Helicity +s = 140 GeV +9 GeV2 < Q2 < 100 GeV2 +20 GeV < W < 100 GeV +0.2 < z < 0.9 or PT > 1 GeV +2 +4 +6 +8 +10 +PT [GeV] +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +CSM +NRQCD (C12) +NRQCD (BK11) +NRQCD (G13) +Figure 3. Estimates for the parameter µ at √s = 140 GeV. Paneling order is the same as in Fig. 1. Integration ranges are +given in the blue legend box. +B. +The µ parameter +Estimates for the µ parameter are again provided both in the Gottfried-Jackson and in the Helicity frames, as a +function of z and PT at √s = 140 GeV, Fig. 3, and √s = 45 GeV, Fig. 4. +From these figures we see that the Gottfried-Jackson frame is the best choice to discern among the CSM and +NRQCD approach. A similar conclusion holds for the parameter ν as well, see the discussion in Sec. III C. Indeed, +in Fig. 3 the separation between the CSM estimates and the corresponding NRQCD ones are remarkably sizeable for +z ≳ 0.5 and PT ≳ 5 GeV. On the contrary, estimates in the Helicity frame both with respect to z and PT are so close +to each other that one cannot draw any conclusion. +The wave/parton decomposition of the W∆ helicity function, that is directly related to the µ numerator, allows us +to get some further insights. The main CO contribution comes from the P-wave term. In particular, differences in +NRQCD predictions as a function of z (left panels of Fig. 3) are driven by the gluon P-wave LDMEs. Moreover, the +gluon P-wave dominates the numerator behavior with respect to PT too (right panels of Fig. 3). In addition, we find +that the NRQCD predictions in the Gottfried-Jackson frame receive a significant contribution from the gluon P-wave +also at low-PT , namely PT ≲ 3 GeV. At variance with the behavior in z, here the quark P-wave channel is relevant +at high PT , especially when considering the Helicity frame. +Moving to the lower cm energy, we see that the CSM µ estimates in the Gottfried-Jackson frame, Fig. 4 (upper +panels), vary significantly for z ≳ 0.5 and PT ≳ 5 GeV, as compared with what happens at √s = 140 GeV. We +remark that this variation can also appear via a proper Q-binning in the higher cm energy case (√s = 140 GeV). +In contrast, estimates within the Helicity frame at lower energies (lower panels of Fig. 4) do not present the same +energy/Q-binning dependence. The only remarkable exception resides in the PT distribution, where CSM predictions +increase up to ∼ 40%, to be compared with the √s = 140 GeV case where the CSM result is at most ∼ 25%. Despite +this, µ estimates in the Helicity frame do not differ enough to discern among different models. +Looking at the wave/parton decomposition, we confirm that also for the µ numerator the role of quarks is enhanced +at lower energies. This is particularly true for the PT dependence. Here we find that NRQCD predictions at the +higher PT values, namely PT ≳ 6 GeV, are mostly driven by the quark P-wave; moreover, in the same PT region we +observe that the 3S[8] +1 +quark wave is non-negligible. + +8 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +Gottfried-Jackson +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.2 +0.4 +0.6 +0.8 +z +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +Helicity +s = 45 GeV +2.5 GeV2 < Q2 < 100 GeV2 +10 GeV < W < 40 GeV +0.2 < z < 0.9 or PT > 1 GeV +2 +4 +6 +8 +10 +PT [GeV] +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +CSM +NRQCD (C12) +NRQCD (BK11) +NRQCD (G13) +Figure 4. Estimates for the parameter µ at √s = 45 GeV. Paneling order is the same as in Fig. 1. Integration ranges are given +in the red legend box. +C. +The ν parameter +We now discuss the parameter ν, which is particularly important in the TMD framework, since it is directly related +to the TMD distribution of linearly polarized gluons inside an unpolarized proton, h⊥g +1 . This could play a role in the +region of moderately low PT , where the two factorization schemes overlap. +Again, we focus initially on the higher cm energy (√s = 140 GeV), Fig. 5, and then we describe the main differences +with respect to the smaller cm energy (√s = 45 GeV), Fig. 6. +Starting from the z-dependent distribution in Fig. 5 (left panels), we see once again that even if the estimated +ν values are potentially sizeable, at least in the Helicity frame, the separation among the different approaches is in +general very poor. Nevertheless, it is worth remarking that at high z we find more sensitivity to the LDME sets in +the NRQCD framework. The situation is slightly different for the PT case (right panels): if the Helicity frame does +not show a promising scenario, in the Gottfried-Jackson case the differences in the medium/high-PT region between +the two approaches are sizeable. +As said, results at high z and/or small PT are in general promising for future analyses regarding the h⊥g +1 +gluon +distribution in the TMD region. Nevertheless, it is important to remark that for the ν parameter the shape functions +and their TMD extensions enter, potentially, in a different way in the numerator and the denominator, and their role +could be important. This requires further investigation, together with a full higher-order description in αs, which is +not available at present. +It is once again interesting to look into the parton and wave decomposition. The z-dependent W∆∆ is dominated, +for almost all z values, by the CS wave; only for z → 0.9 the CS contribution becomes negligible, and the results +are driven by the CO P-wave, in particular by the gluon term. Moving to the PT dependence, we find again some +similarities with the λ case: the CS term is the relevant contribution to the numerator over the whole PT spectrum, +together with the gluon P-wave. At variance with the λ parameter case, the quark contribution to the P-wave term +starts becoming important already at small-PT values. +Moving to the lower cm energy, from Fig. 6 we see that the z distribution is sensitive to the energy change in the +whole spectrum, at variance with the λ case. The differences, particularly noticeable in the Gottfried-Jackson frame, +are mostly in size and not in the general behavior, implying that even in this case it would be difficult to extract any +information. Again, we remark that the rapid variation of ν estimates at low-z values is due to a geometrical factor +(Eq. (A16) of Ref. [25]). The PT -dependent distributions, instead, have a quite different behavior for the two frames + +9 +0.2 +0.0 +0.2 +0.4 +Gottfried-Jackson +s = 140 GeV +9 GeV2 < Q2 < 100 GeV2 +20 GeV < W < 100 GeV +0.2 < z < 0.9 or PT > 1 GeV +0.6 +0.4 +0.2 +0.0 +0.2 +0.2 +0.4 +0.6 +0.8 +z +0.2 +0.0 +0.2 +0.4 +Helicity +2 +4 +6 +8 +10 +PT [GeV] +0.6 +0.4 +0.2 +0.0 +0.2 +CSM +NRQCD (C12) +NRQCD (BK11) +NRQCD (G13) +Figure 5. Estimates for the parameter ν at √s = 140 GeV. Paneling order is the same as in Fig. 1. Integration ranges are +given in the blue legend box. +displayed. The Gottfried-Jackson estimates vary significantly in size, especially if one considers the CSM; moreover all +the LDME sets give similar predictions, compatible with zero, for PT > 5 GeV, while predictions, in both approaches, +are sizeable (up to ∼ 20%) at low-PT values. This could be very promising for further extensions to the TMD region. +The curves in the Helicity frame, instead, do not show the same dependence on the energy. In general, we conclude +that the study of the ν parameter, at least in this frame, is not very effective. Nevertheless it becomes more interesting +when its information is combined with other parameters, as done in the study of the invariant quantities in the next +section, Sec. IV. +Concerning the wave decomposition, we find that both quark and gluon P-wave contributions to the PT and z +distributions are enhanced at lower energies, even if for the latter this is true only at large z values. Notice that +the different (larger) size of the ν parameter at z → 0.9 could also affect the TMD region, increasing the possibility +of extracting information on the linearly polarized gluon distribution. +The main source of this enhancement at +√s = 45 GeV is related once again to the lower photon virtualities explored. In this sense, very similar predictions +might be expected at higher cm energy via a binned analysis with 1.6 GeV < Q < Mψ. +IV. +ROTATIONAL INVARIANTS +The polarization parameters λ, µ and ν, as widely discussed in the previous sections, are frame dependent by +definition, since they are expressed with respect to the solid angle Ω spanned by the l+ particle in the J/ψ decay and +in its rest frame. As already pointed out, the frame choice is not unique and the results appear different from frame +to frame. On the other hand, the relations among the most used reference frames are computable, since they differ +only in the Z-axis direction. +A complementary and powerful tool to study J/ψ polarization, both from the experimental and the phenomeno- +logical points of view, is the use of rotational invariant parameters, that are rest-frame independent by construction. +These can be defined taking into account what follows. +For all the most common choices, the Z- and X-axes, lying in the J/ψ production plane, are defined in terms of +physical momenta in the quarkonium rest frame (see Appendix A of Ref. [25]), with the Y -axis always perpendicular +with respect to this plane and always pointing in the same direction. This implies that two frames (F, F ′) can be +connected by a simple rotation of an angle ψ around the Y -axis, and the corresponding polarization parameters can + +10 +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Gottfried-Jackson +s = 45 GeV +2.5 GeV2 < Q2 < 100 GeV2 +10 GeV < W < 40 GeV +0.2 < z < 0.9 or PT > 1 GeV +0.2 +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +CSM +NRQCD (C12) +NRQCD (BK11) +NRQCD (G13) +0.2 +0.4 +0.6 +0.8 +z +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Helicity +2 +4 +6 +8 +10 +PT [GeV] +0.2 +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +Figure 6. Estimates for the parameter ν at √s = 45 GeV. Paneling order is the same as in Fig. 1. Integration ranges are given +in the red legend box. +be directly related as1 +� +� +λ +µ +ν +� +� +F ′ += +1 +1 + ρ +� +� +1 − 3 +2 sin2 ψ +3 +2 sin 2ψ +3 +4 sin2 ψ +− 1 +2 sin 2ψ +cos 2ψ +1 +4 sin 2ψ +sin2 ψ +− sin 2ψ 1 − 1 +2 sin2 ψ +� +� +� +� +λ +µ +ν +� +� +F +, +(13) +with +ρ = sin2 ψ +2 +� +λF − νF +2 +� +− sin 2ψ µF +2 , +(14) +as given in Eqs. (A.18) and (A.19) of Ref. [25], where we have changed the rotation angle from θ to ψ to avoid any +confusion with the polar angle of the final lepton l+. Notice that the quantity ρ depends on the kinematics, since the +rotation angle itself depends on the partonic Mandelstam variables (see Eqs. (A.14)-(A.16) of Ref. [25] for details). +From Eq. (13), one can construct several quantities which do not change upon rotation around the Y direction. +The following relations are extremely useful in this respect: +3 + λF ′ = +1 +1 + ρ (3 + λF ) , +1 − νF ′ +2 += +1 +1 + ρ +� +1 − νF +2 +� +. +(15) +A group of rotational invariants, as initially proposed in Ref. [36], can be defined in terms of two polarization +parameters, namely λ and ν, +F(ci) = c0(3 + λ) + c1(1 − ν/2) +c2(3 + λ) + c3(1 − ν/2) , +(16) +where ci are suitable free constants. +1 Here µF stands for the µ parameter in a specific frame F, not to be confused with the factorization scale µF defined in the previous +sections. + +11 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +s = 140 GeV +s = 140 GeV +9 GeV2 < Q2 < 100 GeV2 +20 GeV < W < 100 GeV +0.2 < z < 0.9 or PT > 1 GeV +0.25 +0.30 +0.35 +0.40 +0.2 +0.4 +0.6 +0.8 +z +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +s = 45 GeV +CSM +NRQCD (C12) +NRQCD (BK11) +NRQCD (G13) +2 +4 +6 +8 +10 +PT [GeV] +0.25 +0.30 +0.35 +0.40 +s = 45 GeV +2.5 GeV2 < Q2 < 100 GeV2 +10 GeV < W < 40 GeV +0.2 < z < 0.9 or PT > 1 GeV +Figure 7. Estimates for the invariant F, Eq. (17), as a function of z (left panels) and PT (right panels) at two cm energies, +√s = 140 GeV (upper panels) and √s = 45 GeV (lower panels), for different approaches and LDME sets. Kinematic ranges +are given in the legend boxes. +Among all possible combinations, two of them play an important role and have received special attention [37–41] +F ≡ F(1,−2,1,0) = 1 + λ + ν +3 + λ +(17) +and +˜λ ≡ F(1,−3,0,1) = 2 λ + 3 ν +2 − ν +. +(18) +These invariants have been widely studied in pp and heavy-ion processes [42, 43]. +It is worth noticing that both invariants can be similarly defined for Drell-Yan processes, where they acquire a +constant value if the Lam-Tung relation (1 − λ = 2ν) holds [26]: FDY = 1/2 and ˜λDY = +1, as pointed out in +Refs. [38, 41]. +Another interesting feature is that ˜λ = +1(−1) is related to a natural transverse (longitudinal) +polarization [36]. It is important to stress that the constant behavior is purely dynamical, and in particular for the +Drell-Yan case is a consequence of rotational invariance and helicity conservation [44]. Since J/ψ couples differently +in SIDIS processes, the Lam-Tung relation is expected to be broken in this case. +Not all the invariants belong to the previous family. Indeed, one can exploit another relation that involves all +polarization parameters in two frames and that, upon rotation around the Y -axis, reads +(λF ′ − νF ′/2)2 + 4µ2 +F ′ = (λF − νF /2)2 + 4µ2 +F +(1 + ρ)2 +. +(19) +From this, one can construct an invariant quantity involving the polarization parameters squared, as first pointed +out in Ref. [45]. As an example, we recall +˜λ′ = (λ − ν/2)2 + 4µ2 +(3 + λ)2 +, +(20) +as introduced in Ref. [41]. + +12 +The study of rotational invariants has not only a theoretical interest, but it is relevant also from the experimental +point of view, since their expected equality among different frames is an important check of experimental acceptances +and systematics as shown, for instance, by the ATLAS Collaboration [46]. +For these reasons, we consider, as a case of study, one of these quantities at the kinematics explored by the EIC. +In Fig. 7 we show the theoretical estimates in the collinear framework, for the invariant F, Eq. (17), as a function of +z (left panels) and PT (right panels). Once again we compute this quantity at two energies, √s = 140 GeV (upper +panels) and √s = 45 GeV (lower panels) for different approaches and LDME sets. +From Fig. 7 we clearly see that F is not equal to 1/2, as expected from the Lam-Tung relation. Moreover, it is +neither a constant, since its value depends on both z and PT variables. In principle, for some LDME sets a constant +behavior could accidentally appear, but this would be limited to a specific kinematic region. +Another interesting remark is that, while the denominator of F is proportional to the unpolarized cross section, its +numerator is controlled by the relative size of the λ and ν parameters. This can vary significantly, depending on the +frames and approaches adopted, as discussed in the previous Section. +From this preliminary study we can conclude that, even if not easily accessible from the experimental point of view, +these invariant quantities could represent an invaluable tool to learn on the J/ψ polarization mechanism. +V. +CONCLUSIONS +The study of quarkonium polarization, interesting by itself, is also a powerful tool to explore the still challenging +issue of its formation mechanism within QCD. In this spirit, we have presented a phenomenological analysis of J/ψ +polarization in SIDIS at large PT . +More specifically, we have looked at the dilepton angular distribution in the +J/ψ → ℓ+ℓ− decay in terms of the associated polarization parameters, that could be accessed at the future EIC. By +exploiting the theoretical results of Ref. [25], we have computed the parameters, λ, µ and ν, in different frames, trying +to emphasize whether one can use these observables to discriminate among two well consolidated frameworks, still +under investigation: the Color Singlet Model and the NRQCD approach. Moreover, for the latter we have employed +three different LDME sets, based on different extractions and assumptions, highlighting their impact on quarkonium +polarization estimates. +We have shown results both as a function of z and PT , adopting two quite different cm energies, for standard +kinematics at the EIC, together with a detailed analysis in terms of parton and NRQCD wave contributions. +The main findings of our study can be summarized as follows: i) concerning the λ parameter, the large-z region, +both in the Gottfried-Jackson and the Helicity frame, turns out to be very promising, with the only caveat of possible +contributions from (TMD) shape functions (even if expected to be reduced being λ a ratio of helicity structure +functions); similarly its PT distribution, at medium-large values, could be an ideal ground to disentangle the formation +mechanisms, both at high and low energies. ii) The µ parameter displays some interesting features when studied in +the Gottfried-Jackson frame, namely: a clear separation among the estimates in different frameworks at medium-large +z or as a function of PT in the high-energy set-up; a different behavior with respect to the corresponding lower-energy +estimates at medium-large z or at moderate PT . Moreover, in the Helicity frame at low energies one could extract +important information by looking in the large PT region. iii) Similarly, for the ν parameter, relevant also in the +context of the TMD framework, medium-large PT values in the Gottfried-Jackson frame are certainly worth to be +explored. +Finally, we have discussed a selection of frame-independent (rotational invariant) polarization parameters, relevant +not only from the theory point of view, but extremely useful as an important check of experimental acceptances and +systematics. In particular, we have focused on the invariant F, controlled by the relative weight of the λ and ν +parameters, that strongly depend on the frames and frameworks adopted. 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Palestini, “Angular distribution and rotations of frame in vector meson decays into lepton pairs,” Phys. Rev. D 83 +(2011) 031503, arXiv:1012.2485 [hep-ph]. +[46] ATLAS Collaboration, G. Aad et al., “Measurement of the differential cross-sections of inclusive, prompt and +non-prompt J/ψ production in proton-proton collisions at √s = 7 TeV,” Nucl. Phys. B 850 (2011) 387–444, +arXiv:1104.3038 [hep-ex]. + diff --git a/EtFLT4oBgHgl3EQfFS_K/content/tmp_files/load_file.txt b/EtFLT4oBgHgl3EQfFS_K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..68fd9013b5830558c5c04525734523407bcb6461 --- /dev/null +++ b/EtFLT4oBgHgl3EQfFS_K/content/tmp_files/load_file.txt @@ -0,0 +1,1010 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf,len=1009 +page_content='J/ψ polarization in large-PT semi-inclusive deep-inelastic scattering at the EIC Umberto D’Alesio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' ∗ Luca Maxia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' † Francesco Murgia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' ‡ Cristian Pisano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' § and Sangem Rajesh3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' ¶ 1Dipartimento di Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Universit`a di Cagliari,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Cittadella Universitaria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' I-09042 Monserrato (CA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Italy 2INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Sezione di Cagliari,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Cittadella Universitaria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' I-09042 Monserrato (CA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Italy 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' School of Advanced Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Vellore Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Vellore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Tamil Nadu 632014,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' India 4INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Sezione di Perugia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' via A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Pascoli snc, 06123, Perugia, Italy (Dated: January 31, 2023) We present a detailed phenomenological study of J/ψ polarization in semi-inclusive deep inelastic scattering processes, focusing on the kinematics accessible at the future Electron-Ion Collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' We show theoretical estimates for the standard polarization parameters for different frames usually adopted in the literature, in the large PT region, namely PT ≫ ΛQCD, where collinear factorization is expected to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' We adopt both the Color Singlet Model and the Nonrelativistic QCD approach, paying special attention to the role of different sets of Long Distance Matrix Elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Finally we present a preliminary analysis of some frame independent polarization invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' INTRODUCTION Our understanding of the J/ψ production mechanism at high energies has improved significantly since its discovery almost 50 years ago [1, 2], thanks to the combined efforts from both the theoretical and experimental communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' However, there are still major problems in the theoretical analyses of the available data, such as the long-standing J/ψ polarization puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Namely, J/ψ polarization measurements cannot yet be explained in a way entirely consistent with the world experimental results for the unpolarized J/ψ yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The present theoretical frameworks all agree in providing a perturbative description of the creation of the charm quark-antiquark (c¯c) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The charm mass mc plays the role of the hard scale, since it is much larger than the asymptotic scale parameter of QCD, ΛQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' These approaches nonetheless differ in the treatment of the subsequent nonperturbative transition to the hadronic bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' For instance, in the traditional Color-Singlet Model (CSM) [3] the c¯c pair is produced at short distances directly with the quantum numbers of the J/ψ meson, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' in a color-singlet (CS) state with spin one and no orbital angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This is possible by the emission of an additional hard gluon, which implies the suppression of the cross section by one power of the strong coupling constant αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' However, the CSM cannot be considered as a complete theory, since at the next-to-leading order (NLO) P-wave quarkonia are affected by uncanceled infrared singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' These singularities are properly removed in the effective field theory approach of nonrelativistic QCD (NRQCD), based on a rigorous factorization theorem, which was assumed in the original paper by Bodwin, Braaten, and Lep- age [4], and later explicitly proven to next-to-next-to-leading order (NNLO) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' NRQCD therefore implies a sep- aration of process-dependent short-distance coefficients, to be calculated perturbatively as expansions in αs, from long-distance matrix elements (LDMEs), which are expected to be universal and have to be extracted from experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Scaling rules [6] predict each of the LDMEs to scale with a definite power of the relative velocity v of the heavy quark-antiquark pair in the quarkonium rest frame in the limit v ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Observables are hence evaluated by means of a double expansion in αs and in v, with αs ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 and v2 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='3 for charmonium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' An essential feature of this approach is that the c¯c pair at short distance can be produced in any Fock state n = 2S+1L[c] J with definite orbital angular momentum L, spin S, total angular momentum J and color configuration c = 1, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' NRQCD hence predicts the existence of intermediate color-octet (CO) states, which subsequently evolve into physical, CS quarkonia by the emission of soft gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' For S-wave quarkonia, the CSM is recovered in the limit v → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In the specific case of J/ψ production, the CSM prediction is based only on the 3S[1] 1 CS state, while NRQCD includes the leading relativistic corrections as well, which at the relative order O(v4) are given by the CO states 1S[8] 0 , 3S[8] 1 , and 3P [8] J with J = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The values of the CO LDMEs extracted from different fits to data on J/ψ and Υ yields [7–11] are not compatible with each other, even within the large uncertainties [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Therefore, any new method to determine them with better precision is worth exploring [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In this paper we propose to look at the J/ψ polarization parameters in ∗ umberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='dalesio@ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='it † luca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='maxia@ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='it ‡ francesco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='murgia@ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='it § cristian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='pisano@unica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='it ¶ sangem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='rajesh@vit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='11987v1 [hep-ph] 27 Jan 2023 2 semi-inclusive deep-inelastic scattering (SIDIS), e p → e′ J/ψ X, in a kinematic region where the transverse momentum of the J/ψ meson PT is large, namely PT ≫ ΛQCD, and collinear factorization is expected to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Analysing SIDIS at finite values of the exchanged photon virtuality Q2 has certain experimental and theoretical advantages as compared to photoproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Namely, as Q2 increases theoretical uncertainties in the different frameworks decrease and resolved photon contributions are expected to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, background from diffractive J/ψ production is expected to decrease with Q2 faster than the SIDIS cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The distinct signature of the scattered lepton makes the process particularly easy to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Clearly, cross sections are smaller than those expected in the photoproduction case, however, considering the achievable high luminosities, this study should be feasible at the future Electron-Ion Collider (EIC) planned in the United States [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' So far, only a single experimental study of J/ψ polarization in SIDIS has been performed, by the H1 Collaboration at HERA [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Such a measurement is limited to the polarization parameter λ in the helicity frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This result turns out to be compatible with the predictions provided in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [22, 23], but it can hardly discriminate among the different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In analogy with Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [22, 23], our phenomenological analysis has been carried out at the perturbative order α2 s, which has to be considered as the state of the art for these observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Higher-order effects have been calculated very recently only for the unpolarized cross section within the CSM [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Anyway, we expect these effects (at least in the large Q2 region) to be small for the observables we are investigating, because they are ratios of cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' We point out that our estimates include also the polarization parameters µ and ν, not addressed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [22, 23], which are studied in different reference frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Furthermore, we perform a preliminary study of rotational invariant combinations of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In section II we recall the standard SIDIS variables and collect the expressions of the differential cross section for quarkonium production and its leptonic decay in terms of the helicity structure functions and the polarization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In section III we discuss the three polarization parameters λ, µ, ν, showing their estimates in two reference frames and paying special attention to their energy, z and PT dependences as well as to the impact of the LDME set adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' To overcome the intrinsic frame dependence of the polarization parameters, in section IV we present two classes of the so-called rotational invariant quantities, and show, as a case of study, some results for one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Finally in section V we gather our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' KINEMATICS AND FORMALISM In this section we provide the main analytic expressions needed to carry out the phenomenological analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' For more details and the complete formalism we refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' We consider the SIDIS process e(k) + p(P) → e′(k′) + J/ψ(Pψ) + X(PX) , (1) with the subsequent J/ψ decay into a lepton pair J/ψ(Pψ) → l+(l) + l−(l′) , (2) where, in brackets, we have shown the four-momenta of each particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The J/ψ meson is produced via the partonic subprocess γ∗(q) + a(pa) → c¯c[n](Pψ) + a(p′ a) , (3) with q2 = −Q2 and P 2 ψ = M 2 ψ = (2mc)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The initial parton momentum, pa, is related to the parent proton one, P, as pa = ξP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (4) We adopt the following three standard invariant quantities, defined in terms of the photon and hadron momenta xB = Q2 2P · q , y = P · q P · k , z = P · Pψ P · q , (5) where xB is the Bjorken variable, y is the inelasticity and z is the energy fraction carried out by the J/ψ (in the proton rest frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' All these variables are constrained in the region 0 ≤ xB, y, z ≤ 1 and they are connected to other kinematical quantities of the system, like the total center-of-mass (cm) energy √s and the virtual photon-proton cm energy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The cross section that describes the J/ψ formation and its decay into a lepton pair can be written as 1 Bll dσ dxB dy dz d2PT dΩ = α 8 y z Q2 3 8π � WT (1 + cos2 θ) + WL(1 − cos2 θ) + W∆ sin 2θ cos φ + W∆∆ sin2 θ cos 2φ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (6) 3 where PT is the J/ψ transverse momentum in the cm frame of the virtual photon and the proton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Bll is the branching ratio for the decay process J/ψ → ℓ+ℓ− and Ω(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' φ) refers to the solid angle spanned by the lepton ℓ+ in a reference frame where the system formed by ℓ+ and ℓ− is at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, we have introduced the following helicity structure functions WT ≡ W11 = W−1,−1 , WL ≡ W00 , W∆ ≡ 1 √ 2 (W10 + W01) = √ 2 Re [W10] , W∆∆ ≡ W1,−1 = W−1,1 , (7) where the subscripts refer to the J/ψ polarization states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' More specifically, WT and WL are respectively the structure functions for transversely and longitudinally polarized J/ψ mesons, W∆ is the single-helicity flip structure function, and W∆∆ is the double-helicity flip one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Notice that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (6) we have introduced a proper overall constant factor w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='35) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25] to ensure the normalization when integrated over the solid angle, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (8) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This does not affect any conclusion of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25], where all relevant quantities are defined as ratios of helicity structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25], the structure functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (7) can be further decomposed in terms of the contributions coming from the longitudinal ( ) and transverse (⊥) polarizations of the virtual photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, within a collinear factorization scheme, they are given as convolutions of collinear parton distribution functions (PDFs) with partonic helicity structure functions (weighted by proper LDMEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' These, in turn, can be expressed as functions of the partonic Mandelstam invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The unpolarized cross section is obtained by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (6) over the solid angle Ω, 1 Bll dσ dxB dy dz d2PT = α 8 y z Q2 (2WT + WL) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (8) It is then useful to introduce the ratio of polarized and unpolarized cross sections dN dΩ ≡ dσ dxB dy dz d2PT dΩ � dσ dxB dy dz d2PT �−1 , (9) which can be expressed as follows dN dΩ = 3 4π 1 λ + 3 � 1 + λ cos2 θ + µ sin 2θ cos ϕ + 1 2 ν sin2 θ cos 2ϕ � , (10) where we have defined the polarization parameters λ = W11 − W00 W11 + W00 , µ = √ 2 Re [W10] W11 + W00 , ν = W1, −1 W11 + W00 , (11) or alternatively adopting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (7), λ = WT − WL WT + WL , µ = W∆ WT + WL , ν = 2 W∆∆ WT + WL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (12) The parameterizations shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (6) and (10) are standard for the study of the angular distribution of a spin-one particle decay into a lepton pair and, indeed, they are commonly adopted in Drell-Yan processes [26] and in J/ψ photoproduction [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Among the polarization coefficients, λ, µ and ν, the most investigated experimentally is λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, from the phenomenological point of view it has a very intuitive interpretation, with λ = +1(−1) describing a trans- verse(longitudinal) polarization state for the J/ψ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' a J/ψ helicity equal to ±1 or 0), while λ = 0 for an unpolarized one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The main goal of our study is to present estimates for these polarization quantities, within both the CSM and the NRQCD frameworks, focusing on the kinematic region accessible at the future EIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' As we will show in the following, such a detailed phenomenological study could help in disentangling among the production mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 4 LDME Set ⟨O1[ 3S1]⟩ � GeV3� ⟨O8[ 1S0]⟩ � GeV3� ⟨O8[ 3S1]⟩ � GeV3� ⟨O8[ 3P0]⟩ � GeV5� C12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0126 G13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='097 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0046 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0214 BK11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00168 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00908 Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' LDME set (central) values for the J/ψ state: C12 [8], G13 [28] and BK11 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' For the other 3PJ states we use the standard spin-symmetry relation ⟨O8[ 3PJ]⟩ = (2J + 1) ⟨O8[ 3P0]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' ANGULAR DISTRIBUTIONS In this section we analyze the polarization parameters defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (11) showing both their z and PT distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The explicit analytic expressions of the underlying partonic structure functions, calculated at the perturbative order α2 s, are presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25] for the so-called Gottfried-Jackson frame, together with all prescriptions needed to transform them in the other relevant frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' For the predictions based on the NRQCD approach, in addition to the CS contribution, given by a pure gluon fusion channel, we consider the CO channels up to the order v4, which involve both gluon and quark final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The CTEQ6L1 set [30] is used for the unpolarized parton distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, in order to assess the stability of our results against higher order corrections, we produce uncertainty bands by varying the factorization scale µF in the range µ0/2 < µF < 2µ0, around the central value µ0 = � Q2 + M 2 ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Concerning the CO LDME values, three different sets are adopted, see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Here we only recall their main features: the C12 set [8] has been extracted simultaneously from both polarized and unpolarized J/ψ production data in pp collision at PT > 7 GeV, measured by the CDF (Run II) Collaboration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' the G13 set [28] is obtained including only PT > 7 GeV unpolarized data from the CDF and LHCb Collaborations and then used to predict the J/ψ polarization in pp collisions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' it is in agreement with the C12 set if feed-down contribution is negligible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' the BK11 set [29] is based on a fit without any polarization data, but starting from a lower PT value, around 3 GeV, and including both photoproduction and hadroproduction data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The high cm energy kinematical set-ups expected at the EIC are an ideal environment to study J/ψ polarization in electroproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, they will allow to better explore high photon virtualities (Q), avoiding the competing contributions from photoproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Furthermore, since we are interested in the region where collinear factorization holds, our results will be shown only for PT values above PT min = 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Notice that around this value we actually enter the region where the transverse momentum dependent (TMD) factorization could be applied and therefore our estimates are pushed down to the overlapping region of validity of the two factorization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The λ parameter In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 1 we present our predictions for λ at √s = 140 GeV, as a function of both the J/ψ energy fraction z (left panels) and its transverse momentum PT (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Two quarkonium rest frames are explicitly considered: the Gottfried-Jackson (upper panels) and the Helicity (lower panels) ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In this and in the following figures, the kinematical ranges explored are indicated in the legend boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' For completeness we report here the corresponding regions explored in xB and y at √s = 140 GeV, 10−3 ≲ xB ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 and y ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='5 respectively, even if the effectively probed maximum value in xB is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Concerning other typical frames, like the Target and Collins-Soper ones, we only notice that the first one give estimates very close to those in the Helicity frame, while predictions obtained in the second one, at least for the kinematics considered, are in general much smaller than those in the Gottfried-Jackson frame or even close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Notice that for such observable, defined as a ratio of cross sections, the dependence on the scale µF in the range [µ0/2, 2µ0] is barely appreciable and therefore is not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The study of the λ parameter as a function of z presents very interesting features from the phenomenological point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The reasons are manifold: first of all its expected relative large size as compared to the µ and ν parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, it is experimentally under more active investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' On the other hand, theoretical estimates for λ as a function of z (for small and moderate values) do not vary significantly adopting different frameworks (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 1, left panels), which implies that, in order to get information on the quarkonium formation mechanism, one would need highly precise measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The same problem was found in different analyses performed by the HERA Collaborations, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [21, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The situation changes considerably at z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6, which represents a very interesting region from the phenomenological point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' As is well known, NRQCD estimates for the unpolarized cross section manifest a divergent behavior as 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 Gottfried-Jackson 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='8 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 Helicity CSM NRQCD (C12) NRQCD (BK11) NRQCD (G13) 2 4 6 8 10 PT [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 s = 140 GeV 9 GeV2 < Q2 < 100 GeV2 20 GeV < W < 100 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9 or PT > 1 GeV Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Estimates for λ at √s = 140 GeV as a function of z (left panels) and PT (right panels) for different models and LDME sets and two reference frames: Gottfried-Jackson (upper panels) and Helicity (lower panels) frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Integration ranges are given in the light-blue legend box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' z → 1, due to the corresponding ˆt → 0 singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This can potentially spoil the validity of NRQCD factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [31], in order to extend the region of applicability of NRQCD up to 1 − z ∼ v2, one can introduce a new set of functions, the so-called shape functions [32], that allow to improve noticeably the convergence for photoproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' We expect such quantities to be relevant also for the SIDIS process, together with their TMD extensions, which have been adopted in the study of pp collisions in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [33, 34] and whose perturbative tails have been derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [35] for unpolarized and in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25] for polarized J/ψ SIDIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' On the other hand, the impact of the shape functions on λ is expected to be strongly reduced since λ is a ratio of cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This can be tested with future available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' A much more powerful tool to assess the relevance of the CO contributions is the study of the PT distribution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 1, right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In the Gottfried-Jackson frame (upper panel) we see a clear separation as well as a different behavior between the CSM and NRQCD curves, in particular in the region 4 < PT < 7 GeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' similarly in the Helicity frame there is a wide separation between the CSM and the NRQCD curves, while different LDME sets give predictions much closer to each other and closer to λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' It is worth noticing that, even if the unpolarized cross section decreases as PT increases, a good separation can be found already around PT ≃ 5 GeV, which is also far away from the TMD region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Before concluding the analysis of λ at large cm energies, a comment on the contributions from different partonic channels and/or different NRQCD waves can be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Concerning the z distribution, we find that the main con- tribution to the numerator of λ comes from the (gluon) CS wave, while the differences among NRQCD predictions, especially around z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9, are due to the gluon P-wave, modulated by the corresponding LDME parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' For the PT distribution we find, similarly, that the CS term is on the whole the most relevant contribution, followed again by the gluon P-wave one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In particular at PT → 1 GeV the size of the gluon P-wave contribution becomes comparable to (or even larger than) the CS one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' moreover, since the low-PT region dominates the integration over PT , one can also understand why the gluon P-wave is so relevant in our estimates vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' z, with the most visible effects for z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' At medium PT values the quark P-wave starts becoming important and at even higher PT values it is similar in size to the gluon one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' this means that in this region, the full P-wave contribution (gluon+quark) dominates over the CS one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Another interesting possibility given by the future EIC facility is the corresponding analysis at smaller energies: in the following we will adopt √s = 45 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In this case, different integration ranges have been considered for W and Q2, as reported in the legend box of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' These, in turn, correspond to 10−3 ≲ xB ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='5 (with an effective 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 Gottfried-Jackson 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='8 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 Helicity s = 45 GeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='5 GeV2 < Q2 < 100 GeV2 10 GeV < W < 40 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9 or PT > 1 GeV 2 4 6 8 10 PT [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 CSM NRQCD (C12) NRQCD (BK11) NRQCD (G13) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Estimates for λ at cm energy √s = 45 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The integration region, different with respect to the higher-energy case, is given in the red legend box, while curves and panels have the same meaning as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The scale error bands are sizable and explicitly shown only for the CSM prediction as a function of PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' upper limit around xB ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2) and y ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='8, a more valence-like region w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, since at lower energies it is more difficult to reach high photon virtualities, we get contributions mostly from moderately low Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Consistently we adopt a lower limit, Qmin ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 GeV, in the integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Notice that in this kinematic region, at least for the high PT dependence of λ within the CSM, the scale error bands are once again sizeable enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 2 (left panels) we can see that the z distribution does not depend significantly on the energy for z ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6, while at higher z values the estimates are closer to zero, at variance with those at higher cm energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' As said, a polarization study pushed up to this regime can suffer from factorization breaking effects in NRQCD even if data in this region could be relevant from the phenomenological point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' We also observe a rapid variation of all curves in the Helicity frame at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This is due to geometrical factors which are energy dependent (see also Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (A16) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The same variation is also present at higher cm energy, but for z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1 (outside the range shown in the lower-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Concerning the PT dependence, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 2 (right panels), we notice that the CSM results are very different with respect to the corresponding ones in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 1, while the same is not true for the NRQCD cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This is related to the different virtualities explored, on which the CSM estimates depend heavily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This difference can be considered as an extra tool in the quest of discerning among different frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Finally, we briefly comment on how the parton and/or wave contributions vary with the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' While the z distribution manifests almost no energy dependence, the PT spectrum presents interesting features in the two frames considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' For the Gottfried-Jackson one the relative contribution from the quark P-wave is widely increased at this lower energy, making it the leading term in the numerator at medium/high PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Regarding the Helicity frame the situation is, potentially, even more interesting, since the CSM and P-wave (both gluon and quark) contributions are highly suppressed at this energy, especially at large PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The main role is then played by the 3S(8) 1 quark wave, which is responsible for the difference among the predictions based on the LDME sets considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Even if in this region it is quite hard to expect precise enough data to discriminate between models, it is nevertheless worth stressing that it could be very useful in constraining the nonperturbative physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 Gottfried-Jackson 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='8 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 Helicity s = 140 GeV 9 GeV2 < Q2 < 100 GeV2 20 GeV < W < 100 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9 or PT > 1 GeV 2 4 6 8 10 PT [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 CSM NRQCD (C12) NRQCD (BK11) NRQCD (G13) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Estimates for the parameter µ at √s = 140 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Paneling order is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Integration ranges are given in the blue legend box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The µ parameter Estimates for the µ parameter are again provided both in the Gottfried-Jackson and in the Helicity frames, as a function of z and PT at √s = 140 GeV, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 3, and √s = 45 GeV, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' From these figures we see that the Gottfried-Jackson frame is the best choice to discern among the CSM and NRQCD approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' A similar conclusion holds for the parameter ν as well, see the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Indeed, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 3 the separation between the CSM estimates and the corresponding NRQCD ones are remarkably sizeable for z ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='5 and PT ≳ 5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' On the contrary, estimates in the Helicity frame both with respect to z and PT are so close to each other that one cannot draw any conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The wave/parton decomposition of the W∆ helicity function, that is directly related to the µ numerator, allows us to get some further insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The main CO contribution comes from the P-wave term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In particular, differences in NRQCD predictions as a function of z (left panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 3) are driven by the gluon P-wave LDMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, the gluon P-wave dominates the numerator behavior with respect to PT too (right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In addition, we find that the NRQCD predictions in the Gottfried-Jackson frame receive a significant contribution from the gluon P-wave also at low-PT , namely PT ≲ 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' At variance with the behavior in z, here the quark P-wave channel is relevant at high PT , especially when considering the Helicity frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moving to the lower cm energy, we see that the CSM µ estimates in the Gottfried-Jackson frame, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 4 (upper panels), vary significantly for z ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='5 and PT ≳ 5 GeV, as compared with what happens at √s = 140 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' We remark that this variation can also appear via a proper Q-binning in the higher cm energy case (√s = 140 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In contrast, estimates within the Helicity frame at lower energies (lower panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 4) do not present the same energy/Q-binning dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The only remarkable exception resides in the PT distribution, where CSM predictions increase up to ∼ 40%, to be compared with the √s = 140 GeV case where the CSM result is at most ∼ 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Despite this, µ estimates in the Helicity frame do not differ enough to discern among different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Looking at the wave/parton decomposition, we confirm that also for the µ numerator the role of quarks is enhanced at lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This is particularly true for the PT dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Here we find that NRQCD predictions at the higher PT values, namely PT ≳ 6 GeV, are mostly driven by the quark P-wave;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' moreover, in the same PT region we observe that the 3S[8] 1 quark wave is non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 Gottfried-Jackson 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='8 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 Helicity s = 45 GeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='5 GeV2 < Q2 < 100 GeV2 10 GeV < W < 40 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9 or PT > 1 GeV 2 4 6 8 10 PT [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 CSM NRQCD (C12) NRQCD (BK11) NRQCD (G13) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Estimates for the parameter µ at √s = 45 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Paneling order is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Integration ranges are given in the red legend box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The ν parameter We now discuss the parameter ν, which is particularly important in the TMD framework, since it is directly related to the TMD distribution of linearly polarized gluons inside an unpolarized proton, h⊥g 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This could play a role in the region of moderately low PT , where the two factorization schemes overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Again, we focus initially on the higher cm energy (√s = 140 GeV), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 5, and then we describe the main differences with respect to the smaller cm energy (√s = 45 GeV), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Starting from the z-dependent distribution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 5 (left panels), we see once again that even if the estimated ν values are potentially sizeable, at least in the Helicity frame, the separation among the different approaches is in general very poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Nevertheless, it is worth remarking that at high z we find more sensitivity to the LDME sets in the NRQCD framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The situation is slightly different for the PT case (right panels): if the Helicity frame does not show a promising scenario, in the Gottfried-Jackson case the differences in the medium/high-PT region between the two approaches are sizeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' As said, results at high z and/or small PT are in general promising for future analyses regarding the h⊥g 1 gluon distribution in the TMD region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Nevertheless, it is important to remark that for the ν parameter the shape functions and their TMD extensions enter, potentially, in a different way in the numerator and the denominator, and their role could be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This requires further investigation, together with a full higher-order description in αs, which is not available at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' It is once again interesting to look into the parton and wave decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The z-dependent W∆∆ is dominated, for almost all z values, by the CS wave;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' only for z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9 the CS contribution becomes negligible, and the results are driven by the CO P-wave, in particular by the gluon term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moving to the PT dependence, we find again some similarities with the λ case: the CS term is the relevant contribution to the numerator over the whole PT spectrum, together with the gluon P-wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' At variance with the λ parameter case, the quark contribution to the P-wave term starts becoming important already at small-PT values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moving to the lower cm energy, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 6 we see that the z distribution is sensitive to the energy change in the whole spectrum, at variance with the λ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The differences, particularly noticeable in the Gottfried-Jackson frame, are mostly in size and not in the general behavior, implying that even in this case it would be difficult to extract any information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Again, we remark that the rapid variation of ν estimates at low-z values is due to a geometrical factor (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (A16) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The PT -dependent distributions, instead, have a quite different behavior for the two frames 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 Gottfried-Jackson s = 140 GeV 9 GeV2 < Q2 < 100 GeV2 20 GeV < W < 100 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9 or PT > 1 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='8 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 Helicity 2 4 6 8 10 PT [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 CSM NRQCD (C12) NRQCD (BK11) NRQCD (G13) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Estimates for the parameter ν at √s = 140 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Paneling order is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Integration ranges are given in the blue legend box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The Gottfried-Jackson estimates vary significantly in size, especially if one considers the CSM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' moreover all the LDME sets give similar predictions, compatible with zero, for PT > 5 GeV, while predictions, in both approaches, are sizeable (up to ∼ 20%) at low-PT values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This could be very promising for further extensions to the TMD region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The curves in the Helicity frame, instead, do not show the same dependence on the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In general, we conclude that the study of the ν parameter, at least in this frame, is not very effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Nevertheless it becomes more interesting when its information is combined with other parameters, as done in the study of the invariant quantities in the next section, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Concerning the wave decomposition, we find that both quark and gluon P-wave contributions to the PT and z distributions are enhanced at lower energies, even if for the latter this is true only at large z values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Notice that the different (larger) size of the ν parameter at z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9 could also affect the TMD region, increasing the possibility of extracting information on the linearly polarized gluon distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The main source of this enhancement at √s = 45 GeV is related once again to the lower photon virtualities explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In this sense, very similar predictions might be expected at higher cm energy via a binned analysis with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 GeV < Q < Mψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' ROTATIONAL INVARIANTS The polarization parameters λ, µ and ν, as widely discussed in the previous sections, are frame dependent by definition, since they are expressed with respect to the solid angle Ω spanned by the l+ particle in the J/ψ decay and in its rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' As already pointed out, the frame choice is not unique and the results appear different from frame to frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' On the other hand, the relations among the most used reference frames are computable, since they differ only in the Z-axis direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' A complementary and powerful tool to study J/ψ polarization, both from the experimental and the phenomeno- logical points of view, is the use of rotational invariant parameters, that are rest-frame independent by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' These can be defined taking into account what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' For all the most common choices, the Z- and X-axes, lying in the J/ψ production plane, are defined in terms of physical momenta in the quarkonium rest frame (see Appendix A of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25]), with the Y -axis always perpendicular with respect to this plane and always pointing in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This implies that two frames (F, F ′) can be connected by a simple rotation of an angle ψ around the Y -axis, and the corresponding polarization parameters can 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='5 Gottfried-Jackson s = 45 GeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='5 GeV2 < Q2 < 100 GeV2 10 GeV < W < 40 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9 or PT > 1 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 CSM NRQCD (C12) NRQCD (BK11) NRQCD (G13) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='8 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='5 Helicity 2 4 6 8 10 PT [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Estimates for the parameter ν at √s = 45 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Paneling order is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Integration ranges are given in the red legend box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' be directly related as1 � � λ µ ν � � F ′ = 1 1 + ρ � � 1 − 3 2 sin2 ψ 3 2 sin 2ψ 3 4 sin2 ψ − 1 2 sin 2ψ cos 2ψ 1 4 sin 2ψ sin2 ψ − sin 2ψ 1 − 1 2 sin2 ψ � � � � λ µ ν � � F , (13) with ρ = sin2 ψ 2 � λF − νF 2 � − sin 2ψ µF 2 , (14) as given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='18) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='19) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25], where we have changed the rotation angle from θ to ψ to avoid any confusion with the polar angle of the final lepton l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Notice that the quantity ρ depends on the kinematics, since the rotation angle itself depends on the partonic Mandelstam variables (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='14)-(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='16) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (13), one can construct several quantities which do not change upon rotation around the Y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The following relations are extremely useful in this respect: 3 + λF ′ = 1 1 + ρ (3 + λF ) , 1 − νF ′ 2 = 1 1 + ρ � 1 − νF 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (15) A group of rotational invariants, as initially proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [36], can be defined in terms of two polarization parameters, namely λ and ν, F(ci) = c0(3 + λ) + c1(1 − ν/2) c2(3 + λ) + c3(1 − ν/2) , (16) where ci are suitable free constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 1 Here µF stands for the µ parameter in a specific frame F, not to be confused with the factorization scale µF defined in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 s = 140 GeV s = 140 GeV 9 GeV2 < Q2 < 100 GeV2 20 GeV < W < 100 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9 or PT > 1 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='8 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='50 s = 45 GeV CSM NRQCD (C12) NRQCD (BK11) NRQCD (G13) 2 4 6 8 10 PT [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='40 s = 45 GeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='5 GeV2 < Q2 < 100 GeV2 10 GeV < W < 40 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content='9 or PT > 1 GeV Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Estimates for the invariant F, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (17), as a function of z (left panels) and PT (right panels) at two cm energies, √s = 140 GeV (upper panels) and √s = 45 GeV (lower panels), for different approaches and LDME sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Kinematic ranges are given in the legend boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Among all possible combinations, two of them play an important role and have received special attention [37–41] F ≡ F(1,−2,1,0) = 1 + λ + ν 3 + λ (17) and ˜λ ≡ F(1,−3,0,1) = 2 λ + 3 ν 2 − ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (18) These invariants have been widely studied in pp and heavy-ion processes [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' It is worth noticing that both invariants can be similarly defined for Drell-Yan processes, where they acquire a constant value if the Lam-Tung relation (1 − λ = 2ν) holds [26]: FDY = 1/2 and ˜λDY = +1, as pointed out in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [38, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Another interesting feature is that ˜λ = +1(−1) is related to a natural transverse (longitudinal) polarization [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' It is important to stress that the constant behavior is purely dynamical, and in particular for the Drell-Yan case is a consequence of rotational invariance and helicity conservation [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Since J/ψ couples differently in SIDIS processes, the Lam-Tung relation is expected to be broken in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Not all the invariants belong to the previous family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Indeed, one can exploit another relation that involves all polarization parameters in two frames and that, upon rotation around the Y -axis, reads (λF ′ − νF ′/2)2 + 4µ2 F ′ = (λF − νF /2)2 + 4µ2 F (1 + ρ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (19) From this, one can construct an invariant quantity involving the polarization parameters squared, as first pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' As an example, we recall ˜λ′ = (λ − ν/2)2 + 4µ2 (3 + λ)2 , (20) as introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 12 The study of rotational invariants has not only a theoretical interest, but it is relevant also from the experimental point of view, since their expected equality among different frames is an important check of experimental acceptances and systematics as shown, for instance, by the ATLAS Collaboration [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' For these reasons, we consider, as a case of study, one of these quantities at the kinematics explored by the EIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 7 we show the theoretical estimates in the collinear framework, for the invariant F, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' (17), as a function of z (left panels) and PT (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Once again we compute this quantity at two energies, √s = 140 GeV (upper panels) and √s = 45 GeV (lower panels) for different approaches and LDME sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' 7 we clearly see that F is not equal to 1/2, as expected from the Lam-Tung relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, it is neither a constant, since its value depends on both z and PT variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In principle, for some LDME sets a constant behavior could accidentally appear, but this would be limited to a specific kinematic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Another interesting remark is that, while the denominator of F is proportional to the unpolarized cross section, its numerator is controlled by the relative size of the λ and ν parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This can vary significantly, depending on the frames and approaches adopted, as discussed in the previous Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' From this preliminary study we can conclude that, even if not easily accessible from the experimental point of view, these invariant quantities could represent an invaluable tool to learn on the J/ψ polarization mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' CONCLUSIONS The study of quarkonium polarization, interesting by itself, is also a powerful tool to explore the still challenging issue of its formation mechanism within QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In this spirit, we have presented a phenomenological analysis of J/ψ polarization in SIDIS at large PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' More specifically, we have looked at the dilepton angular distribution in the J/ψ → ℓ+ℓ− decay in terms of the associated polarization parameters, that could be accessed at the future EIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' By exploiting the theoretical results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' [25], we have computed the parameters, λ, µ and ν, in different frames, trying to emphasize whether one can use these observables to discriminate among two well consolidated frameworks, still under investigation: the Color Singlet Model and the NRQCD approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, for the latter we have employed three different LDME sets, based on different extractions and assumptions, highlighting their impact on quarkonium polarization estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' We have shown results both as a function of z and PT , adopting two quite different cm energies, for standard kinematics at the EIC, together with a detailed analysis in terms of parton and NRQCD wave contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' The main findings of our study can be summarized as follows: i) concerning the λ parameter, the large-z region, both in the Gottfried-Jackson and the Helicity frame, turns out to be very promising, with the only caveat of possible contributions from (TMD) shape functions (even if expected to be reduced being λ a ratio of helicity structure functions);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' similarly its PT distribution, at medium-large values, could be an ideal ground to disentangle the formation mechanisms, both at high and low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' ii) The µ parameter displays some interesting features when studied in the Gottfried-Jackson frame, namely: a clear separation among the estimates in different frameworks at medium-large z or as a function of PT in the high-energy set-up;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' a different behavior with respect to the corresponding lower-energy estimates at medium-large z or at moderate PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Moreover, in the Helicity frame at low energies one could extract important information by looking in the large PT region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' iii) Similarly, for the ν parameter, relevant also in the context of the TMD framework, medium-large PT values in the Gottfried-Jackson frame are certainly worth to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Finally, we have discussed a selection of frame-independent (rotational invariant) polarization parameters, relevant not only from the theory point of view, but extremely useful as an important check of experimental acceptances and systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' In particular, we have focused on the invariant F, controlled by the relative weight of the λ and ν parameters, that strongly depend on the frames and frameworks adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' As shown, this observable could clearly help in getting information on the J/ψ formation mechanism, both at large z (high- and low-energy set-ups) and as a function of PT (at large energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' We can certainly conclude that a study of the dilepton angular distribution in J/ψ decay in SIDIS at the EIC could be an invaluable tool to shed light on the J/ψ polarization as well as on its formation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Faccioli, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Stebel and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' Venugopalan for clarifying some aspects concerning the rotational invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} +page_content=' This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement STRONG 2020—No 824093.' metadata={'source': 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+page_content='3038 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFLT4oBgHgl3EQfFS_K/content/2301.11987v1.pdf'} diff --git a/FtE3T4oBgHgl3EQfVwq1/content/tmp_files/2301.04463v1.pdf.txt b/FtE3T4oBgHgl3EQfVwq1/content/tmp_files/2301.04463v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5516b2aeeaa9a2a10e773035eee29e43cc055345 --- /dev/null +++ b/FtE3T4oBgHgl3EQfVwq1/content/tmp_files/2301.04463v1.pdf.txt @@ -0,0 +1,3332 @@ +Astronomy & Astrophysics manuscript no. main +© ESO 2023 +January 12, 2023 +New members of the Lupus I cloud based on Gaia astrometry +⋆ +Physical and accretion properties from X-Shooter spectra +F. Z. Majidi1,2, J. M. Alcal´a3, A. Frasca4, S. Desidera2, C. F. Manara5, G. Beccari5, V. D’Orazi2,6, A. +Bayo5,7, K. Biazzo8, R. Claudi2, E. Covino3, G. Mantovan1,2, M. Montalto4, D. Nardiello2,9, G. Piotto1, and +E. Rigliaco2 +1 Dipartimento di Fisica e Astronomia, Universit´a degli Studi di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, +Italy +2 INAF-Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, 35122 Padova, Italy +3 INAF-Osservatorio Astronomico di Capodimonte, via Moiariello 16, 80131 Napoli, Italy +4 INAF-Osservatorio Astrofisico di Catania, via S. Sofia, 78, 95123 Catania, Italy +5 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei M¨unchen, Germany +6 Department of Physics, University of Rome Tor Vergata, via della ricerca scientifica 1, 00133, Rome, Italy +7 Instituto de F´ısica y Astronom´ıa, Facultad de Ciencias, Universidad de Valpara´ıso, Av. Gran Breta˜na 1111, Valpara´ıso, +Chile +8 INAF - Rome Astronomical Observatory, Via di Frascati, 33, I-00044, Monte Porzio Catone, Italy +9 Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France +Received +ABSTRACT +We characterize twelve young stellar objects (YSOs) located in the Lupus I region, spatially overlapping with the Upper +Centaurus Lupus (UCL) sub-stellar association. The aim of this study is to understand whether the Lupus I cloud has +more members than what has been claimed so far in the literature and gain a deeper insight into the global properties +of the region. We selected our targets using Gaia DR2 catalog, based on their consistent kinematic properties with the +Lupus I bona fide members. In our sample of twelve YSOs observed by X-Shooter, we identified ten Lupus I members. +We could not determine the membership status of two of our targets, namely Gaia DR2 6014269268967059840 and +2MASS J15361110-3444473 due to technical issues. We found out that four of our targets are accretors, among them +2MASS J15551027-3455045, with a mass of ∼0.03 M⊙, is one of the least massive accretors in the Lupus complex to +date. Several of our targets (including accretors) are formed in-situ and off-cloud with respect to the main filaments of +Lupus I, hence, our study may hint that there are diffused populations of M-dwarfs around Lupus I main filaments. In +this context, we would like to emphasize that our kinematic analysis with Gaia catalogs played a key role in identifying +the new members of the Lupus I cloud. +Key words. Accretion, Accretion Disks – Stars: activity, atmospheres, chromospheres, low-mass, pre-main sequence +1. Introduction +Observation of young stellar populations in nearby star- +forming regions and comparison of their properties with +more massive and distant ones is a key to understanding +the impact of the environment on the star formation process +and the properties of protoplanetary disks. +The Lupus dark cloud complex is one of the main low- +mass star-forming regions (SFRs) within 200 pc of the Sun. +It consists of a loosely connected group of dark clouds +and low-mass pre-main sequence (PMS) stars. The complex +hosts four active SFRs plus five other looser dark clouds +with signs of moderate star-formation activity (Comer´on +2008). Infrared (IR) and optical surveys (Evans et al. 2009; +Rygl et al. 2012) have shown that objects in all evolution- +ary phases, from embedded Class I objects to evolved Class +III stars, are found majorly concentrated in the Lupus I, II +and III clouds with Lupus III being the richest in YSOs. +⋆ Based on observations collected at the European Southern +Observatory at Paranal, under program 105.20P9.001 +Different distances to the Lupus stellar sub-groups have +been claimed in the past from Hipparcos parallaxes and +extinction star counts (Comer´on 2008), but recent investi- +gations based on Gaia DR2 showed that the vast majority +of YSOs in all Lupus clouds are at a distance of ∼160 pc +(see the Appendix in Alcal´a et al. 2019). Out of the three +main clouds, Lupus III has been recognized as the most +massive and active star-forming region in Lupus by far, +with a great number of young low-mass and very-low mass +stars (Comer´on 2008), while Lupus I, II and IV represent +regions of low star-formation activity, with Lupus V and +VI lacking star-formation (Spezzi et al. 2011; Manara et al. +2018). +In this paper we investigate the Lupus I cloud. This +cloud has less than thirty bona fide members, which from +now on we refer to as Lupus I core members. The main +motivation for selecting this cloud over the others with a +low star-forming activity was the recent discovery of the +star GQ Lup C (Alcal´a et al. 2020; Lazzoni et al. 2020), +which is located on the main filament. +1 +arXiv:2301.04463v1 [astro-ph.SR] 11 Jan 2023 + +Majidi et al.: New members of the Lupus I cloud +This target was specifically selected by our team for +discovering possible wide companions to SPHERE-GTO +targets on Gaia DR2 with a high specific interest in the +presence of planets, brown dwarfs, or spatially resolved cir- +cumstellar disks (Alcal´a et al. 2020; Majidi et al. 2020). GQ +Lup C was proved to be a strong accretor that surprisingly +had escaped detection in previous IR and Hα surveys, sug- +gesting the possibility that many YSOs in the region are +yet to be discovered. This discovery hence motivated us to +conduct a more extended search in Gaia DR2 to select new +YSO candidates in the same region. In this work, we present +the spectroscopic characterization of 12 YSOs in the Lupus +I cloud. +The outline of this paper is as follows: in Sect. 2, we +discuss the target selection criteria, as well as compiling +a complete list of the bona fide Lupus I members, in ad- +dition to the observation and data reduction methods; in +Sect. 3, we discuss the data analysis methods employed for +analyzing the X-Shooter spectra, the membership criteria, +and accreting objects; in Sect. 4, we discuss the results of +our analysis; in Sect. 5, we introduce additional qualities of +our targets in Lupus I, present their spectral energy distri- +butions (SEDs), and evaluate them as potential wide com- +panion candidates; and eventually, Sect. 6 will present our +conclusions. +2. Target selection, observations, and data +reduction +2.1. Target selection +The Gaia astrometric catalog (Gaia Collaboration 2018) +has been recently used to efficiently identify young clus- +ters and associations within 1.5 kpc from the Sun (see +Prisinzano et al. 2022, and references therein). We selected +our sample of YSO candidates based on a statistical anal- +ysis using the Gaia DR2 catalog detailed in the following. +As a first step, we identified the genuine population (core +members) of Lupus I. These core members were gathered +from the catalogs existing in the literature (Hughes et al. +1994; Mer´ın et al. 2008; Mortier et al. 2011; Galli et al. +2013; Alcal´a et al. 2014; Frasca et al. 2017; Benedettini et +al. 2018; Dzib et al. 2018; Comer´on et al. 2013; Galli et al. +2020), and are listed in Table 1. We calculated the member- +ship probability of these targets to Upper Centaurus Lupus +(UCL) with BANYAN Σ (Gagn´e et al. 2018) which are also +quoted in Table 1. It should be noted that the catalog does +not evaluate the Lupus membership. +We then extracted the kinematic properties (i.e., par- +allaxes, ϖ, and proper motions µα∗ and µδ) of these core +members from Gaia DR2, and constrained a range over +these parameters (see Appendix B of Alcal´a et al. 2020). +Using this constrained range, we searched for the objects +with similar kinematic properties to Lupus I core members +in Gaia DR2 in a radius of 3 degrees from the center of +the Lupus I cloud. At this stage, we found 247 objects. We +placed these objects on a color-magnitude diagram (CMD) +with Main Sequence (MS) stars (Pecaut & Mamajek 2013) +and we removed those that were close to the limiting magni- +tude of Gaia (with photometric errors preventing a reliable +classification according to their position on CMD) and we +ended up with 186 targets. For generating this CMD, we +used G magnitudes and Bp − Rp colors. This sample was +then restricted to objects with a parallax within 5.5 to 7.5 +mas (140-170 pc), within the < ϖ > ±4·σϖ parallax range +of Lupus I core members, but we kept both sources lying +close and far from the main filaments of the Lupus I to +be inclusive both with the kinematic properties and spatial +location of the selected targets. We also excluded those ob- +jects which were too faint for X-Shooter to observe (J > 15 +mag) or older than typical YSOs in Lupus I (inconsistent +with the Lupus I core members on our generated CMD). +Taking into account all these constraints, we identi- +fied 43 candidates as potential members of Lupus I. As +shown in the CMD in Fig. 1, all of our eventual candidates +lie above the MS stars identified by Pecaut & Mamajek +(2013) and possess magnitudes and colors very similar to +those of Lupus I members. Among these 43 objects, there +are targets that i) have never been recognized as poten- +tial members of Lupus I (17 objects), ii) were introduced +as candidate members of Lupus I according to their con- +sistent kinematic and/or photometric properties, but need +spectroscopic confirmation (23 objects), iii) were known as +members of Lupus I, but were poorly characterized in the +literature, and, were never observed with X-Shooter (3 ob- +jects). We chose to include all these categories of objects +to be followed up by X-Shooter, and the main reason for +keeping the third category was that with X-Shooter spec- +troscopy we can determine their radial velocity (RV) and +projected radial velocity (v sin i), or further explore their +chromospheric and accretion properties in a more detailed +fashion than previously done. +Targets in this category are Sz 70 (Hughes et al. 1994), +2MASS J15383733-3422022 (Comer´on et al. 2013), and +2MASS J15464664-3210006 (Eisner et al. 2007). Among the +eight objects selected in Lupus I in the unbiased photomet- +ric survey by Comer´on et al. (2013, see their Table 2), only +three were selected by our criteria and are those for which +these authors provide stellar parameters, qualifying them +as genuine YSOs. The other five were suspected to be fore- +ground objects. Indeed, we confirmed that the astrometric +parameters of the latter are out of range of our selection +criteria. +As a final step, we cross-matched our full sample of +43 objects with the OmegaCAM Hα survey in Lupus (see +Beccari et al. 2018, for details of this survey), with only 4 +being recognized as Hα emitters. This confirms that many +potential YSOs may have escaped detection in Hα imag- +ing surveys and motivated us to spectroscopically charac- +terize our full sample, giving a high priority to the four +OmegaCAM Hα emitters as potentially strong accretors. +2.2. Observations +The observations were done with the X-Shooter spectro- +graph (Vernet et al. 2011) at the VLT, within a filler pro- +gram, and terminated at the end of the observing period, +when only ∼28% of the proposed sample was observed. +Hence, of the 43 proposed targets, only 12 were eventu- +ally observed which are fully characterized in this paper, +and are listed in Table 2. The list of the targets that were +not observed is reported in Appendix A. These 12 targets +were selected by ESO staff from the list of our proposed +43 targets, and include all of the Hα emitters. Although +the observed sample is small, all the 12 observed targets +were confirmed to be YSOs whose physical and chromo- +spheric/accretion properties are worth to be investigated. +For two stars the OBs were not validated by ESO observing +2 + +Majidi et al.: New members of the Lupus I cloud +Table 1: Lupus I core members known from the literature (measurement errors are displayed in parenthesis). The column +under Prob stands for the UCL membership probability percentage of the targets calculated by BANYAN Σ (Gagn´e et +al. 2018). +Name +α (J2000) +δ (J2000) +ϖ +µα∗ +µδ +RV +Prob +age +(h:m:s) +(d:m:s) +(mas) +(mas/yr) +(mas/yr) +(km/s) +% +Myr +RX J1529.7-3628 +15 29 47.26 +–36 28 37.41 +6.04(0.09) +–14.69(0.10) +–19.66(0.08) +0.90(0.27)a +98.6 +- +IRAS 15334-3411 +15 36 39.92 +–34 21 42.17 +6.89(0.13) +–11.80(0.19) +–19.84(0.12) +- +91.6 +- +Sz 65/V∗ IK Lup +15 39 27.77 +–34 46 17.21 +6.44(0.05) +–13.27(0.12) +–22.24(0.07) +–2.70(2.00) +98.6 +1.9b +Sz 66 +15 39 28.28 +–34 46 18.09 +6.36(0.09) +–13.60(0.19) +–21.56(0.12) +2.40(1.80) +99.5 +3.9b +RX J1539.7-3450A +15 39 46.38 +–34 51 02.54 +6.40(0.04) +–15.25(0.09) +–22.33(0.05) +7.17(1.28)a +99.6 +- +UCAC4 274-081081 +15 48 06.26 +–35 15 48.13 +6.61(0.09) +–12.12(0.19) +–22.33(0.13) +- +97.4 +- +RX J1539.7-3450B +15 39 46.37 +–34 51 03.66 +6.40(0.13) +–13.52(0.26) +–20.85(0.13) +- +98.2 +- +2MASS J15440096-3531056 +15 44 00.96 +–35 31 05.72 +6.45(0.14) +–11.49(0.26) +–24.07(0.19) +- +89.3 +- +AKC2006 18 +15 41 40.81 +–33 45 18.86 +6.69(0.35) +–18.84(0.33) +–22.06(0.27) +9.10(2.30) +95.3 +8.3 +AKC2006 19 +15 44 57.89 +–34 23 39.36 +6.54(0.14) +–18.94(0.089) +–22.75(0.06) +9.60(2.10) +97.0 +8.0 +Sz 68/HT LUP A-B +15 45 12.87 +–34 17 30.65 +6.49(0.06) +–13.63(0.13) +–21.60(0.08) +–4.3(1.8) +99.1 +0.5b +HT Lup C +15 45 12.67 +–34 17 29.37 +6.55(0.19) +–15.43(0.22) +–20.27(0.15) +1.2(3.9)d +97.8 +- +Sz 69 +15 45 17.41 +–34 18 28.29 +6.47(0.08) +–15.05(0.15) +–22.15(0.11) +5.40(2.90) +99.6 +2.6b +2MASS J15451851-3421246 +15 45 18.52 +–34 21 24.56 +6.59(0.18) +–15.14(0.34) +–21.77(0.22) +4.40(2.90) +99.7 +0.5b +IRAS 15422-3414 +15 45 29.78 +–34 23 38.81 +6.46(0.17) +–15.25(0.31) +–22.52(0.24) +- +99.1 +- +RX J1546.6-3618 +15 46 41.20 +–36 18 47.44 +6.69(0.07) +–17.38(0.12) +–24.29(0.08) +7.20(0.10)c +99.8 +- +Sz 71/GW LUP +15 46 44.73 +–34 30 35.68 +6.41(0.06) +–14.03(0.10) +–23.36(0.07) +–3.30(1.90) +99.0 +2.0b +Sz 72/HM LUP +15 47 50.63 +–35 28 35.40 +6.41(0.05) +–14.26(0.09) +–23.16(0.06) +6.90(2.40) +99.6 +2.9b +Sz 73/THA 15-5 +15 47 56.94 +–35 14 34.79 +6.38(0.06) +–14.20(0.11) +–22.26(0.07) +5.00(2.20) +99.7 +3.7b +GQ LUP/CD-3510525 +15 49 12.11 +–35 39 05.05 +6.59(0.05) +–14.26(0.09) +–23.59(0.07) +–3.60(1.30) +99.4 +0.9b +Sz 76 +15 49 30.74 +–35 49 51.42 +6.27(0.05) +–12.77(0.11) +–23.37(0.08) +1.40(1.00) +99.4 +2.3b +Sz 77 +15 51 46.96 +–35 56 44.11 +6.46(0.05) +–12.42(0.09) +–24.16(0.06) +2.40(1.50) +99.3 +3.0b +RX J1556.0-3655 +15 56 02.09 +–36 55 28.27 +6.33(0.04) +–11.66(0.07) +–22.50(0.05) +2.60(1.20) +99.3 +7.8b +2MASS J15443392-3352540d +15 44 33.92 +–33 52 54.11 +7.48(0.24) +–22.03(0.27) +–24.92(0.16) +0.9(3.8) +96.3 +4.5e +2MASS J15392180-3400195d +15 39 21.81 +–34 00 19.56 +6.39(0.19) +–17.23(0.2) +–20.18(0.15) +1.1(3.8) +97.8 +7.1e +a Gaia Collaboration (2018) +b Both RV and age are obtained by Frasca et al. (2017) +c Torres et al. (2006) +d RV for this YSO candidate is the optimal RV determined by BANYAN Σ as a member of UCL. +e Age obtained by Comer´on et al. (2013). +Fig. 1: CMD of all the potential members of Lupus I in our +original sample of 43 objects (blue dots), with the MS stars +(Pecaut & Mamajek 2013) (orange dots) and the Lupus I +core members (red triangles) included in Table 1. +staff (due to not fulfilling some of our requirements). But +the spectra are nevertheless useful for classification pur- +poses and are used in this work. +X-Shooter spectra are divided into three arms (Vernet +et al. 2011), the UVB (λ ∼ 300–500 nm), VIS (λ ∼ 500- +1050 nm), and NIR (λ ∼ 1000–2500 nm). We decided to +observe all our targets with 1.′′0, 0.′′9, and 0.′′9 slit widths +(for UVB, VIS, and NIR arms respectively) for one or two +cycles based on their J band magnitudes. For our faintest +objects with J > 14 mag, we considered two cycles of ABBA +nodding mode. Among our observed targets, only 2MASS +J15551027-3455045 belongs to this category, and due to its +faintness, the final signal-to-noise ratio (SNR) of its spec- +tra was lower than expected. The exposure time for each +arm and the total execution time taking into account the +overheads are reported for each target in Table 3. For our +brightest target, TYC7335-550-1 with J = 9.65 mag, we +decided that only one cycle of ABBA nodding would be +sufficient for our scientific aims. +For some targets with a higher scientific significance to +our program or because of their faintness, we decided to also +observe telluric standard stars. Only a few of our targets +(analyzed in this work) did not have a telluric star observa- +tion included in their observation block (OB) and these are +UCAC4 273-083363, 2MASS J15414827-3501458 (with J = +11.55 mag and 11.05 mag respectively), UCAC4 269-083981 +(J = 10.72 mag), and Gaia DR2 6014269268967059840 (J += 13.64 mag) which had a lower scientific priority for our +program – either were not lying on the main filament, were +not strong candidates for membership in Lupus I, were not +3 + +OurLupusICandidates +Pecaut and Mamajek Objects +Lupus ICore Members +G +10 +15 +20 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +Bp-RpMajidi et al.: New members of the Lupus I cloud +Table 2: Objects observed with X-Shooter (measurement errors are displayed in parenthesis). The column under Prob +stands for the UCL membership probability percentage of the targets calculated by BANYAN Σ (Gagn´e et al. 2018). +The four candidates detected in the OmegaCAM Hα imaging survey are flagged with ( Hα) right to their names (See +Sect. 2.1). +Name +α (J2000) +δ (J2000) +ϖ +µα∗ +µδ +Prob +G +(h:m:s) +(d:m:s) +(mas) +(mas/yr) +(mas/yr) +% +(mag) +Partially known targets: +2MASS J15383733-3422022 +15 38 37.34 +–34 22 02.26 +6.79(0.15) +–18.25(0.26) +–24.15(0.19) +99.4 +16.78 +Sz 70 +15 46 42.99 +–34 30 11.55 +6.09(0.21) +–12.58(0.39) +–22.16(0.25) +95.7 +14.50 +Candidates: +TYC 7335-550-1a +15 36 11.55 +–34 45 20.54 +6.26(0.07) +–13.93(2.43) +–19.51(1.01) +99.2 +11.31 +2MASS J15361110-3444473b ( Hα) +15 36 11.09 +–34 44 47.82 +5.83(0.29) +–13.56(0.29) +–20.21(0.23) +94.8 +18.92 +2MASS J15523574-3344288c ( Hα) +15 52 35.74 +–33 44 28.87 +5.98(0.17) +–20.06(0.37) +–22.17(0.23) +50.2 +17.06 +2MASS J15551027-3455045d ( Hα) +15 55 10.28 +–34 55 04.67 +6.78(0.26) +–11.09(0.54) +–23.94(0.31) +93.8 +18.23 +2MASS J16011870-3437332e ( Hα) +16 01 18.70 +–34 37 33.20 +7.35(0.07) +–16.59(0.07) +–24.97(0.05) +98.5 +16.46 +UCAC4 269-083981f +15 56 19.06 +–36 13 25.15 +6.095(0.04) +–13.77(0.09) +–22.29(0.06) +98.7 +13.02 +Gaia DR2 6010590577947703936 +15 56 55.36 +–36 11 10.73 +6.83(0.11) +–15.64(0.24) +–25.82(0.15) +98.7 +16.37 +2MASS J15414827-3501458g +15 41 48.28 +–35 01 45.84 +6.74(0.13) +–17.99(0.25) +–25.39(0.18) +99.5 +13.98 +UCAC4 273-083363 +15 46 46.15 +–35 24 11.40 +6.99(0.06) +–18.14(0.11) +–25.04(0.08) +99.6 +14.46 +Gaia DR2 6014269268967059840 +15 36 55.30 +–33 45 22.19 +6.68(0.24) +–16.23(0.37) +–22.29(0.27) +95.3 +17.39 +a Proposed candidate member of Lupus I by Zari et al. (2018). +b aka Gaia DR1 6014141205925321984. +c aka Gaia DR2 6012155767105823616. +d aka Gaia DR2 6011827867821601792, candidate Lupus I member also proposed by Galli et al. (2020). +e Gaia DR3 6011165313293141760. +f Dipper, candidate member of Lupus I also proposed by Nardiello et al. (2020). +g aka SSTc2dJ154148.3-350145, a candidate Lupus I member previously proposed by Comer´on et al. (2009). +Table 3: Observing log of the new candidate members of Lupus I. +Name +Date +Exposure time +Seeing +Ttot +airmass +SNR +J +Grade +(yyyy-mm-dd) +(sec) +(′′) +(hour) +(mag) +2MASS J15383733-3422022 +2021-08-03 +1920/1800/1920 1.72/1.72/1.72 +0.67 +1.04 +5.4/47.1/68.6 +13.39 +A +Sz 70 +2021-07-06 +600/500/600 +0.55/0.52/0.52 +0.33 +1.03 +6.9/67.8/132.4 +10.85 +A +TYC7335-550-1 +2021-06-27 +300/200/300 +0.72/0.77/0.77 +0.33 +1.36 +71.1/117.0/245.6 +9.65 +A +2MASS J15361110-3444473 +2021-06-27 +3600/3400/3840 0.73/0.69/0.70 +1.25 +1.15 +0.1/4.9/21.3 +14.91 +A +2MASS J15523574-3344288 +2021-06-27 +1800/1700/1920 0.72/0.72/0.69 +0.7 +1.43 +0.4/12.2/33.3 +13.49 +A +2MASS J15551027-3455045 +2021-08-01 +1800/1700/1920 1.73/1.79/1.79 +0.62 +1.11 +0.7/15.0/41.2 +13.76 +A +2MASS J16011870-3437332 +2021-08-08 +1800/1700/1920 1.49/1.49/1.49 +0.72 +1.35 +5.6/48.9/76.8 +13.07 +A +UCAC4 269-083981 +2021-08-01 +600/500/600 +2.27/2.27/2.27 +0.33 +1.19 +39.5/108.4/123.2 10.72 +Ca +Gaia DR2 6010590577947703936 +2021-08-06 +1920/1820/1920 2.04/1.92/1.92 +0.67 +1.14 +5.9/51.0/78.9 +13.08 +A +2MASS J15414827-3501458 +2021-07-14 +600/500/600 +1.13/1.13/1.13 +0.33 +1.12 +25.4/100.2/232.3 11.05 +A +UCAC4 273-083363 +2021-07-14 +600/500/600 +1.33/1.29/1.33 +0.33 +1.08 +18.3/73.6/171.0 +11.55 +A +Gaia DR2 6014269268967059840 +2021-08-04 +1800/1700/1800 2.49/2.49/2.49 +0.65 +1.13 +1.5/26.1/50.5 +13.64 +Cb +Notes. Date of observation, exposure time allocated to each arm, mean seeing, and SNR (in order for UVB, VIS, and NIR +wavelengths) as well as the total execution time, mean airmass, and the observation grades (as provided by the ESO observing +staff) are reported. +a UCAC4 269-083981 had an out of constraint seeing (2.′′0 which was exceeded). +b Gaia DR2 6014269268967059840 was reported to have an out of constraint seeing. +Hα emitters, or were not faint for X-shooter to necessitate +the observation of a telluric template. As we will detail +later, we will also adopt a different approach to remove +telluric lines for these objects. For the targets containing +telluric observation in their OBs, the same nodding strat- +egy as those of the targets was employed to minimize noise +4 + +Majidi et al.: New members of the Lupus I cloud +and cosmetics, with an airmass as close as possible to the +targets. The airmass and seeing reported in Table 3 are +averaged over the exposure times for each arm. +2.3. Data reduction +The data used in this work have been reduced with the X- +Shooter pipeline xshoo of version 2.3.12 and higher1, and +hence they have been de-biased, flat-fielded, wavelength- +calibrated, order-merged, extracted, sky-subtracted and +eventually flux-calibrated. The result of this pipeline output +is an ESO one-dimensional standard binary table and the +two-dimensional ancillary files ready for scientific analysis. +Flux calibration based on the photometric data available +in the literature was done later directly on the available +spectra, along with the telluric removal process which is +not done for the distributed spectra reduced by the xshoo +pipeline. +We used the Image Reduction and Analysis Facility +(IRAF, Tody 1986, 1993) to remove the telluric lines from +the target spectra and to flux calibrate them, as well as +to derive the stellar parameters from the spectra, which +we shall discuss in detail in the upcoming sections. Since +the strategy for arranging our observation blocks did not +include wide slit observations, the flux calibration of our +targets totally relies on the photometric data available in +the literature, which have been collected in various surveys +(with the corresponding flux errors of e-16 W.m−2 for the +UVB arm, e-16 W.m−2 for the VIS arm, and 2.5e-15 W.m−2 +for the NIR arm). For some of our faint objects, we only +had access to very limited photometric data and had to cal- +ibrate the UVB portion of the spectra in accordance with +the available photometric data in the VIS range. +For the objects with observations of telluric standard +stars, we removed the telluric lines and molecular bands +using the IRAF task Telluric. For the three targets with- +out telluric star observations in our sample, which namely +are 2MASS J15414827-3501458, UCAC4 273-083363, and +Gaia DR2 6014269268967059840, we used the TelFit +Python code. This code fits the telluric absorption spec- +trum in the observed spectra (Gullikson et al. 2014) using +the LBLRTM code which models the line-by-line radiative +transfer (Clough et al. 2005). Applying TelFit, we cor- +rected the spectra for oxygen and water molecular bands +in the visible range (∼550-1000 nm), as well as for water, +oxygen, and CO2 molecular bands in the NIR (∼1000-2500 +nm) (for the details on the wavelength ranges where these +molecular bands dominate the spectrum the reader is re- +ferred to Smette et al. 2015). +3. Data Analysis +There are several immediate aims that we planned to fulfill +through our program. With the X-Shooter spectra, we can +confirm the youth of the selected candidates through the +presence of the Li i (6708 ˚A) absorption line, in addition to +Hα emission, and other lines of the Balmer series as further +hints. We also determine the spectral type (SpT) classifi- +cation and the determination of stellar physical parameters +such as effective temperature (Teff), luminosity (L), mass +(M) and age. It is also possible that some of our candidates +1 https://www.eso.org/sci/software/pipelines/ +xshooter/ +may belong to Scorpius-Centaurus Association (with an age +10-18 Myr, UCL sub-association) rather than Lupus (1-2 +Myr). We can single out these objects once we have fully +characterized them. The disentanglement between the two +associations would be useful for clarifying their relation- +ship. Using spectral lines of the Balmer series, we will also +measure the accretion luminosity (Lacc) and mass accretion +rate ( ˙Macc) of those objects that we qualify as accretors. In +the following, we describe the methods used for achieving +our immediate goals. +3.1. Spectroscopic analysis methods +3.1.1. Spectral typing and line equivalent widths +To obtain the SpTs of our objects, we first compared the +spectrum obtained with X-Shooter’s VIS arm with a li- +brary of visible spectra of already characterized stars and +brown dwarfs formerly observed by X-Shooter (Manara et +al. 2013). For the quantitative spectral typing of the stars, +we then calculated the spectral indices described in Riddick +et al. (2007) based on the ratios of the average flux of +molecular absorption bands within narrow wavelength re- +gions, yielding in all cases an uncertainty of 0.5 subclasses. +For TYC 7335-550-1 and UCAC4 269-083981, which are +brighter than the rest of the targets and do not show clear +molecular bands in their spectra suitable for measuring the +Riddick’s indices, the SpT is instead estimated through the +Teff obtained by the ROTFIT code (see Sect. 3.1.2). The +results can be found in Table 7. +The EW of the atomic lines reported in Table 5 is mea- +sured by taking an average over i) the direct integration of +the line profiles between two marked pixels and ii) fitting +a Gaussian. The errors associated with these values thus +report the difference between the measurements made with +these methods. There are cases for which we could not de- +tect the Li i line at 6708 ˚A. Hence, for these objects we +only report an upper limit on the measurement of EWLi i. +As suggested by Cayrel (1988), a three-sigma upper limit +on the flux of the lithium line can be calculated as: +dEW = 3 × 1.06 +� +(FWHM)dx/(S/N), +(1) +in which FWHM is the full width at half maximum, S/N is +the signal-to-noise ratio, and the bin size (dx) can be fixed +to 0.2 ˚A for the VIS arm. The values of these measurements +are reported in Table 5 and Table 6 for TYC7335-550-1. +3.1.2. ROTFIT +We used ROTFIT as the basis of our analysis for assessing +the stellar parameters of our targets. Using ROTFIT, we +evaluated their RV, v sin i, and surface gravity (log g). The +version of ROTFIT used for this purpose is the one designed +for the optimal usage of the X-Shooter spectra (Frasca et al. +2017). The stellar parameters obtained with ROTFIT can +be found in Table 4. The fitting process with ROTFIT code +was carried out within a veiling (the UV excess continuum +that influences the entire photosphere of the star from UVB +to NIR) range from 0 to 1. None of our objects showed +significant veiling, hence the veiling parameter for all our +studied targets in this paper is equal to zero. +5 + +Majidi et al.: New members of the Lupus I cloud +Table 4: Physical stellar parameters of the targets obtained with the ROTFIT code. +Name +Teff +log g +vsini +RV +Prob +(K) +(km/s) +(km/s) +% +2MASS J15383733-3422022 +3111±70 +4.75±0.13 +<8 +4.1±2.7 +99.8 +Sz 70 +3038±76 +4.02±0.11 +14.0±14.0 +1.1±2.6 +84.6 +TYC 7335-550-1 +4488±140 +4.06±0.22 +<8 +2.6±2.0 +99.2 +2MASS J15361110-3444473 +2883±104 +4.41±0.12 +13.0±10.0 +6.9±2.6 +97.9 +2MASS J15523574-3344288 +2981±44 +4.54±0.10 +<8 +2.6±2.7 +75.3 +2MASS J15551027-3455045 +2700±103 +3.60±0.11 +19.0±8.0 +0.1±2.9 +97.9 +2MASS J16011870-3437332 +3121±90 +4.73±0.14 +12.0±8.0 +–0.5±2.3 +98.7 +UCAC4 269-083981 +3846±47 +4.53±0.11 +<8 +0.6±2.7 +99.6 +Gaia DR2 6010590577947703936 +3154±72 +4.77±0.13 +40.8±3.6 +0.5±4.7 +99.2 +2MASS J15414827-3501458 +3213±94 +4.52±0.23 +53.3±5.7 +3.4±4.3 +99.8 +UCAC4 273-083363 +3211±56 +4.51±0.15 +<8 +1.3±2.3 +99.8 +Gaia DR2 6014269268967059840 +3019±108 +4.75±0.14 +44.0±12.0 +1.7±4.6 +98.3 +Notes. The column Prob represents the probability of the target to be member of Lupus I according to BANYAN Σ, which is +based on the RVs measured with ROTFIT and the kinematic properties reported by Gaia DR2. +Table 5: EWs of the relevant lines indicating the chromospheric and accretion tracers for our targets. Negative values +indicate the lines that are in emission. +Name +EWLi i +EWHα +EWHβ +EWHγ +EWHδ +WHα(10%) +(˚A) +(˚A) +(˚A) +(˚A) +(˚A) +(km/s) +2MASS J15383733-3422022 +0.74±0.04 +–8.77±0.92 +–7.71±0.04 +–7.99±0.21 +–7.20±0.52 +128±18 +Sz 70 +0.55±0.05 +–43.37±3.97 +–9.97±1.07 +–10.28±1.04 +–11.14±1.51 +366±14 +2MASS J15361110-3444473 +< 0.25a +–71.4±8.77 +. . . +. . . +. . . +292±14 +2MASS J15523574-3344288 +0.81±0.09 +–13.52±0.76 +–10.9±0.88 +–3.9±1.1 +–2.84±0.49 +146±9 +2MASS J15551027-3455045 +-b +–88.9±1.17 +–29.7±0.85 +–6.68±0.24 +–5.09±0.49 +229±14 +2MASS J16011870-3437332 +0.67±0.03 +–21.47±1.59 +–21.61±1.28 +–19.41±0.75 +–13.34±2.18 +274±14 +UCAC4 269-083981 +0.56±0.01 +–1.69±0.07 +–1.63±0.08 +–1.56±0.24 +–1.44±0.21 +174±5 +Gaia DR2 6010590577947703936 +0.68±0.06 +–6.53±0.38 +–6.75±0.25 +–6.97±0.09 +–6.69±0.22 +183±5 +2MASS J15414827-3501458 +< 0.012a +–10.04±0.53 +–9.55±0.61 +–10.64±0.29 +–10.21±0.7 +210±18 +UCAC4 273-083363 +< 0.017a +–11.4±0.94 +–11.12±0.45 +–11.15±1.35 +–8.59±0.67 +155±9 +Gaia DR2 6014269268967059840 +< 0.047a +–17.53±2.20 +. . . +. . . +. . . +219±14 +a Three-sigma upper limits on the measurement (read Subsection for further explanation). +b Li I line was affected by a cosmic ray hit and could not be measured. +Table 6: EWs of the relevant lines indicating the chromospheric and accretion tracers for TYC 7335-550-1. +Name +EWLi i +EWHα +EWHϵ +EW H +Ca ii +EW K +Ca ii +EW 8498 +Ca ii +EW 8542 +Ca ii +EW 8662 +Ca ii +(˚A) +(˚A) +(˚A) +(˚A) +(˚A) +(˚A) +(˚A) +(˚A) +TYC 7335-550-1 +0.39±0.02 +–0.45±0.06 +–0.32±0.16 +–1.07±0.14 +–1.41±0.19 +–0.47±0.03 +–0.78±0.06 +–0.68±0.06 +Notes. The EW of Hα, Hϵ, and Ca ii lines relate to the emission in the cores of these lines obtained by the subtraction of the photospheric +template. +3.1.3. Physical parameters +We used the bolometric correction (BC) relation proposed +by Pecaut & Mamajek (2013, 2016) for evaluating the lu- +minosity in both V and J bands and the radius of can- +didates according to their observed parallaxes and magni- +tudes. This is possible because none of our targets show +significant near-IR excess (Fig. 2) nor strong veiling (Sect. +3.1.2). +For the objects only resolved in Gaia DR2 catalog, the +BC relationship introduced by the Gaia DR2 science team2 +is used. In order to have a correct estimation of the lu- +minosity, we have also taken into account the extinction +2 https://gea.esac.esa.int/archive/documentation/ +GDR2/Data_analysis/chap_cu8par/sec_cu8par_process/ +ssec_cu8par_process_flame.html +of the objects which was determined using the grid of X- +Shooter spectra of zero-extinction non-accreting T Tauri +stars (Manara et al. 2013), as explained in Sect. 3.2 of +Alcal´a et al. (2014). It is evident from Fig. 2 that the targets +have low extinction and little or no NIR excess, probably +except for the rightmost point in the diagram, which corre- +sponds to 2MASS J15361110-3444473. The relatively red- +der H −Ks color of this object in comparison with the oth- +ers, may be due to the presence of an unresolved very late- +type companion. This will be further discussed in Appendix +C. +Once the Teff (from ROTFIT), luminosity, and ra- +dius of the targets are derived, their mass, age, and log g +can be evaluated through various evolutionary tracks and +isochrones available in the literature. The corresponding +values of these parameters, which are reported in Table +6 + +Majidi et al.: New members of the Lupus I cloud +Table 7: Physical stellar parameters of the targets. +Name +SpT +AV +L⋆ +R⋆ +M⋆ +Age +log g +(mag) +(L⊙) +(R⊙) +(M⊙) +(Myr) +2MASS J15383733-3422022 +M5 +0 +0.012±0.006 +0.39±0.01 +0.09±0.05 +10.7±5 +4.20±0.5 +Sz 70 +M5 +0.5 +0.25±0.11 +1.87±0.05 +0.17±0.05 +0.5±0.3 +3.28±0.2 +TYC 7335-550-1 +K4.5 +0.7 +0.94±0.56 +1.60±0.05 +1.1±0.1 +3.50±1 +4.04±0.2 +2MASS J15361110-3444473 +M5.5 +1.75 +0.006±0.003 +0.32±0.01 +0.05±0.05 +9.77±5 +4.13±0.3 +2MASS J15523574-3344288 +M5.5 +0.5 +0.02±0.01 +0.55±0.01 +0.11±0.03 +6.3±3 +4.04±0.4 +2MASS J15551027-3455045 +M7.5 +0.75 +0.0072±0.0034 +0.39±0.02 +0.03±0.02 +1.7±1.5 +3.71±0.3 +2MASS J16011870-3437332 +M5 +0 +0.013±0.006 +0.41±0.01 +0.09±0.04 +9.55±5 +4.16±0.5 +UCAC4 269-083981 +M0 +0.5 +0.30±0.14 +1.23±0.02 +0.6±0.3 +4.2±1 +4.03±0.5 +Gaia DR2 6010590577947703936 +M4.5 +0 +0.017±0.007 +0.45±0.01 +0.11±0.05 +8.8±4 +4.16±0.3 +2MASS J15414827-3501458 +M4 +0 +0.12±0.06 +1.13±0.03 +0.2±0.08 +1.82±1 +3.64±0.4 +UCAC4 273-083363 +M3.5 +0 +0.069±0.032 +0.83±0.01 +0.2±0.04 +3.63±1.5 +3.88±0.3 +Gaia DR2 6014269268967059840 +M6 +0 +0.01±0.005 +0.41±0.02 +0.05±0.03 +6.46±2 +3.93±0.5 +Notes. The methods used for calculating SpT, AV , L⋆, and R⋆ are described in the text. M⋆, log g, and age of the stars are +evaluated according to Baraffe et al. (2015) isochrones, except for TYC 7335-550-1, for which we have used the MIST isochrones. +The SpT for TYC 7335-550-1 and UCAC4 269-083981 (in italic) are obtained using the temperatures derived by the ROTFIT code +(Table 4) and the SpT–Teff calibration of Pecaut & Mamajek (2013). The errors associated with SpT and AV are 0.5 subclasses +and 0.4 mag respectively. The errors associated with mass and age are internal to the tracks and isochrones. +Fig. 2: J − H (mag) vs. H − Ks (mag) diagram of all our +targets. The red dots show the chromospherically-dominant +targets, the cyan dots are the accretors, and the blue line +represents the colors of MS objects, down to spectral type +M9.5. The normal reddening vector, shown with the black +arrow, corresponds to AV = 2 mag. The rightmost target is +2MASS J15361110-3444473 which is suspected to be a bi- +nary, hence, it might have color contribution from a second +target. +7, are derived by the evolutionary models calculated by +Baraffe et al. (2015). The Hertzsprung-Russel (HR) dia- +gram of the Lupus I targets, including the previously known +and the newly discovered members, is displayed Fig. 3. One +of our targets, namely TYC 7335-550-1, is much brighter +than the other stars investigated in the present work, and +falls outside the range covered by the Baraffe et al. (2015) +models. Therefore, to derive its stellar parameters, we used +MESA Isochrones and Stellar Tracks (MIST Paxton et al. +2015; Choi et al. 2016; Dotter 2016). For modeling pur- +poses, we assumed that all targets have solar metallicity +(Baratella et al. 2020). +Some of our objects display strong emission lines which +is a sign of noticeable chromospheric activity (see the EW of +some of the chromospheric activity indicators in Table 5) or +magnetospheric accretion from a circumstellar disk. If the +magnetic activity is relevant, the position of the star in the +HR diagram can be significantly affected by photospheric +starspots and by the changes in the internal structure in- +duced by the magnetic fields (see Gangi et al. 2022, for in- +teresting cases in the Taurus SFR). In this case, isochrones +that do not take into account these effects (such as Baraffe +et al. 2015) may lead to systematic effects in the estimate +of mass and age. In particular, they may indicate an age +half the real age of star (Asensio-Torres et al. 2019; Feiden +2016). This is crucial for our study which also aims at de- +termining the membership of the stars in Lupus I or UCL +associations. Thus, in addition to MIST and the isochrones +provided by Baraffe et al. (2015), we used other isochrones. +A set of evolutionary models that considers the mag- +netic activity of the stars is the Dartmouth magnetic +isochrones (Feiden 2016), which we also use in this work to +estimate the ages of all our targets. These isochrones were +originally developed for estimating the age of the Upper +Scorpius members (11±2 Myr), almost coeval to the UCL +(15±3 Myr), and hence are quite useful to fulfill our sci- +entific aims. In addition to Baraffe et al. (2015) and MIST +models, we used both Dartmouth std and Dartmouth mag +(Feiden 2016, and the references therein) models, as well as +PARSEC + COLIBRI S37 (Bressan et al. 2012; Pastorelli +et al. 2019, 2020). For all our targets, we obtained over- +estimated ages using PARSEC + COLIBRI S37 isochrones +totally inconsistent with the other isochrones, hence, we +do not report our results obtained with this isochrone to +avoid confusion. The results of age estimation with all the +other isochrones are included in Table B.1. For all the mod- +els, we have assumed our targets have solar metallicity. For +PARSEC models, extinction is also a free parameter that +can be fixed and was thus set to the corresponding ex- +tinction of the targets reported in Table 7. Eventually, we +would like to point out that it is not straightforward to +state which targets may have an under-estimated age, par- +ticularly in the case of objects that are as young as the +members of Lupus I and UCL considered in this work. +7 + +1.5 +1 +J-H +0.5 +0 +0 +0.5 +1 +H-KsMajidi et al.: New members of the Lupus I cloud +Fig. 3: log L⋆(L⊙) vs log Teff (K) diagram for all our tar- +gets (cyan and red dots represent accretors and non- +accretors, respectively), together with the previously char- +acterized Lupus members (black dots, Alcal´a et al. 2019, +sub-luminous objects are not plotted). Blue dashed lines +represent evolutionary tracks of Baraffe et al. (2015) for +stars with masses indicated by the number (in M⊙) next +to the top or bottom of each track. The red lines indicate +isochrones calculated with the same models at ages of 1, 3, +30 Myrs, and 10 Gyrs, from the right to the left. +3.2. Lupus I membership criteria +According to the works previously done in the Lupus com- +plex (Alcal´a et al. 2014, and the references therein), in ad- +dition to the kinematical properties expressed by the Gaia +parallax and proper motions, membership criteria in this +star-forming region are: +i) the presence of lithium in their atmospheres, which +is the main signature of youth. Despite the obviousness of +this criterion, there are previously acknowledged members +of the Lupus cloud that lack lithium. An example is rep- +resented by Sz 94 in the Lupus III cloud (Manara et al. +2013; Biazzo et al. 2017; Frasca et al. 2017); ii) an age con- +sistent with the core members of the cloud. Although the +estimated age of the Lupus complex is ∼ 1–2 Myr, there are +previously recognized members of the complex that exceed +this age range. Examples of such targets are AKC2006 18 +and AKC2006 19 in Lupus I, although their apparent old +age may be ascribed to disks seen edge-on that obscure +the central objects making them sub-luminous on the HR +diagram (see other examples in Sect. 7.4 in Alcal´a et al. +2014); iii) an RV consistent with the values of the genuine +members of the Lupus I (Frasca et al. 2017). +If an object does not match the membership criteria +defined above, there are two possibilities. Either it is older +than the UCL (age>20 Myr), and we would hence identify it +as field star; or it has a consistent age with UCL (∼15 Myr) +which would confirm its membership to this sub-cloud of +the Scorpius-Centaurus stellar association. To this aim, we +have used various isochrones to evaluate the age of our tar- +gets. +Fig. 4: |EWHα| vs SpT of our targets with the weak lined T +Tauri stars studied by Manara et al. (2013, blue dots). The +cyan dots represent accretors, and the red dots represent +chromospherically-dominant objects. The horizontal lines +in red represent the thresholds that separate non-accreting +and accreting objects considering their SpTs (White & +Basri 2003). +3.3. Accreting objects +There are several criteria for determining whether an object +is actively accreting matter. Usually, an accreting object is +characterized by strong emission lines, strong UV and NIR +continuum excess emission, or structured line profiles (e.g., +Manara et al. 2013). Here, to establish whether an object is +an accretor, we use the criterion proposed by White & Basri +(2003) which distinguishes the accreting and non-accreting +objects based on the EW of their Hα emission versus SpT. +The method used in this paper for calculating the Lacc (ac- +cretion luminosity) and +˙Macc (mass accretion rate) of our +targets involves measuring the line luminosity of the emis- +sion lines of the accreting targets and using the established +relationships between the Lline (for each emission line) with +Lacc (Alcal´a et al. 2017). We quote the eventual accretion +line luminosity that is obtained this way as log Lacc−line in +Table 8 and Table 9. +The whole procedure that we carried out for this task +can be summarized as follows: we corrected the spectra for +telluric lines and flux-calibrated them, then measured the +flux at Earth of the emission lines by integrating their pro- +file above the local continuum, corrected the flux for ex- +tinction, calculated the luminosity of each emission line by +multiplying the flux at Earth for 4πd (adopting a distance +d = 1000/ϖ pc, with ϖ in mas), and eventually took an +average over all the values of log Lacc−line. We chose Hα, +Hβ, and Hγ emission lines to measure the accretion lumi- +nosity of our targets. After deducing the log Lacc for each +target, we obtained their +˙Macc accordingly (Alcal´a et al. +2017). The results of our measurements are presented in +Table 8. +Among all our targets, only TYC 7335-550-1 does not +show Hydrogen emission lines above the continuum, and +its Hα line is instead in absorption. For this target, we +used ROTFIT to subtract the photospheric template in or- +der to measure the flux of the emission components that +fill the cores of Hydrogen and Ca ii lines. This method has +been successfully used to emphasize chromospheric emis- +sion or a moderate accretion whenever the photospheric +8 + +1.0 +0 +(o) +logL +0.5 +0.05 +0.4 +2 +0.3 +0.2 +0.02 +Y +3.8 +3.7 +3.6 +3.5 +3.4 +logTeff (K)100 +10 +IEWHαl +1 +0.1 +K3 +K4 +K5 +K6 +K7 +K8 +K9 +MO +M1M2 +M3 +M4 +M5 +M6 +M7 +M8M9M10 +SpTMajidi et al.: New members of the Lupus I cloud +flux is large and the emission is only detectable as a filling of +the line core or an emission bump within the photospheric +line wings that do not emerge above the continuum (e.g., +Frasca et al. 2015, 2017, and references therein). The spec- +tral subtraction allows us to recognize and measure the EW +of the emission that fills in the Hα line (Fig. 5). Adopting +the same method, we measured the fluxes of the H&K lines +of the Ca ii and in the cores of the three infrared lines of +the Ca ii IRT at λ =849.8, 854.2, and 866.2 nm (Fig. 6). +We were also able to separate the contribution of the Hϵ +emission from the nearby Ca ii H line. +Fig. 5: X-Shooter spectrum of TYC 7335-550-1 in the Hα +region, normalized to the local continuum (black solid line) +along with the inactive photospheric template (red dotted +line). The latter is produced by ROTFIT with the BT- +Settl synthetic spectrum at the Teff and log g of this target +that is degraded to the resolution of X-Shooter, rotationally +broadened, and wavelength shifted according to the target +RV. The difference target − template is displayed at the +bottom of the box and emphasizes the Hα emission that +fills in the line core (green hatched area), which has been +integrated to obtain the Hα line flux. +4. Results +4.1. Stellar parameters and membership +The physical stellar parameters that we obtained from +the spectral analysis and the HR diagram as described in +Sects. 3.1.1 and 3.1.3 are reported in Table 7. The stellar pa- +rameters obtained with ROTFIT are presented in Table 4, +where the membership probability was recalculated with +the BANYAN Σ using the values of RVs measured with +ROTFIT. Both Teff and log g found with ROTFIT are in +good agreement with those derived from SpT and the HR +diagram and reported in Table 7. +We note that, at the resolution of the X-Shooter VIS +spectra, the minimum value of v sin i that can be measured +is 8 km/s (see, e.g., Frasca et al. 2017) and hence this value +should be considered as an upper limit. With this knowl- +edge, we can classify targets with v sin i < 8 km/s as slow +rotators, and those with v sin i > 40 km/s as fast rotators. +Moreover, the large RV range of the bona fide members of +Lupus I (∼ –5-12 km/s, according to Table 1) denies us to +Fig. 6: a) X-Shooter UVB spectrum of TYC 7335-550-1 in +the Ca ii H&K region (black solid line) along with the in- +active photospheric template (red dotted line). b) and c) +Residual (target − template) spectrum around the Ca ii K +and Ca ii H line, respectively. The hatched green areas mark +the residual H and K emissions that have been integrated to +obtain the EWs and fluxes. The purple-filled area relates to +Hϵ. d) and e) Observed Ca ii IRT line profiles (black solid +lines) with the photospheric template overlaid with red dot- +ted lines. The residual spectra are shown at the bottom of +each panel shifted downward by 0.2 in relative flux units +for clarity. +put a strict constraint on the Lupus I membership of our +targets (Fig. 7). The RVs of the Lupus I members confirmed +in this work, however, are within a smaller range with re- +spect to the previously confirmed core members of the same +region, except for 2MASS J15361110-3444473 which may or +may not be a Lupus I member. +According +to +our +full +characterization, +besides +TYC 7335-550-1 which is a K4.5 type star, all the +others have M spectral types. Three-quarters of our +targets, have spectral types between M4 and M6, which +is in accordance with the previously identified members +of the Lupus complex (Alcal´a et al. 2014; Frasca et al. +2017; Krautter et al. 1997; Herczeg & Hillenbrand 2014; +Comer´on et al. 2013; Galli et al. 2020). The ages of these +targets cover a large range of 0.7-11 Myrs, with masses in +the range of 0.02 to 1.1 M⊙ (as also indicated in Fig. 3). +As discussed in Sect. 2.1, Sz 70 and 2MASS J15383733- +3422022 were partially known in the literature. The phys- +ical parameters that we report here for Sz 70 are in excel- +lent agreement with the results of Hughes et al. (1994). For +9 + +Tyc7335-550- +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +LAW +0.2 +6520 +6540 +6560 +6580 +6600 +x (A)Tyc7335-550-1 +2.0 +1.5 +1.0 +0.5 +3920 +3940 +3960 +3980 +(A) +6 +1.5 +1.5 +0 +Call K +Call H +1.0 +1.0 +0.5 +0.5 +He +0.0 +0.0 +3926 +3929 +39.32 +3935 +3938 +3941 +3962 +3965 +3968 +3971 +3974 +3977 +^ (A) +> (A)1.0 +0.5 +0.5 +0.0 +0'0 +8480 +8500 +B520 +8540 +8560 +8640 8650 8660 8670 8680 8690 +^ (A) +A (A)Majidi et al.: New members of the Lupus I cloud +Table 8: Accretion luminosity of the accretors derived from the line luminosities. The mass accretion rates are derived +from the average of these values (Lacc−average). +Name +log Lacc−Hα +log Lacc−Hβ +log Lacc−Hγ +log Lacc−average +log +˙Macc +(L⊙) +(L⊙) +(L⊙) +(L⊙) +(M⊙yr−1) +Accretors: +Sz 70 +–2.73 +–2.95 +–2.91 +–2.85 +–9.22 +2MASS J15361110-3444473 +–3.62 +. . . +. . . +–3.62 +–10.21 +2MASS J15551027-3455045 +–3.85 +–3.95 +–3.96 +–3.92 +–10.20 +2MASS J16011870-3437332 +–4.04 +–4.29 +–4.20 +–4.16 +–10.91 +Active stars: +2MASS J15383733-3422022 +–5.41 +–5.43 +–5.52 +–5.45 +–12.21 +2MASS J15523574-3344288 +–4.62 +–4.87 +–4.80 +–4.75 +-11.46 +UCAC4 269-083981 +–4.07 +–4.09 +–4.24 +–4.13 +-11.22 +Gaia DR2 6010590577947703936 +–5.12 +–5.09 +–5.03 +–5.08 +–11.86 +2MASS J15414827-3501458 +–3.97 +–3.93 +–4.07 +-3.99 +-10.63 +UCAC4 273-083363 +–4.01 +–4.14 +–4.19 +–4.11 +–10.89 +Gaia DR2 6014269268967059840 +–5.22 +. . . +. . . +–5.22 +–11.07 +Table 9: Accretion luminosity of TYC 7335-550-1 derived from its line luminosities. Its mass accretion rate is derived +from the average of these values (Lacc−average). +Name +log Lacc log Lacc +log Lacc +log Lacc +log Lacc +log Lacc +log Lacc +log Lacc +log +˙ +Macc +Hα +Hϵ +Ca II (H) Ca II (K) Ca II (8498.02) Ca II (8542.09) Ca II (8662.14) average +(L⊙) +(L⊙) +(L⊙) +(L⊙) +(L⊙) +(L⊙) +(L⊙) +(L⊙) +(M⊙yr−1) +TYC 7335-550-1 +–3.43 +–2.82 +–2.31 +–2.19 +–2.01 +–1.94 +–1.88 +–2.16 +–9.40 +Fig. 7: RV of our accretors (cyan dots), chromospherically- +dominant targets (red dots), and the Lupus I core members +(black dots). +2MASS J15383733-3422022, our results are again in good +agreement with those reported by Comer´on et al. (2013), +but their difference emanates from the fact that Comer´on et +al. (2013) measured AV = 1.2 mag for 2MASS J15383733- +3422022, which results in a discrepancy in luminosity, mass, +and radius. +4.2. Equivalent widths +The EWs of several lines are quoted in Table 5, and sepa- +rately for TYC 7335-550-1, in Table 6, as for this star the +flux and EW measurements were performed by subtracting +the photospheric spectrum. +We could not detect the Li i line in the spectra of some +of our targets for various reasons, which can be i) solely +due to the low SNR of their spectra; ii) based on the simu- +lations conducted by Constantino et al. (2021), for initially +lithium-rich stars we know that slow rotators could deplete +their lithium (also considering their SpT) at early ages (< +10 Myr), while fast rotators tend to retain their lithium; iii) +a combination of the low SNR and fast rotation (which may +be especially true for Gaia DR2 6014269268967059840), +which would further complicate the issues associated with +Li i detection; iv) a complex relationship between the ac- +cretion processes, early angular momentum evolution, and +possibly planet formation for young stars (∼ 5 Myr) that +yet needs to be fully explored (Bouvier et al. 2016); v) no +obvious relationship between the rotation of YSOs and the +lithium depletion process (Binks et al. 2022). +The non-detection of Li i in the spectra of some objects +has been reported as a three-sigma upper limit on the flux +of the lithium line which is a sensitive enough threshold for +separating them from objects containing lithium. +4.3. Evolutionary status of the targets +The main properties and final status of all our targets are +summarized in Table 10. Based on all the criteria discussed +in Sect. 3.2, we confirm that all our objects are YSOs, with +ages < 11 Myrs. +The +targets +2MASS +J15414827-3501458 +and +UCAC4 273-083363 do not show the presence of the +lithium line in the spectra, but their effective temperature +is compatible with the possible presence of a large amount +of Li depletion for fully convective pre-main sequence stars +(Bildsten et al. 1997). Lithium depletion was investigated +in several star forming regions, like some sub-groups of +Orion (Palla et al. 2007; Sacco et al. 2007), but also +in Lupus I and III (see, e.g., Biazzo et al. 2017, and +references therein). Due to their very young age (< 4 Myr), +10 + +12 +10 +8 +6 +(s/w>) +-2 +-4 +-6 +-8 +5.8 +6 +6.2 +6.4 +6.6 +6.8 +7 +7.2 +7.4 +7.6 +Parallax (mas)Majidi et al.: New members of the Lupus I cloud +Table 10: Overall status checklist for our targets. The rotation column refers to fast (F) or slow (S) rotators. +Name +Membership +Active +Accreting +Contains Li i +Rotation +Av +Conclusion +(UCL/Lup I) +(yes/no) +(yes/no) +(yes/no) +(F/S) +(mag) +2MASS J15383733-3422022 +Lup I +yes +no +yes +S +0 +Genuine member of Lup I +Sz 70 +Lup I +yes +yes +yes +S +0.5 +Genuine Lup I member + +wide companion candidate +TYC 7335-550-1 +Lup I +yes +no +yes +S +0.7 +Genuine member of Lup I + +wide companion candidate +2MASS J15361110-3444473 +? +yes +yes +no +S +1.75 +Unresolved binary (?) + +wide companion candidate +2MASS J15523574-3344288 +Lup I +yes +no +yes +S +0.5 +New member of Lup I +2MASS J15551027-3455045 +Lup I +yes +yes +? +S +0.75 +Genuine member of Lup I +2MASS J16011870-3437332 +Lup I +yes +yes +yes +S +0 +New member of Lup I +UCAC4 269-083981 +Lup I +yes +no +yes +S +0.5 +Genuine member of Lup I +Gaia DR2 6010590577947703936 +Lup I +yes +no +yes +F +0 +New member of Lup I +2MASS J15414827-3501458 +Lup I +yes +no +no +F +0 +Genuine member of Lup I +UCAC4 273-083363 +Lup I +yes +no +no +S +0 +Genuine member of Lup I +Gaia DR2 6014269268967059840 +? +yes +no +no +F +0 +? +we +therefore +classify +2MASS +J15414827-3501458 +and +UCAC4 273-083363 as Lupus I members. Newly discovered +members of Lupus I in this work are 2MASS J15523574- +3344288, +2MASS +J16011870-3437332, +and +Gaia +DR2 +6010590577947703936. +There are also two objects analyzed in this work that +we could not identify either as a member of Lupus I or +UCL. These are 2MASS J15361110-3444473, whose spec- +trum indicates an unresolved binary star of spectral types +M5.5 (VIS arm) and M8 (NIR arm), and we could not +detect lithium in its spectrum (see Appendix C for more +details on the analysis of this target). However, we would +like to emphasize that 2MASS J15361110-3444473 is an ac- +creting source that has consistent kinematic and physical +properties with the genuine members of Lupus I, hence, +there is a possibility that this target also qualifies as a +new member of Lupus I. The other object is Gaia DR2 +6014269268967059840, for which we acquired a spectrum +with poor SNR (see Sect. 2 for details on the observation +conditions of this target). The poor SNR of its UVB spec- +trum hindered us from carrying out any measurements on +its Hβ and Hγ lines in emission (as reported in Table 5), +which also leads to evaluating its accretion properties only +according to its Hα emission line (as reported in Table 8). +Therefore, the non-detection of lithium in its spectrum can +be purely due the poor SNR in the VIS arm, and we do not +approve nor rule out the possibility of this target being a +member of Lupus I. +We hence confirm that all our targets are YSOs, with +Hydrogen lines in emission above the continuum. Therefore, +this investigation suggests that although only four of our +targets were retrieved as Hα emitters in the OmegaCAM +survey (flagged in Table 2), it is likely that our entire sample +of 43 candidate YSOs could include Hα emitters or objects +with filled Hα profiles, which can only be confirmed by a +high- or mid-resolution spectroscopic study or in deep X- +ray surveys. +As a further investigation to strengthen our argument, +we cross-matched all of the Lupus I core members included +in Table 1 with the OmegaCAM survey. Except for three +objects, they were all retrieved in the survey as Hα emit- +ters. These exceptional three core members are RXJ1529.7- +3628 (which was out of the field of view of the survey), RX +J1539.7-3450B and Sz 68/HT Lup C, for which only one +object was resolved in the survey. Combining this result +with the results of this paper, we emphasize the necessity +of observing all our sample to characterize all the members +of Lupus I that have escaped the Hα surveys. +4.4. Accretion versus chromospheric–dominated objects +We realized that four of our targets in the current sam- +ple are accretors. We measured the Lacc of these tar- +gets, in addition to our chromospherically-dominant objects +(Table 8 and Table 9). The measured Lacc for all our tar- +gets are displayed in Fig. 8. In the same figure, we have +included the limits suggested by Manara et al. (2017b) +for objects with Teff > 4000 K and Teff < 4000 K, be- +low which the chromospheric activity of targets is domi- +nant. All our four accretors exceed this limit for targets +with Teff < 4000 K, confirming that they are accretion- +dominated. The rest of our targets within the same ef- +fective temperature range are below this threshold, which +make them chromospheric-dominated objects, as expected. +2MASS J15523574-3344288, however, lies exactly on the +threshold between these two regimes, which is consistent +with its significant Hα emission. We also emphasize that +this target was retrieved in the OmegaCAM survey as an +Hα emitter. +Fig. 9 shows the +˙Macc versus M∗ for the four accre- +tors in our sample in comparison with the Lupus members. +Among the four accretors, 2MASS J15551027-3455045 is +the least massive target, and has a very high mass accretion +rate in comparison with Lupus members of similar mass. +This target also stands above the double power-law rela- +tionship between +˙Macc and M∗ established by Vorobyov & +Basu (2009), based on modeling self-regulated accretion by +gravitational torques in self-gravitating disks. As concluded +by Alcal´a et al. (2017), only the strongest accretors stand +above this model. Our three other accretors have values of +mass accretion rates typical of Lupus accretors. +Finally, it is worth noting that three of our accretors (Sz +70, 2MASS J15361110-3444473, and 2MASS J16011870- +3437332) have WHα(10%)>270 km/s (see Table 5), which +is expected from accreting stars. Our chromospherically- +dominant targets have much narrower Hα profiles. +11 + +Majidi et al.: New members of the Lupus I cloud +Fig. 8: Log < Lacc/L∗ > vs Teff for all our targets. The +cyan dots represent accretors, and the red dots represent +chromospherically-dominant targets. The lines indicate the +limit below which the chromospheric activity for a star is +dominant (Manara et al. 2017b), for two regimes of stars +with Teff ≤ 4000 K (the diagonal blue line) and those with +Teff ≥ 4000 K (the horizontal orange line). +Fig. 9: Log Macc(M⊙/yr) vs log M∗(M⊙) for the four accre- +tors in our sample (cyan dots), together with the previously +identified members of the Lupus (black dots). The blue +crossed squares represent the substellar accreting compan- +ions detected at wide orbits by Zhou et al. (2014) around +GQ Tau, GSC 06214 00210 and DH Tau as labeled. 2MASS +J15551027-3455045, GQ Lup c and 2MASS J16085953- +3856275 are also labelled. 2MASS J15523574-3344288 is +labelled as red dot. The continuous red line indicates the +double power-law prediction of Vorobyov & Basu (2009), +while the magenta dashed line shows the prediction of disk +fragmentation model by Samatellos & Herczeg (2015). +5. Discussion +In this paper, we analyzed 12 objects observed by X- +Shooter out of our original sample of 43 proposed new +candidate members of Lupus I. We confirm that all these +12 objects are YSOs, and ten out of 12 are members of +Lupus I. We could not determine the membership of two of +our targets, namely 2MASS J15361110-3444473 and Gaia +DR2 6014269268967059840, as explained in the previous +Section. We could not fully measure the accretion prop- +erties of Gaia DR2 6014269268967059840 and hence our +analysis in this regard for this specific target is not reliable. +2MASS J15361110-3444473, on the other hand, is a rather +(intrinsic) faint object to be followed up by any available +spectrographs, but perhaps can be followed up with ALMA +to understand whether it is surrounded by a disk. Although +recognized to have an older age with respect to Lupus I +members (9 Myr), it can be still strongly accreting matter, +consistent with the members of γ Vel with age ∼10 Myr +(Frasca et al. 2015). One of the interesting targets discussed +in this work is TYC 7335-550-1, a lithium-rich K-type star +with Hα in absorption and without IR excess. We would +like to emphasize that YSOs with these particular charac- +teristics would never appear in Hα imaging surveys such as +OmegaCAM, although one of their main aims is to identify +the members of young star forming regions. All the above +points considered, we have fully characterized ten members +of Lupus I in this work. +In the following, we will discuss further qualities of our +targets, which are mainly based on the data available in +the literature in connection with the targets analyzed in +this work. +5.1. Spectral energy distributions / Circumstellar disks +For all our objects, we also investigated whether there are +hints of continuum flux excess suggestive of circumstellar +disks. To this aim, we extracted their SEDs from literature +which are collectively exhibited in Figs. 10 and 11. For this +work, we only concentrate on the morphology and trends +of the SEDs of our targets, as well as their near- to mid- +infrared photometric data (published by 2MASS and WISE +surveys). For generating the SEDs, we have used the follow- +ing WISE filters: W1 (3.4 microns), W2 (4.6 microns), W3 +(12 microns), W4 (22 microns). In a parallel paper (Majidi +et al. in prep), we will study the variability of these stars +and model their disks. +The photometric data for all four accretors significantly +deviate from their BT-Settl spectral model (based on their +Teff, log g, and zero metallicity) in W3 and W4 filters +(with the average flux errors of 5e-17 W.m−2 and 1.7e-16 +W.m−2 respectively). This trend can be observed for our +less massive, stronger accretors 2MASS J15551027-3455045 +and 2MASS J15361110-3444473 in all four WISE filters +(W1, W2, W3, and W4). According to Sicilia-Aguilar et +al. (2014), the morphology of the SEDs of all our four ac- +cretors in addition to 2MASS J15523574-3344288 is com- +patible with objects surrounded by full disks. This is further +confirmed by the disk categorization of Bredall et al. (2020) +based on Ks−W3 and Ks−W4 magnitudes for Lupus dip- +pers, Lupus YSOs, Upper Scorpius and Taurus members. +Hence, also according to Bredall et al. (2020), all our four +accretors in addition to 2MASS J15523574-3344288 are sur- +rounded by a full disk. Note, however, that the “valley” +around W3 in the SED of 2MASS J15361110-3444473 is +typical of those seen in transitional disks. +For the rest of our targets, we have two categories +of circumstellar disks based on the morphology of their +SEDs further approved by their Ks − W3 and Ks − W4 +magnitudes: i) Evolved disks, which are characterized by +only W4 excess with respect to the theoretical BT-Settl +model, and are evident in the SEDs of 2MASS J15383733- +3422022, Gaia DR2 6010590577947703936, and Gaia DR2 +6014269268967059840 (Fig. 11), ii) Debris disks, which are +12 + +-8 +(Mo yr-1) +GQ Lup +区 +GQ/Lup +c +区 +-10 +2MASS15551 +GSC 06214 b +区 +2MASS16085 +-DH Tau +b +-12 +2 +0 +logM* (Mo)-1 +-1.5 +-2 +-2.5 +60 +-3 +-3.5 +-4 +5000 +4500 +4000 +3500 +3000 +2500 +Teff (K)Majidi et al.: New members of the Lupus I cloud +Fig. 10: BT-Settl models (in grey) with the photometric data (red dots) for our accretors. +characterized by little to no mid-infrared excess, and is ev- +ident in the SEDs of TYC 7335-550-1, UCAC4 269-083981, +2MASS J15414827-3501458, and UCAC4 273-083363 (Fig. +11). +5.2. High accretion in the low-mass regime +Deriving +˙Macc for the lowest mass accretors is relevant for +the studies of disk evolution. There is growing evidence +of a change in the slope of the M⋆– ˙Macc relationship for +YSOs with ages of 2-3 Myr at M⋆<0.2 M⊙ (Manara et al. +2017b and Alcal´a et al. 2017, and see Fig. 9). Such a break +could be related to a faster disk evolution at the low-masses +(e.g. Vorobyov & Basu (2009)). To verify this, the +˙Macc– +M⋆ relationship needs to be sampled at much lower M⋆ and +˙Macc values than done so far. +Our target 2MASS J15551027-3455045 is one of the +lowest +mass +accretors +in +Lupus +(see +Fig. +3). +With +M⋆=0.02 M⊙, 2MASS J16085953-3856275 is the accretor +with comparable mass reported in the previous Lupus stud- +ies (Alcal´a et al. 2017, 2019). Considering the very low mass +of this YSO, its accretion rate +˙Macc∼10−11 M⊙/yr (Alcal´a +et al. 2019) is relatively high. Yet the ˙Macc value for 2MASS +J15551027-3455045 is about an order of magnitude higher +(see Fig. 9); hence, it is one of strongest accretors in Lupus +in the mass range 0.02–0.03M⊙, i.e. close to the planetary +mass regime. From modeling of a shock at the surface of +a planetary-mass object, Aoyama et al. (2021) have pre- +dicted much higher Lacc values than what the scaling Lacc– +Lline relations for stars would predict. The relationships by +these authors would yield an even higher ˙Macc value, almost +an order of magnitude higher than our estimate. This ob- +ject falls above the model prediction by Vorobyov & Basu +(2009), in contrast with the idea of faster disk evolution at +very low masses. However, statistics are still rather poor at +this mass regime for a firm conclusion. +Other very low-mass YSOs, companions to T Tauri +stars, have been found to exhibit similar, or even higher +rates of mass accretion (Betti et al. 2022; Zhou et al. 2014, +see Fig. 9). To explain the very high levels of accretion +observed in such sub-stellar and planetary-mass compan- +ions, Samatellos & Herczeg (2015) modeled the accretion +onto very low-mass objects that formed by the fragmenta- +tion of the disk around the hosting star. During the early +evolution the individual disks of sub-stellar companions, +including those at the planetary-mass regime, accrete addi- +tional material from the gas-rich parent disk, hence, their +disks are more massive and their accretion rates are higher +than if they were formed in isolation. Therefore, these very +low-mass objects have disk masses and accretion rates that +are independent of the mass of the central object and are +higher than expected from the scaling relation +˙Macc ∝ M 2 +⋆ +of more massive YSOs. These models predict that +˙Macc is +independent of M⋆. +Using Gaia DR3, we have investigated whether 2MASS +J15551027-3455045 might be a wide companion of another +star, but it is an isolated object. Hence, the high mass ac- +cretion rate cannot be explained in terms of the Samatellos +& Herczeg (2015) scenario. Due to its intrinsic faintness, +2MASS J15551027-3455045 would be an interesting target +to be followed up by CUBES, which is a next-generation +spectrograph suitable for investigating fainter, low-mass ac- +creting YSOs (Alcal´a et al. 2022). +13 + +2MASSJ15551027-3455045 +-10 +Teff = 2700 K, log g = 3.5 +10.5 +PhotometricData + cm-2) +-11 +(erg S-1 +11.5 +-12 +12.5 +-13 +13.5 +-14 +1000 +10000 +入 (nm)2MASSJ15361110-3444473 +Teff = 2900 K, log g = 4.5 +-11 +Photometric Data +L cm-2) +11.5 +12 +-12.5 +-13 +1000 +10000 +入 (nm)Sz 70 +6 +Teff = 3000 K, log g = 4.0 +PhotometricData +9.5 +-10 +-10.5 +log 入 Flux +11 +11.5 +-12 +1000 +10000 +入 (nm)2MASSJ16011870-3437332 +-10 +Teff = 3100 K, log g = 4.5 +Photometric Data +10.5 +-11 +11.5 +log 入Flux +12 +12.5 +13 +1000 +10000 +入 (nm)Majidi et al.: New members of the Lupus I cloud +Fig. 11: BT-Settl models (in grey) with the photometric data (red dots) for our chromospherically-dominant targets. +5.3. Possible wide companions +While studying the kinematic properties of the targets, we +also noticed that a few of our targets and core members +of the Lupus I share similar kinematic properties, and can +be considered as wide companion candidates. These wide +companion candidates are presented in Table 12 and Table +13, divided into two categories of candidates studied in this +14 + +TYC 7335-550-1 +8 +Teff = 4500 K, log g = 4.0 +Photometric Data +(erg s-1 cm-2) +-10 +log 入Flux +-11 +12 +13 +1000 +10000 +入 (nm)2MASSJ15523574-3344288 +-10 +Teff = 3000 K, log g = 4.5 +PhotometricData +10.5 +-11 +-11.5 +log 入Flux +12 +12.5 +-13 +1000 +10000 +入 (nm)UCAC4269-083981 +-9 +Teff = 3800 K, log g = 4.5 +9.5 +Photometric Data + cm-2) +-10 +(erg s-1 +10.5 +-11 +log 入Flux +11.5 +-12 +12.5 +13 +1000 +10000 +入 (nm)2MASSJ15383733-3422022 +-10 +Teff = 3100 K, log g = 4.5 +Photometric Data +-10.5 +-11 +-11.5 +log 入Flux +-12 +12.5 +13 +1000 +10000 +入 (nm)2MASSJ15414827-3501458 +-9 +Teff = 3200 K, log g = 4.5 +9.5 +PhotometricData + cm-2) +-10 +(erg s-1 +10.5 +11 +log 入Flux +11.5 +-12 +12.5 +13 +1000 +10000 +入 (nm)GaiaDR26010590577947703936 +-10 +Teff = 3100 K, log g = 4.5 +Photometric Data +10.5 +-11 +-11.5 +log 入Flux +-12.5 +13 +1000 +10000 +入 (nm)UCAC4273-083363 +-9 +Teff = 3000 K, log g = 4.5 +9.5 +Photometric Data + cm-2) +-10 +10.5 +-11 +log 入Flux +11.5 +-12 +12.5 +13 +1000 +10000 +入 (nm)GaiaDR26014269268967059840 +-10 +Teff = 3000 K, log g = 4.5 +10.5 +Photometric Data +. cm-2) +-11 +(erg s-1 +11.5 +-12 +12.5 +13 +13.5 +-14 +1000 +10000 +入 (nm)Majidi et al.: New members of the Lupus I cloud +Table 11: Disk categorization of all our targets, in addition to their reddest colors available in the 2MASS and WISE +catalogs. +Name +Ks − W3 +Ks − W4 +Bredall et al. (2020) +Sicilia-Aguilar et al. (2014) +mag +mag +Disk type +SED/Disk type +2MASS J15383733-3422022 +0.75 +3.93 +Evolved disk +Sz 70 +2.28 +3.9 +Full disk +Full disk +TYC 7335-550-1 +0.20 +1.14 +Debris disk +2MASS J15361110-3444473 +2.70 +5.04 +Full disk +Full disk +2MASS J15523574-3344288 +2.69 +4.31 +Full disk +Full disk +2MASS J15551027-3455045 +3.24 +5.7 +Full disk +Full disk +2MASS J16011870-3437332 +2.18 +4.09 +Full disk +Full disk +UCAC4 269-083981 +0.13 +1.06 +Debris disk +Gaia DR2 6010590577947703936 +0.61 +3.79 +Evolved disk +2MASS J15414827-3501458 +0.39 +1.16 +Debris disk +UCAC4 273-083363 +0.4 +1.86 +Debris disk +Gaia DR2 6014269268967059840 +0.89 +3.58 +Evolved disk +Notes. The overall SED of 2MASS J15361110-3444473 may be affected by a possible unresolved M8-type companion. +work and the Lupus I core members. In order to understand +whether two objects with similar kinematic properties are +gravitationally bound, we calculated their total velocity dif- +ference (∆v) and compared it with the maximum total ve- +locity difference (∆vmax) as a function of projected sepa- +ration between the two binary components, suggested by +Andrews et al. (2017). If ∆v exceeds ∆vmax, we do not ex- +pect the two targets to be gravitationally bound. It should +be noted, however, that the theoretical maximum velocity +difference modeled by Andrews et al. (2017) is only for bina- +ries of total mass 10 M⊙ in circular orbits. We summarize +our results on identifying wide companions candidates in +the Lupus I cloud as follows: +Sz 70 and Sz 71 – Same as the GQ Lup triple system +(Alcal´a et al. 2020), Sz 70 and Sz 71 (GW Lup) are located +on the main filament of Lupus I. Sz 70 lies at a separation of +32.32 arcseconds from GW Lup, and in between these ob- +jects lies the X-ray source [KWS97] Lupus I 37 (Krautter +et al. 1997) at a separation of 24.23 arcseconds from Sz 70. +We conducted a chance projection study in Alcal´a et al. +(2020, Appendix E), which was focused on understanding +how probable it is to find a field object around a genuine +member of Lupus I, lying on the same filament where GQ +Lup stellar system and Sz 70/Sz 71 are located. The linear +density of this filament is 0.0024 objects/arcsec, or an av- +erage object separation of 418 arcsec, which is 13 times the +observed separation between Sz 70 and Sz 71. As exhibited +in Fig. 12, Sz 70 and Sz 71 do not qualify as gravitation- +ally bound stars, but we would like to emphasize that the +test proposed by Andrews et al. (2017) is only valid for +gravitationally bound binaries, and not systems of higher +multiplicities (if this is the case for this stellar system). +Hence, we would consider this case as a wide companion +candidate that cannot be confirmed or ruled out according +to the available information. +TYC +7335-550-1 +and +2MASS +J15361110- +3444473 – As discussed in Sect. 4, 2MASS J15361110- +3444473 might be an unresolved binary, composed of an +M6 (VIS spectrum) and an M8 (NIR spectrum) star. The +RV calculated for this target based on the ROTFIT code +is obtained by cross-correlations conducted on the VIS +spectrum of this target, which is also used for calculating +the maximum velocity difference between TYC 7335-550-1 +and 2MASS J15361110-3444473. As exhibited in Fig. 12, +the two objects can be gravitationally bound. However, +Fig. 12: Log-log plot of total velocity difference ∆v (km/s) +versus projected separation s (au) for the wide companion +candidates analyzed in this work, in addition to the genuine +wide companions GQ Lup and GQ Lup C. ∆vmax (km/s) +(orange line) indicates the maximum total velocity differ- +ence that bound binaries with a total mass equal to 10 M⊙ +in circular orbits can possess (Andrews et al. 2017). Each +point is marked as one of the wide companion candidates +involved. For the detailed information, see Tables 12 and +13. +TYC 7335-550-1 has an age of ∼ 4 Myr and 2MASS +J15361110-3444473 an age of ∼ 9 Myr, which states +the two stellar systems are probably not coeval. Also, +unlike TYC 7335-550-1, we could not determine whether +2MASS J15361110-3444473 is a member of Lupus I due to +many uncertainties explained earlier. Hence, any further +comments on its physical association with TYC 7335-550-1 +would be misleading and inconclusive. +Sz 65 and Sz 66 – At a separation of 6.45 arcseconds, +with ∆V = 5.26±2.69 km/s, Sz 65 and Sz 66 (although +coeval) according to the test suggested by Andrews et al. +(2017) are not gravitationally bound. There are no other +objects located in a close separation with respect to either +Sz 65 or Sz 66. Hence, we rule out the possibility of Sz 65 +and Sz 66 as wide companion candidates. +HT Lup A-B-C – This stellar system is located in +an over-crowded region on the same filament of Lupus I +as GQ Lup stellar system. In Gaia DR2 catalog, HT Lup +15 + +1.4 +1.2 +1 +(km/s) +0.8 +(△ v) +0.6 +0.4 +0.2 +0 +-0.2 +2.6 +2.8 +3 +3.2 +3.4 +3.6 +3.8 +4 +log s (au) +GQLupC +SZ 66 +HT Lup +Sz 70 +TYC 7335-550-1 +△ Vmax (km/s)Majidi et al.: New members of the Lupus I cloud +Table 12: Kinematic properties of the Lupus I members from this work (measurement errors are displayed in parenthesis). +Name +α (J2000) +δ (J2000) +ϖ +µα∗ +µδ +RV +Age +∆V +δ∆V +S +(h:m:s) +(d:m:s) +(mas) +(mas/yr) +(mas/yr) +(km/s) +(Myr) +(km/s) +(km/s) +(′′) +Sz 71/GW LUP∗ +15 46 44.73 +–34 30 35.68 +6.41(0.06) +–14.03(0.10) +–23.36(0.07) +–3.30(1.90) +2.0 +6.07 +3.24 +32.32 +Sz 70 +15 46 42.99 +–34 30 11.55 +6.09(0.21) +–12.58(0.39) +–22.16(0.25) +1.1(2.6) +0.5 +2MASS J15361110-3444473 +15 36 11.09 +–34 44 47.82 +5.83(0.29) +–13.56(0.29) +–20.21(0.23) +6.9(2.6) +9.77 +4.72 +3.47 +16.28 +TYC 7335-550-1 +15 36 11.55 +–34 45 20.54 +6.26(0.07) +–13.93(2.43) +–19.51(1.01) +2.6(2.0) +3.55 +∗ RV obtained by Frasca et al. (2017). +Table 13: Core members of Lupus I sharing similar kinematic properties (measurement errors are displayed in parenthesis). +Name +α (J2000) +δ (J2000) +ϖ +µα∗ +µδ +RV +Age +∆V +δ∆V +S +(h:m:s) +(d:m:s) +(mas) +(mas/yr) +(mas/yr) +(km/s) +(Myr) +(km/s) +(km/s) +(′′) +Sz 65/V∗ IK Lup∗ +15 39 27.77 +–34 46 17.21 +6.44(0.05) +–13.27(0.12) +–22.24(0.07) +–2.70(2.00) +1.9 +5.26 +2.69 +6.41 +Sz 66∗ +15 39 28.28 +–34 46 18.09 +6.36(0.09) +–13.60(0.19) +–21.56(0.12) +2.40(1.80) +3.9 +Sz 68/HT LUP A-B∗ +15 45 12.87 +–34 17 30.65 +6.49(0.06) +–13.63(0.13) +–21.60(0.08) +–4.30(1.80) +0.5 +6.30 +4.30 +2.82 +CD-33 10685C/HT Lup C∗∗ +15 45 12.67 +–34 17 29.37 +6.55(0.19) +–15.43(0.22) +–20.27(0.15) +1.2(3.9) +– +∗ RV and age obtained by Frasca et al. (2017). +∗∗ RV for this target is adopted from the optimal RV calculated by BANYAN Σ, considering HT Lup C is a member of UCL. +A and B are not resolved separately, hence we assume the +central star to be Sz 68 (or HT Lup A), composed of two +unresolved stars, and adopt its stellar characteristics from +Frasca et al. (2017). As genuine members of Lupus I, we +assume all the components of this triple system to have an +age consistent with the other bona fide members of Lupus I +(≤ 2 Myr), and hence, to be coeval. However, the RVs used +here should be taken with caution, both because HT Lup +A-B are not resolved, and also because we have adopted +the optimal RV calculated by BANYAN σ for HT Lup C +considered as a member of UCL. With a separation of 2.82 +arc seconds, we have shown in Fig. 12 that as expected, this +triple system is possibly gravitationally bound. +We thus conclude that the possibility of Sz 70 & Sz 71 +being wide companions is rather low and for TYC 7335- +550-1 & 2MASS J15361110-344447, follow-up studies on +2MASS J15361110-344447 are required. As for the previ- +ously identified members of Lupus I, we understood that +Sz 65 and Sz 66 are not gravitationally bound, and HT +Lup A-B-C are the components of a triple system. +6. Conclusion +The main conclusions of this paper can be summarized as +follows: +– Out of the 12 objects fully characterized in this work, +ten are recognized as genuine members of Lupus I, and +two remain ambiguous in terms of stellar properties. +– Out of the ten members of Lupus I analyzed in this +work, three were recognized to be accretors (Sz 70, +2MASS J15551027-3455045, and 2MASS J16011870- +3437332), and Sz 70 and 2MASS J15551027-3455045 are +likely to be surrounded by full disks. 2MASS J15551027- +3455045 is among the least massive accretors discovered +so far in the Lupus complex, formed in full isolation and +is an off-cloud member of Lupus I. +– All of the three off-cloud targets included in our +program +turned +out +to +be +genuine +members +of +Lupus I. These targets are 2MASS J15523574-3344288, +2MASS J15551027-3455045, and 2MASS J16011870- +3437332, with 2MASS J15551027-3455045 and 2MASS +J16011870-3437332 +actively +accreting +matter, +and +2MASS J15523574-3344288 mildly accreting matter. +Further investigation in this area may reveal a diffused +population of M dwarfs close to the main filament of +Lupus I. We thus would like to acknowledge that this +work also contributes to revealing the diffused popula- +tions of M-dwarfs around the Lupus cloud by Comer´on +(2008). +– Although the sample studied in this work is small, we +proved that many interesting targets in young star form- +ing regions can escape Hα surveys due to various rea- +sons. Hence, using the kinematic properties of candi- +date YSOs can play a key role in identifying the gen- +uine members of the young stellar associations. This is +specifically true for genuine members such as TYC 7335- +550-1 that have Hα in absorption, and hence would not +appear in Hα surveys. +– We have identified a plausible binary system among +the targets analyzed in this work, namely, TYC 7335- +550-1 and 2MASS J15361110-3444473. It is noteworthy, +however, that 2MASS J15361110-3444473 might be an +unresolved binary, and its kinematic properties (espe- +cially RV) should be revised with next-generation spec- +trographs (due to its intrinsic faintness). +– All the above points considered, we conclude that char- +acterizing only a small portion of our sample has proved +to have a high success rate for discovering the new mem- +bers of Lupus I. This shows that the spectroscopy of our +entire sample of 43 objects could have resulted in a far +more solid investigation of the region in terms of de- +termining the disk fraction, stellar properties, and the +number of new members of Lupus I. +Acknowledgements. FZM is grateful to Eugene Vasiliev for fruitful +discussions on how to use Gaia catalogs. AFR is grateful to Giovanni +Catanzaro for helping us with the analysis of TYC 7335-550-1. FZM is +funded by ”Bando per il Finanziamento di Assegni di Ricerca Progetto +Dipartimenti di Eccellenza Anno 2020” and is co-funded in agree- +ment with ASI-INAF n.2019-29-HH.0 from 26 Nov/2019 for ”Italian +participation in the operative phase of CHEOPS mission” (DOR - +Prof. Piotto). A.B. acknowledges partial funding by the Deutsche +Forschungsgemeinschaft Excellence Strategy - EXC 2094 - 390783311 +and the ANID BASAL project FB210003. JMA, AFR, CFM, KBI +and ECO acknowledge financial support from the project PRIN- +INAF 2019 “Spectroscopically Tracing the Disk Dispersal Evolution” +16 + +Majidi et al.: New members of the Lupus I cloud +(STRADE). CFM is funded by the European Union under the +European Union’s Horizon Europe Research & Innovation Programme +101039452 (WANDA). This work has also been supported by the +PRIN-INAF 2019 ”Planetary systems at young ages (PLATEA)” and +ASI-INAF agreement n.2018-16-HH.0. Views and opinions expressed +are however those of the author(s) only and do not necessarily re- +flect those of the European Union or the European Research Council. +Neither the European Union nor the granting authority can be held +responsible for them. +This work has made use of data from the European Space +Agency +(ESA) +mission +Gaia +(https://www.cosmos.esa.int/gaia), +processed by the Gaia Data Processing and Analysis Consortium +(DPAC, +https://www.cosmos.esa.int/web/gaia/dpac/consortium). +Funding for the DPAC has been provided by national institutions, +in particular, the institutions participating in the Gaia Multilateral +Agreement. +This research has made use of the SIMBAD database and Vizier +services, operated at CDS, Strasbourg, France. 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J., Kraus, A. L. et al. 2014, ApJ, 783, 17 +17 + +Majidi et al.: New members of the Lupus I cloud +Appendix A: Candidate members of Lupus I +As we explained in Sect. 2, we proposed 43 objects to be ob- +served with X-Shooter. Twelve out of these 43 objects were +observed during a filler program, and in this work we fully +characterized them. The rest of our targets in this sam- +ple that were not observed are listed in Table A.1. Among +these targets, only 2MASS J15464664-3210006 (Eisner et +al. 2007) is partly characterized, and 20 objects are identi- +fied as candidate YSOs using Gaia DR2 (Zari et al. 2018). +Appendix B: Age estimation and isochrones +For estimating the age of our targets we used multiple +isochrones for the reasons explained in Sect. 3.2. In this +Appendix, we present the ages of our targets using various +isochrones. We repeat that the ages estimated for all our +targets were overestimated by PARSEC models in compar- +ison with all the other models with a considerable gap. We +thus decided to remove the results achieved by the PARSEC +models to avoid confusion. This is, however, a well-known +problem of PARSEC isochrones that they overestimate the +age of cool stars, and all our targets fall in this category. +Appendix C: 2MASS J15361110-3444473 +Fig. C.1: Flux-calibrated, extinction-corrected NIR spec- +trum of 2MASS J15361110-3444473 (in black) with its BT- +Settl model (Teff = 2500 K and log g = 4.5, in grey). +2MASS J15361110-3444473 is an M5.5 star according to +its VIS spectrum (as we quantitatively indicated) and an +M8 star based on its NIR spectrum (based on the fitting +done with the BT-Settl model Teff = 2500 K and log g += 4.5, as exhibited in Fig. C.1), with a total extinction +of AV = 1.75 mag. All the spectral typing and analysis +that we have performed in this paper are based on the VIS +spectrum of this target, especially the ROTFIT results are +all based on the VIS spectrum. Hence, although we keep our +analysis limited to the spectroscopy conducted on the VIS +spectrum, we would like to emphasize that the possibility +of this target being an unresolved binary (composed of two +M dwarfs) with SpTs of M5.5 and M8 is viable. Considering +the available data, we also cannot rule out the possibility +that the star is heavily spotted instead of being a binary. +Appendix D: Updates with Gaia DR3 +As stated in Sect. 2, we used the Gaia DR2 catalog to select +our targets. Very recently, Gaia DR3 (Gaia Collaboration +2021) became public and gave us the opportunity to check +the catalog for any possible changes or updates on the +kinematic or stellar properties of our objects analyzed in +this work. We did not find any considerable difference be- +tween the kinematic properties reported in both catalogs. +However, we report the highlights of our search using these +two catalogs in the following: +TYC 7335-550-1 – as obtained in this work, for TYC +7335-550-1 we obtained Teff = 4488 K, while in both Gaia +DR2 and Gaia DR3 its reported temperature is 5000 K. +The reported RV for TYC 7335-550-1 in Gaia DR2 is +1.20±1.65 km/s, which is better constrained than the RV +we report here (2.6±2.0 km/s). As the wide companion can- +didate of 2MASS J15361110-3444473, we recalculated their +∆v using the Gaia DR3 kinematic properties of TYC 7335- +550-1, and it resulted in ∆v = 5.34±3.30 (km/s) which is +consistent with the previous ∆v = 4.72±3.47 (km/s). For +both of these calculations, we use the RVs calculated by +ROTFIT. +Sz 70 – has a high RUWE in both catalogs (4.86), but +we saw no signs of binarity in the spectrum of Sz 70. Using +the kinematic properties of Sz 70 reported in Gaia DR3 +and those of Sz 71 (which is also updated in Gaia DR3), +we recalculated their maximum velocity difference, and it +resulted in ∆v = 8.36±3.24 (km/s), which is consistent with +the ∆v = 6.07±3.24 (km/s) calculated based on Gaia DR2. +2MASS J15414827-3501458 – has a high RUWE +(4.198) in both Gaia DR2 and Gaia DR3 catalogs, but we +detected no signs of binarity in the spectrum of the object. +We report that the kinematic properties of all our tar- +gets (parallax and proper motions) are consistent within 3σ +in the two catalogs. Also according to Manara et al. (2022), +we do not expect the stellar physical parameters of our core +sample to be changed with the astrometry reported in Gaia +DR3. +18 + +-11 +Teff = 2500, log g = 4.5 +2MASS|15361110-3444473 +-11.2 +nm-1) + (erg s-1 cm-2 I +11.4 +11.6 +log 入Flux +11.8 +-12 +-12.2 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +入 (nm)Majidi et al.: New members of the Lupus I cloud +Table A.1: Astrometric properties of the candidate Lupus I members that were not observed by X-Shooter, with their +errors in parentheses. +Name +α (J2000) +δ (J2000) +ϖ +µα∗ +µδ +J +(h:m:s) +(d:m:s) +(mas) +(mas/yr) +(mas/yr) +(mag) +2MASS J15464664-3210006a +15 46 46.64 +–32 10 00.62 +7.05(0.021) +–19.47(0.023) +–23.76(0.014) +11.22 +Gaia DR2 6013000844869745664 +15 39 24.47 +–35 58 50.88 +6.62(0.039) +–18.00(0.081) +–22.23(0.057) +10.11 +Gaia DR2 6013065853493820416b +15 43 15.62 +–35 39 38.18 +6.88(0.015) +–17.68(0.018) +–24.51(0.012) +10.20 +Gaia DR2 6011737574730221568c +15 50 46.50 +–34 22 38.49 +6.69(0.019) +–16.20(0.020) +–22.52(0.015) +10.74 +Gaia DR2 6012258330925877632d +15 53 36.13 +–33 31 02.60 +6.92(0.016) +–16.97(0.018) +–24.57(0.016) +10.75 +Gaia DR2 6039383622075982848e +15 57 09.76 +–32 04 33.91 +6.72(0.02) +–14.24(0.023) +–23.58(0.015) +10.56 +Gaia DR2 6011518462675791872f +15 48 13.16 +–35 43 31.08 +6.62(0.023) +–16.65(0.028) +–24.31(0.023) +11.48 +Gaia DR2 6011797738632729216g +15 57 20.96 +–35 00 01.21 +6.71(0.027) +–16.29(0.033) +–24.21(0.024) +11.65 +Gaia DR2 6014049985115937408 +15 34 59.21 +–34 58 16.16 +6.83(0.097) +–17.76(0.16) +–24.03(0.11) +12.16 +Gaia DR2 6014830844535625344h +15 47 58.08 +–33 46 59.53 +6.84(0.027) +–17.73(0.031) +–24.48(0.025) +11.31 +Gaia DR2 6014224051546189568 +15 34 42.05 +–34 17 48.09 +6.66(0.098) +–17.36(0.134) +–23.67(0.094) +11.94 +Gaia DR2 6009936093645659136 +15 43 49.43 +–36 48 38.64 +6.94(0.13) +–20.45(0.28) +–22.89(0.19) +10.92 +Gaia DR2 6039633559115225344i +15 52 59.02 +–31 38 33.57 +6.59(0.03) +–18.34(0.036) +–22.89(0.029) +11.93 +Gaia DR2 6013187040287810944j +15 37 53.31 +–35 55 12.42 +6.74(0.027) +–17.9(0.03) +–24.08(0.024) +11.95 +Gaia DR2 6016139332082870272 +15 39 25.88 +–32 10 04.68 +6.42(0.40) +–20.32(0.54) +–23.65(0.37) +10.81 +Gaia DR2 6013126738951338624k +15 43 28.48 +–35 17 27.40 +6.77(0.032) +–17.67(0.035) +–24.48(0.022) +11.91 +Gaia DR2 6013190201383772288 +15 37 53.00 +–35 52 28.70 +6.75(0.055) +–19.08(0.13) +–22.62(0.087) +12.22 +Gaia DR2 6013077192207599232m +15 43 11.42 +–35 26 34.43 +6.78(0.032) +–17.32(0.034) +–24.29(0.025) +11.82 +Gaia DR2 6015181897983193728m +15 51 57.84 +–33 29 33.17 +6.74(0.032) +–16.22(0.039) +–22.37(0.026) +12.03 +Gaia DR2 6014590429442468096m +15 45 06.91 +–35 06 21.73 +6.99(0.036) +–16.97(0.042) +–23.09(0.029) +11.82 +Gaia DR2 6009995742152335232m +15 44 26.97 +–36 25 42.75 +6.52(0.034) +–18.30(0.043) +–23.21(0.031) +11.82 +Gaia DR2 6011607694917034112m +15 50 00.76 +–35 29 19.71 +7.23(0.044) +–20.18(0.052) +–25.32(0.034) +12.37 +Gaia DR2 6011695690208264320m +15 47 59.03 +–34 56 38.36 +6.99(0.06) +–17.93(0.069) +–25.07(0.045) +12.69 +Gaia DR2 6011261726715424128 +15 50 29.19 +–36 25 11.80 +6.92(0.11) +–17.08(0.23) +–23.52(0.16) +13.32 +Gaia DR2 6015222957871475584 +15 48 46.12 +–33 18 35.48 +6.69(0.13) +–19.21(0.26) +–23.77(0.17) +13.77 +Gaia DR2 6013030875279571328 +15 41 55.22 +–35 59 35.36 +6.97(0.12) +–17.12(0.24) +–25.52(0.14) +13.17 +Gaia DR2 6014112107523072640m +15 34 35.79 +–34 36 01.54 +6.88(0.084) +–16.89(0.087) +–24.841(0.066) +13.14 +Gaia DR2 6012977136650130560m +15 39 48.47 +–36 13 48.07 +6.94(0.10) +–20.069(0.11) +–23.61(0.069) +12.81 +Gaia DR2 6015141830223216640 +15 50 19.17 +–33 50 07.12 +6.84(0.15) +–17.29(0.29) +–26.46(0.19) +13.92 +Gaia DR2 6011581856393988352n +15 48 06.26 +–35 15 48.15 +6.05(0.07) +–12.22(0.084) +–21.04(0.057) +10.56 +Gaia DR2 6016191485871670400 +15 38 35.63 +–32 02 37.66 +6.53(0.26) +–18.90(0.39) +–23.38(0.28) +14.35 +a 2MASS J15464664-3210006 is an M2, T Tauri star (Eisner et al. 2007). +b aka UCAC4 272-080482, this target is a YSO candidate (Zari et al. 2018). +c aka UCAC4 279-083370, this target is a YSO candidate (Zari et al. 2018). +d aka UCAC4 283-086052, this target is a YSO candidate (Zari et al. 2018). +e aka RX J1557.1-3204A, this target is a YSO candidate (Zari et al. 2018). +f aka UCAC4 272-081081, this target is a YSO candidate (Zari et al. 2018). +g aka UCAC4 275-083957, this target is a YSO candidate (Zari et al. 2018). +h aka UCAC4 282-082547, this target is a YSO candidate (Zari et al. 2018). +i aka UCAC4 292-084899, this target is a YSO candidate (Zari et al. 2018). +j aka UCAC4 271-080669, this target is a YSO candidate (Zari et al. 2018). +k aka UCAC4 274-080590, this target is a YSO candidate (Zari et al. 2018). +l aka UCAC4 274-080590, this target is a YSO candidate (Zari et al. 2018). +m This target is a YSO candidate (Zari et al. 2018). +n aka UCAC4 274-081081, this target is a YSO candidate (Zari et al. 2018). +19 + +Majidi et al.: New members of the Lupus I cloud +Table B.1: Ages of our targets estimated using various isochrones. The ages are all in Myr. +Name +Dartmouth +Dartmouth +MIST +Baraffe +std +mag +models +2MASS J15383733-3422022 +11 +20 +12.6 +10.7 +Sz 70 +<1 +1 +<0.25 +0.5 +TYC 7335-550-1 +3 +5 +3.5 +3.55 +2MASS J15361110-3444473 +9 +20 +9 +9.77 +2MASS J15523574-3344288 +8 +13 +8 +6.3 +2MASS J15551027-3455045 +- +- +-a +1.7 +2MASS J16011870-3437332 +9.5 +14 +9.5 +9.55 +UCAC4 269-083981 +4.5 +8 +3.5 +4.2 +Gaia DR2 6010590577947703936 +8 +14 +8 +8.8 +2MASS J15414827-3501458 +2.5 +3 +1.78 +1.82 +UCAC4 273-083363 +4.5 +8 +3.5 +3.63 +Gaia DR2 6014269268967059840 +8 +13 +8 +6.46 +a None of the three isochrones used here were able to reproduce the stellar parameters of this target due to its dimness. +20 + diff --git a/FtE3T4oBgHgl3EQfVwq1/content/tmp_files/load_file.txt b/FtE3T4oBgHgl3EQfVwq1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0a37c3b5a4e4c4d3db8a1256f09a366365721b11 --- /dev/null +++ b/FtE3T4oBgHgl3EQfVwq1/content/tmp_files/load_file.txt @@ 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Bayo5,7, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Biazzo8, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Claudi2, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Covino3, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Mantovan1,2, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Montalto4, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Nardiello2,9, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Piotto1, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Rigliaco2 1 Dipartimento di Fisica e Astronomia, Universit´a degli Studi di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy 2 INAF-Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, 35122 Padova, Italy 3 INAF-Osservatorio Astronomico di Capodimonte, via Moiariello 16, 80131 Napoli, Italy 4 INAF-Osservatorio Astrofisico di Catania, via S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Sofia, 78, 95123 Catania, Italy 5 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei M¨unchen, Germany 6 Department of Physics, University of Rome Tor Vergata, via della ricerca scientifica 1, 00133, Rome, Italy 7 Instituto de F´ısica y Astronom´ıa, Facultad de Ciencias, Universidad de Valpara´ıso, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Gran Breta˜na 1111, Valpara´ıso, Chile 8 INAF - Rome Astronomical Observatory, Via di Frascati, 33, I-00044, Monte Porzio Catone, Italy 9 Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France Received ABSTRACT We characterize twelve young stellar objects (YSOs) located in the Lupus I region, spatially overlapping with the Upper Centaurus Lupus (UCL) sub-stellar association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The aim of this study is to understand whether the Lupus I cloud has more members than what has been claimed so far in the literature and gain a deeper insight into the global properties of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We selected our targets using Gaia DR2 catalog, based on their consistent kinematic properties with the Lupus I bona fide members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In our sample of twelve YSOs observed by X-Shooter, we identified ten Lupus I members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We could not determine the membership status of two of our targets, namely Gaia DR2 6014269268967059840 and 2MASS J15361110-3444473 due to technical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We found out that four of our targets are accretors, among them 2MASS J15551027-3455045, with a mass of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='03 M⊙, is one of the least massive accretors in the Lupus complex to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Several of our targets (including accretors) are formed in-situ and off-cloud with respect to the main filaments of Lupus I, hence, our study may hint that there are diffused populations of M-dwarfs around Lupus I main filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In this context, we would like to emphasize that our kinematic analysis with Gaia catalogs played a key role in identifying the new members of the Lupus I cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Accretion, Accretion Disks – Stars: activity, atmospheres, chromospheres, low-mass, pre-main sequence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Introduction Observation of young stellar populations in nearby star- forming regions and comparison of their properties with more massive and distant ones is a key to understanding the impact of the environment on the star formation process and the properties of protoplanetary disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The Lupus dark cloud complex is one of the main low- mass star-forming regions (SFRs) within 200 pc of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' It consists of a loosely connected group of dark clouds and low-mass pre-main sequence (PMS) stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The complex hosts four active SFRs plus five other looser dark clouds with signs of moderate star-formation activity (Comer´on 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Infrared (IR) and optical surveys (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Rygl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2012) have shown that objects in all evolution- ary phases, from embedded Class I objects to evolved Class III stars, are found majorly concentrated in the Lupus I, II and III clouds with Lupus III being the richest in YSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' ⋆ Based on observations collected at the European Southern Observatory at Paranal, under program 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='20P9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='001 Different distances to the Lupus stellar sub-groups have been claimed in the past from Hipparcos parallaxes and extinction star counts (Comer´on 2008), but recent investi- gations based on Gaia DR2 showed that the vast majority of YSOs in all Lupus clouds are at a distance of ∼160 pc (see the Appendix in Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Out of the three main clouds, Lupus III has been recognized as the most massive and active star-forming region in Lupus by far, with a great number of young low-mass and very-low mass stars (Comer´on 2008), while Lupus I, II and IV represent regions of low star-formation activity, with Lupus V and VI lacking star-formation (Spezzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In this paper we investigate the Lupus I cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This cloud has less than thirty bona fide members, which from now on we refer to as Lupus I core members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The main motivation for selecting this cloud over the others with a low star-forming activity was the recent discovery of the star GQ Lup C (Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Lazzoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2020), which is located on the main filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04463v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='SR] 11 Jan 2023 Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud This target was specifically selected by our team for discovering possible wide companions to SPHERE-GTO targets on Gaia DR2 with a high specific interest in the presence of planets, brown dwarfs, or spatially resolved cir- cumstellar disks (Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' GQ Lup C was proved to be a strong accretor that surprisingly had escaped detection in previous IR and Hα surveys, sug- gesting the possibility that many YSOs in the region are yet to be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This discovery hence motivated us to conduct a more extended search in Gaia DR2 to select new YSO candidates in the same region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In this work, we present the spectroscopic characterization of 12 YSOs in the Lupus I cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The outline of this paper is as follows: in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2, we discuss the target selection criteria, as well as compiling a complete list of the bona fide Lupus I members, in ad- dition to the observation and data reduction methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3, we discuss the data analysis methods employed for analyzing the X-Shooter spectra, the membership criteria, and accreting objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 4, we discuss the results of our analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 5, we introduce additional qualities of our targets in Lupus I, present their spectral energy distri- butions (SEDs), and evaluate them as potential wide com- panion candidates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' and eventually, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 6 will present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Target selection, observations, and data reduction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Target selection The Gaia astrometric catalog (Gaia Collaboration 2018) has been recently used to efficiently identify young clus- ters and associations within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 kpc from the Sun (see Prisinzano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2022, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We selected our sample of YSO candidates based on a statistical anal- ysis using the Gaia DR2 catalog detailed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As a first step, we identified the genuine population (core members) of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' These core members were gathered from the catalogs existing in the literature (Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Mer´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Mortier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Benedettini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Dzib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Comer´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2020), and are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We calculated the member- ship probability of these targets to Upper Centaurus Lupus (UCL) with BANYAN Σ (Gagn´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018) which are also quoted in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' It should be noted that the catalog does not evaluate the Lupus membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We then extracted the kinematic properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=', par- allaxes, ϖ, and proper motions µα∗ and µδ) of these core members from Gaia DR2, and constrained a range over these parameters (see Appendix B of Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Using this constrained range, we searched for the objects with similar kinematic properties to Lupus I core members in Gaia DR2 in a radius of 3 degrees from the center of the Lupus I cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' At this stage, we found 247 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We placed these objects on a color-magnitude diagram (CMD) with Main Sequence (MS) stars (Pecaut & Mamajek 2013) and we removed those that were close to the limiting magni- tude of Gaia (with photometric errors preventing a reliable classification according to their position on CMD) and we ended up with 186 targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For generating this CMD, we used G magnitudes and Bp − Rp colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This sample was then restricted to objects with a parallax within 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 mas (140-170 pc), within the < ϖ > ±4·σϖ parallax range of Lupus I core members, but we kept both sources lying close and far from the main filaments of the Lupus I to be inclusive both with the kinematic properties and spatial location of the selected targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We also excluded those ob- jects which were too faint for X-Shooter to observe (J > 15 mag) or older than typical YSOs in Lupus I (inconsistent with the Lupus I core members on our generated CMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Taking into account all these constraints, we identi- fied 43 candidates as potential members of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As shown in the CMD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 1, all of our eventual candidates lie above the MS stars identified by Pecaut & Mamajek (2013) and possess magnitudes and colors very similar to those of Lupus I members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Among these 43 objects, there are targets that i) have never been recognized as poten- tial members of Lupus I (17 objects), ii) were introduced as candidate members of Lupus I according to their con- sistent kinematic and/or photometric properties, but need spectroscopic confirmation (23 objects), iii) were known as members of Lupus I, but were poorly characterized in the literature, and, were never observed with X-Shooter (3 ob- jects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We chose to include all these categories of objects to be followed up by X-Shooter, and the main reason for keeping the third category was that with X-Shooter spec- troscopy we can determine their radial velocity (RV) and projected radial velocity (v sin i), or further explore their chromospheric and accretion properties in a more detailed fashion than previously done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Targets in this category are Sz 70 (Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 1994), 2MASS J15383733-3422022 (Comer´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2013), and 2MASS J15464664-3210006 (Eisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Among the eight objects selected in Lupus I in the unbiased photomet- ric survey by Comer´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2013, see their Table 2), only three were selected by our criteria and are those for which these authors provide stellar parameters, qualifying them as genuine YSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The other five were suspected to be fore- ground objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Indeed, we confirmed that the astrometric parameters of the latter are out of range of our selection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As a final step, we cross-matched our full sample of 43 objects with the OmegaCAM Hα survey in Lupus (see Beccari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018, for details of this survey), with only 4 being recognized as Hα emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This confirms that many potential YSOs may have escaped detection in Hα imag- ing surveys and motivated us to spectroscopically charac- terize our full sample, giving a high priority to the four OmegaCAM Hα emitters as potentially strong accretors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Observations The observations were done with the X-Shooter spectro- graph (Vernet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2011) at the VLT, within a filler pro- gram, and terminated at the end of the observing period, when only ∼28% of the proposed sample was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Hence, of the 43 proposed targets, only 12 were eventu- ally observed which are fully characterized in this paper, and are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The list of the targets that were not observed is reported in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' These 12 targets were selected by ESO staff from the list of our proposed 43 targets, and include all of the Hα emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Although the observed sample is small, all the 12 observed targets were confirmed to be YSOs whose physical and chromo- spheric/accretion properties are worth to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For two stars the OBs were not validated by ESO observing 2 Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Table 1: Lupus I core members known from the literature (measurement errors are displayed in parenthesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The column under Prob stands for the UCL membership probability percentage of the targets calculated by BANYAN Σ (Gagn´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name α (J2000) δ (J2000) ϖ µα∗ µδ RV Prob age (h:m:s) (d:m:s) (mas) (mas/yr) (mas/yr) (km/s) % Myr RX J1529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7-3628 15 29 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='26 –36 28 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09) –14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='69(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='10) –19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='66(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='08) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='90(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='27)a 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 IRAS 15334-3411 15 36 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='92 –34 21 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='89(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13) –11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='80(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='19) –19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='84(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='12) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 Sz 65/V∗ IK Lup 15 39 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='77 –34 46 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='44(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05) –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='27(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='12) –22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='24(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07) –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='70(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='00) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9b Sz 66 15 39 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='28 –34 46 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='36(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09) –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='60(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='19) –21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='56(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='12) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='40(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='80) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9b RX J1539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7-3450A 15 39 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='38 –34 51 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='54 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='40(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04) –15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2) –20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='18(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='15) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1e a Gaia Collaboration (2018) b Both RV and age are obtained by Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2017) c Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2006) d RV for this YSO candidate is the optimal RV determined by BANYAN Σ as a member of UCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' e Age obtained by Comer´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 1: CMD of all the potential members of Lupus I in our original sample of 43 objects (blue dots), with the MS stars (Pecaut & Mamajek 2013) (orange dots) and the Lupus I core members (red triangles) included in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' staff (due to not fulfilling some of our requirements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' But the spectra are nevertheless useful for classification pur- poses and are used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' X-Shooter spectra are divided into three arms (Vernet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2011), the UVB (λ ∼ 300–500 nm), VIS (λ ∼ 500- 1050 nm), and NIR (λ ∼ 1000–2500 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We decided to observe all our targets with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='′′0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='′′9, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='′′9 slit widths (for UVB, VIS, and NIR arms respectively) for one or two cycles based on their J band magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For our faintest objects with J > 14 mag, we considered two cycles of ABBA nodding mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Among our observed targets, only 2MASS J15551027-3455045 belongs to this category, and due to its faintness, the final signal-to-noise ratio (SNR) of its spec- tra was lower than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The exposure time for each arm and the total execution time taking into account the overheads are reported for each target in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For our brightest target, TYC7335-550-1 with J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='65 mag, we decided that only one cycle of ABBA nodding would be sufficient for our scientific aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For some targets with a higher scientific significance to our program or because of their faintness, we decided to also observe telluric standard stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Only a few of our targets (analyzed in this work) did not have a telluric star observa- tion included in their observation block (OB) and these are UCAC4 273-083363, 2MASS J15414827-3501458 (with J = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55 mag and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05 mag respectively), UCAC4 269-083981 (J = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='72 mag), and Gaia DR2 6014269268967059840 (J = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='64 mag) which had a lower scientific priority for our program – either were not lying on the main filament, were not strong candidates for membership in Lupus I, were not 3 OurLupusICandidates Pecaut and Mamajek Objects Lupus ICore Members G 10 15 20 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 5 Bp-RpMajidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Table 2: Objects observed with X-Shooter (measurement errors are displayed in parenthesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The column under Prob stands for the UCL membership probability percentage of the targets calculated by BANYAN Σ (Gagn´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The four candidates detected in the OmegaCAM Hα imaging survey are flagged with ( Hα) right to their names (See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name α (J2000) δ (J2000) ϖ µα∗ µδ Prob G (h:m:s) (d:m:s) (mas) (mas/yr) (mas/yr) % (mag) Partially known targets: 2MASS J15383733-3422022 15 38 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='34 –34 22 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='26 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='79(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='15) –18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='25(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='26) –24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='15(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='19) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='78 Sz 70 15 46 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='99 –34 30 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='21) –12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='58(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='39) –22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='16(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='25) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='50 Candidates: TYC 7335-550-1a 15 36 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55 –34 45 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='54 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='26(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07) –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='93(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='43) –19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='51(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='31 2MASS J15361110-3444473b ( Hα) 15 36 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09 –34 44 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='82 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='83(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='29) –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='56(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='29) –20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='21(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='23) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='92 2MASS J15523574-3344288c ( Hα) 15 52 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='74 –33 44 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='87 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='98(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='17) –20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='37) –22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='17(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='23) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06 2MASS J15551027-3455045d ( Hα) 15 55 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='28 –34 55 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='67 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='78(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='26) –11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='54) –23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='94(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='31) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='23 2MASS J16011870-3437332e ( Hα) 16 01 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='70 –34 37 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='35(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02 Gaia DR2 6010590577947703936 15 56 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='36 –36 11 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='73 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='83(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='11) –15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='64(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='24) –25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='82(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='15) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='37 2MASS J15414827-3501458g 15 41 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} 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+page_content='18) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='98 UCAC4 273-083363 15 46 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='15 –35 24 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='99(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06) –18.' metadata={'source': 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22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='68(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='24) –16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='23(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='37) –22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='29(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='27) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='39 a Proposed candidate member of Lupus I by Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' b aka Gaia DR1 6014141205925321984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' c aka Gaia DR2 6012155767105823616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' d aka Gaia DR2 6011827867821601792, candidate Lupus I member also proposed by Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' e Gaia DR3 6011165313293141760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' f Dipper, candidate member of Lupus I also proposed by Nardiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' g aka SSTc2dJ154148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3-350145, a candidate Lupus I member previously proposed by Comer´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Table 3: Observing log of the new candidate members of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name Date Exposure time Seeing Ttot airmass SNR J Grade (yyyy-mm-dd) (sec) (′′) (hour) (mag) 2MASS J15383733-3422022 2021-08-03 1920/1800/1920 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='72/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='72/1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='52/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='03 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9/67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8/132.' metadata={'source': 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+page_content='14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9/51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='08 A 2MASS J15414827-3501458 2021-07-14 600/500/600 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='12 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2/232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05 A UCAC4 273-083363 2021-07-14 600/500/600 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='33/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='29/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='08 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6/171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55 A Gaia DR2 6014269268967059840 2021-08-04 1800/1700/1800 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='49/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='49/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5/26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1/50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='64 Cb Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Date of observation, exposure time allocated to each arm, mean seeing, and SNR (in order for UVB, VIS, and NIR wavelengths) as well as the total execution time, mean airmass, and the observation grades (as provided by the ESO observing staff) are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' a UCAC4 269-083981 had an out of constraint seeing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='′′0 which was exceeded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' b Gaia DR2 6014269268967059840 was reported to have an out of constraint seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Hα emitters, or were not faint for X-shooter to necessitate the observation of a telluric template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As we will detail later, we will also adopt a different approach to remove telluric lines for these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For the targets containing telluric observation in their OBs, the same nodding strat- egy as those of the targets was employed to minimize noise 4 Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud and cosmetics, with an airmass as close as possible to the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The airmass and seeing reported in Table 3 are averaged over the exposure times for each arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Data reduction The data used in this work have been reduced with the X- Shooter pipeline xshoo of version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='12 and higher1, and hence they have been de-biased, flat-fielded, wavelength- calibrated, order-merged, extracted, sky-subtracted and eventually flux-calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The result of this pipeline output is an ESO one-dimensional standard binary table and the two-dimensional ancillary files ready for scientific analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Flux calibration based on the photometric data available in the literature was done later directly on the available spectra, along with the telluric removal process which is not done for the distributed spectra reduced by the xshoo pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We used the Image Reduction and Analysis Facility (IRAF, Tody 1986, 1993) to remove the telluric lines from the target spectra and to flux calibrate them, as well as to derive the stellar parameters from the spectra, which we shall discuss in detail in the upcoming sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Since the strategy for arranging our observation blocks did not include wide slit observations, the flux calibration of our targets totally relies on the photometric data available in the literature, which have been collected in various surveys (with the corresponding flux errors of e-16 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='m−2 for the UVB arm, e-16 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='m−2 for the VIS arm, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5e-15 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='m−2 for the NIR arm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For some of our faint objects, we only had access to very limited photometric data and had to cal- ibrate the UVB portion of the spectra in accordance with the available photometric data in the VIS range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For the objects with observations of telluric standard stars, we removed the telluric lines and molecular bands using the IRAF task Telluric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For the three targets with- out telluric star observations in our sample, which namely are 2MASS J15414827-3501458, UCAC4 273-083363, and Gaia DR2 6014269268967059840, we used the TelFit Python code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This code fits the telluric absorption spec- trum in the observed spectra (Gullikson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2014) using the LBLRTM code which models the line-by-line radiative transfer (Clough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Applying TelFit, we cor- rected the spectra for oxygen and water molecular bands in the visible range (∼550-1000 nm), as well as for water, oxygen, and CO2 molecular bands in the NIR (∼1000-2500 nm) (for the details on the wavelength ranges where these molecular bands dominate the spectrum the reader is re- ferred to Smette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Data Analysis There are several immediate aims that we planned to fulfill through our program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' With the X-Shooter spectra, we can confirm the youth of the selected candidates through the presence of the Li i (6708 ˚A) absorption line, in addition to Hα emission, and other lines of the Balmer series as further hints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We also determine the spectral type (SpT) classifi- cation and the determination of stellar physical parameters such as effective temperature (Teff), luminosity (L), mass (M) and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' It is also possible that some of our candidates 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='org/sci/software/pipelines/ xshooter/ may belong to Scorpius-Centaurus Association (with an age 10-18 Myr, UCL sub-association) rather than Lupus (1-2 Myr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We can single out these objects once we have fully characterized them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The disentanglement between the two associations would be useful for clarifying their relation- ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Using spectral lines of the Balmer series, we will also measure the accretion luminosity (Lacc) and mass accretion rate ( ˙Macc) of those objects that we qualify as accretors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In the following, we describe the methods used for achieving our immediate goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Spectroscopic analysis methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Spectral typing and line equivalent widths To obtain the SpTs of our objects, we first compared the spectrum obtained with X-Shooter’s VIS arm with a li- brary of visible spectra of already characterized stars and brown dwarfs formerly observed by X-Shooter (Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For the quantitative spectral typing of the stars, we then calculated the spectral indices described in Riddick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2007) based on the ratios of the average flux of molecular absorption bands within narrow wavelength re- gions, yielding in all cases an uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 subclasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For TYC 7335-550-1 and UCAC4 269-083981, which are brighter than the rest of the targets and do not show clear molecular bands in their spectra suitable for measuring the Riddick’s indices, the SpT is instead estimated through the Teff obtained by the ROTFIT code (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The results can be found in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The EW of the atomic lines reported in Table 5 is mea- sured by taking an average over i) the direct integration of the line profiles between two marked pixels and ii) fitting a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The errors associated with these values thus report the difference between the measurements made with these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' There are cases for which we could not de- tect the Li i line at 6708 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Hence, for these objects we only report an upper limit on the measurement of EWLi i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As suggested by Cayrel (1988), a three-sigma upper limit on the flux of the lithium line can be calculated as: dEW = 3 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06 � (FWHM)dx/(S/N), (1) in which FWHM is the full width at half maximum, S/N is the signal-to-noise ratio, and the bin size (dx) can be fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 ˚A for the VIS arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The values of these measurements are reported in Table 5 and Table 6 for TYC7335-550-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' ROTFIT We used ROTFIT as the basis of our analysis for assessing the stellar parameters of our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Using ROTFIT, we evaluated their RV, v sin i, and surface gravity (log g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The version of ROTFIT used for this purpose is the one designed for the optimal usage of the X-Shooter spectra (Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The stellar parameters obtained with ROTFIT can be found in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The fitting process with ROTFIT code was carried out within a veiling (the UV excess continuum that influences the entire photosphere of the star from UVB to NIR) range from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' None of our objects showed significant veiling, hence the veiling parameter for all our studied targets in this paper is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 5 Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Table 4: Physical stellar parameters of the targets obtained with the ROTFIT code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name Teff log g vsini RV Prob (K) (km/s) (km/s) % 2MASS J15383733-3422022 3111±70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13 <8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 Sz 70 3038±76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='11 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 TYC 7335-550-1 4488±140 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='22 <8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 2MASS J15361110-3444473 2883±104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='12 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9 2MASS J15523574-3344288 2981±44 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='10 <8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 2MASS J15551027-3455045 2700±103 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='60±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='11 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9 2MASS J16011870-3437332 3121±90 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='14 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 UCAC4 269-083981 3846±47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='11 <8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 Gaia DR2 6010590577947703936 3154±72 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 2MASS J15414827-3501458 3213±94 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='23 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 UCAC4 273-083363 3211±56 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='15 <8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 Gaia DR2 6014269268967059840 3019±108 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='14 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The column Prob represents the probability of the target to be member of Lupus I according to BANYAN Σ, which is based on the RVs measured with ROTFIT and the kinematic properties reported by Gaia DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Table 5: EWs of the relevant lines indicating the chromospheric and accretion tracers for our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Negative values indicate the lines that are in emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name EWLi i EWHα EWHβ EWHγ EWHδ WHα(10%) (˚A) (˚A) (˚A) (˚A) (˚A) (km/s) 2MASS J15383733-3422022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04 –8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='92 –7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='71±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04 –7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='99±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='21 –7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='52 128±18 Sz 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05 –43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='37±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='97 –9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='97±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07 –10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='28±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04 –11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='14±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='51 366±14 2MASS J15361110-3444473 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='25a –71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='77 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 292±14 2MASS J15523574-3344288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09 –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='76 –10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='88 –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='49 146±9 2MASS J15551027-3455045 b –88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='17 –29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='85 –6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='24 –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='49 229±14 2MASS J16011870-3437332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='03 –21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='47±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='59 –21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='61±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='28 –19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='75 –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='34±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='18 274±14 UCAC4 269-083981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='56±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='63±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='08 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='56±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='24 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='21 174±5 Gaia DR2 6010590577947703936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06 –6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='38 –6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='25 –6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='97±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09 –6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='22 183±5 2MASS J15414827-3501458 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='012a –10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='53 –9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='61 –10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='29 –10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 210±18 UCAC4 273-083363 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='017a –11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='94 –11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='45 –11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='15±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='35 –8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='67 155±9 Gaia DR2 6014269268967059840 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='047a –17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='53±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 219±14 a Three-sigma upper limits on the measurement (read Subsection for further explanation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' b Li I line was affected by a cosmic ray hit and could not be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Table 6: EWs of the relevant lines indicating the chromospheric and accretion tracers for TYC 7335-550-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name EWLi i EWHα EWHϵ EW H Ca ii EW K Ca ii EW 8498 Ca ii EW 8542 Ca ii EW 8662 Ca ii (˚A) (˚A) (˚A) (˚A) (˚A) (˚A) (˚A) (˚A) TYC 7335-550-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='16 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='14 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='19 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='03 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The EW of Hα, Hϵ, and Ca ii lines relate to the emission in the cores of these lines obtained by the subtraction of the photospheric template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Physical parameters We used the bolometric correction (BC) relation proposed by Pecaut & Mamajek (2013, 2016) for evaluating the lu- minosity in both V and J bands and the radius of can- didates according to their observed parallaxes and magni- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This is possible because none of our targets show significant near-IR excess (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2) nor strong veiling (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For the objects only resolved in Gaia DR2 catalog, the BC relationship introduced by the Gaia DR2 science team2 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In order to have a correct estimation of the lu- minosity, we have also taken into account the extinction 2 https://gea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='int/archive/documentation/ GDR2/Data_analysis/chap_cu8par/sec_cu8par_process/ ssec_cu8par_process_flame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='html of the objects which was determined using the grid of X- Shooter spectra of zero-extinction non-accreting T Tauri stars (Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2013), as explained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 of Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' It is evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2 that the targets have low extinction and little or no NIR excess, probably except for the rightmost point in the diagram, which corre- sponds to 2MASS J15361110-3444473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The relatively red- der H −Ks color of this object in comparison with the oth- ers, may be due to the presence of an unresolved very late- type companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This will be further discussed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Once the Teff (from ROTFIT), luminosity, and ra- dius of the targets are derived, their mass, age, and log g can be evaluated through various evolutionary tracks and isochrones available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The corresponding values of these parameters, which are reported in Table 6 Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Table 7: Physical stellar parameters of the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name SpT AV L⋆ R⋆ M⋆ Age log g (mag) (L⊙) (R⊙) (M⊙) (Myr) 2MASS J15383733-3422022 M5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='012±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7±5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 Sz 70 M5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 TYC 7335-550-1 K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='60±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='50±1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 2MASS J15361110-3444473 M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 1.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='77±5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 2MASS J15523574-3344288 M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='03 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3±3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 2MASS J15551027-3455045 M7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='71±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 2MASS J16011870-3437332 M5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='013±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55±5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 UCAC4 269-083981 M0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2±1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 Gaia DR2 6010590577947703936 M4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='017±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8±4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 2MASS J15414827-3501458 M4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='82±1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 UCAC4 273-083363 M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='069±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='63±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 Gaia DR2 6014269268967059840 M6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='03 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='46±2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The methods used for calculating SpT, AV , L⋆, and R⋆ are described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' M⋆, log g, and age of the stars are evaluated according to Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2015) isochrones, except for TYC 7335-550-1, for which we have used the MIST isochrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The SpT for TYC 7335-550-1 and UCAC4 269-083981 (in italic) are obtained using the temperatures derived by the ROTFIT code (Table 4) and the SpT–Teff calibration of Pecaut & Mamajek (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The errors associated with SpT and AV are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 subclasses and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 mag respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The errors associated with mass and age are internal to the tracks and isochrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2: J − H (mag) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' H − Ks (mag) diagram of all our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The red dots show the chromospherically-dominant targets, the cyan dots are the accretors, and the blue line represents the colors of MS objects, down to spectral type M9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The normal reddening vector, shown with the black arrow, corresponds to AV = 2 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The rightmost target is 2MASS J15361110-3444473 which is suspected to be a bi- nary, hence, it might have color contribution from a second target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 7, are derived by the evolutionary models calculated by Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The Hertzsprung-Russel (HR) dia- gram of the Lupus I targets, including the previously known and the newly discovered members, is displayed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' One of our targets, namely TYC 7335-550-1, is much brighter than the other stars investigated in the present work, and falls outside the range covered by the Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2015) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Therefore, to derive its stellar parameters, we used MESA Isochrones and Stellar Tracks (MIST Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Dotter 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For modeling pur- poses, we assumed that all targets have solar metallicity (Baratella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Some of our objects display strong emission lines which is a sign of noticeable chromospheric activity (see the EW of some of the chromospheric activity indicators in Table 5) or magnetospheric accretion from a circumstellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' If the magnetic activity is relevant, the position of the star in the HR diagram can be significantly affected by photospheric starspots and by the changes in the internal structure in- duced by the magnetic fields (see Gangi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2022, for in- teresting cases in the Taurus SFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In this case, isochrones that do not take into account these effects (such as Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2015) may lead to systematic effects in the estimate of mass and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In particular, they may indicate an age half the real age of star (Asensio-Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Feiden 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This is crucial for our study which also aims at de- termining the membership of the stars in Lupus I or UCL associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Thus, in addition to MIST and the isochrones provided by Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2015), we used other isochrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' A set of evolutionary models that considers the mag- netic activity of the stars is the Dartmouth magnetic isochrones (Feiden 2016), which we also use in this work to estimate the ages of all our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' These isochrones were originally developed for estimating the age of the Upper Scorpius members (11±2 Myr), almost coeval to the UCL (15±3 Myr), and hence are quite useful to fulfill our sci- entific aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In addition to Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2015) and MIST models, we used both Dartmouth std and Dartmouth mag (Feiden 2016, and the references therein) models, as well as PARSEC + COLIBRI S37 (Bressan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Pastorelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For all our targets, we obtained over- estimated ages using PARSEC + COLIBRI S37 isochrones totally inconsistent with the other isochrones, hence, we do not report our results obtained with this isochrone to avoid confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The results of age estimation with all the other isochrones are included in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For all the mod- els, we have assumed our targets have solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For PARSEC models, extinction is also a free parameter that can be fixed and was thus set to the corresponding ex- tinction of the targets reported in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Eventually, we would like to point out that it is not straightforward to state which targets may have an under-estimated age, par- ticularly in the case of objects that are as young as the members of Lupus I and UCL considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 1 J-H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 1 H-KsMajidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3: log L⋆(L⊙) vs log Teff (K) diagram for all our tar- gets (cyan and red dots represent accretors and non- accretors, respectively), together with the previously char- acterized Lupus members (black dots, Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2019, sub-luminous objects are not plotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Blue dashed lines represent evolutionary tracks of Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2015) for stars with masses indicated by the number (in M⊙) next to the top or bottom of each track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The red lines indicate isochrones calculated with the same models at ages of 1, 3, 30 Myrs, and 10 Gyrs, from the right to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Lupus I membership criteria According to the works previously done in the Lupus com- plex (Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2014, and the references therein), in ad- dition to the kinematical properties expressed by the Gaia parallax and proper motions, membership criteria in this star-forming region are: i) the presence of lithium in their atmospheres, which is the main signature of youth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Despite the obviousness of this criterion, there are previously acknowledged members of the Lupus cloud that lack lithium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' An example is rep- resented by Sz 94 in the Lupus III cloud (Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Biazzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' ii) an age con- sistent with the core members of the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Although the estimated age of the Lupus complex is ∼ 1–2 Myr, there are previously recognized members of the complex that exceed this age range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Examples of such targets are AKC2006 18 and AKC2006 19 in Lupus I, although their apparent old age may be ascribed to disks seen edge-on that obscure the central objects making them sub-luminous on the HR diagram (see other examples in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 in Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' iii) an RV consistent with the values of the genuine members of the Lupus I (Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' If an object does not match the membership criteria defined above, there are two possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Either it is older than the UCL (age>20 Myr), and we would hence identify it as field star;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' or it has a consistent age with UCL (∼15 Myr) which would confirm its membership to this sub-cloud of the Scorpius-Centaurus stellar association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' To this aim, we have used various isochrones to evaluate the age of our tar- gets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 4: |EWHα| vs SpT of our targets with the weak lined T Tauri stars studied by Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2013, blue dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The cyan dots represent accretors, and the red dots represent chromospherically-dominant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The horizontal lines in red represent the thresholds that separate non-accreting and accreting objects considering their SpTs (White & Basri 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Accreting objects There are several criteria for determining whether an object is actively accreting matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Usually, an accreting object is characterized by strong emission lines, strong UV and NIR continuum excess emission, or structured line profiles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=', Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Here, to establish whether an object is an accretor, we use the criterion proposed by White & Basri (2003) which distinguishes the accreting and non-accreting objects based on the EW of their Hα emission versus SpT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The method used in this paper for calculating the Lacc (ac- cretion luminosity) and ˙Macc (mass accretion rate) of our targets involves measuring the line luminosity of the emis- sion lines of the accreting targets and using the established relationships between the Lline (for each emission line) with Lacc (Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We quote the eventual accretion line luminosity that is obtained this way as log Lacc−line in Table 8 and Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The whole procedure that we carried out for this task can be summarized as follows: we corrected the spectra for telluric lines and flux-calibrated them,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' then measured the flux at Earth of the emission lines by integrating their pro- file above the local continuum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' corrected the flux for ex- tinction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' calculated the luminosity of each emission line by multiplying the flux at Earth for 4πd (adopting a distance d = 1000/ϖ pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' with ϖ in mas),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' and eventually took an average over all the values of log Lacc−line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We chose Hα, Hβ, and Hγ emission lines to measure the accretion lumi- nosity of our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' After deducing the log Lacc for each target, we obtained their ˙Macc accordingly (Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The results of our measurements are presented in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Among all our targets, only TYC 7335-550-1 does not show Hydrogen emission lines above the continuum, and its Hα line is instead in absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For this target, we used ROTFIT to subtract the photospheric template in or- der to measure the flux of the emission components that fill the cores of Hydrogen and Ca ii lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This method has been successfully used to emphasize chromospheric emis- sion or a moderate accretion whenever the photospheric 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 0 (o) logL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02 Y 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 logTeff (K)100 10 IEWHαl 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1 K3 K4 K5 K6 K7 K8 K9 MO M1M2 M3 M4 M5 M6 M7 M8M9M10 SpTMajidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud flux is large and the emission is only detectable as a filling of the line core or an emission bump within the photospheric line wings that do not emerge above the continuum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=', Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2015, 2017, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The spec- tral subtraction allows us to recognize and measure the EW of the emission that fills in the Hα line (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Adopting the same method, we measured the fluxes of the H&K lines of the Ca ii and in the cores of the three infrared lines of the Ca ii IRT at λ =849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8, 854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2, and 866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We were also able to separate the contribution of the Hϵ emission from the nearby Ca ii H line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 5: X-Shooter spectrum of TYC 7335-550-1 in the Hα region, normalized to the local continuum (black solid line) along with the inactive photospheric template (red dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The latter is produced by ROTFIT with the BT- Settl synthetic spectrum at the Teff and log g of this target that is degraded to the resolution of X-Shooter, rotationally broadened, and wavelength shifted according to the target RV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The difference target − template is displayed at the bottom of the box and emphasizes the Hα emission that fills in the line core (green hatched area), which has been integrated to obtain the Hα line flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Stellar parameters and membership The physical stellar parameters that we obtained from the spectral analysis and the HR diagram as described in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 are reported in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The stellar pa- rameters obtained with ROTFIT are presented in Table 4, where the membership probability was recalculated with the BANYAN Σ using the values of RVs measured with ROTFIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Both Teff and log g found with ROTFIT are in good agreement with those derived from SpT and the HR diagram and reported in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We note that, at the resolution of the X-Shooter VIS spectra, the minimum value of v sin i that can be measured is 8 km/s (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=', Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017) and hence this value should be considered as an upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' With this knowl- edge, we can classify targets with v sin i < 8 km/s as slow rotators, and those with v sin i > 40 km/s as fast rotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Moreover, the large RV range of the bona fide members of Lupus I (∼ –5-12 km/s, according to Table 1) denies us to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 6: a) X-Shooter UVB spectrum of TYC 7335-550-1 in the Ca ii H&K region (black solid line) along with the in- active photospheric template (red dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' b) and c) Residual (target − template) spectrum around the Ca ii K and Ca ii H line, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The hatched green areas mark the residual H and K emissions that have been integrated to obtain the EWs and fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The purple-filled area relates to Hϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' d) and e) Observed Ca ii IRT line profiles (black solid lines) with the photospheric template overlaid with red dot- ted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The residual spectra are shown at the bottom of each panel shifted downward by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 in relative flux units for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' put a strict constraint on the Lupus I membership of our targets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The RVs of the Lupus I members confirmed in this work, however, are within a smaller range with re- spect to the previously confirmed core members of the same region, except for 2MASS J15361110-3444473 which may or may not be a Lupus I member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' According to our full characterization, besides TYC 7335-550-1 which is a K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 type star, all the others have M spectral types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Three-quarters of our targets, have spectral types between M4 and M6, which is in accordance with the previously identified members of the Lupus complex (Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Krautter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Herczeg & Hillenbrand 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Comer´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The ages of these targets cover a large range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7-11 Myrs, with masses in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1 M⊙ (as also indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1, Sz 70 and 2MASS J15383733- 3422022 were partially known in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The phys- ical parameters that we report here for Sz 70 are in excel- lent agreement with the results of Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For 9 Tyc7335-550- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 LAW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 6520 6540 6560 6580 6600 x (A)Tyc7335-550-1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 3920 3940 3960 3980 (A) 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0 Call K Call H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 He 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 3926 3929 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='32 3935 3938 3941 3962 3965 3968 3971 3974 3977 ^ (A) > (A)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content="0 0'0 8480 8500 B520 8540 8560 8640 8650 8660 8670 8680 8690 ^ (A) A (A)Majidi et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Table 8: Accretion luminosity of the accretors derived from the line luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The mass accretion rates are derived from the average of these values (Lacc−average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name log Lacc−Hα log Lacc−Hβ log Lacc−Hγ log Lacc−average log ˙Macc (L⊙) (L⊙) (L⊙) (L⊙) (M⊙yr−1) Accretors: Sz 70 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='73 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='95 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='91 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='85 –9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='22 2MASS J15361110-3444473 –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='62 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='62 –10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='21 2MASS J15551027-3455045 –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='85 –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='95 –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='96 –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='92 –10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='20 2MASS J16011870-3437332 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='29 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='20 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='16 –10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='91 Active stars: 2MASS J15383733-3422022 –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='41 –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='43 –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='52 –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='45 –12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='21 2MASS J15523574-3344288 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='62 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='87 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='80 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='75 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='46 UCAC4 269-083981 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='24 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='22 Gaia DR2 6010590577947703936 –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='12 –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09 –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='03 –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='08 –11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='86 2MASS J15414827-3501458 –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='97 –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='93 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='99 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='63 UCAC4 273-083363 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='14 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='19 –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='11 –10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='89 Gaia DR2 6014269268967059840 –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='22 –11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07 Table 9: Accretion luminosity of TYC 7335-550-1 derived from its line luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Its mass accretion rate is derived from the average of these values (Lacc−average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name log Lacc log Lacc log Lacc log Lacc log Lacc log Lacc log Lacc log Lacc log ˙ Macc Hα Hϵ Ca II (H) Ca II (K) Ca II (8498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02) Ca II (8542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09) Ca II (8662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='14) average (L⊙) (L⊙) (L⊙) (L⊙) (L⊙) (L⊙) (L⊙) (L⊙) (M⊙yr−1) TYC 7335-550-1 –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='43 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='82 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='31 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='19 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='94 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='88 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='16 –9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 7: RV of our accretors (cyan dots), chromospherically- dominant targets (red dots), and the Lupus I core members (black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2MASS J15383733-3422022, our results are again in good agreement with those reported by Comer´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2013), but their difference emanates from the fact that Comer´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2013) measured AV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 mag for 2MASS J15383733- 3422022, which results in a discrepancy in luminosity, mass, and radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Equivalent widths The EWs of several lines are quoted in Table 5, and sepa- rately for TYC 7335-550-1, in Table 6, as for this star the flux and EW measurements were performed by subtracting the photospheric spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We could not detect the Li i line in the spectra of some of our targets for various reasons, which can be i) solely due to the low SNR of their spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' ii) based on the simu- lations conducted by Constantino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2021), for initially lithium-rich stars we know that slow rotators could deplete their lithium (also considering their SpT) at early ages (< 10 Myr), while fast rotators tend to retain their lithium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' iii) a combination of the low SNR and fast rotation (which may be especially true for Gaia DR2 6014269268967059840), which would further complicate the issues associated with Li i detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' iv) a complex relationship between the ac- cretion processes, early angular momentum evolution, and possibly planet formation for young stars (∼ 5 Myr) that yet needs to be fully explored (Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' v) no obvious relationship between the rotation of YSOs and the lithium depletion process (Binks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The non-detection of Li i in the spectra of some objects has been reported as a three-sigma upper limit on the flux of the lithium line which is a sensitive enough threshold for separating them from objects containing lithium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Evolutionary status of the targets The main properties and final status of all our targets are summarized in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Based on all the criteria discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2, we confirm that all our objects are YSOs, with ages < 11 Myrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The targets 2MASS J15414827-3501458 and UCAC4 273-083363 do not show the presence of the lithium line in the spectra, but their effective temperature is compatible with the possible presence of a large amount of Li depletion for fully convective pre-main sequence stars (Bildsten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Lithium depletion was investigated in several star forming regions, like some sub-groups of Orion (Palla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Sacco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2007), but also in Lupus I and III (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=', Biazzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Due to their very young age (< 4 Myr), 10 12 10 8 6 (s/w>) 2 4 6 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 Parallax (mas)Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Table 10: Overall status checklist for our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The rotation column refers to fast (F) or slow (S) rotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name Membership Active Accreting Contains Li i Rotation Av Conclusion (UCL/Lup I) (yes/no) (yes/no) (yes/no) (F/S) (mag) 2MASS J15383733-3422022 Lup I yes no yes S 0 Genuine member of Lup I Sz 70 Lup I yes yes yes S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 Genuine Lup I member + wide companion candidate TYC 7335-550-1 Lup I yes no yes S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 Genuine member of Lup I + wide companion candidate 2MASS J15361110-3444473 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' yes yes no S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='75 Unresolved binary (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=') + wide companion candidate 2MASS J15523574-3344288 Lup I yes no yes S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 New member of Lup I 2MASS J15551027-3455045 Lup I yes yes ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='75 Genuine member of Lup I 2MASS J16011870-3437332 Lup I yes yes yes S 0 New member of Lup I UCAC4 269-083981 Lup I yes no yes S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 Genuine member of Lup I Gaia DR2 6010590577947703936 Lup I yes no yes F 0 New member of Lup I 2MASS J15414827-3501458 Lup I yes no no F 0 Genuine member of Lup I UCAC4 273-083363 Lup I yes no no S 0 Genuine member of Lup I Gaia DR2 6014269268967059840 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' yes no no F 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' we therefore classify 2MASS J15414827-3501458 and UCAC4 273-083363 as Lupus I members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Newly discovered members of Lupus I in this work are 2MASS J15523574- 3344288, 2MASS J16011870-3437332, and Gaia DR2 6010590577947703936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' There are also two objects analyzed in this work that we could not identify either as a member of Lupus I or UCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' These are 2MASS J15361110-3444473, whose spec- trum indicates an unresolved binary star of spectral types M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 (VIS arm) and M8 (NIR arm), and we could not detect lithium in its spectrum (see Appendix C for more details on the analysis of this target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' However, we would like to emphasize that 2MASS J15361110-3444473 is an ac- creting source that has consistent kinematic and physical properties with the genuine members of Lupus I, hence, there is a possibility that this target also qualifies as a new member of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The other object is Gaia DR2 6014269268967059840, for which we acquired a spectrum with poor SNR (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2 for details on the observation conditions of this target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The poor SNR of its UVB spec- trum hindered us from carrying out any measurements on its Hβ and Hγ lines in emission (as reported in Table 5), which also leads to evaluating its accretion properties only according to its Hα emission line (as reported in Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Therefore, the non-detection of lithium in its spectrum can be purely due the poor SNR in the VIS arm, and we do not approve nor rule out the possibility of this target being a member of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We hence confirm that all our targets are YSOs, with Hydrogen lines in emission above the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Therefore, this investigation suggests that although only four of our targets were retrieved as Hα emitters in the OmegaCAM survey (flagged in Table 2), it is likely that our entire sample of 43 candidate YSOs could include Hα emitters or objects with filled Hα profiles, which can only be confirmed by a high- or mid-resolution spectroscopic study or in deep X- ray surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As a further investigation to strengthen our argument, we cross-matched all of the Lupus I core members included in Table 1 with the OmegaCAM survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Except for three objects, they were all retrieved in the survey as Hα emit- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' These exceptional three core members are RXJ1529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7- 3628 (which was out of the field of view of the survey), RX J1539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7-3450B and Sz 68/HT Lup C, for which only one object was resolved in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Combining this result with the results of this paper, we emphasize the necessity of observing all our sample to characterize all the members of Lupus I that have escaped the Hα surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Accretion versus chromospheric–dominated objects We realized that four of our targets in the current sam- ple are accretors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We measured the Lacc of these tar- gets, in addition to our chromospherically-dominant objects (Table 8 and Table 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The measured Lacc for all our tar- gets are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In the same figure, we have included the limits suggested by Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2017b) for objects with Teff > 4000 K and Teff < 4000 K, be- low which the chromospheric activity of targets is domi- nant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' All our four accretors exceed this limit for targets with Teff < 4000 K, confirming that they are accretion- dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The rest of our targets within the same ef- fective temperature range are below this threshold, which make them chromospheric-dominated objects, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2MASS J15523574-3344288, however, lies exactly on the threshold between these two regimes, which is consistent with its significant Hα emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We also emphasize that this target was retrieved in the OmegaCAM survey as an Hα emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 9 shows the ˙Macc versus M∗ for the four accre- tors in our sample in comparison with the Lupus members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Among the four accretors, 2MASS J15551027-3455045 is the least massive target, and has a very high mass accretion rate in comparison with Lupus members of similar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This target also stands above the double power-law rela- tionship between ˙Macc and M∗ established by Vorobyov & Basu (2009), based on modeling self-regulated accretion by gravitational torques in self-gravitating disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As concluded by Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2017), only the strongest accretors stand above this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Our three other accretors have values of mass accretion rates typical of Lupus accretors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Finally, it is worth noting that three of our accretors (Sz 70, 2MASS J15361110-3444473, and 2MASS J16011870- 3437332) have WHα(10%)>270 km/s (see Table 5), which is expected from accreting stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Our chromospherically- dominant targets have much narrower Hα profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 11 Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 8: Log < Lacc/L∗ > vs Teff for all our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The cyan dots represent accretors, and the red dots represent chromospherically-dominant targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The lines indicate the limit below which the chromospheric activity for a star is dominant (Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017b), for two regimes of stars with Teff ≤ 4000 K (the diagonal blue line) and those with Teff ≥ 4000 K (the horizontal orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 9: Log Macc(M⊙/yr) vs log M∗(M⊙) for the four accre- tors in our sample (cyan dots), together with the previously identified members of the Lupus (black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The blue crossed squares represent the substellar accreting compan- ions detected at wide orbits by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2014) around GQ Tau, GSC 06214 00210 and DH Tau as labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2MASS J15551027-3455045, GQ Lup c and 2MASS J16085953- 3856275 are also labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2MASS J15523574-3344288 is labelled as red dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The continuous red line indicates the double power-law prediction of Vorobyov & Basu (2009), while the magenta dashed line shows the prediction of disk fragmentation model by Samatellos & Herczeg (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Discussion In this paper, we analyzed 12 objects observed by X- Shooter out of our original sample of 43 proposed new candidate members of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We confirm that all these 12 objects are YSOs, and ten out of 12 are members of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We could not determine the membership of two of our targets, namely 2MASS J15361110-3444473 and Gaia DR2 6014269268967059840, as explained in the previous Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We could not fully measure the accretion prop- erties of Gaia DR2 6014269268967059840 and hence our analysis in this regard for this specific target is not reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2MASS J15361110-3444473, on the other hand, is a rather (intrinsic) faint object to be followed up by any available spectrographs, but perhaps can be followed up with ALMA to understand whether it is surrounded by a disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Although recognized to have an older age with respect to Lupus I members (9 Myr), it can be still strongly accreting matter, consistent with the members of γ Vel with age ∼10 Myr (Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' One of the interesting targets discussed in this work is TYC 7335-550-1, a lithium-rich K-type star with Hα in absorption and without IR excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We would like to emphasize that YSOs with these particular charac- teristics would never appear in Hα imaging surveys such as OmegaCAM, although one of their main aims is to identify the members of young star forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' All the above points considered, we have fully characterized ten members of Lupus I in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In the following, we will discuss further qualities of our targets, which are mainly based on the data available in the literature in connection with the targets analyzed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Spectral energy distributions / Circumstellar disks For all our objects, we also investigated whether there are hints of continuum flux excess suggestive of circumstellar disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' To this aim, we extracted their SEDs from literature which are collectively exhibited in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For this work, we only concentrate on the morphology and trends of the SEDs of our targets, as well as their near- to mid- infrared photometric data (published by 2MASS and WISE surveys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For generating the SEDs, we have used the follow- ing WISE filters: W1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 microns), W2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 microns), W3 (12 microns), W4 (22 microns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In a parallel paper (Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' in prep), we will study the variability of these stars and model their disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The photometric data for all four accretors significantly deviate from their BT-Settl spectral model (based on their Teff, log g, and zero metallicity) in W3 and W4 filters (with the average flux errors of 5e-17 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='m−2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7e-16 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='m−2 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This trend can be observed for our less massive, stronger accretors 2MASS J15551027-3455045 and 2MASS J15361110-3444473 in all four WISE filters (W1, W2, W3, and W4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' According to Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2014), the morphology of the SEDs of all our four ac- cretors in addition to 2MASS J15523574-3344288 is com- patible with objects surrounded by full disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This is further confirmed by the disk categorization of Bredall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2020) based on Ks−W3 and Ks−W4 magnitudes for Lupus dip- pers, Lupus YSOs, Upper Scorpius and Taurus members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Hence, also according to Bredall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2020), all our four accretors in addition to 2MASS J15523574-3344288 are sur- rounded by a full disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Note, however, that the “valley” around W3 in the SED of 2MASS J15361110-3444473 is typical of those seen in transitional disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For the rest of our targets, we have two categories of circumstellar disks based on the morphology of their SEDs further approved by their Ks − W3 and Ks − W4 magnitudes: i) Evolved disks, which are characterized by only W4 excess with respect to the theoretical BT-Settl model, and are evident in the SEDs of 2MASS J15383733- 3422022, Gaia DR2 6010590577947703936, and Gaia DR2 6014269268967059840 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 11), ii) Debris disks, which are 12 8 (Mo yr-1) GQ Lup 区 GQ/Lup c 区 10 2MASS15551 GSC 06214 b 区 2MASS16085 DH Tau b 12 2 0 logM* (Mo)-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 60 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 4 5000 4500 4000 3500 3000 2500 Teff (K)Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 10: BT-Settl models (in grey) with the photometric data (red dots) for our accretors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' characterized by little to no mid-infrared excess, and is ev- ident in the SEDs of TYC 7335-550-1, UCAC4 269-083981, 2MASS J15414827-3501458, and UCAC4 273-083363 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' High accretion in the low-mass regime Deriving ˙Macc for the lowest mass accretors is relevant for the studies of disk evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' There is growing evidence of a change in the slope of the M⋆– ˙Macc relationship for YSOs with ages of 2-3 Myr at M⋆<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 M⊙ (Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017b and Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017, and see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Such a break could be related to a faster disk evolution at the low-masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Vorobyov & Basu (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' To verify this, the ˙Macc– M⋆ relationship needs to be sampled at much lower M⋆ and ˙Macc values than done so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Our target 2MASS J15551027-3455045 is one of the lowest mass accretors in Lupus (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' With M⋆=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02 M⊙, 2MASS J16085953-3856275 is the accretor with comparable mass reported in the previous Lupus stud- ies (Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Considering the very low mass of this YSO, its accretion rate ˙Macc∼10−11 M⊙/yr (Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2019) is relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Yet the ˙Macc value for 2MASS J15551027-3455045 is about an order of magnitude higher (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' hence, it is one of strongest accretors in Lupus in the mass range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='02–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='03M⊙, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' close to the planetary mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' From modeling of a shock at the surface of a planetary-mass object, Aoyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2021) have pre- dicted much higher Lacc values than what the scaling Lacc– Lline relations for stars would predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The relationships by these authors would yield an even higher ˙Macc value, almost an order of magnitude higher than our estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This ob- ject falls above the model prediction by Vorobyov & Basu (2009), in contrast with the idea of faster disk evolution at very low masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' However, statistics are still rather poor at this mass regime for a firm conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Other very low-mass YSOs, companions to T Tauri stars, have been found to exhibit similar, or even higher rates of mass accretion (Betti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2014, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' To explain the very high levels of accretion observed in such sub-stellar and planetary-mass compan- ions, Samatellos & Herczeg (2015) modeled the accretion onto very low-mass objects that formed by the fragmenta- tion of the disk around the hosting star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' During the early evolution the individual disks of sub-stellar companions, including those at the planetary-mass regime, accrete addi- tional material from the gas-rich parent disk, hence, their disks are more massive and their accretion rates are higher than if they were formed in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Therefore, these very low-mass objects have disk masses and accretion rates that are independent of the mass of the central object and are higher than expected from the scaling relation ˙Macc ∝ M 2 ⋆ of more massive YSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' These models predict that ˙Macc is independent of M⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Using Gaia DR3, we have investigated whether 2MASS J15551027-3455045 might be a wide companion of another star, but it is an isolated object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Hence, the high mass ac- cretion rate cannot be explained in terms of the Samatellos & Herczeg (2015) scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Due to its intrinsic faintness, 2MASS J15551027-3455045 would be an interesting target to be followed up by CUBES, which is a next-generation spectrograph suitable for investigating fainter, low-mass ac- creting YSOs (Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 13 2MASSJ15551027-3455045 10 Teff = 2700 K, log g = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 PhotometricData cm-2) 11 (erg S-1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 14 1000 10000 入 (nm)2MASSJ15361110-3444473 Teff = 2900 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 11 Photometric Data L cm-2) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13 1000 10000 入 (nm)Sz 70 6 Teff = 3000 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 PhotometricData 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 log 入 Flux 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 12 1000 10000 入 (nm)2MASSJ16011870-3437332 10 Teff = 3100 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 Photometric Data 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 log 入Flux 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13 1000 10000 入 (nm)Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 11: BT-Settl models (in grey) with the photometric data (red dots) for our chromospherically-dominant targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Possible wide companions While studying the kinematic properties of the targets, we also noticed that a few of our targets and core members of the Lupus I share similar kinematic properties, and can be considered as wide companion candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' These wide companion candidates are presented in Table 12 and Table 13, divided into two categories of candidates studied in this 14 TYC 7335-550-1 8 Teff = 4500 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 Photometric Data (erg s-1 cm-2) 10 log 入Flux 11 12 13 1000 10000 入 (nm)2MASSJ15523574-3344288 10 Teff = 3000 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 PhotometricData 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 log 入Flux 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13 1000 10000 入 (nm)UCAC4269-083981 9 Teff = 3800 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 Photometric Data cm-2) 10 (erg s-1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 11 log 入Flux 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13 1000 10000 入 (nm)2MASSJ15383733-3422022 10 Teff = 3100 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 Photometric Data 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 log 入Flux 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13 1000 10000 入 (nm)2MASSJ15414827-3501458 9 Teff = 3200 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 PhotometricData cm-2) 10 (erg s-1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 11 log 入Flux 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13 1000 10000 入 (nm)GaiaDR26010590577947703936 10 Teff = 3100 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 Photometric Data 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 log 入Flux 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13 1000 10000 入 (nm)UCAC4273-083363 9 Teff = 3000 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 Photometric Data cm-2) 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 11 log 入Flux 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13 1000 10000 入 (nm)GaiaDR26014269268967059840 10 Teff = 3000 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 Photometric Data .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' cm-2) 11 (erg s-1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 14 1000 10000 入 (nm)Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Table 11: Disk categorization of all our targets, in addition to their reddest colors available in the 2MASS and WISE catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name Ks − W3 Ks − W4 Bredall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2020) Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2014) mag mag Disk type SED/Disk type 2MASS J15383733-3422022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='93 Evolved disk Sz 70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9 Full disk Full disk TYC 7335-550-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='14 Debris disk 2MASS J15361110-3444473 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='70 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='04 Full disk Full disk 2MASS J15523574-3344288 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='69 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='31 Full disk Full disk 2MASS J15551027-3455045 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 Full disk Full disk 2MASS J16011870-3437332 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09 Full disk Full disk UCAC4 269-083981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06 Debris disk Gaia DR2 6010590577947703936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='79 Evolved disk 2MASS J15414827-3501458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='16 Debris disk UCAC4 273-083363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='86 Debris disk Gaia DR2 6014269268967059840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='58 Evolved disk Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The overall SED of 2MASS J15361110-3444473 may be affected by a possible unresolved M8-type companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' work and the Lupus I core members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In order to understand whether two objects with similar kinematic properties are gravitationally bound, we calculated their total velocity dif- ference (∆v) and compared it with the maximum total ve- locity difference (∆vmax) as a function of projected sepa- ration between the two binary components, suggested by Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' If ∆v exceeds ∆vmax, we do not ex- pect the two targets to be gravitationally bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' It should be noted, however, that the theoretical maximum velocity difference modeled by Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2017) is only for bina- ries of total mass 10 M⊙ in circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We summarize our results on identifying wide companions candidates in the Lupus I cloud as follows: Sz 70 and Sz 71 – Same as the GQ Lup triple system (Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2020), Sz 70 and Sz 71 (GW Lup) are located on the main filament of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Sz 70 lies at a separation of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='32 arcseconds from GW Lup, and in between these ob- jects lies the X-ray source [KWS97] Lupus I 37 (Krautter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 1997) at a separation of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='23 arcseconds from Sz 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We conducted a chance projection study in Alcal´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2020, Appendix E), which was focused on understanding how probable it is to find a field object around a genuine member of Lupus I, lying on the same filament where GQ Lup stellar system and Sz 70/Sz 71 are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The linear density of this filament is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0024 objects/arcsec, or an av- erage object separation of 418 arcsec, which is 13 times the observed separation between Sz 70 and Sz 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 12, Sz 70 and Sz 71 do not qualify as gravitation- ally bound stars, but we would like to emphasize that the test proposed by Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2017) is only valid for gravitationally bound binaries, and not systems of higher multiplicities (if this is the case for this stellar system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Hence, we would consider this case as a wide companion candidate that cannot be confirmed or ruled out according to the available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' TYC 7335-550-1 and 2MASS J15361110- 3444473 – As discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 4, 2MASS J15361110- 3444473 might be an unresolved binary, composed of an M6 (VIS spectrum) and an M8 (NIR spectrum) star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The RV calculated for this target based on the ROTFIT code is obtained by cross-correlations conducted on the VIS spectrum of this target, which is also used for calculating the maximum velocity difference between TYC 7335-550-1 and 2MASS J15361110-3444473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 12, the two objects can be gravitationally bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' However, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 12: Log-log plot of total velocity difference ∆v (km/s) versus projected separation s (au) for the wide companion candidates analyzed in this work, in addition to the genuine wide companions GQ Lup and GQ Lup C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' ∆vmax (km/s) (orange line) indicates the maximum total velocity differ- ence that bound binaries with a total mass equal to 10 M⊙ in circular orbits can possess (Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Each point is marked as one of the wide companion candidates involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For the detailed information, see Tables 12 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' TYC 7335-550-1 has an age of ∼ 4 Myr and 2MASS J15361110-3444473 an age of ∼ 9 Myr, which states the two stellar systems are probably not coeval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Also, unlike TYC 7335-550-1, we could not determine whether 2MASS J15361110-3444473 is a member of Lupus I due to many uncertainties explained earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Hence, any further comments on its physical association with TYC 7335-550-1 would be misleading and inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Sz 65 and Sz 66 – At a separation of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='45 arcseconds, with ∆V = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='26±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='69 km/s, Sz 65 and Sz 66 (although coeval) according to the test suggested by Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2017) are not gravitationally bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' There are no other objects located in a close separation with respect to either Sz 65 or Sz 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Hence, we rule out the possibility of Sz 65 and Sz 66 as wide companion candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' HT Lup A-B-C – This stellar system is located in an over-crowded region on the same filament of Lupus I as GQ Lup stellar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In Gaia DR2 catalog, HT Lup 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 1 (km/s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 (△ v) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 4 log s (au) GQLupC SZ 66 HT Lup Sz 70 TYC 7335-550-1 △ Vmax (km/s)Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Table 12: Kinematic properties of the Lupus I members from this work (measurement errors are displayed in parenthesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name α (J2000) δ (J2000) ϖ µα∗ µδ RV Age ∆V δ∆V S (h:m:s) (d:m:s) (mas) (mas/yr) (mas/yr) (km/s) (Myr) (km/s) (km/s) (′′) Sz 71/GW LUP∗ 15 46 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='73 –34 30 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='68 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='41(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06) –14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='03(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='10) –23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='36(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07) –3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='30(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='90) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='24 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='32 Sz 70 15 46 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='99 –34 30 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='21) –12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='58(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='39) –22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='16(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='25) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 2MASS J15361110-3444473 15 36 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09 –34 44 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='82 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='83(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='29) –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='56(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='29) –20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='21(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='23) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='77 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='47 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='28 TYC 7335-550-1 15 36 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55 –34 45 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='54 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='26(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07) –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='93(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='43) –19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='51(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='01) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55 ∗ RV obtained by Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Table 13: Core members of Lupus I sharing similar kinematic properties (measurement errors are displayed in parenthesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name α (J2000) δ (J2000) ϖ µα∗ µδ RV Age ∆V δ∆V S (h:m:s) (d:m:s) (mas) (mas/yr) (mas/yr) (km/s) (Myr) (km/s) (km/s) (′′) Sz 65/V∗ IK Lup∗ 15 39 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='77 –34 46 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='44(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05) –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='27(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='12) –22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='24(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07) –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='70(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='69 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='41 Sz 66∗ 15 39 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='28 –34 46 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='36(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='09) –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='60(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='19) –21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='56(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='12) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='40(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='80) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9 Sz 68/HT LUP A-B∗ 15 45 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='87 –34 17 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='65 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='49(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='06) –13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='63(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13) –21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='60(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='08) –4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='30(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='80) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='82 CD-33 10685C/HT Lup C∗∗ 15 45 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='67 –34 17 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='19) –15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='43(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='22) –20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='27(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='15) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='9) – ∗ RV and age obtained by Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' ∗∗ RV for this target is adopted from the optimal RV calculated by BANYAN Σ, considering HT Lup C is a member of UCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' A and B are not resolved separately, hence we assume the central star to be Sz 68 (or HT Lup A), composed of two unresolved stars, and adopt its stellar characteristics from Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As genuine members of Lupus I, we assume all the components of this triple system to have an age consistent with the other bona fide members of Lupus I (≤ 2 Myr), and hence, to be coeval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' However, the RVs used here should be taken with caution, both because HT Lup A-B are not resolved, and also because we have adopted the optimal RV calculated by BANYAN σ for HT Lup C considered as a member of UCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' With a separation of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='82 arc seconds, we have shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 12 that as expected, this triple system is possibly gravitationally bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We thus conclude that the possibility of Sz 70 & Sz 71 being wide companions is rather low and for TYC 7335- 550-1 & 2MASS J15361110-344447, follow-up studies on 2MASS J15361110-344447 are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As for the previ- ously identified members of Lupus I, we understood that Sz 65 and Sz 66 are not gravitationally bound, and HT Lup A-B-C are the components of a triple system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Conclusion The main conclusions of this paper can be summarized as follows: – Out of the 12 objects fully characterized in this work, ten are recognized as genuine members of Lupus I, and two remain ambiguous in terms of stellar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' – Out of the ten members of Lupus I analyzed in this work, three were recognized to be accretors (Sz 70, 2MASS J15551027-3455045, and 2MASS J16011870- 3437332), and Sz 70 and 2MASS J15551027-3455045 are likely to be surrounded by full disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2MASS J15551027- 3455045 is among the least massive accretors discovered so far in the Lupus complex, formed in full isolation and is an off-cloud member of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' – All of the three off-cloud targets included in our program turned out to be genuine members of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' These targets are 2MASS J15523574-3344288, 2MASS J15551027-3455045, and 2MASS J16011870- 3437332, with 2MASS J15551027-3455045 and 2MASS J16011870-3437332 actively accreting matter, and 2MASS J15523574-3344288 mildly accreting matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Further investigation in this area may reveal a diffused population of M dwarfs close to the main filament of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We thus would like to acknowledge that this work also contributes to revealing the diffused popula- tions of M-dwarfs around the Lupus cloud by Comer´on (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' – Although the sample studied in this work is small, we proved that many interesting targets in young star form- ing regions can escape Hα surveys due to various rea- sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Hence, using the kinematic properties of candi- date YSOs can play a key role in identifying the gen- uine members of the young stellar associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This is specifically true for genuine members such as TYC 7335- 550-1 that have Hα in absorption, and hence would not appear in Hα surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' – We have identified a plausible binary system among the targets analyzed in this work, namely, TYC 7335- 550-1 and 2MASS J15361110-3444473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' It is noteworthy, however, that 2MASS J15361110-3444473 might be an unresolved binary, and its kinematic properties (espe- cially RV) should be revised with next-generation spec- trographs (due to its intrinsic faintness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' – All the above points considered, we conclude that char- acterizing only a small portion of our sample has proved to have a high success rate for discovering the new mem- bers of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This shows that the spectroscopy of our entire sample of 43 objects could have resulted in a far more solid investigation of the region in terms of de- termining the disk fraction, stellar properties, and the number of new members of Lupus I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' FZM is grateful to Eugene Vasiliev for fruitful discussions on how to use Gaia catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' AFR is grateful to Giovanni Catanzaro for helping us with the analysis of TYC 7335-550-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' FZM is funded by ”Bando per il Finanziamento di Assegni di Ricerca Progetto Dipartimenti di Eccellenza Anno 2020” and is co-funded in agree- ment with ASI-INAF n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2019-29-HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 from 26 Nov/2019 for ”Italian participation in the operative phase of CHEOPS mission” (DOR - Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Piotto).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' acknowledges partial funding by the Deutsche Forschungsgemeinschaft Excellence Strategy - EXC 2094 - 390783311 and the ANID BASAL project FB210003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' JMA, AFR, CFM, KBI and ECO acknowledge financial support from the project PRIN- INAF 2019 “Spectroscopically Tracing the Disk Dispersal Evolution” 16 Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud (STRADE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' CFM is funded by the European Union under the European Union’s Horizon Europe Research & Innovation Programme 101039452 (WANDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This work has also been supported by the PRIN-INAF 2019 ”Planetary systems at young ages (PLATEA)” and ASI-INAF agreement n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2018-16-HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Views and opinions expressed are however those of the author(s) only and do not necessarily re- flect those of the European Union or the European Research Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Neither the European Union nor the granting authority can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='int/web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institutions, in particular, the institutions participating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This research has made use of the SIMBAD database and Vizier services, operated at CDS, Strasbourg, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This research has made use of the services of the ESO Science Archive Facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Finally, we would like to thank the anonymous referee who also contributed to this paper with his/her valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' References Alcal´a, J.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=', Comeron, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018, A&A, 615, L1 Mer´ın, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=', Jørgensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=', Spezzi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=', et al.' metadata={'source': 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17 17 Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Appendix A: Candidate members of Lupus I As we explained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2, we proposed 43 objects to be ob- served with X-Shooter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Twelve out of these 43 objects were observed during a filler program, and in this work we fully characterized them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The rest of our targets in this sam- ple that were not observed are listed in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Among these targets, only 2MASS J15464664-3210006 (Eisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2007) is partly characterized, and 20 objects are identi- fied as candidate YSOs using Gaia DR2 (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Appendix B: Age estimation and isochrones For estimating the age of our targets we used multiple isochrones for the reasons explained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' In this Appendix, we present the ages of our targets using various isochrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We repeat that the ages estimated for all our targets were overestimated by PARSEC models in compar- ison with all the other models with a considerable gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We thus decided to remove the results achieved by the PARSEC models to avoid confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' This is, however, a well-known problem of PARSEC isochrones that they overestimate the age of cool stars, and all our targets fall in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Appendix C: 2MASS J15361110-3444473 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1: Flux-calibrated, extinction-corrected NIR spec- trum of 2MASS J15361110-3444473 (in black) with its BT- Settl model (Teff = 2500 K and log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5, in grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2MASS J15361110-3444473 is an M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 star according to its VIS spectrum (as we quantitatively indicated) and an M8 star based on its NIR spectrum (based on the fitting done with the BT-Settl model Teff = 2500 K and log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5, as exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1), with a total extinction of AV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='75 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' All the spectral typing and analysis that we have performed in this paper are based on the VIS spectrum of this target, especially the ROTFIT results are all based on the VIS spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Hence, although we keep our analysis limited to the spectroscopy conducted on the VIS spectrum, we would like to emphasize that the possibility of this target being an unresolved binary (composed of two M dwarfs) with SpTs of M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 and M8 is viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Considering the available data, we also cannot rule out the possibility that the star is heavily spotted instead of being a binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Appendix D: Updates with Gaia DR3 As stated in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2, we used the Gaia DR2 catalog to select our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Very recently, Gaia DR3 (Gaia Collaboration 2021) became public and gave us the opportunity to check the catalog for any possible changes or updates on the kinematic or stellar properties of our objects analyzed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We did not find any considerable difference be- tween the kinematic properties reported in both catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' However, we report the highlights of our search using these two catalogs in the following: TYC 7335-550-1 – as obtained in this work, for TYC 7335-550-1 we obtained Teff = 4488 K, while in both Gaia DR2 and Gaia DR3 its reported temperature is 5000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The reported RV for TYC 7335-550-1 in Gaia DR2 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='20±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='65 km/s, which is better constrained than the RV we report here (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='0 km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' As the wide companion can- didate of 2MASS J15361110-3444473, we recalculated their ∆v using the Gaia DR3 kinematic properties of TYC 7335- 550-1, and it resulted in ∆v = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='34±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='30 (km/s) which is consistent with the previous ∆v = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='72±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='47 (km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' For both of these calculations, we use the RVs calculated by ROTFIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Sz 70 – has a high RUWE in both catalogs (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='86), but we saw no signs of binarity in the spectrum of Sz 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Using the kinematic properties of Sz 70 reported in Gaia DR3 and those of Sz 71 (which is also updated in Gaia DR3), we recalculated their maximum velocity difference, and it resulted in ∆v = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='36±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='24 (km/s), which is consistent with the ∆v = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='07±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='24 (km/s) calculated based on Gaia DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2MASS J15414827-3501458 – has a high RUWE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='198) in both Gaia DR2 and Gaia DR3 catalogs, but we detected no signs of binarity in the spectrum of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' We report that the kinematic properties of all our tar- gets (parallax and proper motions) are consistent within 3σ in the two catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Also according to Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' (2022), we do not expect the stellar physical parameters of our core sample to be changed with the astrometry reported in Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 18 11 Teff = 2500, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 2MASS|15361110-3444473 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 nm-1) (erg s-1 cm-2 I 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 log 入Flux 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 500 1000 1500 2000 2500 3000 3500 4000 入 (nm)Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1: Astrometric properties of the candidate Lupus I members that were not observed by X-Shooter, with their errors in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name α (J2000) δ (J2000) ϖ µα∗ µδ J (h:m:s) (d:m:s) (mas) (mas/yr) (mas/yr) (mag) 2MASS J15464664-3210006a 15 46 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='64 –32 10 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='62 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='05(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='021) –19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='47(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='023) –23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='76(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='014) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='22 Gaia DR2 6013000844869745664 15 39 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='47 –35 58 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='88 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='62(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='039) –18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='081) –22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='23(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='057) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='11 Gaia DR2 6013065853493820416b 15 43 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='62 –35 39 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='88(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='015) –17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='68(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='018) –24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='51(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='012) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='20 Gaia DR2 6011737574730221568c 15 50 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='50 –34 22 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='49 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='69(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='019) –16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='20(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='020) –22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='52(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='015) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='74 Gaia DR2 6012258330925877632d 15 53 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='13 –33 31 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='60 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='92(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='016) –16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='97(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='018) –24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='57(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='016) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='75 Gaia DR2 6039383622075982848e 15 57 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='76 –32 04 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='91 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+page_content='38(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='28) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='35 a 2MASS J15464664-3210006 is an M2, T Tauri star (Eisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' b aka UCAC4 272-080482, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' c aka UCAC4 279-083370, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' d aka UCAC4 283-086052, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' e aka RX J1557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1-3204A, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' f aka UCAC4 272-081081, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' g aka UCAC4 275-083957, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' h aka UCAC4 282-082547, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' i aka UCAC4 292-084899, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' j aka UCAC4 271-080669, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' k aka UCAC4 274-080590, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' l aka UCAC4 274-080590, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' m This target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' n aka UCAC4 274-081081, this target is a YSO candidate (Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 19 Majidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' : New members of the Lupus I cloud Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='1: Ages of our targets estimated using various isochrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' The ages are all in Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' Name Dartmouth Dartmouth MIST Baraffe std mag models 2MASS J15383733-3422022 11 20 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 Sz 70 <1 1 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 TYC 7335-550-1 3 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55 2MASS J15361110-3444473 9 20 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='77 2MASS J15523574-3344288 8 13 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='3 2MASS J15551027-3455045 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='7 2MASS J16011870-3437332 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 14 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='55 UCAC4 269-083981 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='2 Gaia DR2 6010590577947703936 8 14 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='8 2MASS J15414827-3501458 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='82 UCAC4 273-083363 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='63 Gaia DR2 6014269268967059840 8 13 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content='46 a None of the three isochrones used here were able to reproduce the stellar parameters of this target due to its dimness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQfVwq1/content/2301.04463v1.pdf'} diff --git a/JNFRT4oBgHgl3EQfzTgG/content/tmp_files/2301.13649v1.pdf.txt b/JNFRT4oBgHgl3EQfzTgG/content/tmp_files/2301.13649v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..22cec8f58e37cef0f28c88511911e895b92c2361 --- /dev/null +++ b/JNFRT4oBgHgl3EQfzTgG/content/tmp_files/2301.13649v1.pdf.txt @@ -0,0 +1,636 @@ +Studies of New Physics in 𝑩0 +𝒒 − ¯𝑩0 +𝒒 Mixing and +Implications for Leptonic Decays +Kristof De Bruyn,𝑎,𝑏 Robert Fleischer,𝑎,𝑐 Eleftheria Malami𝑎,𝑑,∗ and Philine van Vliet𝑒 +𝑎Nikhef, +Science Park 105, 1098 XG Amsterdam, Netherlands +𝑏Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, +9747 Groningen, Netherlands +𝑐Faculty of Science, Vrije Universiteit Amsterdam, +1081 HV Amsterdam, Netherlands +𝑑Center for Particle Physics Siegen (CPPS), Theoretische Physik 1, Universität Siegen, +D-57068 Siegen, Germany +𝑒Deutsches Elektronen-Synchrotron DESY, +Notkestr. 85, 22607 Hamburg, Germany +E-mail: Eleftheria.Malami@uni-siegen.de +The phenomenon of 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing (𝑞 = 𝑑, 𝑠) provides a sensitive probe for physics beyond the +Standard Model. We have a careful look at the determination of the Unitarity Triangle apex, which +is needed for the Standard Model predictions of the 𝐵𝑞 mixing parameters, and explore how much +space for New Physics is left through the current data. We study the impact of tensions between +inclusive and exclusive determinations of the CKM matrix elements |𝑉𝑢𝑏| and |𝑉𝑐𝑏|, and focus on +the 𝛾 angle extraction. We present various future scenarios and discuss the application of these +results for leptonic rare 𝐵 decays, which allows us to minimise the CKM parameter impact in +the New Physics searches. Performing future projections, we explore and illustrate the impact of +increased precision on key input quantities. It will be exciting to see how more precise data in the +future high-precision era of flavour physics can lead to a much sharper picture. +8th Symposium on Prospects in the Physics of Discrete Symmetries (DISCRETE 2022) +7-11 November, 2022 +Baden-Baden, Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.13649v1 [hep-ph] 31 Jan 2023 + +Studies of New Physics in 𝐵0 +𝑞 − ¯𝐵0 +𝑞 Mixing and Implications for Leptonic Decays +Eleftheria Malami +1. +Introduction +The phenomenon of 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing (where 𝑞 = 𝑑, 𝑠) arises only from loop processes in the +Standard Model (SM) and is sensitive to possible New Physics (NP) contributions, which could +enter the loop topologies or even at the tree level, for instance in 𝑍 ′ models. Associated to the mixing +phenomenon are the mixing parameters and the CP-violating phases for which we have impressive +experimental data. In this presentation, we follow Ref. [1] and explore the space allowed for NP +by current measurements and the state-of-the-art parameters. In addition, we point out interesting +connections to the studies of leptonic rare 𝐵 decays. +In order to determine the parameter space of possible NP effects to 𝐵0 +𝑞– ¯𝐵0 +𝑞 mixing, we have to +compare the SM predictions of the mixing parameters with the corresponding experimental values. +For these SM predictions, a careful analysis of the Unitarity Triangle (UT) apex is required. We +pay special attention to the different determinations of the Cabibbo-Kobayashi-Maskawa (CKM) +parameters and the tensions that arise between the extractions of the |𝑉𝑢𝑏| and |𝑉𝑐𝑏| matrix elements +through inclusive and exclusive semileptonic 𝐵 meson decays. These longstanding tensions have a +profound impact on the whole analysis. +2. +Unitarity Triangle +Using the parametrisation of the Particle Data Group (PDG), the UT apex is given as [2]: +𝑅𝑏 𝑒𝑖𝛾 = ¯𝜌 + 𝑖 ¯𝜂 , +¯𝜌 ≡ +� +1 − (𝜆2/2) +� +𝜌 , +¯𝜂 ≡ +� +1 − (𝜆2/2) +� +𝜂 . +(1) +Here, 𝜌, 𝜂 and 𝜆 are the Wolfenstein parameters [3, 4], 𝑅𝑏 is the side from the origin to the apex of +the UT, defined with the help of the CKM matrix elements 𝜆 ≡ |𝑉𝑢𝑠|, |𝑉𝑢𝑏| and |𝑉𝑐𝑏| as: +𝑅𝑏 ≡ +� +1 − 𝜆2 +2 +� 1 +𝜆 +���� +𝑉𝑢𝑏 +𝑉𝑐𝑏 +���� = +√︃ +¯𝜌 2 + ¯𝜂 2 , +(2) +and 𝛾 ≡ arg �−𝑉𝑢𝑑𝑉∗ +𝑢𝑏/𝑉𝑐𝑑𝑉∗ +𝑐𝑏 +� is the angle between the 𝑅𝑏 side and the UT basis. +2.1 Determining the UT Apex Utilising 𝛾 and 𝑅𝑏 +In this subsection, we work in the SM and are interested in obtaining the UT apex in a way +that is not affected by possible NP in 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing. One way of determining the apex is utilising +the side 𝑅𝑏 and the angle 𝛾, which can both be determined from decays that proceed only via tree +decays. The value of 𝛾 can be determined either from 𝐵 → 𝐷𝐾 decays or from a 𝐵 → 𝜋𝜋, 𝜌𝜋, 𝜌𝜌 +isospin analysis. +More specifically, one option is to use the time-dependent 𝐵0 +𝑠 → 𝐷∓ +𝑠 𝐾± system, where mixing- +induced CP violation plays a key role. Through interference effects caused by 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing, the +CP asymmetry parameters allow the determination of 𝜙𝑠 + 𝛾, where 𝜙𝑠 is the 𝐵0 +𝑠- ¯𝐵0 +𝑠 mixing phase. +Since 𝜙𝑠 is determined through the 𝐵0 +𝑠 → 𝐽/𝜓𝜙 channel, including penguin corrections [5, 6], 𝛾 +can be obtained in a theoretically clean way [7, 8]. However, the surprisingly large value arising in +this case still needs to be further explored. An alternative way of getting the 𝛾 value is using the +time-independent 𝐵 → 𝐷𝐾 transitions, where the sensitivity to 𝛾 comes from direct CP violation +[9]. Last but not least, another interesting system is provided by 𝐵 → 𝜋𝜋, 𝜌𝜋, 𝜌𝜌 modes [10, 11], +2 + +Studies of New Physics in 𝐵0 +𝑞 − ¯𝐵0 +𝑞 Mixing and Implications for Leptonic Decays +Eleftheria Malami +which usually are used to determine 𝛼 from an isospin analysis. Actually this value corresponds to +𝛾 when we use the 𝐵0 +𝑑- ¯𝐵0 +𝑑 mixing phase 𝜙𝑑, determined from 𝐵0 +𝑑 → 𝐽/𝜓𝐾0 [5, 6], taking penguin +effects into account. Thus, we can convert the result 𝜙𝑑 + 2𝛾 into 𝛾. The value from the latter case +is in good agreement with the one coming from 𝐵 → 𝐷𝐾 modes. Therefore, for our analysis, we +average these two results [1]: +𝛾avg = (68.4 ± 3.4)◦. +(3) +Regarding 𝑅𝑏 there are tensions between the various theoretical and experimental approaches. +Even though there are different determinations of the |𝑉𝑢𝑠| element and the tensions between them +are intriguing, they only have a negligible impact on NP studies in neutral 𝐵𝑞 mixing. Thus, we +choose to work with the value |𝑉𝑢𝑠| = 0.22309 ± 0.00056 [12, 13]. Contrary to the |𝑉𝑢𝑠| case, the +deviations between determinations of |𝑉𝑢𝑏| and |𝑉𝑐𝑏| from inclusive and exclusive semileptonic 𝐵 +decays, which are given as follows [14, 15]: +|𝑉𝑢𝑏|incl = (4.19 ± 0.17) × 10−3 , +|𝑉𝑢𝑏|excl = (3.51 ± 0.12) × 10−3 , +differing by 3.9 𝜎, +(4) +|𝑉𝑐𝑏|incl = (42.16 ± 0.50) × 10−3 , +|𝑉𝑐𝑏|excl = (39.10 ± 0.50) × 10−3 , +differing by 4.3 𝜎, +(5) +have a significant impact on the allowed parameter space for NP in 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing. Trying to +understand and resolve these tensions, another case is studied in the literature [15–18], which is a +hybrid scenario combining the exclusive |𝑉𝑢𝑏| with the inclusive |𝑉𝑐𝑏| determination. Therefore, +we consider for the rest of our analysis all these three cases. The corresponding 𝑅𝑏 results are: +𝑅𝑏,incl = 0.434 ± 0.018 , +𝑅𝑏,excl = 0.392 ± 0.014 , +𝑅𝑏,hybrid = 0.364 ± 0.013 . +(6) +Making a fit to 𝑅𝑏 and 𝛾, the UT apex is determined [1]: +Incl. +¯𝜌 = 0.160 ± 0.025 , +¯𝜂 = 0.404 ± 0.022 , +(7) +Excl. +¯𝜌 = 0.144 ± 0.022 , +¯𝜂 = 0.365 ± 0.018 , +(8) +Hybrid +¯𝜌 = 0.134 ± 0.021 , +¯𝜂 = 0.338 ± 0.017 . +(9) +The results are illustrated in Fig. 1. The plot also shows the hyperbola coming from the |𝜀𝐾 | +observable, which is related to indirect CP violation in the neutral kaon system and is highly +sensitive to the |𝑉𝑐𝑏| numerical value. The hybrid case gives the most consistent picture of the +UT apex within the SM, which illustrates the strong dependence on |𝑉𝑐𝑏|. In the future, this could +help us to understand the inclusive-exclusive puzzle, if NP in the kaon system can be controlled or +ignored. +2.2 Determining the UT Apex Utilising 𝑅𝑏 and 𝑅𝑡 +An alternative way of determining the UT apex is utilising the 𝑅𝑡 side, which is defined as: +𝑅𝑡 ≡ |𝑉𝑡𝑑𝑉𝑡𝑏/𝑉𝑐𝑑𝑉𝑐𝑏| = +√︃ +(1 − ¯𝜌)2 + ¯𝜂 2. +(10) +3 + +Studies of New Physics in 𝐵0 +𝑞 − ¯𝐵0 +𝑞 Mixing and Implications for Leptonic Decays +Eleftheria Malami +0 +0.2 +0.4 +0.6 +0.8 +1 +ρ +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +η +avg +γ +b +R +Fit Solution +| +K +ε| +contours hold 39%, 87% CL +| from Kl3 +us + & |V +b +Incl. R +0 +0.2 +0.4 +0.6 +0.8 +1 +ρ +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +η +avg +γ +b +R +Fit Solution +| +K +ε| +contours hold 39%, 87% CL +| from Kl3 +us + & |V +b +Excl. R +0 +0.2 +0.4 +0.6 +0.8 +1 +ρ +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +η +avg +γ +b +R +Fit Solution +| +K +ε| +contours hold 39%, 87% CL +| from Kl3 +us + & |V +b +Hybrid R +Figure 1: Determination of the UT apex from the 𝑅𝑏 and 𝛾 measurements for the inclusive (left), exclusive +(right) and hybrid (botttom) case [1]. +In this case, only information on the two UT sides 𝑅𝑏 and 𝑅𝑡 is required without needing any +information from 𝛾. However, in order to get the 𝑅𝑡, we have to assume SM expressions for the +mixing parameters Δ𝑚𝑑 and Δ𝑚𝑠. The numerical predictions are given in [1]. +The side 𝑅𝑡 can be written as +𝑅𝑡 = 1 +𝜆 +���� +𝑉𝑡𝑑 +𝑉𝑡𝑠 +���� +� +1 − 𝜆2 +2 (1 − 2 ¯𝜌) +� ++ O +� +𝜆4� +, +(11) +where +���� +𝑉𝑡𝑑 +𝑉𝑡𝑠 +���� = 𝜉 +√︄ +𝑚𝐵𝑠Δ𝑚SM +𝑑 +𝑚𝐵𝑑Δ𝑚SM +𝑠 +. +(12) +Here the SU(3)-breaking parameter 𝜉 is the ratio of bag parameters and decay constants of the +𝐵𝑑 and the 𝐵𝑠 systems that can be calculated on the lattice. The advantage of the ratio is that +uncertainties cancel, making it cleaner than using individual results. +Making a fit to the 𝑅𝑏 and 𝑅𝑡 sides, we obtain [1]: +Incl. +¯𝜌 = 0.180 ± 0.014 , +¯𝜂 = 0.395 ± 0.020 , +(13) +Excl. +¯𝜌 = 0.163 ± 0.013 , +¯𝜂 = 0.357 ± 0.017 , +(14) +Hybrid +¯𝜌 = 0.153 ± 0.013 , +¯𝜂 = 0.330 ± 0.016 . +(15) +We note that the UT apex determinations relying on 𝛾 are a factor 2 less precise than those without +information from 𝛾. However, the determination through 𝑅𝑏 and 𝑅𝑡 requires the SM expressions +of Δ𝑚𝑑 and Δ𝑚𝑠, thus ignores possible NP contributions in 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing. +4 + +Studies of New Physics in 𝐵0 +𝑞 − ¯𝐵0 +𝑞 Mixing and Implications for Leptonic Decays +Eleftheria Malami +0 +50 +100 +150 +200 +250 +300 +350 +]° + [ +σ +0 +0.1 +0.2 +0.3 +0.4 +0.5 +κ + System (Scenario I) +d +B + System (Scenario I) +s +B +FUNP (Scenario II) +contours hold 39%, 87% CL +| from Kl3 +us + & |V +b +Incl. R +0 +50 +100 +150 +200 +250 +300 +350 +]° + [ +σ +0 +0.1 +0.2 +0.3 +0.4 +0.5 +κ + System (Scenario I) +d +B + System (Scenario I) +s +B +FUNP (Scenario II) +contours hold 39%, 87% CL +| from Kl3 +us + & |V +b +Excl. R +0 +50 +100 +150 +200 +250 +300 +350 +]° + [ +σ +0 +0.1 +0.2 +0.3 +0.4 +0.5 +κ + System (Scenario I) +d +B + System (Scenario I) +s +B +FUNP (Scenario II) +contours hold 39%, 87% CL +| from Kl3 +us + & |V +b +Hybrid R +Figure 2: Comparing Scenario I and Scenario II fits for 𝜅𝑞 and 𝜎𝑞 for the inclusive (left), exclusive (right) +and hybrid (bottom) case [1]. +3. +NP in 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing +The neutral 𝐵𝑞-meson mixing is a sensitive phenomenon for NP. In order to quantify its impact, +we introduce NP parameters 𝜅𝑞, which describes the size of the NP effects, and 𝜎𝑞, which is a +complex phase accounting for additional CP-violating effects. The generalised expressions of the +mixing parameters take the following form [19]: +Δ𝑚𝑞 = Δ𝑚SM +𝑞 +��1 + 𝜅𝑞𝑒𝑖𝜎𝑞�� , +(16) +𝜙𝑞 = 𝜙SM +𝑞 ++ 𝜙NP +𝑞 += 𝜙SM +𝑞 ++ arg �1 + 𝜅𝑞𝑒𝑖𝜎𝑞� . +(17) +This is a model independent parametrization. Utilising these relations, we explore two different NP +scenarios; the first one is the most general case and the second one assumes Flavour Universal NP +(FUNP) [1]. +Let us firstly discuss the general case, namely Scenario I. The only assumption here is that there +is no NP in the angle 𝛾 and 𝑅𝑏. The determination from 𝑅𝑏 and 𝛾 does not rely on information from +mixing. We make use of this determination to obtain the UT apex, which we then need for getting +the SM predictions for the mixing parameters Δ𝑚𝑞 and 𝜙𝑞. Comparing them with their measured +values, we can constrain the NP parameters. Here, the NP parameters (𝜅𝑑, 𝜎𝑑) and (𝜅𝑠, 𝜎𝑠) are +determined independently from each other. +In the second case, Scenario II, we have the FUNP assumption where we consider that the NP +contributions are equal in the 𝐵𝑑 and 𝐵𝑠 systems, thus (𝜅𝑑, 𝜎𝑑) = (𝜅𝑠, 𝜎𝑠). This is not a Minimal +Flavour Violation scenario but it can be realised in NP models with 𝑈(2) symmetry [20, 21]. The +UT apex fit relies on 𝑅𝑏 and 𝑅𝑡, without using 𝛾 information, therefore possible NP in the angle 𝛾 +5 + +Studies of New Physics in 𝐵0 +𝑞 − ¯𝐵0 +𝑞 Mixing and Implications for Leptonic Decays +Eleftheria Malami +will not affect the findings. Comparing the two scenarios, we have a test of the FUNP assumption +and we see the impact of the assumptions on the constraints on the parameter space of NP in mixing. +Fig. 2 illustrates this comparison of the two fits for 𝜅𝑞 and 𝜎𝑞 for the inclusive, the exclusive and +the hybrid cases. +4. +Rare Leptonic Decays 𝐵0 +𝑞 → 𝜇+𝜇− +The tensions between the CKM matrix elements have an impact not only on the UT apex +determination and possible NP in 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing but also on the branching ratios of rare decays. A +key example is the leptonic 𝐵0 +𝑞 → 𝜇+𝜇− transition. These modes are pure loop processes and helicity +suppressed in the SM. This helicity suppression could be lifted by new scalar and pseudoscalar +conttributions, therefore putting these decays in an outstanding position to probe NP in this sector. +As these are decays of neutral 𝐵 mesons, 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing enters and leads to subtleties concerning the +measurement of the experimental branching ratio and comparison with the theoretical prediction +[22]. However, NP in 𝐵0 +𝑠- ¯𝐵0 +𝑠 mixing is included through the experimental values of the mixing +parameters. +The SM predictions require information on |𝑉𝑡𝑠| which we determine through |𝑉𝑐𝑏|, which +again depends on inclusive and exclusive determinations. In order to minimise the dependence on +|𝑉𝑐𝑏| and the UT apex, we create the following ratio with the 𝐵𝑠 mass difference Δ𝑚𝑠 [23–25]: +R𝑠𝜇 ≡ ¯B(𝐵𝑠 → 𝜇+𝜇−)/Δ𝑚𝑠 . +(18) +Using this ratio, we can eliminate the leading dependence on the CKM elements but we have to +correct for the possible NP contributions to 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing. This is now possible following our +analysis in [1]. +So, we include NP effects in Δ𝑚𝑠 and then we can use the ratio R𝑠𝜇 to constrain NP in the +scalar and pseudoscalar sector. We obtain the generalised expression: +R𝑠𝜇 = RSM +𝑠𝜇 × +1 + A𝜇𝜇 +ΔΓ𝑠 𝑦𝑠 +1 + 𝑦𝑠 +|𝑃𝑠 +𝜇𝜇|2 + |𝑆𝑠 +𝜇𝜇|2 +√︁ +1 + 2𝜅𝑠 cos 𝜎𝑠 + 𝜅2𝑠 +, +(19) +with 𝑃𝑠 +𝜇𝜇 ≡ |𝑃𝑠 +𝜇𝜇|𝑒𝑖𝜑𝑃, 𝑆𝑠 +𝜇𝜇 ≡ |𝑆𝑠 +𝜇𝜇|𝑒𝑖𝜑𝑆, where 𝜑𝑃, 𝜑𝑆 are CP-violating phases, and the observable +A𝜇𝜇 +ΔΓ𝑠 in terms of the NP phase 𝜙NP +𝑠 : +A𝜇𝜇 +ΔΓ = +|𝑃𝑠 +𝜇𝜇|2 cos(2𝜑𝑃 − 𝜙NP +𝑠 ) − |𝑆𝑠 +𝜇𝜇|2 cos(2𝜑𝑆 − 𝜙NP +𝑠 ) +|𝑃𝑠𝜇𝜇|2 + |𝑆𝑠𝜇𝜇|2 +. +(20) +The R𝑠𝜇 has only a dependence on the CKM matrix elements through the NP parameters 𝜅𝑞 +and 𝜎𝑞, determined as described above. Therefore, we have another constraint on the scalar and +pseudoscalar contributions. The same strategy can be applied to the 𝐵0 +𝑑 → 𝜇+𝜇− channel once in +the future accurate measurements of the branching ratio will become available. +5. +Future Prospects and Final Remarks +It will be important in the future to achieve improved precision on the NP parameters 𝜅𝑞 and +𝜎𝑞. In order to get a feeling of the prospects, we assume a hypothetical reduction of 50% on each +6 + +Studies of New Physics in 𝐵0 +𝑞 − ¯𝐵0 +𝑞 Mixing and Implications for Leptonic Decays +Eleftheria Malami +one of the three input parameters, which are the |𝑉𝑐𝑏|, the lattice calculations and the UT apex [1]. +We obtain interesting findings, which of course depend on these assumptions. In our studies, we +demonstrate that in the 𝐵𝑑-system the apex plays a limiting factor and in order to fully explore the +potentials of this system, progress on the UT apex has to be made. On the other hand, in the 𝐵𝑠- +system we do not have this situation as the SM prediction of 𝜙𝑠 is more robust. Therefore, searches +of NP in 𝐵0 +𝑠- ¯𝐵0 +𝑠 mixing are more promising than in the 𝐵𝑑-system but it is of key importance to +constrain NP in both systems as much as possible. +Another essential future prospect is related to the angle 𝛾. Improved precision on the input +measurements might lead to significant discrepancies between the different 𝛾 determinations due +to NP effects. In this case, averaging over the different results, as we did in this analysis, would +no longer be justified. Therefore, the UT should then be revisited. Independent information from +additional observables would be necessary to resolve such a situation. Exciting new opportunities +might come up to search for NP, both in 𝛾 and in 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing, which is strongly correlated with +the UT apex coordinates. +Last but not least, the branching ratios of the 𝐵0 +𝑞 → 𝜇+𝜇− decays might offer interesting +opportunities. The ratio of the branching fractions between 𝐵0 +𝑑 → 𝜇+𝜇− and 𝐵0 +𝑠 → 𝜇+𝜇− can +provide an alternative way to determine the UT side 𝑅𝑡. Another useful application for the ratio of +the branching fractions between these channels is the quantity [26]: +𝑈𝑑𝑠 +𝜇𝜇 ∝ +����� +𝑉𝑡𝑠 +𝑉𝑡𝑑 +���� +2 ¯B(𝐵𝑑 → 𝜇+𝜇−) +¯B(𝐵𝑠 → 𝜇+𝜇−) +�1/2 +(21) +which requires knowledge of 𝑅𝑡 and offers a very powerful test of the SM, where 𝑈𝑑𝑠 +𝜇𝜇 = 1. +In the future, 𝐵0 +𝑞- ¯𝐵0 +𝑞 mixing will remain a key element for constraining NP. It will be exciting +to see how more precise data in the high-precision era of flavour physics ahead of us can lead to a +much sharper picture. +Acknowledgements +We would like to thank the DISCRETE 2022 organisers for the invitation and for giving us the +opportunity to present our studies. This research has been supported by the Netherlands Organisation +for Scientific Research (NWO). PvV acknowledges support from the DFG through the Emmy +Noether research project 400570283, and through the German-Israeli Project Cooperation (DIP). +References +[1] K. De Bruyn, R. Fleischer, E. Malami and P. van Vliet, 2022 J. Phys. G: Nucl. Part. Phys. +https://doi.org/10.1088/1361-6471/acab1d +[2] R. L. Workman et al. [Particle Data Group], PTEP 2022 (2022), 083C01 +[3] L. Wolfenstein, Phys. Rev. Lett. 51 (1983), 1945 doi:10.1103/PhysRevLett.51.1945 +[4] A. J. Buras, M. E. Lautenbacher and G. Ostermaier, Phys. Rev. D 50 (1994), 3433-3446 +7 + +Studies of New Physics in 𝐵0 +𝑞 − ¯𝐵0 +𝑞 Mixing and Implications for Leptonic Decays +Eleftheria Malami +[5] M. Z. Barel, K. De Bruyn, R. Fleischer and E. Malami, [arXiv:2203.14652 [hep-ph]]. +[6] M. Z. Barel, K. De Bruyn, R. Fleischer and E. Malami, J. Phys. G 48 (2021) no.6, 065002 +[7] R. Fleischer and E. Malami, Phys. Rev. D 106 (2022) no.5, 056004 +[8] R. Fleischer and E. Malami, [arXiv:2110.04240 [hep-ph]]. +[9] R. Aaij et al. 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Tetlalmatzi-Xolocotzi, JHEP 05 (2017), 156 +8 + diff --git a/JNFRT4oBgHgl3EQfzTgG/content/tmp_files/load_file.txt b/JNFRT4oBgHgl3EQfzTgG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..37b92df22c05bbda7a71d728a6421d6c6d877fde --- /dev/null +++ b/JNFRT4oBgHgl3EQfzTgG/content/tmp_files/load_file.txt @@ -0,0 +1,387 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf,len=386 +page_content='Studies of New Physics in 𝑩0 𝒒 − ¯𝑩0 𝒒 Mixing and Implications for Leptonic Decays Kristof De Bruyn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='𝑏 Robert Fleischer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='𝑐 Eleftheria Malami𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='∗ and Philine van Vliet𝑒 𝑎Nikhef,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Science Park 105,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 1098 XG Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Netherlands 𝑏Van Swinderen Institute for Particle Physics and Gravity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' University of Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 9747 Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Netherlands 𝑐Faculty of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Vrije Universiteit Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 1081 HV Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Netherlands 𝑑Center for Particle Physics Siegen (CPPS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Theoretische Physik 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Universität Siegen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' D-57068 Siegen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Germany 𝑒Deutsches Elektronen-Synchrotron DESY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 85, 22607 Hamburg, Germany E-mail: Eleftheria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='Malami@uni-siegen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='de The phenomenon of 𝐵0 𝑞- ¯𝐵0 𝑞 mixing (𝑞 = 𝑑, 𝑠) provides a sensitive probe for physics beyond the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' We have a careful look at the determination of the Unitarity Triangle apex, which is needed for the Standard Model predictions of the 𝐵𝑞 mixing parameters, and explore how much space for New Physics is left through the current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' We study the impact of tensions between inclusive and exclusive determinations of the CKM matrix elements |𝑉𝑢𝑏| and |𝑉𝑐𝑏|, and focus on the 𝛾 angle extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' We present various future scenarios and discuss the application of these results for leptonic rare 𝐵 decays, which allows us to minimise the CKM parameter impact in the New Physics searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Performing future projections, we explore and illustrate the impact of increased precision on key input quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' It will be exciting to see how more precise data in the future high-precision era of flavour physics can lead to a much sharper picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 8th Symposium on Prospects in the Physics of Discrete Symmetries (DISCRETE 2022) 7-11 November, 2022 Baden-Baden, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='13649v1 [hep-ph] 31 Jan 2023 Studies of New Physics in 𝐵0 𝑞 − ¯𝐵0 𝑞 Mixing and Implications for Leptonic Decays Eleftheria Malami 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Introduction The phenomenon of 𝐵0 𝑞- ¯𝐵0 𝑞 mixing (where 𝑞 = 𝑑, 𝑠) arises only from loop processes in the Standard Model (SM) and is sensitive to possible New Physics (NP) contributions, which could enter the loop topologies or even at the tree level, for instance in 𝑍 ′ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Associated to the mixing phenomenon are the mixing parameters and the CP-violating phases for which we have impressive experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In this presentation, we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' [1] and explore the space allowed for NP by current measurements and the state-of-the-art parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In addition, we point out interesting connections to the studies of leptonic rare 𝐵 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In order to determine the parameter space of possible NP effects to 𝐵0 𝑞– ¯𝐵0 𝑞 mixing, we have to compare the SM predictions of the mixing parameters with the corresponding experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' For these SM predictions, a careful analysis of the Unitarity Triangle (UT) apex is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' We pay special attention to the different determinations of the Cabibbo-Kobayashi-Maskawa (CKM) parameters and the tensions that arise between the extractions of the |𝑉𝑢𝑏| and |𝑉𝑐𝑏| matrix elements through inclusive and exclusive semileptonic 𝐵 meson decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' These longstanding tensions have a profound impact on the whole analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Unitarity Triangle Using the parametrisation of the Particle Data Group (PDG), the UT apex is given as [2]: 𝑅𝑏 𝑒𝑖𝛾 = ¯𝜌 + 𝑖 ¯𝜂 , ¯𝜌 ≡ � 1 − (𝜆2/2) � 𝜌 , ¯𝜂 ≡ � 1 − (𝜆2/2) � 𝜂 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' (1) Here, 𝜌, 𝜂 and 𝜆 are the Wolfenstein parameters [3, 4], 𝑅𝑏 is the side from the origin to the apex of the UT, defined with the help of the CKM matrix elements 𝜆 ≡ |𝑉𝑢𝑠|, |𝑉𝑢𝑏| and |𝑉𝑐𝑏| as: 𝑅𝑏 ≡ � 1 − 𝜆2 2 � 1 𝜆 ���� 𝑉𝑢𝑏 𝑉𝑐𝑏 ���� = √︃ ¯𝜌 2 + ¯𝜂 2 , (2) and 𝛾 ≡ arg �−𝑉𝑢𝑑𝑉∗ 𝑢𝑏/𝑉𝑐𝑑𝑉∗ 𝑐𝑏 � is the angle between the 𝑅𝑏 side and the UT basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='1 Determining the UT Apex Utilising 𝛾 and 𝑅𝑏 In this subsection, we work in the SM and are interested in obtaining the UT apex in a way that is not affected by possible NP in 𝐵0 𝑞- ¯𝐵0 𝑞 mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' One way of determining the apex is utilising the side 𝑅𝑏 and the angle 𝛾, which can both be determined from decays that proceed only via tree decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The value of 𝛾 can be determined either from 𝐵 → 𝐷𝐾 decays or from a 𝐵 → 𝜋𝜋, 𝜌𝜋, 𝜌𝜌 isospin analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' More specifically, one option is to use the time-dependent 𝐵0 𝑠 → 𝐷∓ 𝑠 𝐾± system, where mixing- induced CP violation plays a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Through interference effects caused by 𝐵0 𝑞- ¯𝐵0 𝑞 mixing, the CP asymmetry parameters allow the determination of 𝜙𝑠 + 𝛾, where 𝜙𝑠 is the 𝐵0 𝑠- ¯𝐵0 𝑠 mixing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Since 𝜙𝑠 is determined through the 𝐵0 𝑠 → 𝐽/𝜓𝜙 channel, including penguin corrections [5, 6], 𝛾 can be obtained in a theoretically clean way [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' However, the surprisingly large value arising in this case still needs to be further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' An alternative way of getting the 𝛾 value is using the time-independent 𝐵 → 𝐷𝐾 transitions, where the sensitivity to 𝛾 comes from direct CP violation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Last but not least, another interesting system is provided by 𝐵 → 𝜋𝜋, 𝜌𝜋, 𝜌𝜌 modes [10, 11], 2 Studies of New Physics in 𝐵0 𝑞 − ¯𝐵0 𝑞 Mixing and Implications for Leptonic Decays Eleftheria Malami which usually are used to determine 𝛼 from an isospin analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Actually this value corresponds to 𝛾 when we use the 𝐵0 𝑑- ¯𝐵0 𝑑 mixing phase 𝜙𝑑, determined from 𝐵0 𝑑 → 𝐽/𝜓𝐾0 [5, 6], taking penguin effects into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Thus, we can convert the result 𝜙𝑑 + 2𝛾 into 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The value from the latter case is in good agreement with the one coming from 𝐵 → 𝐷𝐾 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Therefore, for our analysis, we average these two results [1]: 𝛾avg = (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='4)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' (3) Regarding 𝑅𝑏 there are tensions between the various theoretical and experimental approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Even though there are different determinations of the |𝑉𝑢𝑠| element and the tensions between them are intriguing, they only have a negligible impact on NP studies in neutral 𝐵𝑞 mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Thus, we choose to work with the value |𝑉𝑢𝑠| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='22309 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='00056 [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Contrary to the |𝑉𝑢𝑠| case, the deviations between determinations of |𝑉𝑢𝑏| and |𝑉𝑐𝑏| from inclusive and exclusive semileptonic 𝐵 decays, which are given as follows [14, 15]: |𝑉𝑢𝑏|incl = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='17) × 10−3 , |𝑉𝑢𝑏|excl = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='12) × 10−3 , differing by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='9 𝜎, (4) |𝑉𝑐𝑏|incl = (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='50) × 10−3 , |𝑉𝑐𝑏|excl = (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='50) × 10−3 , differing by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='3 𝜎, (5) have a significant impact on the allowed parameter space for NP in 𝐵0 𝑞- ¯𝐵0 𝑞 mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Trying to understand and resolve these tensions, another case is studied in the literature [15–18], which is a hybrid scenario combining the exclusive |𝑉𝑢𝑏| with the inclusive |𝑉𝑐𝑏| determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Therefore, we consider for the rest of our analysis all these three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The corresponding 𝑅𝑏 results are: 𝑅𝑏,incl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='434 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='018 , 𝑅𝑏,excl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='392 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='014 , 𝑅𝑏,hybrid = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='364 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='013 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' (6) Making a fit to 𝑅𝑏 and 𝛾, the UT apex is determined [1]: Incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='025 , ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='404 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='022 , (7) Excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='144 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='022 , ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='365 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='018 , (8) Hybrid ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='134 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='021 , ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='338 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='017 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' (9) The results are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The plot also shows the hyperbola coming from the |𝜀𝐾 | observable, which is related to indirect CP violation in the neutral kaon system and is highly sensitive to the |𝑉𝑐𝑏| numerical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The hybrid case gives the most consistent picture of the UT apex within the SM, which illustrates the strong dependence on |𝑉𝑐𝑏|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In the future, this could help us to understand the inclusive-exclusive puzzle, if NP in the kaon system can be controlled or ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='2 Determining the UT Apex Utilising 𝑅𝑏 and 𝑅𝑡 An alternative way of determining the UT apex is utilising the 𝑅𝑡 side, which is defined as: 𝑅𝑡 ≡ |𝑉𝑡𝑑𝑉𝑡𝑏/𝑉𝑐𝑑𝑉𝑐𝑏| = √︃ (1 − ¯𝜌)2 + ¯𝜂 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' (10) 3 Studies of New Physics in 𝐵0 𝑞 − ¯𝐵0 𝑞 Mixing and Implications for Leptonic Decays Eleftheria Malami 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='8 1 ρ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='7 η avg γ b R Fit Solution | K ε| contours hold 39%, 87% CL | from Kl3 us & |V b Incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' R 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='8 1 ρ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='7 η avg γ b R Fit Solution | K ε| contours hold 39%, 87% CL | from Kl3 us & |V b Excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' R 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='8 1 ρ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='7 η avg γ b R Fit Solution | K ε| contours hold 39%, 87% CL | from Kl3 us & |V b Hybrid R Figure 1: Determination of the UT apex from the 𝑅𝑏 and 𝛾 measurements for the inclusive (left), exclusive (right) and hybrid (botttom) case [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In this case, only information on the two UT sides 𝑅𝑏 and 𝑅𝑡 is required without needing any information from 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' However, in order to get the 𝑅𝑡, we have to assume SM expressions for the mixing parameters Δ𝑚𝑑 and Δ𝑚𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The numerical predictions are given in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The side 𝑅𝑡 can be written as 𝑅𝑡 = 1 𝜆 ���� 𝑉𝑡𝑑 𝑉𝑡𝑠 ���� � 1 − 𝜆2 2 (1 − 2 ¯𝜌) � + O � 𝜆4� , (11) where ���� 𝑉𝑡𝑑 𝑉𝑡𝑠 ���� = 𝜉 √︄ 𝑚𝐵𝑠Δ𝑚SM 𝑑 𝑚𝐵𝑑Δ𝑚SM 𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' (12) Here the SU(3)-breaking parameter 𝜉 is the ratio of bag parameters and decay constants of the 𝐵𝑑 and the 𝐵𝑠 systems that can be calculated on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The advantage of the ratio is that uncertainties cancel, making it cleaner than using individual results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Making a fit to the 𝑅𝑏 and 𝑅𝑡 sides, we obtain [1]: Incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='180 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='014 , ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='395 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='020 , (13) Excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='163 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='013 , ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='357 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='017 , (14) Hybrid ¯𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='153 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='013 , ¯𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='330 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='016 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' (15) We note that the UT apex determinations relying on 𝛾 are a factor 2 less precise than those without information from 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' However, the determination through 𝑅𝑏 and 𝑅𝑡 requires the SM expressions of Δ𝑚𝑑 and Δ𝑚𝑠, thus ignores possible NP contributions in 𝐵0 𝑞- ¯𝐵0 𝑞 mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 4 Studies of New Physics in 𝐵0 𝑞 − ¯𝐵0 𝑞 Mixing and Implications for Leptonic Decays Eleftheria Malami 0 50 100 150 200 250 300 350 ]° [ σ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='5 κ System (Scenario I) d B System (Scenario I) s B FUNP (Scenario II) contours hold 39%, 87% CL | from Kl3 us & |V b Incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' R 0 50 100 150 200 250 300 350 ]° [ σ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='5 κ System (Scenario I) d B System (Scenario I) s B FUNP (Scenario II) contours hold 39%, 87% CL | from Kl3 us & |V b Excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' R 0 50 100 150 200 250 300 350 ]° [ σ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content='5 κ System (Scenario I) d B System (Scenario I) s B FUNP (Scenario II) contours hold 39%, 87% CL | from Kl3 us & |V b Hybrid R Figure 2: Comparing Scenario I and Scenario II fits for 𝜅𝑞 and 𝜎𝑞 for the inclusive (left), exclusive (right) and hybrid (bottom) case [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' NP in 𝐵0 𝑞- ¯𝐵0 𝑞 mixing The neutral 𝐵𝑞-meson mixing is a sensitive phenomenon for NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In order to quantify its impact, we introduce NP parameters 𝜅𝑞, which describes the size of the NP effects, and 𝜎𝑞, which is a complex phase accounting for additional CP-violating effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The generalised expressions of the mixing parameters take the following form [19]: Δ𝑚𝑞 = Δ𝑚SM 𝑞 ��1 + 𝜅𝑞𝑒𝑖𝜎𝑞�� , (16) 𝜙𝑞 = 𝜙SM 𝑞 + 𝜙NP 𝑞 = 𝜙SM 𝑞 + arg �1 + 𝜅𝑞𝑒𝑖𝜎𝑞� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' (17) This is a model independent parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Utilising these relations, we explore two different NP scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' the first one is the most general case and the second one assumes Flavour Universal NP (FUNP) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Let us firstly discuss the general case, namely Scenario I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The only assumption here is that there is no NP in the angle 𝛾 and 𝑅𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The determination from 𝑅𝑏 and 𝛾 does not rely on information from mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' We make use of this determination to obtain the UT apex, which we then need for getting the SM predictions for the mixing parameters Δ𝑚𝑞 and 𝜙𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Comparing them with their measured values, we can constrain the NP parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Here, the NP parameters (𝜅𝑑, 𝜎𝑑) and (𝜅𝑠, 𝜎𝑠) are determined independently from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In the second case, Scenario II, we have the FUNP assumption where we consider that the NP contributions are equal in the 𝐵𝑑 and 𝐵𝑠 systems, thus (𝜅𝑑, 𝜎𝑑) = (𝜅𝑠, 𝜎𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' This is not a Minimal Flavour Violation scenario but it can be realised in NP models with 𝑈(2) symmetry [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The UT apex fit relies on 𝑅𝑏 and 𝑅𝑡, without using 𝛾 information, therefore possible NP in the angle 𝛾 5 Studies of New Physics in 𝐵0 𝑞 − ¯𝐵0 𝑞 Mixing and Implications for Leptonic Decays Eleftheria Malami will not affect the findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Comparing the two scenarios, we have a test of the FUNP assumption and we see the impact of the assumptions on the constraints on the parameter space of NP in mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 2 illustrates this comparison of the two fits for 𝜅𝑞 and 𝜎𝑞 for the inclusive, the exclusive and the hybrid cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Rare Leptonic Decays 𝐵0 𝑞 → 𝜇+𝜇− The tensions between the CKM matrix elements have an impact not only on the UT apex determination and possible NP in 𝐵0 𝑞- ¯𝐵0 𝑞 mixing but also on the branching ratios of rare decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' A key example is the leptonic 𝐵0 𝑞 → 𝜇+𝜇− transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' These modes are pure loop processes and helicity suppressed in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' This helicity suppression could be lifted by new scalar and pseudoscalar conttributions, therefore putting these decays in an outstanding position to probe NP in this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' As these are decays of neutral 𝐵 mesons, 𝐵0 𝑞- ¯𝐵0 𝑞 mixing enters and leads to subtleties concerning the measurement of the experimental branching ratio and comparison with the theoretical prediction [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' However, NP in 𝐵0 𝑠- ¯𝐵0 𝑠 mixing is included through the experimental values of the mixing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The SM predictions require information on |𝑉𝑡𝑠| which we determine through |𝑉𝑐𝑏|, which again depends on inclusive and exclusive determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In order to minimise the dependence on |𝑉𝑐𝑏| and the UT apex, we create the following ratio with the 𝐵𝑠 mass difference Δ𝑚𝑠 [23–25]: R𝑠𝜇 ≡ ¯B(𝐵𝑠 → 𝜇+𝜇−)/Δ𝑚𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' (18) Using this ratio, we can eliminate the leading dependence on the CKM elements but we have to correct for the possible NP contributions to 𝐵0 𝑞- ¯𝐵0 𝑞 mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' This is now possible following our analysis in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' So, we include NP effects in Δ𝑚𝑠 and then we can use the ratio R𝑠𝜇 to constrain NP in the scalar and pseudoscalar sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' We obtain the generalised expression: R𝑠𝜇 = RSM 𝑠𝜇 × 1 + A𝜇𝜇 ΔΓ𝑠 𝑦𝑠 1 + 𝑦𝑠 |𝑃𝑠 𝜇𝜇|2 + |𝑆𝑠 𝜇𝜇|2 √︁ 1 + 2𝜅𝑠 cos 𝜎𝑠 + 𝜅2𝑠 , (19) with 𝑃𝑠 𝜇𝜇 ≡ |𝑃𝑠 𝜇𝜇|𝑒𝑖𝜑𝑃, 𝑆𝑠 𝜇𝜇 ≡ |𝑆𝑠 𝜇𝜇|𝑒𝑖𝜑𝑆, where 𝜑𝑃, 𝜑𝑆 are CP-violating phases, and the observable A𝜇𝜇 ΔΓ𝑠 in terms of the NP phase 𝜙NP 𝑠 : A𝜇𝜇 ΔΓ = |𝑃𝑠 𝜇𝜇|2 cos(2𝜑𝑃 − 𝜙NP 𝑠 ) − |𝑆𝑠 𝜇𝜇|2 cos(2𝜑𝑆 − 𝜙NP 𝑠 ) |𝑃𝑠𝜇𝜇|2 + |𝑆𝑠𝜇𝜇|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' (20) The R𝑠𝜇 has only a dependence on the CKM matrix elements through the NP parameters 𝜅𝑞 and 𝜎𝑞, determined as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Therefore, we have another constraint on the scalar and pseudoscalar contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The same strategy can be applied to the 𝐵0 𝑑 → 𝜇+𝜇− channel once in the future accurate measurements of the branching ratio will become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Future Prospects and Final Remarks It will be important in the future to achieve improved precision on the NP parameters 𝜅𝑞 and 𝜎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In order to get a feeling of the prospects, we assume a hypothetical reduction of 50% on each 6 Studies of New Physics in 𝐵0 𝑞 − ¯𝐵0 𝑞 Mixing and Implications for Leptonic Decays Eleftheria Malami one of the three input parameters, which are the |𝑉𝑐𝑏|, the lattice calculations and the UT apex [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' We obtain interesting findings, which of course depend on these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In our studies, we demonstrate that in the 𝐵𝑑-system the apex plays a limiting factor and in order to fully explore the potentials of this system, progress on the UT apex has to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' On the other hand, in the 𝐵𝑠- system we do not have this situation as the SM prediction of 𝜙𝑠 is more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Therefore, searches of NP in 𝐵0 𝑠- ¯𝐵0 𝑠 mixing are more promising than in the 𝐵𝑑-system but it is of key importance to constrain NP in both systems as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Another essential future prospect is related to the angle 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Improved precision on the input measurements might lead to significant discrepancies between the different 𝛾 determinations due to NP effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In this case, averaging over the different results, as we did in this analysis, would no longer be justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Therefore, the UT should then be revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Independent information from additional observables would be necessary to resolve such a situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Exciting new opportunities might come up to search for NP, both in 𝛾 and in 𝐵0 𝑞- ¯𝐵0 𝑞 mixing, which is strongly correlated with the UT apex coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Last but not least, the branching ratios of the 𝐵0 𝑞 → 𝜇+𝜇− decays might offer interesting opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' The ratio of the branching fractions between 𝐵0 𝑑 → 𝜇+𝜇− and 𝐵0 𝑠 → 𝜇+𝜇− can provide an alternative way to determine the UT side 𝑅𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Another useful application for the ratio of the branching fractions between these channels is the quantity [26]: 𝑈𝑑𝑠 𝜇𝜇 ∝ ����� 𝑉𝑡𝑠 𝑉𝑡𝑑 ���� 2 ¯B(𝐵𝑑 → 𝜇+𝜇−) ¯B(𝐵𝑠 → 𝜇+𝜇−) �1/2 (21) which requires knowledge of 𝑅𝑡 and offers a very powerful test of the SM, where 𝑈𝑑𝑠 𝜇𝜇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' In the future, 𝐵0 𝑞- ¯𝐵0 𝑞 mixing will remain a key element for constraining NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' It will be exciting to see how more precise data in the high-precision era of flavour physics ahead of us can lead to a much sharper picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Acknowledgements We would like to thank the DISCRETE 2022 organisers for the invitation and for giving us the opportunity to present our studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' This research has been supported by the Netherlands Organisation for Scientific Research (NWO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' PvV acknowledges support from the DFG through the Emmy Noether research project 400570283, and through the German-Israeli Project Cooperation (DIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' De Bruyn, R.' metadata={'source': 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R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Fleischer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Jaarsma and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} +page_content=' Tetlalmatzi-Xolocotzi, JHEP 05 (2017), 156 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFRT4oBgHgl3EQfzTgG/content/2301.13649v1.pdf'} diff --git a/K9E5T4oBgHgl3EQfYQ9F/content/tmp_files/2301.05572v1.pdf.txt b/K9E5T4oBgHgl3EQfYQ9F/content/tmp_files/2301.05572v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f218421f65f04034a8e636f0a01009c07e34fa79 --- /dev/null +++ b/K9E5T4oBgHgl3EQfYQ9F/content/tmp_files/2301.05572v1.pdf.txt @@ -0,0 +1,2878 @@ +DESIGNING TRANSLATIONAL ANIMAL EXPERIMENTS BY +BAYESIAN META-ANALYTIC PREDICTIVE APPROACHES +A PREPRINT +Theresa Unseld∗ +Department of Epidemiology and Medical Biometry +Ulm University +Ulm, Germany +January 16, 2023 +ABSTRACT +The planning and conduct of animal experiments in the European Union is subject to strict legal +conditions. Still, many preclinical animal experiments are only poorly designed. As a consequence, +discoveries that are made in one animal experiment, cannot be reproduced in another animal experi- +ment or discoveries in translational animal research fail to be translated to humans. When designing +new experiments in a classical frequentist framework, the sample size for the new experiment is +chosen with the goal to achieve at least a certain statistical power, given a statistical test for a null +hypothesis, a significance threshold and a minimally relevant effect size. The statistical test is a +function of the data and the test is used to make statistical inference concerning the data’s underlying, +unobserved parameters of interest. In a Bayesian framework, inference is made by a combination of +both the information from newly observed data and also by a prior distribution, that represents a priori +information on the parameters. In translational animal experiments, a priori information is present in +previously conducted experiments to the same outcome in similar animals. The prior information +can be incorporated in a systematic way in the design and analysis of a new animal experiment by +summarizing the historical data in a (Bayesian) meta-analysis model and using the meta-analysis +model to make predictions for the data in the new experiment. This is called meta-analytic predictive +(MAP) approach. In this work, concepts of how to design translational animal experiments by +MAP approaches are introduced and compared to classical frequentist power-oriented sample size +planning. Current chances and challenges, that exist in the practical application of these approaches +in translational animal research, are discussed. Special emphasis is put on the construction of prior +distributions and sample size calculation by design analysis. The considerations are motivated by a +real world translational research example. +Keywords Translational research, Bayesian statistics, meta-analysis, design analysis, sample size calculation +1 +Introduction +Translational research constitutes a key element to the development of new methods and therapies in human medicine. +Situated in the late stage of preclinical research, its aim is to translate findings from basic laboratory preclinical research +into the clinical application as potential treatments of human diseases. In translational animal research biological +pathways concerning clinically relevant phenotypes or pathologies are examined. These pathways are typically +complex constructs whose mechanisms cannot be fully revealed in a single research experiment. Hence, the success of +translational research fundamentally depends on the sensitive contemplation of new insights from an experiment in light +of past insights and gained expertise knowledge. In Bayesian analysis, prior knowledge is formally incorporated as +probability distribution into the analysis of newly observed data and updated to a posterior distribution that is then used +∗theresa.unseld@uni-ulm.de +arXiv:2301.05572v1 [stat.ME] 13 Jan 2023 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +for Bayesian inference. Several authors have already emphasized the value of Bayesian statistics in preclinical research +(see for example Spiegelhalter et al. (2004), Walley et al. (2016), Kramer & Font (2017), Bonapersona et al. (2019), +Yang & Novick (2019), Gelman & Vákár (2019), Novick & Zhang (2021)). Nonetheless, practical applications of +Bayesian methods for planning and analyzing preclinical animal experiments are rare (see Walley et al. (2016), Kramer +& Font (2017), Bonapersona et al. (2019)). A challenge in the application of Bayesian methods is the specification of +a prior distribution. Translational animal experiments are typically characterized by the fact that the sample sizes in +the experiment’s groups are kept low for animal welfare purposes (see Mayer et al. (2018)), like outlined in the 3R +concept by Russel & Burch (1959). Especially in this situation, the choice of prior distribution can have a major impact +on the posterior inference. Setting up a good prior distribution, that accurately reflects a priori information, can help +to stabilize posterior inference, derive more precise estimates, reduce the impact of single extreme observations and +to indicate if there might be something wrong or unexpected in the measurements of the new data that results in a +wide posterior distribution. However, without a transparent justification for the choice of a prior and without a good +understanding of the impact of the prior distribution on posterior inference, there is a risk that the final conclusion of a +Bayesian analysis might be not sensitive (enough) to evidence in the newly observed data. On the other hand, if the +prior distribution is chosen to have essentially no weight on posterior inference as compared to the data in the new +experiment, then a major aspect of Bayesian inference and its associated benefits over frequentist inference are missing. +One systematic way of setting prior distributions is to specify them based on a meta-analysis of relevant literature or +databases as suggested by Neuenschwander et al. (2010), Pullenayegum (2011), Rhodes et al. (2015), Turner et al. +(2015), Bartoš et al. (2021). A meta-analysis is a popular tool for summarizing information from several statistical +experiments in a common statistical model and quantifying the variability in different experiments. As a requirement, the +experiments all have to address the same question. This assumption is often reasonable for control groups that stem from +a series of experiments that address the same phenotypical outcome. The meta-analysis model can be used to estimate +predictive distributions for a new experiment. These predictive distributions can be used to derive prior distributions for +the analysis of the new experiment. This approach to a prior specification and Bayesian panning and analysis of the new +experiment is termed meta-analytic predictive approach (MAP) Neuenschwander et al. (2010). These MAP priors fall +into a class of data-based priors and are the focus in this article. Other methods for prior specification are summarized +by Mikkola et al. (2021) and are briefly discussed in the discussion. The MAP approach is illustrated for the mean in +control groups of clinical studies by Neuenschwander et al. (2010). However, they don’t use the historical data to derive +prior distributions for other model parameters like the variance. Furthermore, animal experiments are characterized +by their own challenges and methods suggested in human clinical trials cannot be transferred in a straight-forward +manner to the application in animal experiments without considering possible adaptations (see Walley et al. (2016), +Kramer & Font (2017)). Firstly, due to the small sample sizes in the experiments’ groups, the estimates obtained from +animal experiments are usually characterized by large uncertainties. A further challenge is that, although many animal +experiments are conducted, the results from the analysis are usually unorganized and restricted to limited access (see +Kramer & Font (2017), Bonapersona et al. (2019), Novick & Zhang (2021)). It has been the idea to initiate search +tools to perform systematic reviews of preclinical studies (see Bahor et al. (2021)) or launch common big databases to +gather the information from more institutions (see Keenan et al. (2009), Beckers et al. (2009), Maddatu et al. (2012), +McEntyre et al. (2015), Steger-Hartmann et al. (2020), Pognan et al. (2021)). Until such databases are fully developed, +the remaining alternative option is to construct prior distributions form the little historical information that is available +and be aware of the present uncertainties about the model parameters’ true values. The usage of Bayesian methods +allows to fit meta-analysis models also in the challenging situation a of few, small previous experiments. Methods for +Bayesian meta-analysis in the context of few, small studies have recently been proposed for the application in humans +by Friede et al. (2017a). Furthermore, the MAP approach has been discussed in the context of animal experiments by +Walley et al. (2016). Still, there exist only few applications of how such Bayesian meta-analytic predictive methods are +used to analyze real-world examples from preclinical translational research and even fewer examples of how to conduct +sample size calculations for animal experiments in a Bayesian framework (see Kramer & Font (2017)). +The planning and conduct of animal experiments in the European Union (EU) is subject to strict legal conditions as +outlined for example in the EU directive 2010/63/EU or for Germany in the legislation for animal welfare (Tierschutz- +Versuchstierverordnung) (see Bundesministerium der Justitz (2010)). In particular, animal experiments are subject to +an authorization by the respective competent authorities. Researchers can apply for this authorization by submitting +an animal experiment proposal. If the animal experiments are set up to proof hypothesis, the proposals must be +submitted together with a form or with a biometric review that provide a description of the analysis methods as well as +a justification of the proposed sample size. Classically, sample size calculations are done in a frequentist framework by +choosing the minimal number of animals that reaches a predefined power of a statistical test for a null hypothesis, given +a data model, including its parameters. However, the classical approach to null hypothesis significance testing (NHST) +and power analysis has been criticized for several reasons (see Gelman & Carlin (2014), Kruschke (2015), Schönbrodt +& Wagenmakers (2018), Stefan et al. (2019), Gelman, Hill & Vehtari (2020)). Claiming statistical significance by a +frequentist test for a certain hypothesis is equivalent to claiming statistical significance based on the corresponding +2 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +p-value of the test or the confidence interval (see Lehmann & Romano (2006)). These decision rules for a null hypothesis +have been criticized for hypothesis testing, for example by Wagenmakers et al. (2020), Gelman, Hill & Vehtari (2020), +Schad et al. (2022). One critique is connected to the principle of predictive irrelevance (see Wagenmakers et al. (2020)) +and Bayes factors are proposed and as an alternative. Gelman & Carlin (2014), Gelman, Hill & Vehtari (2020) point out +that, by choosing an experimental design with the goal of reaching a certain power of the statistical (while limiting the +probability of type I errors of the test), other important goals of the experimental design are missed. More specifically, +they argue that, by focusing on power and statistical significance, the reported estimates are systematically biased. +This is because actually small effects are only reported for those data sets where the observed effect size happens +to be large (by chance) and those cases are highly subjective to being overestimated and to being of the wrong sign. +Sample size calculations in a classical framework are based on fixed estimates of an effect size and of the additional +model parameters. One option is to derive these estimates from a researcher’s previous experiment. This is problematic +because the small sample sizes in animal experiments result in parameter estimates that are accompanied with large +uncertainties (see Mayer & Muche (2013)) and these uncertainties are not well reflected by single point estimates. As a +consequence, the sample size calculations in animal experiments are often only of limited use. As an alternative to +power-oriented sample size calculation, design analysis using fake-data simulation has been suggested by Kruschke +(2015), Gelman, Hill & Vehtari (2020). +In this paper the above aspects of Bayesian analysis, meta-analysis, prior specification using predictive approaches +and design analysis using fake-data simulation are considered jointly in the context of sample size calculation for +translational animal experiments. To the author’s knowledge, there exists so far no work that illustrates sample size +calculation for translational animal experiments using these approaches jointly. After the introduction in this section, +section two introduces sample size calculations in a classical frequentist framework and introduces fake-data design +analysis as its alternative. Furthermore, a Bayesian MAP approach to designing and analysing new experiments based +on historical data is explained. A distributional Bayesian model is used to deal with unequal group variances in the +historical and new data. The considerations are applied to a real-world application example section three. In section +four the main points are summarized and possible extensions are discussed. +2 +Methods +2.1 +Statistical hypothesis tests and power-oriented sample size calculation +For sample size calculations, assumptions have to be made concerning the distribution models for all experimental +groups and assumptions concerning the experimental design that specifies which comparisons are made. Furthermore, +estimable effects of interest, δ, have to be defined that typically represent the differences in the means or effects of two +or several groups. Additionally, for making binary decisions whether there is sufficient evidence in the data that the true +effect of interest δ is different from a null effect δ0 or not, a decision function is required. The decision function in the +classical classical frequentist framework is a statistical hypothesis test ϕ. Statistical hypothesis tests ϕ compare test +statistics T to a critical value c to make a decision whether or not to reject a null hypothesis +H0 : {δ = δ0}vs. H1 : {δ ̸= δ0} +(1) +If H0 is rejected, the alternative hypothesis H1 is said to be accepted. The tests statistic T is constructed as a function of +the data whose probability distribution under H0 is known or can be approximated by a known distribution. A common +situation is the comparison of two independent groups, an experimental (E) and control (C) group (e.g. knockout +animals vs. wildtyp animals), under assumption of normally distributed data: +yik = θi + ϵik, +ϵik ∼ N(0, σ2 +i ), i = C, E, k = 1, . . . , ni. +(2) +2.1.1 +Hypothesis tests in a frequentist analysis +As frequentist hypothesis test on the mean difference δ = |θE − θC| the two-sample t-test can be used in case of equal +variances σE = σC (homoscedasticity) (see Lehmann & Romano (2022)). Often, however, the residual variance is +greater in the experimental than the control group due to varying treatment effects and the Welch test (Satterthwaite +test) by Welch (1938), Satterthwaite (1941) is more appropriate since it does not make a homoscedasticity assumption: +TWelch = +y1. − y2. +� +σ2 +C +n1 + σ2 +E +n2 +(3) +3 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +Under H0, the distribution of the test statistic TWelch is approximated by a t-distribution t˜ν with modified number of +degrees of freedom ˜ν (for details see Welch (1938, 1947)). The variances σ2 +E and σ2 +C can be estimated as empirical +variances from observed data. The corresponding two-sided hypothesis test is then +ϕWelch = 1{|TWelch| > t˜ν;1− α +2 } +(4) +where t˜ν;1− α +2 denotes the 1 − α +2 quantile of the t-distribution with modified degrees of freedom. There are two standard +types of errors that can be made by a hypothesis test: +1. Type I error: ϕ rejects H0 although it is true +2. Type II error: ϕ doesn’t reject H0 although it is not true +Both errors cannot be minimized simultaneously. Since the type I error is generally regarded as worse then the type II +error, a classical hypothesis test ϕ = ϕα is set up to limit the probability of type I error by a significance level α and +then finding the optimal test among all the tests at level α by minimizing the probability of type II errors (see Lehmann +& Romano (2006)). This gives a test ϕ∗ +α that has maximal power P(ϕ = 1|H1) while controlling type I errors at level +α. The sample size is then classically calculated as minimal number for which the chosen hypothesis test ϕ∗ +α detects a +clinically relevant effect size δ∗ at least with probability 1 − β. More specifically, in applications with simple analytic +distribution form of the test statistic, the sample size can be calculated by solving the power inequality +P(ϕWelch,α,nE,nC = 1|δ ≥ δ∗) ≥ 1 − β. +(5) +For the Welsh test, the critical value t˜ν;1− α +2 itself is a functions of the sample size through the (approximated) degrees +of freedom. Hence, the inequality cannot be solved directly. Alternative approaches are to approximate the critical +values by a normal distribution or to use an iterative approach. +2.1.2 +Hypothesis tests in a Bayesian analysis +“Statistically significant" in a frequentist context means that the test statistic T is greater than the critical value c (here: +c = tν;1− α +2 ) or equivalently that the p-value of the test is smaller than α or the confidence interval at level 1 − α for the +tested effect δ does not include the null value δ0 (see Lehmann & Romano (2006)). Similar decision rules or “tests" +can be established in a Bayesian framework. In a full Bayesian model, all parameters are modelled as probability +distributions with their own prior distributions. The idea of Bayesian analysis is to update these prior distributions to +posterior distributions by the information in new data using Bayes’ rule. Details on prior distributions are given in +section 2.4. Endowing all model parameters with probability distribution rather than fixing parameters to concrete +values leads on the one hand to better representation of uncertainties than in the frequentist framework. On the other +hand it often leads to complicated posterior density functions that require the evaluation of high-dimensional integrals +and can no longer be expressed in analytic form. Numerical computation using Markov chain Monte Carlo (MCMC) +methods is a common alternative. In MCMC methods the posterior of the parameter distribution is approximated by a +series of MCMC draws. For one-sided test problems, like the upper test problem H0 : {δ ≤ δ0} vs. H1 : {δ > δ0}, the +estimated posterior probability p(δ|y) of δ being greater than a null value δ0, given the data y = (yC, yE), or than a +clinically relevant value δ∗ can be computed. To transform this idea into a decision rule this posterior probability can be +compared to a predefined critical value like suggested by Weber et al. (2021). This approach cannot be used for the +two-sided problem (1), when the tested effect δ is modeled by a continuous probability distribution, like it is the case +when δ = |θE − θC| reflects the difference in two continuous means. In this case the posterior probability of a single +point event P({δ = δ0}) is equal to zero. Alternatives are the consideration of two-sided Bayes factors or Bayesian +credible intervals. +Credible intervals are Bayesian versions of frequentist confidence intervals with slightly different interpretation. By +definition, a frequentist 1 − α confidence intervals for δ is an interval with random bounds that covers the unknown +(but regarded as fixed) parameter δ with probability 1 − α. Hence, if one would generate data from the corresponding +probability model and calculate a confidence interval for each data set, one would expect that (1−α)% of these intervals +would cover (the fixed) δ. The interpretation of a 1 − α Bayesian credible interval is instead that it is set up (with +bounds regarded as fixed) so the probability that a random realization of δ falls within the credible interval is 1 − α. +Equivalently it can be stated that the Bayesian 1 − α credible interval includes (1 − α)% of the posterior probability +mass of δ, given the data y (see Held & Sabanés Bové (2014)). Yet, this definition does not uniquely define the interval. +There are two common types of Bayesian credible intervals: quantile intervals and highest density intervals. The bounds +4 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +of a 1 − α quantile interval for a parameter are given by the α +2 and 1 − α +2 quantiles of its posterior distribution and +is also called equal-tail interval. In praxis, the bounds of a quantile interval can be approximated by the quantiles of +the MCMC draws. A highest density interval (HDI) is defined as a credible interval for which the posterior density +of each point inside the interval is higher than the posterior density for an arbitrary point outside the interval. This is +a desirable property as a summary of the distribution and is especially relevant for skewed distributions where, for +the quantile interval, it is possible that parameter values inside the quantile interval are less probable than parameter +values outside the interval. Moreover, the HDI is the smallest among all possible credible intervals which is a desirable +property when it is used for posterior inference. On the other hand, an advantage of the quantile interval is, that it is +easier to interpret for transformed parameters than the HDI. This is because one can derive the quantile interval of the +transformed parameter simply by back-transforming the intervals that where derived for the transformed parameter. +In contrast, the HDI of the untransformed parameter cannot be simply derived by back-transforming the HDI of the +transformed parameter. A further difference is, that the quantile interval always includes the median of the posterior +distribution, whereas the HDI always includes the mode(s) of the posterior distribution (Kruschke (2015)). In symmetric +distributions, the quantile interval and the HDI return similar results. Moreover, under certain conditions Bayesian +credible intervals also coincidence frequentist confidence intervals coincidence (see Jaynes & Kempthorne (1976)). +With Bayesian credible intervals one can set up a decision rule by testing if δ0 falls withing the 1 − α posterior credible +interval of δ. A more meaningful approach may be to test whether the posterior interval excludes a region of practical +equivalence (ROPE) which may be defined as all parameter values that are smaller (in absolute value) than the minimal +clinically relevant effect size δrel, as suggested by Kruschke (2015)). As a further alternative, Kruschke suggests to set +as goal not to reach a certain power but rather a certain precision. An according decision rule can be implemented by +deciding if the width of the credible intervals is smaller than a threshold value representing a target precision. +Classical decision rules for a null hypothesis have been criticized for null hypothesis testing, for example by Wagen- +makers et al. (2020), Gelman, Hill & Vehtari (2020), Schad et al. (2022). One critique is connected to the principle +of predictive irrelevance stating that data that are predicted equally well by both a null model M0 (corresponding to +H0) and an alternative model M1 (corresponding to H1), data which the authors in Wagenmakers et al. (2020) call +uninformative or irrelevant, should not lead to favor one model above the other. However, this can happen in the above +described scenario of null hypothesis significance testing (NHST) when intervals or p-values are estimated only under +one of both models. Instead, to quantify the evidence for an effect that differs from the null hypothesis, Bayes factors +are proposed. Bayes factors compare the probability of the observed data under (at least) two models M0, M1: +BF10 = p(y|M1) +p(y|M0) +(6) +This Bayes factor gives an impression of how much more likely the data was generated by the model under H1 over the +model under H0. It is related to the odds of posterior model probabilities p(M1|y) +p(M0|y) by being the factor by which the ratio +of prior model odds p(M1) +p(M0) changes after observing the data: +p(M1|y) +p(M0|y) = BF10 · p(M1) +p(M0) +If the Bayes factor is greater than one this indicates that, after observing the data, the odds for the model under H1 over +H0 have increased as compared to the a priori expectations. Schad et al. (2022) show that, to make this an approach +that is sensible to the observed evidence in the data, it is essential to chose appropriate prior distributions for the model +parameters. Typical methods for estimating Bayes factors based on the prior distributions and the data are bridge +sampling (Bennett (1976)) or the Savage-Dickey method (Dickey & Lientz (1970)). To set up decision rules that are +based on Bayes factor, a threshold has to be defined as to when the null hypothesis is rejected. Lee & Wagenmakers +(2014) provide a rough interpretation scheme for Bayes factors that is adjusted from Jeffreys (1961). They declare +moderate evidence for H1 if the Bayes factor BF10 is greater than 3. This can be used to calculate the percentage +with at least moderate evidence for H1 as a binary decision function. This classification and decision rule present a +convenient overview and method to get something like a power estimate from the Bayes factors, but should not be +used as a strict rule (see Schad et al. (2022), Schönbrodt & Wagenmakers (2018)). Instead, the whole distribution of +the Bayes factors should be considered and ideally simulation based calibration and sensitivity analysis, as presented +by Schad et al. (2022), should be carried out to ensure a correct interpretation of the Bayes factors. Alternatively, a +heuristic decision rule can also be defined by making a decision in favor of H1 if this is the model with the higher +posterior model probability p(Mi|y), i = 0, 1 (see Schad et al. (2022)). This decision rule however does not include +the outcome of no evidence for either hypothesis and is only optimal if both errors of the corresponding decision rule +(deciding for H1 when the data in fact corresponds to model M0 and deciding for H0 when the data in fact corresponds +5 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +to model M1) are equally bad. This is typically not the case in clinical or preclinical research. A more principle scheme +for deriving decision rules is by the definition and optimization of utility functions. Utilities define the cost or value of +decisions, conditionally on the null or alternative hypothesis being true and are necessary to judge the performance +of a decision function. In the frequentist framework utilities are defined in terms of type I and type II error rates and +decision functions are constructed by bounding type I errors and optimization with regard to type II errors. Differences +in the Bayes factor oriented decision rules and the frequentist hypothesis tests are, for example, that the Bayes factors +can distinguish between no evidence and evidence for H0, whereas frequentist tests can not. +2.2 +Simulation based design analysis +Gelman & Carlin (2014), Gelman & Vákár (2019), Gelman, Hill & Vehtari (2020) propose design analysis using +fake-data simulation as an alternative to classical power-oriented sample size calculations. Using fake-data simulation, +the effect of varying parameters on the predefined decision functions can be examined in combination with relevant +candidates for the sample size. To determine if predefined statistical goals are met, models are fitted to the data, +statistical analysis are performed and the previously defined decision functions are evaluated. The statistical goals +are formalized in utility functions as introduced in the previous section. Here, utilities are calculated as type I and +type II error rates or false discovery rates (FDR) and true discovery rates (TDR) by determining the percentages how +often the data were simulated with a δ equal to the null effect but the decision functions decided against the null effect +and how often the decision function decide against the null effect when the data were simulated with δ corresponding +to increasing effect sizes. The goals are then to (find a sample size to) reach certain TDRs or a certain power while +limiting the FDR or type I error rates by a certain α. Additional model characteristics are examined as the type S (sign) +errors, as the probability that the estimate of the true effect has the wrong sign, given that is statistically significant and +type M (magnitude) errors, as the probability that the effect estimate is greater in absolute value than the absolute value +of the true effect, given that it is statistically significant Gelman & Carlin (2014). Moreover, the the mean-squared-error +(MSE) of the estimate of the true effect size is examined in simulated data under varying true effect sizes by Gelman & +Vákár (2019). Also the distribution of Bayes factors is visualized and it is examined how often the Bayes factors are +falsely greater or smaller than one, if the lower bound of the 95% interval of the posterior model probability for the +model under H1 exceeds the value of 50% and what percentage of the Bayes factors lies within the categories defined +by Jeffreys (1961). Given the utility function(s) and decision rules, a minimal sample size can be chosen that reaches +one or several of these predefined goals. +R = 10000 simulated data sets are constructed in accordance with model (2) as sum of a random control group, a +treatment effect and group-specific residuals. In this work the data in the new experiment is generated in a frequentist +framework with fixed values for δ, θC, ψ and λ and randomness in the simulated data comes only from the group-specific +residuals. +y∗ +ik,r = θ∗ +C,r + δ∗ +r1(i = E) + ϵ∗ +il,r +ϵ∗ +ik,r ∼ N(0, σ∗2 +i ), σ∗ +i = exp(η∗ +σi,r), where η∗ +σi,r = ψ∗ +r + λ∗ +r1(i = E) +(7) +for animal k = 1, . . . , ˜ni in group i = C,E of data set r = 1, . . . , 10000. Alternatively, the new data could also be +generated in a fully Bayesian framework as discussed in the discussion. +For each simulated data set a Bayesian model is fit with the brms function of the brms package (Bürkner (2021)) +in R Core Team (2021), using the default MCMC parameters. Frequentist point estimates and confidence intervals +for the population effects are estimated with the lm function in base R. The steps and decisions that are commonly +made in a Bayesian framework are described as a Bayesian workflow by Gelman, Vehtari, Simpson, Margossian, +Carpenter, Yao, Kennedy, Gabry, Bürkner & Modrák (2020), Schad et al. (2021). After fitting a Bayesian model the +next step in a Bayesian workflow is to validate the computations. To decide whether the MCMC draws are likely to have +converged against the target posterior distribution, characteristics of the MCMC chains are examined in convergence +diagnostics. More specifically, MCMC trace plots, auto correlation plots, the effective sample size and the ˆR statistic are +examined (for details see Gelman et al. (2013)). Bayesian credible intervals are estimated as highest density intervals +with the bayestestR package (Makowski et al. (2019)) and quantile intervals computed as empirical quantiles of +the posterior MCMC draws. Bayes factors are computed with the hypothesis function of the brms package. To +account for heteroscedasticity, a distributional model is fit in brms and in the frequentist model, heteroscedasticity +is accommodated by the estimation of robust confidence intervals with the sandwich (Zeileis (2006)) package that +estimates heteroscedasticity consistent (HC) variances. More specifically, a HC3 type estimator is chosen in the +frequentist design that is also appropriate for smaller sample sizes (see Long & Ervin (2000)). Additionally, frequentist +p-values in the Welsh test are calculated. +6 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +2.3 +Meta-analysis model for the historical data +The distributions in the sampling model (7) as well as the prior distributions for a Bayesian analysis of the simulated +(and new) data are based on a Bayesian meta-analysis model of relevant historic data. The hope is that, by the usage +of Bayesian estimation and historical evidence, a prior knowledge and uncertainties are better reflected and a sample +size can be found that is more likely to actually achieve the predefined statistical goals than with classical frequentist +methods. In the best case, and if the prior distributions reflect (major aspects of) the simulated data correctly, using +the historical information as prior distribution in the Bayesian analysis of the new data can even reduce the number of +animals that are needed to reach the predefined statistical goals. +2.3.1 +Normal-Normal hierarchical model +The historical data are modeled as data from G different animal experiments with ng animals each, g = 1, . . . , G. As +simplest and most commonly assumed case the data are assumed to be normally distributed as +ygk = θg + ϵgk, +ϵgk ∼ N(0, σ2 +g) k = 1, . . . , ng, g = 1, . . . , G. +(8) +with residuals ϵgk, g = 1, . . . , G, k = 1, . . . , ng, and experiment specific means θg, g = 1, . . . , G estimated as +arithmetic means yg = +1 +ng +�g +k=1 ygk, g = 1, . . . , G. Data, for which a normal assumption is inappropriate, such as +skew and non-continuous data, can often be transformed to resemble a normal distribution and be handled by this +model (see Hedges & Olkin (1985), Hartung & Knapp (2001), Higgins & Green (2011)). As an assumption that +allows the usage of the historical data for the analysis of the new experiment, the new data and the historical data are +assumed to be exchangeable. This means that there are assumed to be no systematic differences in the new and the +historic data. This assumption is modeled by a random-effects meta-analysis for the historical data and the usage of this +model’s mean prior predictive distribution to construct a prior distribution for the parameters in the new experiment +as outlined by Neuenschwander et al. (2010). Heterogeneity as variance of the historical experiments may occur due +to different ages, animals strains, laboratory conditions or measuring instruments. Such heterogeneity is modeled +by heterogeneity components γg that represent deviation from a common mean µ in the experiment specific means +θg = µ + γg, g = 1, . . . , G. The parameters µ and τ in the distribution of the experiment specific parameters θg of +interest are referred to as hyperparameters. +A Normal-Normal hierarchical model (NNHM) for the historical data is formulated as +ygk|θg, σg ∼ N(θg, σ2 +g) +θg ∼ N(µ, τ 2). +(9) +This model is termed hierarchical since it includes connected sampling distributions on two levels and Normal-Normal +hierarchical model since a normal assumption is assumed for both the residuals ϵgk on the level of the individual +data and the heterogeneity components γg on the level of the experiment specific means. In a frequentist framework +this model with normally distributed means θg ∼ N(µ, τ 2) is also referred to as random effects model. Fitting a +meta-analysis in a Bayesian instead of a frequentist framework has proven to be beneficial especially in the case of +small, few studies by Friede et al. (2017a,b). The estimation of the group-specific means in a hierarchical model +with a common mean as pooled effect leads to so called shrinkage estimates that are shrunken towards the common +mean µ as compared to the estimation of independent means in a fixed effect model. The advantage of the shrinkage +estimation is that single extreme values are relativized and the uncertainty in single groups can be reduced by borrowing +information from other groups. The degree of shrinkage of one group g ∈ {1, . . . , G} depends on its sample size ng (or +the associated standard error sg) and the between-group heterogeneity τ 2. Shrinkage is higher for smaller ng (bigger +sg) and smaller τ 2 (for details see Gelman et al. (2013), Schmidli et al. (2014), Neuenschwander et al. (2016), Wandel +et al. (2017), Röver & Friede (2021)). +2.4 +Prior distributions +In a Bayesian framework further probability distribution models as prior distributions are set up for the additional +parameters (σg, µ and τ in the meta-analysis in equation (9) model and θC, δ, ψ and λ in model (7)). A popular choice +for a parameter’s prior distribution is a so called conjugate prior from the distribution family that is conjugate to the +family of the modeled likelihood distribution of the data Gelman (2006). Choosing such a conjugate prior distribution +for a model parameter implies that also the parameter’s posterior distribution is from the same family as the prior +distribution. This has interpretational and computational advantages as outlined in Gelman (2006), Gelman et al. +7 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +(2013). A prior distribution can be categorized according to its information content to be either non-informative, weakly +informative or informative. This categorization can be controlled by the prior distribution’s variance parameter and has +to be interpreted in context of the likelihood distribution the data (for details see Gelman et al. (2013)). Non-informative +priors have minimal impact on the posterior distribution as compared to the impact of the data. They are constructed +so that their probability density is flat relative to the probability density of the data. In contrast, informative prior +distributions are designed to represent the full available a priori knowledge as accurately as possible and to have a +major impact on the parameters’ posterior distribution together with the impact of the data. Weakly-informative prior +distributions constitute a compromise between non-informative and informative distributions. They are intentionally +designed to be flatter than informative prior distributions. Weakly-informative priors can be chosen to have a regularizing +functionality on the posterior distribution by restricting the posterior distribution to a plausible parameter range (for +details see for example Gelman et al. (2013), McElreath (2018)). Generally, context specific weakly informative or +informative priors are preferred over non-informative priors, especially when Bayes factors are estimated (see Gelman +(2006), Betancourt (2017), Seaman III et al. (2012), Betancourt (2017), Schad et al. (2021), Lemoine (2019)). +2.4.1 +Variance parameters +With few studies, special care hast to paid to the specification of a prior distribution for the heterogeneity parameter τ +in model (10) (see Gelman (2006), Friede et al. (2017a)). Friede et al. (2017a) recommend a prior distribution that +puts most of their probability mass to areas that represent small to large heterogeneity and leave only a little fraction to +values that represent a larger heterogeneity. The interpretation of the heterogeneity degree depends on the scale of the +modelled parameter. Spiegelhalter et al. (2004) suggest to classify heterogeneity in context with the residual standard +deviation σ of the data model into the following classes: +Heterogeneity +r := τ +σ +small +0.0625 +moderate +0.125 +substantial +0.25 +large +0.5 +very large +1.0 +Table 1: Classification of heterogeneity in relation to the standard deviation of the data according to Spiegelhalter et al. +(2004). +Recommended prior distributions are then from the family of of folded non-central t-distributions with special cases of +the half-t and half-Normal distribution. The half-Normal distribution HN(ϕ) has a scale parameter ϕ and is related to +a standard normal distribution with mean zero and variance ϕ2 by taking the standard normal distribution’s absolute +values. Compared to the half-t distribution the tail of a half-normal distribution is smaller which puts less weight on +extreme heterogeneity values (see Spiegelhalter et al. (2004)). The distributions parameters in this applications example +are set up to reflect the assumptions on the heterogeneity as classified by table 1. +2.4.2 +Intercept parameters +As prior distribution for the intercept parameters normal distributions are chosen which is, conditionally on all other +model parameters, the conjugate family to the normal distribution of the data. More specifically, as starting point +for the intercept’s prior, a unit information prior (UIP) is recommended (see Kass & Wasserman (1995), Röver et al. +(2021)). A unit information prior has the information content (variance) that corresponds to one single typical data +point. The unit information prior for the historical data is set up to be centered at the mean of the historical data. +Strictly speaking orienting the prior distribution on information from the data implies to use the same information twice, +once for setting up the prior and once through the likelihood of the observed data that both go into the estimation of +the posterior distribution. This contradicts the Bayesian concept of a prior distribution as representation of a priori +knowledge before seeing the analyzed (historic) data. It can nonetheless serve as a starting point to ensure that the +prior distributions are centered at some reasonable area. By choosing a variance parameter that leads to a wide enough +but not unrealistically wide prior distribution, it reflects only a rough guess and lets the data contribute more exact +information to refine the parameter’s posterior distribution while having a regularizing functionality McElreath (2018). +This again can be examine in sensitivity analysis. A unit information prior can also be used as reference prior for testing +Bayesian hypothesis or for an assessment of the effects on the posterior compared to more informative priors (see Kass +& Wasserman (1995), Raftery (1998), Neuenschwander & Schmidli (2020), Li (2021), Röver et al. (2021)). +8 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +For the mean θC and the standard deviation σC in the control group in the new experiment, MAP priors are computed. +Therefore the posterior_epred function of the brms package is used to estimate the expected posterior predictive +distribution in the operation groups. The expected posterior predictive distribution of the group without operation is +used to derive parametric prior distributions for the mean and standard deviation θC and σC = exp(ψ) in the new +experiment, as explained in the next section. The priors for θE and σE = exp(ψ + λ) are set up to be centered around +the same values like the priors θC and σC = exp(ψ), but with higher variance parameters, corresponding to unit +information priors. The higher variance parameters reduce the weight of the prior distributions as compared to the data. +The intention for centering the prior distributions of the parameters in the control and experimental group around the +same value is that, a priori, the two groups are expected to be equal on average. Meanwhile, the intention for the higher +variance in the priors of the parameters in the experimental group is, that for the control group there is historical data +available, so the prior distribution should have larger influence than for the experimental group, where no (directly +related) historical data is available. +2.4.3 +Approximation of the MCMC draws by parametric distributions +To incorporate the historical data as proper prior distributions, the non-parametric estimates of the posterior predictive +distribution, as represented by the MCMC draws, are approximated by parametric distributions. Röver et al. (2021, +2022) illustrate three general methods for fitting parametric distributions to MCMC draws. In each method the first +step is to specify a distribution family. Choosing a (to the data distribution) conditionally conjugate normal prior +distribution for the mean parameter in the historical data model ensures that the posterior distribution is from the +same parameter family, conditionally on fixed values of all other model parameters. However, when the other model +parameters are not fixed, the posterior distribution is not necessarily a simple normal distribution. In case of the NNHM +from equation (9), Röver (2017) show that, with a normal prior distribution for µ and a half-normal prior distribution +for τ, the posterior predictive distribution for the man parameter µ and the parameter θ∗ in the new experiment are +normal-mixture distributions. Thus, also more complex distribution families have to be considered for the approximation +of the posterior MCMC draws. After a distribution family is specified, a simple method to estimate its parameters is by +taking point estimates, like the mean of median, from the posterior draws. This approximation of the posterior draws as +point estimates however is a reduction of information and does not always reflect all important characteristics of the +posterior distribution. An alternative method is to approximate the posterior draws by marginal distributions (see Röver +et al. (2021)). A third alternative is to choose a model family and determine its parameters by maximum likelihood +(ML) estimation, the expectation-maximization (EM) algorithm or moment matching (MM). Different candidates for +distribution families can be compared by using estimators of the model fit or of their predictive performance like the +Akaike information criterion (AIC) Weber et al. (2021). In the Bayesian context, the Watanabe-Akaike information +criterion (WAIC) and Leave-one-out cross-validation (LOO-CV) are preferred over AIC, as outlined by Gelman et al. +(2014), Vehtari et al. (2017). But since the parametric distributions are fit with frequentist and not with Bayesian +methods, WAIC and LOO-CV are not considered here. In this work, the parametric distribution candidates for the +MAP priors are normal, normal mixture and t-distributions. For fitting a mixture distribution, the automixfit function +of the RBesT R package (Weber et al. (2021)) is used. This function uses the EM algorithm to fit a series of normal +mixture distributions with increasing number of mixture components and selects the best fit according to the AIC value +(which penalizes model complexity). For comparison, simple normal and non-centered t-distributions are fit using ML +estimation. Prior predictive checks, as described in section 2.4, are perfomred to ensure the priors are reasonable. +The approach to formulate a predictive distribution as prior distribution in the new experiment is termed meta-analytic +predictive (MAP) approach (Neuenschwander et al. (2010)) and requires the explicit formulation of a prior distribution +as predictive distribution, given the historical data (MAP prior). An alternative to this sequential approach for the +analysis of the historical and new data is the meta-analytic-combined (MAC) approach where both historic an new data +are analyzed in a common analysis. Schmidli et al. (2014) show that the MAP approach is theoretically equivalent to +the MAC approach. In practice, the MAC approach has the advantage to be more direct and easier since it requires +only to fit one single Bayesian model instead of one model for the historical data, one for the derivation of a MAP +prior from the historic data and one for the analysis of the new data. In the context of design analysis and sample +size determination, however, the MAP approach has the advantage of allowing better judgment and control over the +influence of the historic data on the estimation results in the new analysis. Furthermore, an effective sample size (ESS) +of the MAP prior neff can be calculated that quantifies the influence and information content of the historical data as +ratio of the variance of the MAP prior in a heterogeneous sample to the variance of the MAP prior in a homogeneous, +pooled sample. neff gives an estimate of the number of animals that can be saved in the new experiment by using the +information in the historical data (given that the new and historical data are exchangeable) (see Neuenschwander et al. +(2010, 2020)). +9 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +3 +Application example +As an application the simulation based design analysis and sample size determination is carried out for an animal +experiment from translational preclinical research that aims to examine the role of the C5aR1 receptor on bone quality +under postmenopausal osteoporosis in mice. The pathological condition of postmenopausal osteoporosis is realized by +ovariectomy in mice. Bone quality is measured (among other indicators) by the (unit-less) relative bone volume (bone +volume to tissue volume (BV/TV)) in a µCT scan. This experiment is referred to as C5aR1 experiment in the following. +The aim of the planned animal experiment is to test whether or not there is evidence for an association between the +C5aR1 knockout and the relative bone volume in mice. If there exists such an associations, then a C5aR1 knockout may +serve as basis for a potential treatment that can be examined in humans in clinical trials. The planned experiment shall +consist of data from twelve week old female mice of the C57BL/6J strain, which is the standard mouse strain from the +Jackson laboratory research institution. +3.1 +Data and models in the application example +3.1.1 +Historical data +Internal historical data from a previous experiment is available from the proposer for planning the new experiment. +This internal data comprises data from twelve ovariectomized mice and ten mice with Sham operation. Yet, the +internal historic data from the proposer represents only a fraction of the information that has been collected so far +regarding bone quality in mice since bone quality is an active research topic in preclinical research (see for example +Ignatius, Schoengraf, Kreja, Liedert, Recknagel, Kandert, Brenner, Schneider, Lambris & Huber-Lang (2011), Ignatius, +Ehrnthaller, Brenner, Kreja, Schoengraf, Lisson, Blakytny, Recknagel, Claes, Gebhard et al. (2011), Mödinger et al. +(2018) and the citations therein). The inclusion of additional data has the potential of providing more information about +the distribution of the relative bone volume in mice, the variability among and between different experimental groups +and upon which effect sizes are realistic. However most of the available literature cannot be used as further historical +data, since the animals characteristics (such as age and the animal strain) of the external data and the experimental +conditions under which external data have been collected, deviate from the internal historic data or the outcomes are +measured with different methods, have different definitions or scales. An easy and comprehensive option for retrieving +control data is the Mouse Phenotype Database (MPD) (Grubb et al. (2004), Bogue et al. (2018)). It is an open-access +database that collects phenotype data for the characterization of inbred mice. The data can be used for characterizing +the correlations of complex traits and as control data or for characterization of mutation effects (see Consortium (2007)). +The MPD data comes in a tidy, standardized format and also the experiment protocols and tools for data analysis are +provided. The data is made available by worldwide researchers and managed by employees of the MPD, who also +endow the data with a common public ontology like the Mammalian Phenotype Ontology, that was introduced by Smith +et al. (2005), what makes it easier identify and compare relevant data for the outcome of interest. +The search term “BV/TV" on the MPD web-page leads data from 31 strains of Collaborative Cross (CC) wildtyp mice +from an experiment of Levy et al. (2015). CC mice are recombinant inbred mice from eight genetically divergent strains. +They are characterized by a high degree of genetic diversity that represent on average 90% of the allelic diversity in +the whole mouse genome Chesler et al. (2008). In the planned experiment the mice shall undergo ovariectomy. Since +ovariectomy is expected to have an impact on the relative bone volume, the MPD mice cannot be used on its own to +estimate a posterior predictive distribution for the mean relative bone volume in the new experiment. Still, the estimation +of the distribution for the mean in the wildtyp mice can give an idea for the range of plausible effect sizes that are +examined in the design analysis. For example, one might expect that the best one can expect from the C5aR1 knockout +in the experimental group in the new experiment is to completely reverse the negative effect of the ovariectomy and +bring the relative bone volume back to the basic level in wildtyp mice. Moreover, the additional consideration of the +external data can help to quantify heterogeneity in the outcome in different strains and ideally, it could also give an +impression of how representative the internal mice strains are for the mouse genome in general with respect to the +outcome relative bone volume by comparing internal wildtyp mice to the MPD wildtyp mice. However, int his case +there is no internal data from wildtyp mice but only ovariectomized and Sham mice. +The external data and internal data are represented in table 2. Since the hypothesis refers only to female animals, all +male animals from the external MPD data set are excluded from the meta-analysis. The relative bone volume can by +definition only take positive values. In the historical data most of the values were close to zero with a couple of very big +values. A logarithmic transformation was applied to the data to transform the scale of the outcome to the real numbers +and to make it resemble more a normal distribution. +10 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +Experiment +strain +OP +n +� +E(BV/TV) +� +SD(EI) +� +E(log(EI)) +� +SD(log(EI)) +Internal +C57BL/6 +Ovx +12 +1.5 +1.10 +0.1 +0.93 +Internal +C57BL/6 +Sham +10 +4.2 +1.19 +1.4 +0.36 +MPD +PreCC1061/Tau +None +2 +9.2 +0.74 +2.2 +0.08 +MPD +PreCC111/Tau +None +9 +14.4 +3.23 +2.6 +0.22 +MPD +PreCC1156/Tau +None +2 +15.8 +0.21 +2.8 +0.01 +MPD +PreCC1513/Tau +None +7 +29.2 +6.47 +3.4 +0.21 +MPD +PreCC188/Tau +None +9 +6.9 +2.11 +1.9 +0.29 +MPD +PreCC1912/Tau +None +6 +12.8 +1.17 +2.5 +0.10 +MPD +PreCC2126/Tau +None +6 +2.4 +2.14 +0.6 +0.76 +MPD +PreCC2156/Tau +None +8 +7.7 +2.74 +2.0 +0.34 +MPD +PreCC2391/Tau +None +2 +2.9 +1.76 +0.9 +0.66 +MPD +PreCC2573/Tau +None +3 +12.5 +9.42 +2.4 +0.70 +MPD +PreCC2680/Tau +None +7 +5.8 +2.29 +1.6 +0.53 +MPD +PreCC2689/Tau +None +6 +6.5 +2.24 +1.8 +0.33 +MPD +PreCC2750/Tau +None +5 +6.6 +3.91 +1.8 +0.61 +MPD +PreCC3348/Tau +None +7 +15.1 +3.91 +2.7 +0.24 +MPD +PreCC3438/Tau +None +5 +7.2 +3.46 +1.9 +0.49 +MPD +PreCC3480/Tau +None +3 +6.8 +7.06 +1.4 +1.40 +MPD +PreCC3912/Tau +None +10 +11.3 +2.32 +2.4 +0.20 +MPD +PreCC4052/Tau +None +12 +13.5 +11.47 +2.3 +0.81 +MPD +PreCC4141/Tau +None +6 +12.8 +5.47 +2.5 +0.46 +MPD +PreCC4438/Tau +None +3 +8.5 +0.67 +2.1 +0.08 +MPD +PreCC4457/Tau +None +7 +14.9 +3.83 +2.7 +0.30 +MPD +PreCC519/Tau +None +6 +15.6 +7.55 +2.6 +0.48 +MPD +PreCC521/Tau +None +7 +6.1 +1.47 +1.8 +0.23 +MPD +PreCC557/Tau +None +4 +5.3 +1.72 +1.6 +0.35 +MPD +PreCC611/Tau +None +3 +8.7 +0.16 +2.2 +0.02 +MPD +PreCC670/Tau +None +3 +14.9 +3.53 +2.7 +0.23 +MPD +PreCC711/Tau +None +3 +12.3 +1.01 +2.5 +0.08 +MPD +PreCC72/Tau +None +6 +8.1 +1.88 +2.1 +0.23 +Table 2: Sample size (n), mean (ˆE) and empirical standard deviation ( ˆ +SD) of the relative bone volume (BV/TV) in the +historical data on the original and the logarithmic scale. The estimates are calculated by the group factors experiment +(internal, MPD), mouse strain and operation group (OP as ovariectomy (Ovx), Sham operation and no operation). +3.1.2 +Meta-analysis model +A hierarchical or meta-analysis model with individual data (IP-MA, individual patient meta-analysis) (see for example +Lyman & Kuderer (2005), Michiels et al. (2005), van Walraven (2010)) is fit in a Bayesian framework by using the brm +function in the brms package (Bürkner (2021)). For comparison, a frequentist model is fit in R with the lme of the +nlme package (Pinheiro et al. (2022)) to accommodate the random strain effects, where restricted maximum-likelihood +(REML) estimates are derived by maximization of the log-restricted maximum-likelihood method (see Pinheiro & Bates +(2006)). An operation group variable OP is included as predictor to indicate if the mice had underdone ovariectomy, a +Sham operation or no operation. The ovariectomy and Sham operation may not have the same impact on all animals +in the respective operation group whereas the data in the animals without an operation is expected to have smaller +variance. To allow the mice in the ovariectomy group to have a different residual variance than the residual variance in +the Sham operation and allow the residual variance to be yet different than the residual variance in the group without +operation, a distributional model (like explained in Bürkner (2020)) is fit, in which the residual variance is modeled by +its own operation group specific predictor term, similar to the one in equation (7). Ideally, since bone quality is known +to depend on age, this variable should be included as predictor term. But since each operation group is from a different +age interval (the youngest are the animals without operation, and the animals with Sham operation and ovariectomy are +all of an older age), the effects of age and the operation group cannot be untangled by the inclusion of the age variable. +The same problematic applies to the experiment where the data came from (internal, MPD). Also this variable should +ideally be modeled as fixed or random effects parameter. But since both Sham and ovariectomized animals are internal +data and the animals without operation are external data from the MPD also this effect cannot be estimated. +11 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +In summary, the the following meta-analysis model is fit: +yijk = α + β11(i = 1) + β21(i = 2) + νj + ϵijk, +with operation group specific residuals +ϵijk ∼ N(0, σi), σi = exp(ησi), +ησi = ψ + λ11(i = 1) + λ21(i = 2) +and additional independent variance components +νj ∼ N(0, τ 2 +ν ) +(10) +where k = 1, . . . , nij indices the mouse in operation group i = 0, 1, 2 (none, ovariectomy (Ovx),Sham) and strain +j = 1, . . . , 22. The choice of the prior distributions is explained in the next section. +3.1.3 +Prior distributions for the historical data +In the brms package non-informative or weakly informative prior distributions are specified as default. Different default +priors are chosen for the intercept (intercept of the mean, α, and intercept of the logarithmic residual standard deviation, +ψ), than for the population parameters that apply to the whole data (fixed effects in a frequentist framework, here βi, +λi, i = 1, 2 and γ), and for group-specific variance parameters that model the heterogeneity between groups (random +effects in a frequentist framework, here νj, j = 1, . . . , 22). +In table 3 the manually chosen prior distributions for parameters from model (10) are summarized and contrasted with +the default priors in the brm function. For the intercept, an unit information prior (UIP) is considered as reference as +explained in section 2.4. The intercept α represents the mean relative bone volume in the animals without operation and +at age of eight weeks on the logarithmic scale. An UIP is set up with information content corresponding to a single +observation. Therefore the historical animals without operation are considered. Using ML estimations to fit a normal +distribution to the group without operation, the residual standard deviation is estimated to be 0.72. To make this a bit +less informative, the standard deviation for the prior of α is increased to the value 1. The mean of α’s prior, is set +according to the estimated mean in the group without operation which is 2. These choices result in a 95% interval +of [0.1, 4] on the logarithmic scale and [1.1, 53] on the original scale. The operation-group specific residual standard +deviations σi, i = 0, 1, 2 (no operation, Ovx, Sham), in model (9) are modelled as exponent of a normally distributed +linear predictor ησi, that is centered at mean 0. With a scale parameter of 0.5, σ0 has a 95% quantile of 2.3, which is +considered as more realistic than the default t3(0, 2.5) distribution in brms that leads to a 95% quantile of 360 of σ0. +For the heterogeneity parameter τν of the variance parameters νj, j = 1, . . . , 22 (representing the variance due to the +different strains) a half-normal prior is chosen as discussed in section 2.4. The scale parameter of τν is set to 1 +2, which +corresponds to large heterogeneity when classified with table 1 and the mean of the prior σ0 as reference scale. A +priori, large heterogeneity is expected to exist in the mice strains since they include CC mice from genetically diverse +backgrounds as explained above. The population effects βi, and λi, i = 1, 2 are kept at their default values in brms and +only modified if there are indications of convergence problems or if the prior predictive checks indicate that they are +unrealistic. Neither is the case here. +Parameter +Default +Manual +α +t3(1.7, 2.5) +N(2, 12) +ψ +t3(0, 2.5) +N(0, 0.52) +βi +U(−∞, ∞) +U(−∞, ∞) +λi +U(−∞, ∞) +U(−∞, ∞) +τν +U(−∞, ∞) +HN(0, 0.52) +Table 3: Prior distributions (default in brms and manual choices) for the Bayesian meta-analysis model of the historical +data on logarithmic scale. +3.1.4 +Prior predictive checks +The model including the candidates for the prior distributions are tested in prior predictive checks, introduced by Good +(1950). In prior predictive checks a large amount of replication data is simulated from the given model and prior +distributions with the aim to decide if the model and parameter distributions are (biologically) plausible. Although +plausibility statements can already be made by reviewing the model and parameter distribution definitions, the interplay +of all model components is most easily viewed and judged in such prior predictive checks where the generated data +12 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +reflects all these model components and can be compared to a researchers prior expectation of how typical data in this +context should look like. To perform the prior predictive checks data sets are generated from model (10) with prior +distributions as specified in 3. For the population effects βi, γ, λi, i = 1, 2, that are necessary to construct the data +in the ovariectomy and Sham operation groups, improper, non-informative priors U(−∞, ∞) were chosen. Since no +sampling is possible from this distribution and to reduce the scope of generated plots, the prior predictive checks are +only presented for the group without operation. Hypothetical data for the other groups could be generated with priors +that are approximating the U(−∞, ∞) distribution, for example with a normal distribution N(0, σ2) with very large +σ. 1000 prior predictive data sets are generated. For each data set the model parameters are simulated with random +number generating functions in R and then the hypothetical data are constructed by the model described in (10). For the +animals in the group without operation this corresponds to +˜yr1jk = ˜αr + ˜νr,j[l] + ˜ϵr1jk +(11) +for mouse k = 1, . . . , K from mouse strain j[k] = 1, . . . , 22 in the hypothetical data set r = 1, . . . , 1000. The number +of mice per simulated data set, K, is chosen to correspond to the number of mice in the historical data which is K = 72. +However, the aim is not to choose prior distributions that exactly reflect the distribution of the historical data but rather +to select weakly-informative prior distributions that lead to a prior predictive distribution that is less informative (flatter) +compared to the actual observed historical data distribution but that excludes values that seem implausibly high in +context of the historical data. The theory is that, for a big number of generated data sets, the empirical distribution of +the simulated data approximates the prior predictive distribution of the data (see Good (1950)). +3.2 +Results from the application example +3.2.1 +Fitting the Bayesian Normal-normal hierarchical model to the historic data +The results of the prior predictive checks for the prior distributions of the meta-analysis model in the historical data are +presented in figure 1. The distribution of the histograms indicates that the default prior distributions in brms lead to +very big values of the logarithmic relative bone volume in the group without ovariectomy. The 10% and 90% interval +boundaries are almost at −25 and 25, what corresponds to values about zero and 7.2·1010 on the original scale. With the +weakly-informative prior instead the 10% and 90% interval boundaries are at −7.5 and 7.5, corresponding still to quite +low and high values 5.5 · 10−4 and 1808 on the original scale. From a biological perspective, the weakly-informative +prior distributions seem more reasonable but still allow the actually observed data to have a major impact on the +posterior distributions. Since no convergence problems in the model fit occur and since the posterior estimates seem +reasonable and since the meta-analysis model is not the main model but rather is intended mainly for building a prior +itself for the analysis of the new experiment, no further fine tuning of the prior distributions for the meta-analysis +model is performed. However, additional prior distributions that result in overall smaller relative bone volumes could +be examined. The plausibility of the estimated meta-analysis model is examined in posterior predictive checks as +presented below. Furthermore, MCMC diagnostics are examined. The diagnostics showed no signs of divergence or +high auto-correlations. +The estimated population effects (fixed effects) and standard deviations of the variance components (random effects) and +residuals in the meta-analysis model of the historical data are presented in table 4 as means with 95% quantile intervals +and compared with the estimates from a frequentist analysis with REML estimators and approximate confidence +intervals under normal assumption. Overall, the estimations from the Bayesian MCMC and frequentist REML method +are similar. Slightly bigger differences exist in the estimation of the residual standard deviations. Notably larger +differences exist for the residuals standard deviations of the Sham and Ovx group that have only very few observations. +A forest plot of the strain effects is represented in figure 2. The hierarchical (random effects) model leads to group- +specific estimates that are shrunken towards the common mean (pooled effect) and have smaller variance than in an +independent estimation as fixed effects model, as discussed in section 2.3.1. The point estimates of the Bayesian and +frequentist analysis are again quite similar. +As part of a Bayesian workflow, posterior predictive checks are performed to ensure that the model generates reasonable +data in light of the original data (for details see Gelman et al. (2013)). The results are presented in figure 3. The +posterior distributions seam reasonable in light of the observed data. Compared to the respective prior distributions +(here only shown for the group without operation in figure 3), the posterior distributions have smaller tails due to the +updated information by the historic data. +13 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +0 +5 +10 +−50 +−25 +0 +25 +50 +log(BV/TV) +Count +Default +a +0 +100 +200 +300 +−25 +0 +25 +50 +Mean(log(BV/TV)) +Count +b +0 +100 +200 +0 +10 +20 +30 +40 +50 +SD(log(BV/TV)) +Count +c +0 +10 +20 +30 +−10 +0 +10 +log(BV/TV) +Count +Weakly informativ +a +0 +25 +50 +75 +100 +−2 +0 +2 +4 +6 +Mean(log(BV/TV)) +Count +b +0 +50 +100 +150 +0 +1 +2 +3 +4 +5 +SD(log(BV/TV)) +Count +c +Figure 1: Graphical prior predictive checks adapted from Schad et al. (2021) for the relative bone volume in animals +without operation on a logarithmic scale with different prior distributions. Left column: Default prior distribution in +the R package brms. Right column: weakly-informative prior distribution from table 3. The predictive distributions +were calculated over 1000 simulated data sets. a) Distribution of histograms calculated per simulated data set. The +colored areas correspond in the order of increasing intensity to 10-90, 20-80, 30-70 and 40-60 percent intervals over all +histogram frequencies of the simulated data sets. The dark curve in the middle of the intervals represents the distribution +of the median over all simulate data sets. b) Distribution of arithmetic means. c) Distribution of standard deviations. +Extreme log(BV/TV) values < −50 or > 50 are represented as −50 and 50 for representation. +3.2.2 +Approximation of the MCMC draws and definition of prior predictive distributions +The results of the approximations of normal distributions by the ML method and of (according to AIC best) normal +mixture distributions by the EM method and selection by the AIC (as desribed in section 2.4.3) are presented in table 5. +The approximations with both methods look quite similar and hence the approximation by a simple normal distribution +with the ML method is selected as prior in place of a more complicated mixture distribution to avoid overfitting and for +simplification, since it requires less parameter than the mixture distribution with more than one component. +An effective sample size of the normally distributed prior p(θC) for the control (Ovx) group is calculated with the ess +function of the RBesT package by Weber et al. (2021). The calculation requires the specification a reference scale as an +estimate of the (within-group) residual standard deviation in the historic and new control group animals. This residual +standard deviation estimate is set to the mean of the estimated posterior distribution of the residual standard deviation in +the historical ovariectomized animals. The resulting estimate of the effective sample size of the prior is quite low with +neff = 2 indicating that, in this example, the benefit in using the information of the historical data in the analysis of the +14 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +Variable +Bayes +Frequ. +Intercept +2 [1.7,2.2] +2 [1.7,2.3] +Ovx +-1.9 [-3.2,-0.46] +-1.9 [-3.3,-0.51] +Sham +-0.53 [-1.8,0.78] +-0.57 [-1.9,0.74] +Strain +0.64 [0.47,0.87] +0.63 [0.45,0.88] +SD(Sham) +0.41 [0.254,0.689] +0.36 +SD(Ovx) +1 [0.666,1.62] +0.93 +SD(None) +0.37 [0.301,0.451] +0.35 +Table 4: Estimated population effects (fixed effects) and standard deviations of the variance components (random +effects) and group-specific residuals. Bayes: arithmetic means with 95% quantile intervals of the MCMC simulations. +Frequ.: frequentist REML estimators where the confidence intervals of the random effects are approximate confidence +intervals under normal assumption. +Variable +Distribution +Method +Component +Weight +E +SD +Median (95% interval) +µC +Normal-Mix +EM +1 +1.00 +0.10 +0.69 +0.1 [-1.2,1.5] +µC +Normal +ML +1 +1.00 +0.10 +0.69 +0.1 [-1.3,1.5] +σC +Normal-Mix +EM +1 +0.53 +1.10 +0.27 +1.1 [0.71,1.7] +σC +Normal-Mix +EM +2 +0.47 +0.90 +0.15 +0.9 [0.65,1.2] +σC +Normal +ML +1 +1.00 +1.00 +0.24 +1 [0.64,1.6] +−β1 +Normal-Mix +EM +1 +1.00 +1.90 +0.70 +1.9 [0.49,3.2] +−β1 +Normal +ML +1 +1.00 +1.90 +0.70 +1.9 [0.53,3.3] +Table 5: Results from the approximation of the MCMC posterior distribution in the ovariectomized animals by +parametric distributions. EM: (according to the AIC best) fit normal-mixture approximation of a series of models fitted +by the expectation-maximization algorithm. ML: normal distribution fit by the maximum-likelihood method. E and SD: +mean and standard deviation of the respective parametric distribution. 95% interval: 0.025 and 0.975 quantiles of the +parametric distribution. +new experiment is only small. More informative prior distributions and models for the design analysis could be derived +if there was more historical data available. Methods to promote the availability of historical data are described in the +discussion. +3.2.3 +Design analysis and sample size determination +The candidates for the true δ in the simulated new data are taken from a range that spans from the minimum zero +(corresponding to no effect, i.e. the mean in the control and experimental group are equal on average) to a maximum +that corresponds to the mean of the parametric distribution that was fit to the negative difference in predicted means of +the ovariectomized animals and the animals without operation (β1 in table 5). The mean of the parametric distribution +was estimated to be 1.9 and its standard deviation to be 0.7. With respect to the estimated mean standard deviation in +the ovariectomized group ˆσC = exp( +ˆ +log(σC)) = 1, this estimate corresponds to a Cohen effect by Cohen (1988) size +of d = 1.9 +1 = 1.9 (very large to huge according to the classification heuristics by Ferguson (2016) and Sawilowsky +(2009)). As further options, mean effect sizes of 1 +3 · 1.9 ≈ 0.6 and 2 +3 · 1.9 ≈ 1.3 are modelled that correspond to +medium and large effect sizes with respect to ˆσC = 1. Additional designs with treatment effect δ = 0 (no effect) are +evaluated for investigating type I errors and false discovery rates. Furthermore, heteroscedastic designs with larger +residual standard deviations in the experimental group than the control group (that might occur due to varying effects +of the treatment) are examined. Therefore, the coefficient λ is set to log(1.5) to simulate a standard deviation in the +experimental group that is 1.5 times the standard deviation in the control group (σE = exp(ψ + λ) = 1.5σC). As +sample size candidates typical sample sizes from translational animal experiments are chosen as five and ten animals in +either control our experimental group. If in the experimental designs five animals in either group seem to be to few for +achieving a certain statistic goal and ten animals seem too much, a more finely-tuned set of candidate sample sizes in an +in-between range of five and ten can be examined. The designs are summarized in table 6. The parameters for the prior +and data distribution of the mean θC and the residual standard deviation σC in the new experiment’s control group are +set according to the estimated parameters from the ML fit of the normal distributions in table 5. The prior for θE is +chosen to be weakly informative unit information prior (UIP) with a mean that equals the mean of θC and a standard +deviation that corresponds to the estimated mean of the standard deviation in the historical ovariectomized animals. +Also the mean of the prior for ησE = log(σE) is set equal to the mean of ησC. The standard deviation of the prior for +ησE is set higher than that of ησC, to the value one. With these parameters, the prior for σC is centered around the same +15 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +value as the prior for σE, but has larger tails that allows also for more extreme values. In prior predictive checks these +priors seem reasonable and weakly-informative enough to not overrule the new data. +Model +nE +nC +µ +δ +log(σC) +σE +σC +1 +5 +5 +0.1 +0.0 +0.0 +1.0 +2 +10 +5 +0.1 +0.0 +0.0 +1.0 +3 +10 +10 +0.1 +0.0 +0.0 +1.0 +4 +5 +5 +0.1 +0.6 +0.0 +1.0 +5 +10 +5 +0.1 +0.6 +0.0 +1.0 +6 +10 +10 +0.1 +0.6 +0.0 +1.0 +7 +5 +5 +0.1 +1.3 +0.0 +1.0 +8 +10 +5 +0.1 +1.3 +0.0 +1.0 +9 +10 +10 +0.1 +1.3 +0.0 +1.0 +10 +5 +5 +0.1 +1.9 +0.0 +1.0 +11 +10 +5 +0.1 +1.9 +0.0 +1.0 +12 +10 +10 +0.1 +1.9 +0.0 +1.0 +13 +5 +5 +0.1 +0.0 +0.0 +1.5 +14 +10 +5 +0.1 +0.0 +0.0 +1.5 +15 +10 +10 +0.1 +0.0 +0.0 +1.5 +16 +5 +5 +0.1 +0.6 +0.0 +1.5 +17 +10 +5 +0.1 +0.6 +0.0 +1.5 +18 +10 +10 +0.1 +0.6 +0.0 +1.5 +19 +5 +5 +0.1 +1.3 +0.0 +1.5 +20 +10 +5 +0.1 +1.3 +0.0 +1.5 +21 +10 +10 +0.1 +1.3 +0.0 +1.5 +22 +5 +5 +0.1 +1.9 +0.0 +1.5 +23 +10 +5 +0.1 +1.9 +0.0 +1.5 +24 +10 +10 +0.1 +1.9 +0.0 +1.5 +Table 6: Designs (parameter for the simulated data and sample size candidates) for the design analysis for the outcome +relative bone volume on a logarithmic scale. +10000 data sets are simulated with the chosen designs under model (7). The results of the design analysis are presented in +figures 4, 5, 6, 7 and 8. The simulations were run on the High Performance Computing cluster of Baden-Wuerrtemberg +(bwHPC). Using parallel computation, array jobs and the update functionality in brms, the computations took about 7 +hours. In figure 4 a) the number of p-values of the Welch test that are smaller than 0.5, 95% frequentist confidence +intervals and 95% Bayesian quantile credible intervals that don’t include the null effect δ = 0 are shown, for the +different experimental designs, as a function of the total sample size (number of animals in both the new experimental +and control group, nE + nC). The results with the frequentist regression based decision and the p-values are rather +similar with slightly higher rejection percentages with the Welch test. Comparing the frequentist and the Bayesian +curves in those designs that were simulated with unequal group means (θC ̸= θE), the additional information in the +prior distribution leads to a higher percentage of rejections for the Bayesian model than the frequentist model. Power +in this context can be defined as percentage of 95% confidence or rather credible interval that exclude the value null. +In those designs, that were simulated with equal standard deviations (σC = σE), such a power of at least 80% is +reached with nE = 10 and nC = 5 or nC = 10 in those cases that were the effect was large with δ = 1.9 and also in +the Bayesian model with δ = 1.3. In those designs, that were simulated with unequal standard deviations, a power +of at least 80% is only reached with nE = 10 and nC = 10 for δ = 1.9 and for nE = 10 and nC = 5 also for the +Bayesian model. Figure 4 b) compares the curves of the Bayesian quantile intervals from a) to those derived from +Bayesian highest posterior density intervals (HDI) (for details on HDI and quantile intervals see for example Held & +Sabanés Bové (2014)). +Figure 5 illustrates the precision for the designs as alternative goal for experimental planning. It shows the widths of a +random sample of confidence or credible intervals as suggested by Kruschke (2015) and Elsey (2021). Using the type +HC3 sandwich estimator to account for possibly different standard deviations in the experimental and the control group, +many of the frequentist confidence intervals are much wider than the Bayesian ones from the distributional model. This +can be observed especially for the smaller sample sizes with five animals in the control or experimental group and for +actually unequal residual standard deviations σE +σC = 1.5. In contrast, for ten animals in both groups and for actually +equal residual standard deviations, the width of the frequentist intervals is more similar to that of the Bayesian intervals. +If the designs are analyzed with regard to the goal to reach a certain precision instead of power, then a target precision +has to be defined in terms of a threshold for the desired interval widths. For example, if the goal was that with a high +16 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +probability (e.g. 95%) all intervals in the designs with equal standard deviations are not wider than a threshold of 1.1, +then this would be achieved for the larger sample sizes nE = 10 and nC = 5 or nC = 10 in the Bayesian model and +for none of the sample sizes in the frequentist model. +Figure 6 shows that, for those designs with no to moderate effect δ, the mean squared error (MSE) of the frequentist +estimate is on average bigger than the MSE in the Bayesian model. For bigger sample sizes, the MSE in the frequentist +model gets constantly smaller. In contrast, in the Bayesian model, the MSE only decreases slightly with sample size in +the designs with the larger effect sizes. +Figure 7 shows the average type M and type S error rate for the different designs. In most designs, the type S error rates +are quite similar in the Bayesian models compared to the frequentist models. Large differences exist however in the +type M error rate of the designs with larger δ = 1.3 and δ = 1.9 where the type M error rate is notably larger in the +frequentist models than in the Bayesian models. For the frequentist model, the type M error is very large in all designs, +whereas for the Bayesian model it gets smaller with the larger effects since the choice of prior distribution results in +posterior distributions for δ that are pulled towards zero. This indicates that, if the new experiment is conducted with +a frequentist analysis and either of these designs (and if the new data is actually reflected by the model used for the +fake data generation), then orienting the design choice and statements on statistical significance (in terms of whether or +not the confidence or credible intervals didn’t include the null value) almost always leads to an overestimation of the +treatment effect. The type S error is small in all designs, but around 10% with the models with the small effect δ = 0.3. +The analysis of the estimated Bayes factors gives an impression of how much evidence there is for the null and the +alternative hypothesis. The distributions of the estimated Bayes factors in the different designs are presented in figure +8. In the designs with no effect (δ = 0) or small effect (δ = 0.3) the distribution of the Bayes factor has a clear peak +and the majority of its probability mass below the value one, suggesting that based on the conservative choice of equal +priors for the group means and the evidence from the data, H0 is more likely than H1. While for the smaller sample +size (nC = nE = 5) the peak of the distribution is closer to the value one, the peak moves further towards zero for +bigger sample sizes since then there is more evidence for H0 as compared to the case of smaller sample sizes. As +compared to the null effect, the distribution of Bayes factors gets flatter for larger values of δ since then the information +in the data starts to rule out the tendency of the prior evidence ratio to support the null hypothesis that states equality in +the group means. For the very large effect δ = 1.9, the distribution of the Bayes factors has its peak above the value +one, indicating that there is more evidence for H1, while for the effects that are only of size δ = 1.3 the peak of the +distribution is still very close to the value one, especially for the smaller sample sizes and the case of unequal standard +deviations. For bigger sample sizes with nE = 10 the distribution of the Bayes factors becomes very flat with very +extreme values. Table 7 shows that in those designs, where the data was simulated under the null hypothesis with δ = 0, +there is on average more evidence for H0 and the posterior median model probability of H1 is smaller than 50%. A +goal for design analysis could be to find a sample size where the 95& quantile interval of posterior model probability +for H1 does exceed the value 50%. This goal would be achieved in those designs with the very large effect of δ = 1.9 +and with equal standard deviations in both groups, for sample sizes of ten animals in the experimental group and five +or ten animals in the control group. Another goal for design analysis could be to find a sample size that leads to a +probability of the Bayes factor indicating at least moderate evidence for H1 of at least 80%. For this goal a sample size +of ten animals would be enough in those designs with δ = 1.9 that correspond to reversing the negative ovariectomy +effect in the knockout animals on average. +Further designs were evaluated that are not represented here. More specifically, the effect of decreasing the standard +deviation in the prior for θC to the half of its previous size was examined with the aim to make it more informative. +However, this did not have a noticeable effect. Additionally, the effect of increasing the standard deviation for the prior +of δ to two times its previous value and twenty times its previous value was examined, with the aim to make it less +informative. This lead to a higher FDR since the prior had less effect on the posterior and extreme observations in the +small data set could lead to false positive claims. Furthermore, the prior distribution with a standard deviation of twenty +times its previous value (0.7 · 20 = 14) lead to a very flat distribution of Bayes factors even for those designs that were +simulated with a truly large or very large effect size. Of note, non-informative priors are not recommended to be used in +the context with Bayes factors Schad et al. (2022). Moreover, fake-data was simulated under a Bayesian design with +pobability distributions for all parameter. There, the curves of the designs that where simulated with a null effect on +average (E(δ) = 0) showed, that, if the effect has a large standard deviation (like SD(δ) = 1.2), the percentage of +intervals that doesn’t include the null effect gets much larger than 5%. Hence, one would make many more type I error +as usually intended in the frequentist framework in the cases with total sample sizes 15 and 20 and in groups that have +on average the same relative bone volume but with the experimental group having larger standard deviations. Since in +the Bayesian simulation framework neither the frequentist nor the Bayesian interval based decision rule was designed +for having (asymptotically) such a type I error rate smaller than 5%, this error rate may get much larger than 5%. +17 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +σE/σC +δ +nC +nE +p(M1|y) +BF10 +BF10 > 3 (%) +1.0 +1.9 +10.0 +10 +0.99 [0.73,1] +92 [2.5,2.4e+15] +96.6 +1.0 +1.9 +5.0 +10 +0.98 [0.62,1] +49 [1.6,1400000] +93.6 +1.5 +1.9 +10.0 +10 +0.92 [0.47,1] +12 [0.86,1000] +81.8 +1.0 +1.3 +10.0 +10 +0.91 [0.36,1] +9.6 [0.56,1400] +74.1 +1.5 +1.9 +5.0 +10 +0.9 [0.43,1] +9.1 [0.75,580] +76.1 +1.0 +1.9 +5.0 +5 +0.86 [0.45,0.99] +6 [0.82,150] +72.4 +1.0 +1.3 +5.0 +10 +0.85 [0.34,1] +5.8 [0.51,420] +64.9 +1.5 +1.3 +10.0 +10 +0.74 [0.31,0.99] +2.9 [0.46,170] +49.3 +1.5 +1.9 +5.0 +5 +0.71 [0.38,0.98] +2.5 [0.61,43] +43.1 +1.5 +1.3 +5.0 +10 +0.71 [0.33,0.99] +2.4 [0.49,98] +43.9 +1.0 +1.3 +5.0 +5 +0.7 [0.33,0.98] +2.3 [0.49,57] +41.7 +1.5 +1.3 +5.0 +5 +0.58 [0.34,0.96] +1.4 [0.51,23] +24.0 +1.0 +0.6 +10.0 +10 +0.45 [0.25,0.97] +0.81 [0.32,31] +18.4 +1.0 +0.6 +5.0 +10 +0.43 [0.27,0.94] +0.77 [0.38,16] +14.5 +1.5 +0.6 +5.0 +5 +0.43 [0.31,0.89] +0.76 [0.45,7.8] +8.1 +1.0 +0.6 +5.0 +5 +0.42 [0.29,0.91] +0.74 [0.4,9.8] +10.4 +1.5 +0.6 +5.0 +10 +0.42 [0.3,0.92] +0.73 [0.42,11] +10.8 +1.5 +0.6 +10.0 +10 +0.42 [0.27,0.94] +0.71 [0.37,17] +12.5 +1.5 +0.3 +5.0 +5 +0.41 [0.3,0.84] +0.69 [0.44,5.1] +5.0 +1.5 +0.0 +5.0 +5 +0.4 [0.3,0.81] +0.67 [0.44,4.3] +3.8 +1.0 +0.3 +5.0 +5 +0.38 [0.28,0.83] +0.6 [0.39,4.7] +4.7 +1.5 +0.3 +5.0 +10 +0.37 [0.29,0.83] +0.6 [0.41,4.8] +4.6 +1.0 +0.0 +5.0 +5 +0.36 [0.28,0.76] +0.57 [0.38,3.2] +2.8 +1.5 +0.0 +5.0 +10 +0.36 [0.29,0.76] +0.57 [0.41,3.3] +2.8 +1.5 +0.3 +10.0 +10 +0.36 [0.26,0.87] +0.55 [0.36,6.5] +5.5 +1.0 +0.3 +5.0 +10 +0.35 [0.27,0.83] +0.53 [0.36,4.9] +4.6 +1.5 +0.0 +10.0 +10 +0.34 [0.26,0.79] +0.53 [0.36,3.8] +3.3 +1.0 +0.0 +5.0 +10 +0.33 [0.26,0.72] +0.49 [0.36,2.6] +2.0 +1.0 +0.3 +10.0 +10 +0.33 [0.23,0.88] +0.48 [0.31,7.3] +6.0 +1.0 +0.0 +10.0 +10 +0.3 [0.23,0.76] +0.43 [0.3,3.1] +2.6 +Table 7: Alternative quantification of evidence for the model under H1 in the different designs (as represented by the +four columns to the left). p(M1|y): posterior probability for model M1 as model under H1. BF10: Median Bayes +factor BF10 with 95% quantile interval. BF10 > 3: percentage with at least moderate evidence for H1 as categorized +by Jeffreys (1961). The distributions are calculated over 10000 simulated data sets- +For comparison, classical sample size calculation by solving power equalities is carried out with the frequentist Welch +test and the power_t_test() function in the R MESS package (Ekstrøm (2020)). As candidates for the effect sizes, for +the group-specific standard deviations and for allocation ratio to the control and experimental group ( nE +nC ) the same +design settings as in table 6 are examined. The significance level and target power are set to the conventional values of +0.05 and 0.8. The results are presented in table 8 According to this calculation, a sample size of six animals in both +experimental and control group would be sufficient to detect an effect (as difference in the means) of at least the size 1.9 +with a power of at least 80% in this test, if the residual standard deviations in both groups are equal (σE/σC = 1) and +the allocation ratio to both groups is also equal (nE/nC = 1) (setting 13). If about twice the animals shall be assigned +to the treatment group, then nC = 5 animals for the control group and nE = 9 animals for the experimental group +are calculated for detecting at least this effect size and equal standard deviations (setting 14). For the same δrel = 1.9, +but a greater standard deviation in the experimental group (σE = 1.5), nine animals are calculated for both groups +for an equal allocation ratio (setting 15) and six and eleven animals for an allocation ratio of twice the amount to the +experimental group (setting 16). +4 +Discussion +4.1 +Summary +In this work aspects of sample size determination and analysis of translational animal experiments in a Bayesian +framework were discussed and compared to the classical frequentist procedure in a null hypothesis significance testing +(NHST) framework. The considerations where illustrated on a real-world animal experiment examining the knockout +effect of the C5aR1 receptor in osteoclasts and osteoblasts on the relative bone volume (C5aR1 example). The +determination of a sample size depends on the model assumptions on the new experiment and on the statistical goal +of the analysis and model fit. In the Bayesian framework these assumptions include prior distributions for the model +18 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +Setting +δrel +σC +σE +Alloc. ( nE +nC ) +nC +nE +1 +0.3 +1.0 +1.0 +1 +176 +176 +2 +0.3 +1.0 +1.0 +2 +132 +264 +3 +0.3 +1.0 +1.5 +1 +285 +285 +4 +0.3 +1.0 +1.5 +2 +187 +373 +5 +0.6 +1.0 +1.0 +1 +45 +45 +6 +0.6 +1.0 +1.0 +2 +34 +68 +7 +0.6 +1.0 +1.5 +1 +72 +72 +8 +0.6 +1.0 +1.5 +2 +48 +95 +9 +1.3 +1.0 +1.0 +1 +11 +11 +10 +1.3 +1.0 +1.0 +2 +9 +17 +11 +1.3 +1.0 +1.5 +1 +17 +17 +12 +1.3 +1.0 +1.5 +2 +11 +22 +13 +1.9 +1.0 +1.0 +1 +6 +6 +14 +1.9 +1.0 +1.0 +2 +5 +9 +15 +1.9 +1.0 +1.5 +1 +9 +9 +16 +1.9 +1.0 +1.5 +2 +6 +11 +Table 8: Results from a classical sample size calculation with a two-sided Welsh test, a power of 0.8 and a significance +level of 0.05 calculated with the power_t_test() function in the R MESS package. δrel: Minimal clinically relevant effect +size that shall be detected by the test. σC, σE: (estimated) standard deviations in the new experiment. Alloc. ( nE +nC ): +allocation ratio for the mice in the new experiment to the experimental and control group. nC, nE: resulting sample +sizes for the new experiment. +parameters. As basis for setting up the prior distributions, a Bayesian meta-analysis model was estimated to available +historical data, consisting of internal data from the applicant of the new animal experiment and from external data +from the Mouse Phenome Database (MPD). For comparison, also a frequentist model was fit that gave quite similar +point estimates. Design analysis was performed with prior distributions and fake-data that was based on the fitted +meta-analysis model. The estimate of the effective sample size of the meta-analytic predictive prior for the control group +in the new experiment indicated that the historical control data was only worth two animals, which corresponds to the +general impression that sample size planning in translational animal experiments often comes with large uncertainties +(see Mayer & Muche (2013)). As sample size candidates for the design analysis, the minimum and maximum of +the range of typical sample sizes for preclinical translational animal experiments were considered. The range of the +candidates for the treatment effect was chosen based on what seemed realistic according to the knowledge from the +meta-analysis model that was fitted to the historical data. The simulations required large computational resources, +especially when Bayes factors are evaluated (here the High Performance Computing cluster of Baden-Wuerrtemberg +(bwHPC) was used and computations in the 30 designs with each 10000 simulated data sets took about 7 hours using +parallel computation, array jobs and the update functionality in the brms package). In this example the power-based +sample size calculation (here done with a Welch test) suggested that eleven or less animals in both groups would be +enough to detect differences in the means of at least 1.3 for an equal allocation ratio of the animals to both groups and +residual standard deviations of one in both groups and nine or less for differences of size 1.9 or greater or rather six and +eleven animals for an unequal allocation ration and larger standard deviations in the experimental group. However, +the analysis the type M error rates showed that the design and analysis with classical frequentist can lead to a high +percentage of overestimated effect sizes in those cases where the analysis of the data in the new experiment results in a +test decision against the null hypothesis. Using as Bayes factors oriented goal that 95% of the posterior probability of +the model under H1 is above the value 50% (representing equal model probability for both the model under H0 and H1), +only in those designs with equal standard deviations of one and an effect of size 1.9, ten animals in the experimental +group (and five animals or then in the control group) would be enough. Using goal that the Bayes factor exceeds +with 80% probability the heuristic threshold value for moderate evidence for H1 defined by Jeffreys (1961), Lee & +Wagenmakers (2014), then a sample size of ten animals in both groups would also be enough for unequal standard +deviations with a standard deviation in the experimental group of 1.5 in the new experiment and a standard deviation in +the control group of 1. +Several chances and challenges were identified in this Bayesian meta-analytic predictive framework as compared +to the classical frequentist framework. If the goal of planning the new experiment is to achieve a certain statistical +power, the use of the historical data did not lead to a lower sample size in a Bayesian analysis with prior predictive +distributions from the historical data than with classical frequentist power calculations. However, the use of fake-data +design analysis based on the historical data and the evaluation of additional model characteristics and statistical allowed +a better representation and estimation of present uncertainties in the model parameters. More specifically, the Bayesian +19 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +model framework allows to formally incorporate a priori knowledge (deduced from historical data) as prior distribution +in the analysis of a new experiment. Secondly, uncertainties, that are almost always present in research stage as early as +preclinical translational research, are better represented by modeling all of the model parameters as random variables +instead of fixed parameters. Thirdly, the estimation with MCMC methods allows also for more complex models, that +might represent some data more accurately. As an example, different residual standard deviations were modeled for +different experimental groups in the historic and the new data of the C5aR1 example and the estimates seemed more +precise than frequentist sandwich estimators. Fitting a meta-analysis model to the historic data provides a quantitative +summary and can be used to define the prior distributions for the new experiment. With Bayesian methods heterogeneity +can be reflected in the means of different historic experiments, also in the case of only few experiments or groups. +In contrast, frequentist models can deal less well with fitting meta-analysis or hierarchical models in heterogeneous +data in the situation of with few, small experiments Gelman (2006), Friede et al. (2017b,a). Meta-analysis of similar +historical experiments not only provides an initial guess, that can be used for prior specification, but also a tool for +quality control. More specifically, flaws in the experimental design or analysis may stick out or statements may have +to be relativized when some measurements originating from a common mean hierarchical model differ significantly +from supposedly related measures Walley et al. (2016). Planning and analyzing the new experiment’s in the bigger of +the estimated meta-analysis model of the historical data may help to make more appropriate statements and may lead +to more reproducible results as a step out of the reproducibility crisis in animal research Ioannidis (2005), Begley & +Ellis (2012), Begley & Ioannidis (2015), Loken & Gelman (2017), Goodman et al. (2016), Jilka (2016), Freedman +et al. (2015), Macleod & Mohan (2019), Voelkl et al. (2020). The consideration of not only internal but also external +data like from the MPD gives a broader picture of the natural variation of the outcome of interest. This helps to make +more generalizable statements that are potentially more likely to being translated to the application in humans or to the +reproduction of experiment results in other animals as suggested by Voelkl & Würbel (2016), Voelkl et al. (2020). The +point estimates of the Bayesian and frequentist meta-analysis model in this application example were quite similar, but +the Bayesian approach allowed an easier estimation of the confidence intervals as quantiles of the posterior draws. The +frequentist and Bayesian estimates might differ more if heterogeneity was model for a grouping variable with smaller +number of groups. This could be the case if also the data laboratory (MPD, internal) was modelled. In this case the +Bayesian approach has proven to be superior Friede et al. (2017a,b). Furthermore, determine sample size by design +analysis using fake-data simulation instead of the classical determination by power inequalities, addresses several +problems that are common in translational research. In particular, the variation in an experimental design and data may +be better represented by a continuous value as the Bayes factor instead of the outcome of a binary decision rule and +hence it may be of more value to just report the Bayes factors associated with the different designs. In particular, Schad +et al. (2022) show in the context of a standard cognitive experiment that many standard designs don’t have sufficient +evidence for making conclusive decisions and support the idea of increasing the sample sizes by sharing data across +different researchers and laboratories. +Concerning the challenges, fitting Bayesian models with MCMC methods requires at least a basic understanding of +the additional convergence diagnostics to ensure a proper model fit. Other recommended steps are prior and posterior +predictive checks to make sure that the specified prior and posterior distributions are reasonable. The steps and decisions +that are commonly made in a Bayesian framework are described as a Bayesian workflow Gelman, Vehtari, Simpson, +Margossian, Carpenter, Yao, Kennedy, Gabry, Bürkner & Modrák (2020) and may become quite complex. In particular, +Bayesian inference has typically many more determining factors than frequentist inference through the specification of +all model parameters’ prior distributions and MCMC parameters. The problem is even worse when Bayesian methods +are considered in an design analysis framework where additional determining factors come with different design +options. This abundance of determining factors makes it hard to understand the effect of changing single determining +factors for the posterior inference and makes the investigation of all combinations of determining factors becomes soon +incomprehensible. Furthermore, there is so far no consensus in literature about which procedure to use for sample +size determination in Bayesian framework (if and what decision functions and thresholds should be used etc.). In the +case where only few, small previous experiments are available, special attention has to be paid to the assumptions on +the prior model and its hyperparameters. Especially, setting up a reasonable model for the heterogeneity parameter +is important to properly reflect the variation in the background population under focus and prevents from driving +overly-confident claims that only apply to standardized animals in a single experiment. Concerning the use of Bayes +factors for design analysis and sample size determination, challenges are firstly the definition of a threshold. Secondly, +Bayes factors are highly sensitive to the choice of prior distributions as shown by Schad et al. (2021) and the usual +estimation method by bridge sampling or the Savage-Dickey method requires a large number of MCMC iterations +to be stable Gronau et al. (2020) and preferably several repeated estimations. This makes the estimation of Bayes +factors also computationally challenging. Finally, possible bias in the Bayes factors estimate should be examined in +simulation based calibrations (SBC) Schad et al. (2021). These Bayes factor workflow procedures again increase the +already high manual and computational burden of the simulation based design analysis which might also constitute an +obstacle for the application in translational research where the resources of the researchers are often quite limited. With +20 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +regard to meta-analysis, challenges are that it is difficult to find the relevant historic information since the rate of annual +publications in preclinical research is very high Bannach-Brown et al. (2021) and the published estimates are often +subject to bias like publication bias Sena et al. (2010), ter Riet et al. (2012), Conradi & Joffe (2017), of Health at Charité +(BIH). Furthermore, the relevant literature is often unorganized and outcomes do not follow a unique terminology +what makes it hard to compare results from different experiments Smith et al. (2005). These facts make it currently +challenging for researchers of translational animal experiments to understand and correctly apply Bayesian methods +and make sample size determination too extensive for practical applications in this context without the development of +routines and applications that facilitate their use. +4.2 +Extensions +There are several extensions to the here presented methodology for planning and analysing translational animal +experiments using Bayesian meta-analytic predictive approaches and fake-data design analysis. In this work the +distribution of the Bayes factors were used to visualize the evidence ratio for the null and alternative hypothesis under +different models. To transform the distribution into a decision function, a heuristic threshold was defined based on a +classification scheme of Jeffreys Jeffreys (1961) or by checking if the 95% posterior model probability for the model +under H1 (p(M1|y)) did exceed the value 50% (representing equal probability for model the model undeer H1 and +under H0. A more systematic approach to setting a threshold for the Bayes factors is by the definition of utility +functions as illustrated by Schad et al. (2021) and Schönbrodt & Wagenmakers (2018). Bayes factors compare the +“out-of-sample" predictive performance of the two contrasting models (here the model under H0 and under H1). A +further approach to making a decision whether or not there is evidence in the data that the effect δ differs from that +stated by the null hypothesis is to compare the out-of-sample predictive performance by the investigation of posterior +predictions. A common utility function that measures the out-of-sample predictive performance of a model is the +expected log pointwise predictive density (ELPD) (for details see Gelman et al. (2014) and for practical estimation in a +Bayesian framework see Vehtari et al. (2017)). +In this work a Bayesian meta-analysis model was fit to the historical data with the purpose to get prior distributions +and to define a reasonable design analysis setups. Instead of the Bayesian meta-analysis model, also the estimates +from a frequentist meta-analysis model can be used to set up parametric distributions for definition prior distributions +and sampling distributions for the fake-data design analysis. This was done for example by Schad et al. (2022) in the +context of simulations for the examination of the behavior of Bayes factors under different hypothesis. In this work age +and the historical data’s experiment/ laboratory effects could not be incorporated since necessary data was missing. +If only few historical data is available, it is difficult to check assumptions corresponding to a normal distribution and +non-parametric models bay be more appropriate Konietschke et al. (2021). Burr and Doss Burr & Doss (2005) suggest +a Bayesian semi-parametric meta-analysis model that models the experiment-specific effects through a version of the +Dirichlet process prior and implement it in the R package bspmma Burr (2012). This model could be used if the normal +assumption is in doubt or it can be compared to a parametric model using empirical Bayes to decide which model seems +more appropriate Burr (2012). There are general efforts for facilitating the use of systematic reviews and meta-analysis +in animal research. Examples of such efforts include CAMARADES (Collaborative Approach to Meta-Analysis and +Review of Animal Data from Experimental Studies) of Ediburgh (2021) which offers methodological advise and tools +for conducting systematic reviews and meta-analysis in animal trials or the online review platform SyRF (Systematic +Review Facility) Bahor et al. (2021). Moreover, there are efforts for collecting animal data in common big databases +Eppig et al. (2005), Blake et al. (2006), Consortium (2007), Hancock et al. (2008), Hannover (2022), Pognan et al. +(2021). Furthermore, there are advancements towards a mandatory (pre)registration for animal trials which could +increase the amount and quality of public available data, reduce bias Chamuleau et al. (2018), Bert et al. (2019), +Heinl et al. (2019), Baker (2019), van der Naald et al. (2020). With these advancements it is realistic, that in future +better meta-analysis models of historical evidence can be fit. The simulated fake-data in the design analysis represent +assumptions on the data in the new experiments. The assumptions are based on historical data that is summarized in +a meta-analysis model. In the ideal case of prior distributions that match the true distribution of the future data, the +prior represents just additional information that makes the posterior estimates more precise. If, however, the prior +predictive distribution of the data places major parts of its probability mass to regions that are unlikely according to +the empirical distribution of the observed data, there is so-called prior-data conflict. The effects of prior-data conflict +can be a problem when they invalidate the inference based on the posterior draws (see for example Box (1980), Evans +& Moshonov (2006)). To examine the effects of prior-data conflict for the chosen prior distributions on the posterior +inference, data that deviates from the prior predictive distribution can be simulated. Further steps are necessary to get a +better understanding and better visualizations of all determining factors that affect the posterior inference in the design +analysis with Bayesian models. Recently, the R package priorsense Kallioinen et al. (2021) has been developed +which intends to help investigate the impact of the prior with respect to observed data. Packages like this might help to +get a better feeling of how the chosen prior distributions affect the necessary sample size and test decisions. If prior-data +21 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +conflict seems to be a problem with the chosen prior distributions, the prior distributions can for example be robustified +by adding less informative mixture components to the respective distributions, as suggested by Schmidli et al. (2014). +Instead of using data-based priors for the design and analysis of the the new experiment, the priors can also be chosen +by prior elicitation Garthwaite et al. (2005), O’Hagan et al. (2006). In a prior elicitation approach, the analyst does +not specify the prior directly (like here based on historical data). Instead, there is an subject expert that describes +properties of the outcome of interest and the task of the analyst is to formalize this description as a prior distribution. +Prior elicitation could be an appealing alternative for the here used data-based priors, especially When there is no useful +historical data available. However, at the current state, technical, practical and societal challenges hinder the use of +prior elicitation Mikkola et al. 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(2006), ‘Object-oriented computation of sandwich estimators’, Journal of Statistical Software 16(9), 1–16. +27 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +OP None +OP Ovx +OP Sham +−2 +0 +2 +4 +Pooled effect +C57BL/6 +PreCC1061/Tau +PreCC111/Tau +PreCC1513/Tau +PreCC188/Tau +PreCC1912/Tau +PreCC2126/Tau +PreCC2156/Tau +PreCC2680/Tau +PreCC2689/Tau +PreCC2750/Tau +PreCC3438/Tau +PreCC3480/Tau +PreCC3912/Tau +PreCC4052/Tau +PreCC4141/Tau +PreCC4438/Tau +PreCC4457/Tau +PreCC519/Tau +PreCC521/Tau +PreCC557/Tau +PreCC711/Tau +PreCC72/Tau +Pooled effect +C57BL/6 +PreCC1061/Tau +PreCC111/Tau +PreCC1513/Tau +PreCC188/Tau +PreCC1912/Tau +PreCC2126/Tau +PreCC2156/Tau +PreCC2680/Tau +PreCC2689/Tau +PreCC2750/Tau +PreCC3438/Tau +PreCC3480/Tau +PreCC3912/Tau +PreCC4052/Tau +PreCC4141/Tau +PreCC4438/Tau +PreCC4457/Tau +PreCC519/Tau +PreCC521/Tau +PreCC557/Tau +PreCC711/Tau +PreCC72/Tau +Pooled effect +C57BL/6 +PreCC1061/Tau +PreCC111/Tau +PreCC1513/Tau +PreCC188/Tau +PreCC1912/Tau +PreCC2126/Tau +PreCC2156/Tau +PreCC2680/Tau +PreCC2689/Tau +PreCC2750/Tau +PreCC3438/Tau +PreCC3480/Tau +PreCC3912/Tau +PreCC4052/Tau +PreCC4141/Tau +PreCC4438/Tau +PreCC4457/Tau +PreCC519/Tau +PreCC521/Tau +PreCC557/Tau +PreCC711/Tau +PreCC72/Tau +log(BV/TV) +Strain +Data +Bayes +Frequ. +Observed +Figure 2: Forest plot with strain specific means of the historical animals and their common mean (pooled effect) +stratified by the operation group (none, ovariectomy (Ovx), Sham). Bayes: Bayesian estimates as means of the posterior +MCMC draws (points). Frequ.: frequentist REML estimates. Observed: strain-specific means in the observed data. The +intervals are presented as 80% (thick lines) and 85% (thin lines) quantile intervals. +28 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +None +Ovx +Sham +−2.5 0.0 2.5 5.0 +0 +10 +20 +30 +40 +0 +2 +4 +6 +0 +2 +4 +6 +log(BV/TV) +Count +a +None +Ovx +Sham +0 +1 +2 +0 +20 +40 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +0 +5 +10 +15 +20 +Mean(log(BV/TV)) +Count +b +None +Ovx +Sham +0.0 0.5 1.0 1.5 2.0 +0 +10 +20 +30 +40 +0 +5 +10 +0 +5 +10 +15 +20 +25 +SD(log(BV/TV)) +Count +c +None +Ovx +Sham +−2 +0 +2 +4−2 +0 +2 +4−2 +0 +2 +4 +0 +5 +10 +log(BV/TV) +Count +Data +Original +Replication +Figure 3: Graphical posterior predictive checks adapted from Schad et al. (2021) for the relative bone volume in animals +without operation on a logarithmic scale. Grey: original data. Blue: replicates from the MCMC fit in 1000 simulated +data sets. a) Distribution of histograms calculated per simulated data set. The colored areas correspond in the order of +increasing intensity to 10-90, 20-80, 30-70 and 40-60 percent intervals over all histogram frequencies of the simulated +data sets. The dark curve in the middle of the intervals represents the distribution of the median over all simulate data +sets. b) Distribution of arithmetic means. c) Distribution of standard deviations. Extreme log(BV/TV) values < −50 or +> 50 are represented as −50 and 50 for representation. +29 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +σE / σC = 1 +σE / σC = 1.5 +10 +15 +20 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +Total sample size +Percentage of intervals that don't include 0 +Model +Frequ. (Welsh−Test) +Frequ. (Sandwich) +Bayes (HDI) +δ +0 +0.3 +0.6 +1.3 +1.9 +(a) +σE / σC = 1 +σE / σC = 1.5 +10 +15 +20 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +Total sample size +Percentage of intervals that don't include 0 +δ +0 +0.3 +0.6 +1.3 +1.9 +Interval +Bayes (Quantil) +Bayes (HDI) +(b) +Figure 4: Proportion of the simulated data set in which the decision criteria for "success” is met that the 95% confidence +or credible interval does not include the null value 0 or that the p-value of the Welch test is smaller than 0.05. The +distributions are calculated over 10000 simulated data sets. (a): Dotted line: proportion of simulated data sets with +a 2-sided Welch test p-value< 0.05. Dashed line: proportion of simulated data set where the frequentist confidence +interval with the HC3 sandwich estimator doesn’t include the value 0. Solid line: proportion of simulated data set where +the Bayesian highest density intervals (HDI) doesn’t include the value 0. (b): Comparison of the Bayesian quantile +interval and HDI. Dotted lines: Conventional boundaries for the type I error rate or bower of 0.05 and 0.8. +30 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +δ = 0 +δ = 0.3 +δ = 0.6 +δ = 1.3 +δ = 1.9 +5 +10 +10 +5 +5 +10 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +nE +nC +Width of the 95% interval +σE σC = 1 +δ = 0 +δ = 0.3 +δ = 0.6 +δ = 1.3 +δ = 1.9 +5 +10 +10 +5 +5 +10 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +nE +nC +Width of the 95% interval +σE σC = 1.5 +Model +Bayes (HDI) +Frequ. (Sandwich) +Figure 5: Widths of the 95% Bayesian highest density intervals (HDI)s and the 95% frequentist confidence intervals +with the type HC3 sandwich estimator for a sub-sample of 500 of the simulated data sets in different design setups. +31 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +σE / σC = 1 +σE / σC = 1.5 +δ = 0 +δ = 0.3 +δ = 0.6 +δ = 1.3 +δ = 1.9 +10 +15 +20 +10 +15 +20 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +Total sample size +RMSE +Approach +Bayes (Mean) +Bayes (Median) +Frequ. (REML) +Figure 6: Root mean squared error (RMSE) calculated as root of the average squared difference of the point estimate of +the treatment effect and the true mean of the treatment effect (E(δ)). The RMSEs are represented as average over all +simulated data sets together with 95% quantile intervals. In the Bayesian model the point estimates of δ are arithmetic +means and medians of the posterior MCMC draws of δ and in the frequentist model the estimates of δ are calculated as +difference of the means in the treatment and control group. +32 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +δ = 0.3 +δ = 0.6 +δ = 1.3 +δ = 1.9 +10.0 +12.5 +15.0 +17.5 +20.0 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +nC +Type M error rate +Type M error +δ = 0.3 +δ = 0.6 +δ = 1.3 +δ = 1.9 +10.0 +12.5 +15.0 +17.5 +20.0 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +nC +Type S error rate +Type S error +σE σC$ +1 +1.5 +Model +Bayes (HDI) +Bayes (Quantil) +Frequ. (Sandwich) +Figure 7: Type M (magnitude) error rate: Percentage of the simulated data sets where the effect estimate is bigger +than the true treatment δ in absolute value, calculated in those data sets where the confidence/ credible interval did not +include the value 0. Type S (sign) error rate: Percentage of the simulated data sets where the effect estimate had a +different sign than the true treatment δ in absolute value, calculated in those data sets where the confidence/ credible +interval did not include the value 0. The distributions are calculated over 10000 simulated data sets. +33 + +Designing translational animal experiments by Bayesian MAP approaches +A PREPRINT +σE σC=1 +σE σC=1.5 +δ = 0 +δ = 0.3 +δ = 0.6 +δ = 1.3 +δ = 1.9 +1 +10 +100 +1 +10 +100 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +BF10_print +Frequency (%) +nC + nE +10 +15 +20 +Figure 8: Distribution of the estimated Bayes factors BF10 for evidence for the alternative model that the treatment +effect δ is different from zero over the null model where it is equal to zero. The distributions are calculated over 10000 +simulated data sets for different designs regarding the simulated true distribution in the data where for each design +10000 data sets are simulated. Extreme values of ˆ +BF10 > 10000 are presented as 10000 for representation. +34 + diff --git a/K9E5T4oBgHgl3EQfYQ9F/content/tmp_files/load_file.txt b/K9E5T4oBgHgl3EQfYQ9F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cc527e9bad7e11c3ece093c7282e0511396e196e --- /dev/null +++ b/K9E5T4oBgHgl3EQfYQ9F/content/tmp_files/load_file.txt @@ -0,0 +1,2652 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf,len=2651 +page_content='DESIGNING TRANSLATIONAL ANIMAL EXPERIMENTS BY BAYESIAN META-ANALYTIC PREDICTIVE APPROACHES A PREPRINT Theresa Unseld∗ Department of Epidemiology and Medical Biometry Ulm University Ulm, Germany January 16, 2023 ABSTRACT The planning and conduct of animal experiments in the European Union is subject to strict legal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Still, many preclinical animal experiments are only poorly designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As a consequence, discoveries that are made in one animal experiment, cannot be reproduced in another animal experi- ment or discoveries in translational animal research fail to be translated to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' When designing new experiments in a classical frequentist framework, the sample size for the new experiment is chosen with the goal to achieve at least a certain statistical power, given a statistical test for a null hypothesis, a significance threshold and a minimally relevant effect size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The statistical test is a function of the data and the test is used to make statistical inference concerning the data’s underlying, unobserved parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In a Bayesian framework, inference is made by a combination of both the information from newly observed data and also by a prior distribution, that represents a priori information on the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In translational animal experiments, a priori information is present in previously conducted experiments to the same outcome in similar animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The prior information can be incorporated in a systematic way in the design and analysis of a new animal experiment by summarizing the historical data in a (Bayesian) meta-analysis model and using the meta-analysis model to make predictions for the data in the new experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This is called meta-analytic predictive (MAP) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In this work, concepts of how to design translational animal experiments by MAP approaches are introduced and compared to classical frequentist power-oriented sample size planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Current chances and challenges, that exist in the practical application of these approaches in translational animal research, are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Special emphasis is put on the construction of prior distributions and sample size calculation by design analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The considerations are motivated by a real world translational research example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Keywords Translational research, Bayesian statistics, meta-analysis, design analysis, sample size calculation 1 Introduction Translational research constitutes a key element to the development of new methods and therapies in human medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Situated in the late stage of preclinical research, its aim is to translate findings from basic laboratory preclinical research into the clinical application as potential treatments of human diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In translational animal research biological pathways concerning clinically relevant phenotypes or pathologies are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' These pathways are typically complex constructs whose mechanisms cannot be fully revealed in a single research experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Hence, the success of translational research fundamentally depends on the sensitive contemplation of new insights from an experiment in light of past insights and gained expertise knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In Bayesian analysis, prior knowledge is formally incorporated as probability distribution into the analysis of newly observed data and updated to a posterior distribution that is then used ∗theresa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='unseld@uni-ulm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='de arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='05572v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='ME] 13 Jan 2023 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT for Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Several authors have already emphasized the value of Bayesian statistics in preclinical research (see for example Spiegelhalter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2004), Walley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2016), Kramer & Font (2017), Bonapersona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2019), Yang & Novick (2019), Gelman & Vákár (2019), Novick & Zhang (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Nonetheless, practical applications of Bayesian methods for planning and analyzing preclinical animal experiments are rare (see Walley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2016), Kramer & Font (2017), Bonapersona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A challenge in the application of Bayesian methods is the specification of a prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Translational animal experiments are typically characterized by the fact that the sample sizes in the experiment’s groups are kept low for animal welfare purposes (see Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2018)), like outlined in the 3R concept by Russel & Burch (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Especially in this situation, the choice of prior distribution can have a major impact on the posterior inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Setting up a good prior distribution, that accurately reflects a priori information, can help to stabilize posterior inference, derive more precise estimates, reduce the impact of single extreme observations and to indicate if there might be something wrong or unexpected in the measurements of the new data that results in a wide posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, without a transparent justification for the choice of a prior and without a good understanding of the impact of the prior distribution on posterior inference, there is a risk that the final conclusion of a Bayesian analysis might be not sensitive (enough) to evidence in the newly observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' On the other hand, if the prior distribution is chosen to have essentially no weight on posterior inference as compared to the data in the new experiment, then a major aspect of Bayesian inference and its associated benefits over frequentist inference are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' One systematic way of setting prior distributions is to specify them based on a meta-analysis of relevant literature or databases as suggested by Neuenschwander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2010), Pullenayegum (2011), Rhodes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2015), Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2015), Bartoš et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A meta-analysis is a popular tool for summarizing information from several statistical experiments in a common statistical model and quantifying the variability in different experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As a requirement, the experiments all have to address the same question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This assumption is often reasonable for control groups that stem from a series of experiments that address the same phenotypical outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The meta-analysis model can be used to estimate predictive distributions for a new experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' These predictive distributions can be used to derive prior distributions for the analysis of the new experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This approach to a prior specification and Bayesian panning and analysis of the new experiment is termed meta-analytic predictive approach (MAP) Neuenschwander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' These MAP priors fall into a class of data-based priors and are the focus in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Other methods for prior specification are summarized by Mikkola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021) and are briefly discussed in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The MAP approach is illustrated for the mean in control groups of clinical studies by Neuenschwander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, they don’t use the historical data to derive prior distributions for other model parameters like the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, animal experiments are characterized by their own challenges and methods suggested in human clinical trials cannot be transferred in a straight-forward manner to the application in animal experiments without considering possible adaptations (see Walley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2016), Kramer & Font (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Firstly, due to the small sample sizes in the experiments’ groups, the estimates obtained from animal experiments are usually characterized by large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A further challenge is that, although many animal experiments are conducted, the results from the analysis are usually unorganized and restricted to limited access (see Kramer & Font (2017), Bonapersona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2019), Novick & Zhang (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' It has been the idea to initiate search tools to perform systematic reviews of preclinical studies (see Bahor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021)) or launch common big databases to gather the information from more institutions (see Keenan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2009), Beckers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2009), Maddatu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2012), McEntyre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2015), Steger-Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2020), Pognan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Until such databases are fully developed, the remaining alternative option is to construct prior distributions form the little historical information that is available and be aware of the present uncertainties about the model parameters’ true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The usage of Bayesian methods allows to fit meta-analysis models also in the challenging situation a of few, small previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Methods for Bayesian meta-analysis in the context of few, small studies have recently been proposed for the application in humans by Friede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, the MAP approach has been discussed in the context of animal experiments by Walley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Still, there exist only few applications of how such Bayesian meta-analytic predictive methods are used to analyze real-world examples from preclinical translational research and even fewer examples of how to conduct sample size calculations for animal experiments in a Bayesian framework (see Kramer & Font (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The planning and conduct of animal experiments in the European Union (EU) is subject to strict legal conditions as outlined for example in the EU directive 2010/63/EU or for Germany in the legislation for animal welfare (Tierschutz- Versuchstierverordnung) (see Bundesministerium der Justitz (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In particular, animal experiments are subject to an authorization by the respective competent authorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Researchers can apply for this authorization by submitting an animal experiment proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' If the animal experiments are set up to proof hypothesis, the proposals must be submitted together with a form or with a biometric review that provide a description of the analysis methods as well as a justification of the proposed sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Classically, sample size calculations are done in a frequentist framework by choosing the minimal number of animals that reaches a predefined power of a statistical test for a null hypothesis, given a data model, including its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, the classical approach to null hypothesis significance testing (NHST) and power analysis has been criticized for several reasons (see Gelman & Carlin (2014), Kruschke (2015), Schönbrodt & Wagenmakers (2018), Stefan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2019), Gelman, Hill & Vehtari (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Claiming statistical significance by a frequentist test for a certain hypothesis is equivalent to claiming statistical significance based on the corresponding 2 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT p-value of the test or the confidence interval (see Lehmann & Romano (2006)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' These decision rules for a null hypothesis have been criticized for hypothesis testing, for example by Wagenmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2020), Gelman, Hill & Vehtari (2020), Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' One critique is connected to the principle of predictive irrelevance (see Wagenmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2020)) and Bayes factors are proposed and as an alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Gelman & Carlin (2014), Gelman, Hill & Vehtari (2020) point out that, by choosing an experimental design with the goal of reaching a certain power of the statistical (while limiting the probability of type I errors of the test), other important goals of the experimental design are missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' More specifically, they argue that, by focusing on power and statistical significance, the reported estimates are systematically biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This is because actually small effects are only reported for those data sets where the observed effect size happens to be large (by chance) and those cases are highly subjective to being overestimated and to being of the wrong sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Sample size calculations in a classical framework are based on fixed estimates of an effect size and of the additional model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' One option is to derive these estimates from a researcher’s previous experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This is problematic because the small sample sizes in animal experiments result in parameter estimates that are accompanied with large uncertainties (see Mayer & Muche (2013)) and these uncertainties are not well reflected by single point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As a consequence, the sample size calculations in animal experiments are often only of limited use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As an alternative to power-oriented sample size calculation, design analysis using fake-data simulation has been suggested by Kruschke (2015), Gelman, Hill & Vehtari (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In this paper the above aspects of Bayesian analysis, meta-analysis, prior specification using predictive approaches and design analysis using fake-data simulation are considered jointly in the context of sample size calculation for translational animal experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' To the author’s knowledge, there exists so far no work that illustrates sample size calculation for translational animal experiments using these approaches jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' After the introduction in this section, section two introduces sample size calculations in a classical frequentist framework and introduces fake-data design analysis as its alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, a Bayesian MAP approach to designing and analysing new experiments based on historical data is explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A distributional Bayesian model is used to deal with unequal group variances in the historical and new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The considerations are applied to a real-world application example section three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In section four the main points are summarized and possible extensions are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 2 Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 Statistical hypothesis tests and power-oriented sample size calculation For sample size calculations, assumptions have to be made concerning the distribution models for all experimental groups and assumptions concerning the experimental design that specifies which comparisons are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, estimable effects of interest, δ, have to be defined that typically represent the differences in the means or effects of two or several groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Additionally, for making binary decisions whether there is sufficient evidence in the data that the true effect of interest δ is different from a null effect δ0 or not, a decision function is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The decision function in the classical classical frequentist framework is a statistical hypothesis test ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Statistical hypothesis tests ϕ compare test statistics T to a critical value c to make a decision whether or not to reject a null hypothesis H0 : {δ = δ0}vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' H1 : {δ ̸= δ0} (1) If H0 is rejected, the alternative hypothesis H1 is said to be accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The tests statistic T is constructed as a function of the data whose probability distribution under H0 is known or can be approximated by a known distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A common situation is the comparison of two independent groups, an experimental (E) and control (C) group (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' knockout animals vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' wildtyp animals), under assumption of normally distributed data: yik = θi + ϵik, ϵik ∼ N(0, σ2 i ), i = C, E, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 Hypothesis tests in a frequentist analysis As frequentist hypothesis test on the mean difference δ = |θE − θC| the two-sample t-test can be used in case of equal variances σE = σC (homoscedasticity) (see Lehmann & Romano (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Often, however, the residual variance is greater in the experimental than the control group due to varying treatment effects and the Welch test (Satterthwaite test) by Welch (1938), Satterthwaite (1941) is more appropriate since it does not make a homoscedasticity assumption: TWelch = y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' − y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' � σ2 C n1 + σ2 E n2 (3) 3 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT Under H0, the distribution of the test statistic TWelch is approximated by a t-distribution t˜ν with modified number of degrees of freedom ˜ν (for details see Welch (1938, 1947)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The variances σ2 E and σ2 C can be estimated as empirical variances from observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The corresponding two-sided hypothesis test is then ϕWelch = 1{|TWelch| > t˜ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1− α 2 } (4) where t˜ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1− α 2 denotes the 1 − α 2 quantile of the t-distribution with modified degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' There are two standard types of errors that can be made by a hypothesis test: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Type I error: ϕ rejects H0 although it is true 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Type II error: ϕ doesn’t reject H0 although it is not true Both errors cannot be minimized simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Since the type I error is generally regarded as worse then the type II error, a classical hypothesis test ϕ = ϕα is set up to limit the probability of type I error by a significance level α and then finding the optimal test among all the tests at level α by minimizing the probability of type II errors (see Lehmann & Romano (2006)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This gives a test ϕ∗ α that has maximal power P(ϕ = 1|H1) while controlling type I errors at level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The sample size is then classically calculated as minimal number for which the chosen hypothesis test ϕ∗ α detects a clinically relevant effect size δ∗ at least with probability 1 − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' More specifically, in applications with simple analytic distribution form of the test statistic, the sample size can be calculated by solving the power inequality P(ϕWelch,α,nE,nC = 1|δ ≥ δ∗) ≥ 1 − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (5) For the Welsh test, the critical value t˜ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1− α 2 itself is a functions of the sample size through the (approximated) degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Hence, the inequality cannot be solved directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Alternative approaches are to approximate the critical values by a normal distribution or to use an iterative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 Hypothesis tests in a Bayesian analysis “Statistically significant" in a frequentist context means that the test statistic T is greater than the critical value c (here: c = tν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1− α 2 ) or equivalently that the p-value of the test is smaller than α or the confidence interval at level 1 − α for the tested effect δ does not include the null value δ0 (see Lehmann & Romano (2006)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Similar decision rules or “tests" can be established in a Bayesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In a full Bayesian model, all parameters are modelled as probability distributions with their own prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The idea of Bayesian analysis is to update these prior distributions to posterior distributions by the information in new data using Bayes’ rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Details on prior distributions are given in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Endowing all model parameters with probability distribution rather than fixing parameters to concrete values leads on the one hand to better representation of uncertainties than in the frequentist framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' On the other hand it often leads to complicated posterior density functions that require the evaluation of high-dimensional integrals and can no longer be expressed in analytic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Numerical computation using Markov chain Monte Carlo (MCMC) methods is a common alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In MCMC methods the posterior of the parameter distribution is approximated by a series of MCMC draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For one-sided test problems, like the upper test problem H0 : {δ ≤ δ0} vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' H1 : {δ > δ0}, the estimated posterior probability p(δ|y) of δ being greater than a null value δ0, given the data y = (yC, yE), or than a clinically relevant value δ∗ can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' To transform this idea into a decision rule this posterior probability can be compared to a predefined critical value like suggested by Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This approach cannot be used for the two-sided problem (1), when the tested effect δ is modeled by a continuous probability distribution, like it is the case when δ = |θE − θC| reflects the difference in two continuous means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In this case the posterior probability of a single point event P({δ = δ0}) is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Alternatives are the consideration of two-sided Bayes factors or Bayesian credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Credible intervals are Bayesian versions of frequentist confidence intervals with slightly different interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' By definition, a frequentist 1 − α confidence intervals for δ is an interval with random bounds that covers the unknown (but regarded as fixed) parameter δ with probability 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Hence, if one would generate data from the corresponding probability model and calculate a confidence interval for each data set, one would expect that (1−α)% of these intervals would cover (the fixed) δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The interpretation of a 1 − α Bayesian credible interval is instead that it is set up (with bounds regarded as fixed) so the probability that a random realization of δ falls within the credible interval is 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Equivalently it can be stated that the Bayesian 1 − α credible interval includes (1 − α)% of the posterior probability mass of δ, given the data y (see Held & Sabanés Bové (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Yet, this definition does not uniquely define the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' There are two common types of Bayesian credible intervals: quantile intervals and highest density intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The bounds 4 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT of a 1 − α quantile interval for a parameter are given by the α 2 and 1 − α 2 quantiles of its posterior distribution and is also called equal-tail interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In praxis, the bounds of a quantile interval can be approximated by the quantiles of the MCMC draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A highest density interval (HDI) is defined as a credible interval for which the posterior density of each point inside the interval is higher than the posterior density for an arbitrary point outside the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This is a desirable property as a summary of the distribution and is especially relevant for skewed distributions where, for the quantile interval, it is possible that parameter values inside the quantile interval are less probable than parameter values outside the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Moreover, the HDI is the smallest among all possible credible intervals which is a desirable property when it is used for posterior inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' On the other hand, an advantage of the quantile interval is, that it is easier to interpret for transformed parameters than the HDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This is because one can derive the quantile interval of the transformed parameter simply by back-transforming the intervals that where derived for the transformed parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In contrast, the HDI of the untransformed parameter cannot be simply derived by back-transforming the HDI of the transformed parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A further difference is, that the quantile interval always includes the median of the posterior distribution, whereas the HDI always includes the mode(s) of the posterior distribution (Kruschke (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In symmetric distributions, the quantile interval and the HDI return similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Moreover, under certain conditions Bayesian credible intervals also coincidence frequentist confidence intervals coincidence (see Jaynes & Kempthorne (1976)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' With Bayesian credible intervals one can set up a decision rule by testing if δ0 falls withing the 1 − α posterior credible interval of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A more meaningful approach may be to test whether the posterior interval excludes a region of practical equivalence (ROPE) which may be defined as all parameter values that are smaller (in absolute value) than the minimal clinically relevant effect size δrel, as suggested by Kruschke (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As a further alternative, Kruschke suggests to set as goal not to reach a certain power but rather a certain precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' An according decision rule can be implemented by deciding if the width of the credible intervals is smaller than a threshold value representing a target precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Classical decision rules for a null hypothesis have been criticized for null hypothesis testing, for example by Wagen- makers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2020), Gelman, Hill & Vehtari (2020), Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' One critique is connected to the principle of predictive irrelevance stating that data that are predicted equally well by both a null model M0 (corresponding to H0) and an alternative model M1 (corresponding to H1), data which the authors in Wagenmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2020) call uninformative or irrelevant, should not lead to favor one model above the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, this can happen in the above described scenario of null hypothesis significance testing (NHST) when intervals or p-values are estimated only under one of both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Instead, to quantify the evidence for an effect that differs from the null hypothesis, Bayes factors are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Bayes factors compare the probability of the observed data under (at least) two models M0, M1: BF10 = p(y|M1) p(y|M0) (6) This Bayes factor gives an impression of how much more likely the data was generated by the model under H1 over the model under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' It is related to the odds of posterior model probabilities p(M1|y) p(M0|y) by being the factor by which the ratio of prior model odds p(M1) p(M0) changes after observing the data: p(M1|y) p(M0|y) = BF10 · p(M1) p(M0) If the Bayes factor is greater than one this indicates that, after observing the data, the odds for the model under H1 over H0 have increased as compared to the a priori expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2022) show that, to make this an approach that is sensible to the observed evidence in the data, it is essential to chose appropriate prior distributions for the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Typical methods for estimating Bayes factors based on the prior distributions and the data are bridge sampling (Bennett (1976)) or the Savage-Dickey method (Dickey & Lientz (1970)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' To set up decision rules that are based on Bayes factor, a threshold has to be defined as to when the null hypothesis is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Lee & Wagenmakers (2014) provide a rough interpretation scheme for Bayes factors that is adjusted from Jeffreys (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' They declare moderate evidence for H1 if the Bayes factor BF10 is greater than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This can be used to calculate the percentage with at least moderate evidence for H1 as a binary decision function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This classification and decision rule present a convenient overview and method to get something like a power estimate from the Bayes factors, but should not be used as a strict rule (see Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2022), Schönbrodt & Wagenmakers (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Instead, the whole distribution of the Bayes factors should be considered and ideally simulation based calibration and sensitivity analysis, as presented by Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2022), should be carried out to ensure a correct interpretation of the Bayes factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Alternatively, a heuristic decision rule can also be defined by making a decision in favor of H1 if this is the model with the higher posterior model probability p(Mi|y), i = 0, 1 (see Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This decision rule however does not include the outcome of no evidence for either hypothesis and is only optimal if both errors of the corresponding decision rule (deciding for H1 when the data in fact corresponds to model M0 and deciding for H0 when the data in fact corresponds 5 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT to model M1) are equally bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This is typically not the case in clinical or preclinical research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A more principle scheme for deriving decision rules is by the definition and optimization of utility functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Utilities define the cost or value of decisions, conditionally on the null or alternative hypothesis being true and are necessary to judge the performance of a decision function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In the frequentist framework utilities are defined in terms of type I and type II error rates and decision functions are constructed by bounding type I errors and optimization with regard to type II errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Differences in the Bayes factor oriented decision rules and the frequentist hypothesis tests are, for example, that the Bayes factors can distinguish between no evidence and evidence for H0, whereas frequentist tests can not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 Simulation based design analysis Gelman & Carlin (2014), Gelman & Vákár (2019), Gelman, Hill & Vehtari (2020) propose design analysis using fake-data simulation as an alternative to classical power-oriented sample size calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Using fake-data simulation, the effect of varying parameters on the predefined decision functions can be examined in combination with relevant candidates for the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' To determine if predefined statistical goals are met, models are fitted to the data, statistical analysis are performed and the previously defined decision functions are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The statistical goals are formalized in utility functions as introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Here, utilities are calculated as type I and type II error rates or false discovery rates (FDR) and true discovery rates (TDR) by determining the percentages how often the data were simulated with a δ equal to the null effect but the decision functions decided against the null effect and how often the decision function decide against the null effect when the data were simulated with δ corresponding to increasing effect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The goals are then to (find a sample size to) reach certain TDRs or a certain power while limiting the FDR or type I error rates by a certain α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Additional model characteristics are examined as the type S (sign) errors, as the probability that the estimate of the true effect has the wrong sign, given that is statistically significant and type M (magnitude) errors, as the probability that the effect estimate is greater in absolute value than the absolute value of the true effect, given that it is statistically significant Gelman & Carlin (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Moreover, the the mean-squared-error (MSE) of the estimate of the true effect size is examined in simulated data under varying true effect sizes by Gelman & Vákár (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Also the distribution of Bayes factors is visualized and it is examined how often the Bayes factors are falsely greater or smaller than one, if the lower bound of the 95% interval of the posterior model probability for the model under H1 exceeds the value of 50% and what percentage of the Bayes factors lies within the categories defined by Jeffreys (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Given the utility function(s) and decision rules, a minimal sample size can be chosen that reaches one or several of these predefined goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' R = 10000 simulated data sets are constructed in accordance with model (2) as sum of a random control group, a treatment effect and group-specific residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In this work the data in the new experiment is generated in a frequentist framework with fixed values for δ, θC, ψ and λ and randomness in the simulated data comes only from the group-specific residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' y∗ ik,r = θ∗ C,r + δ∗ r1(i = E) + ϵ∗ il,r ϵ∗ ik,r ∼ N(0, σ∗2 i ), σ∗ i = exp(η∗ σi,r), where η∗ σi,r = ψ∗ r + λ∗ r1(i = E) (7) for animal k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , ˜ni in group i = C,E of data set r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Alternatively, the new data could also be generated in a fully Bayesian framework as discussed in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For each simulated data set a Bayesian model is fit with the brms function of the brms package (Bürkner (2021)) in R Core Team (2021), using the default MCMC parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Frequentist point estimates and confidence intervals for the population effects are estimated with the lm function in base R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The steps and decisions that are commonly made in a Bayesian framework are described as a Bayesian workflow by Gelman, Vehtari, Simpson, Margossian, Carpenter, Yao, Kennedy, Gabry, Bürkner & Modrák (2020), Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' After fitting a Bayesian model the next step in a Bayesian workflow is to validate the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' To decide whether the MCMC draws are likely to have converged against the target posterior distribution, characteristics of the MCMC chains are examined in convergence diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' More specifically, MCMC trace plots, auto correlation plots, the effective sample size and the ˆR statistic are examined (for details see Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Bayesian credible intervals are estimated as highest density intervals with the bayestestR package (Makowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2019)) and quantile intervals computed as empirical quantiles of the posterior MCMC draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Bayes factors are computed with the hypothesis function of the brms package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' To account for heteroscedasticity, a distributional model is fit in brms and in the frequentist model, heteroscedasticity is accommodated by the estimation of robust confidence intervals with the sandwich (Zeileis (2006)) package that estimates heteroscedasticity consistent (HC) variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' More specifically, a HC3 type estimator is chosen in the frequentist design that is also appropriate for smaller sample sizes (see Long & Ervin (2000)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Additionally, frequentist p-values in the Welsh test are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 6 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 Meta-analysis model for the historical data The distributions in the sampling model (7) as well as the prior distributions for a Bayesian analysis of the simulated (and new) data are based on a Bayesian meta-analysis model of relevant historic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The hope is that, by the usage of Bayesian estimation and historical evidence, a prior knowledge and uncertainties are better reflected and a sample size can be found that is more likely to actually achieve the predefined statistical goals than with classical frequentist methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In the best case, and if the prior distributions reflect (major aspects of) the simulated data correctly, using the historical information as prior distribution in the Bayesian analysis of the new data can even reduce the number of animals that are needed to reach the predefined statistical goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 Normal-Normal hierarchical model The historical data are modeled as data from G different animal experiments with ng animals each, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As simplest and most commonly assumed case the data are assumed to be normally distributed as ygk = θg + ϵgk, ϵgk ∼ N(0, σ2 g) k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , ng, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (8) with residuals ϵgk, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , G, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , ng, and experiment specific means θg, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , G estimated as arithmetic means yg = 1 ng �g k=1 ygk, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Data, for which a normal assumption is inappropriate, such as skew and non-continuous data, can often be transformed to resemble a normal distribution and be handled by this model (see Hedges & Olkin (1985), Hartung & Knapp (2001), Higgins & Green (2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As an assumption that allows the usage of the historical data for the analysis of the new experiment, the new data and the historical data are assumed to be exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This means that there are assumed to be no systematic differences in the new and the historic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This assumption is modeled by a random-effects meta-analysis for the historical data and the usage of this model’s mean prior predictive distribution to construct a prior distribution for the parameters in the new experiment as outlined by Neuenschwander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Heterogeneity as variance of the historical experiments may occur due to different ages, animals strains, laboratory conditions or measuring instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Such heterogeneity is modeled by heterogeneity components γg that represent deviation from a common mean µ in the experiment specific means θg = µ + γg, g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The parameters µ and τ in the distribution of the experiment specific parameters θg of interest are referred to as hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A Normal-Normal hierarchical model (NNHM) for the historical data is formulated as ygk|θg, σg ∼ N(θg, σ2 g) θg ∼ N(µ, τ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (9) This model is termed hierarchical since it includes connected sampling distributions on two levels and Normal-Normal hierarchical model since a normal assumption is assumed for both the residuals ϵgk on the level of the individual data and the heterogeneity components γg on the level of the experiment specific means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In a frequentist framework this model with normally distributed means θg ∼ N(µ, τ 2) is also referred to as random effects model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Fitting a meta-analysis in a Bayesian instead of a frequentist framework has proven to be beneficial especially in the case of small, few studies by Friede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2017a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The estimation of the group-specific means in a hierarchical model with a common mean as pooled effect leads to so called shrinkage estimates that are shrunken towards the common mean µ as compared to the estimation of independent means in a fixed effect model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The advantage of the shrinkage estimation is that single extreme values are relativized and the uncertainty in single groups can be reduced by borrowing information from other groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The degree of shrinkage of one group g ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , G} depends on its sample size ng (or the associated standard error sg) and the between-group heterogeneity τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Shrinkage is higher for smaller ng (bigger sg) and smaller τ 2 (for details see Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2013), Schmidli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2014), Neuenschwander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2016), Wandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2017), Röver & Friede (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4 Prior distributions In a Bayesian framework further probability distribution models as prior distributions are set up for the additional parameters (σg, µ and τ in the meta-analysis in equation (9) model and θC, δ, ψ and λ in model (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A popular choice for a parameter’s prior distribution is a so called conjugate prior from the distribution family that is conjugate to the family of the modeled likelihood distribution of the data Gelman (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Choosing such a conjugate prior distribution for a model parameter implies that also the parameter’s posterior distribution is from the same family as the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This has interpretational and computational advantages as outlined in Gelman (2006), Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 7 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A prior distribution can be categorized according to its information content to be either non-informative, weakly informative or informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This categorization can be controlled by the prior distribution’s variance parameter and has to be interpreted in context of the likelihood distribution the data (for details see Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Non-informative priors have minimal impact on the posterior distribution as compared to the impact of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' They are constructed so that their probability density is flat relative to the probability density of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In contrast, informative prior distributions are designed to represent the full available a priori knowledge as accurately as possible and to have a major impact on the parameters’ posterior distribution together with the impact of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Weakly-informative prior distributions constitute a compromise between non-informative and informative distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' They are intentionally designed to be flatter than informative prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Weakly-informative priors can be chosen to have a regularizing functionality on the posterior distribution by restricting the posterior distribution to a plausible parameter range (for details see for example Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2013), McElreath (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Generally, context specific weakly informative or informative priors are preferred over non-informative priors, especially when Bayes factors are estimated (see Gelman (2006), Betancourt (2017), Seaman III et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2012), Betancourt (2017), Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021), Lemoine (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 Variance parameters With few studies, special care hast to paid to the specification of a prior distribution for the heterogeneity parameter τ in model (10) (see Gelman (2006), Friede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2017a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Friede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2017a) recommend a prior distribution that puts most of their probability mass to areas that represent small to large heterogeneity and leave only a little fraction to values that represent a larger heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The interpretation of the heterogeneity degree depends on the scale of the modelled parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Spiegelhalter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2004) suggest to classify heterogeneity in context with the residual standard deviation σ of the data model into the following classes: Heterogeneity r := τ σ small 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0625 moderate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='125 substantial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='25 large 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 very large 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 Table 1: Classification of heterogeneity in relation to the standard deviation of the data according to Spiegelhalter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Recommended prior distributions are then from the family of of folded non-central t-distributions with special cases of the half-t and half-Normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The half-Normal distribution HN(ϕ) has a scale parameter ϕ and is related to a standard normal distribution with mean zero and variance ϕ2 by taking the standard normal distribution’s absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Compared to the half-t distribution the tail of a half-normal distribution is smaller which puts less weight on extreme heterogeneity values (see Spiegelhalter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2004)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The distributions parameters in this applications example are set up to reflect the assumptions on the heterogeneity as classified by table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 Intercept parameters As prior distribution for the intercept parameters normal distributions are chosen which is, conditionally on all other model parameters, the conjugate family to the normal distribution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' More specifically, as starting point for the intercept’s prior, a unit information prior (UIP) is recommended (see Kass & Wasserman (1995), Röver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A unit information prior has the information content (variance) that corresponds to one single typical data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The unit information prior for the historical data is set up to be centered at the mean of the historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Strictly speaking orienting the prior distribution on information from the data implies to use the same information twice, once for setting up the prior and once through the likelihood of the observed data that both go into the estimation of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This contradicts the Bayesian concept of a prior distribution as representation of a priori knowledge before seeing the analyzed (historic) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' It can nonetheless serve as a starting point to ensure that the prior distributions are centered at some reasonable area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' By choosing a variance parameter that leads to a wide enough but not unrealistically wide prior distribution, it reflects only a rough guess and lets the data contribute more exact information to refine the parameter’s posterior distribution while having a regularizing functionality McElreath (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This again can be examine in sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A unit information prior can also be used as reference prior for testing Bayesian hypothesis or for an assessment of the effects on the posterior compared to more informative priors (see Kass & Wasserman (1995), Raftery (1998), Neuenschwander & Schmidli (2020), Li (2021), Röver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 8 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT For the mean θC and the standard deviation σC in the control group in the new experiment, MAP priors are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Therefore the posterior_epred function of the brms package is used to estimate the expected posterior predictive distribution in the operation groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The expected posterior predictive distribution of the group without operation is used to derive parametric prior distributions for the mean and standard deviation θC and σC = exp(ψ) in the new experiment, as explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The priors for θE and σE = exp(ψ + λ) are set up to be centered around the same values like the priors θC and σC = exp(ψ), but with higher variance parameters, corresponding to unit information priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The higher variance parameters reduce the weight of the prior distributions as compared to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The intention for centering the prior distributions of the parameters in the control and experimental group around the same value is that, a priori, the two groups are expected to be equal on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Meanwhile, the intention for the higher variance in the priors of the parameters in the experimental group is, that for the control group there is historical data available, so the prior distribution should have larger influence than for the experimental group, where no (directly related) historical data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 Approximation of the MCMC draws by parametric distributions To incorporate the historical data as proper prior distributions, the non-parametric estimates of the posterior predictive distribution, as represented by the MCMC draws, are approximated by parametric distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Röver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021, 2022) illustrate three general methods for fitting parametric distributions to MCMC draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In each method the first step is to specify a distribution family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Choosing a (to the data distribution) conditionally conjugate normal prior distribution for the mean parameter in the historical data model ensures that the posterior distribution is from the same parameter family, conditionally on fixed values of all other model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, when the other model parameters are not fixed, the posterior distribution is not necessarily a simple normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In case of the NNHM from equation (9), Röver (2017) show that, with a normal prior distribution for µ and a half-normal prior distribution for τ, the posterior predictive distribution for the man parameter µ and the parameter θ∗ in the new experiment are normal-mixture distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Thus, also more complex distribution families have to be considered for the approximation of the posterior MCMC draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' After a distribution family is specified, a simple method to estimate its parameters is by taking point estimates, like the mean of median, from the posterior draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This approximation of the posterior draws as point estimates however is a reduction of information and does not always reflect all important characteristics of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' An alternative method is to approximate the posterior draws by marginal distributions (see Röver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A third alternative is to choose a model family and determine its parameters by maximum likelihood (ML) estimation, the expectation-maximization (EM) algorithm or moment matching (MM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Different candidates for distribution families can be compared by using estimators of the model fit or of their predictive performance like the Akaike information criterion (AIC) Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In the Bayesian context, the Watanabe-Akaike information criterion (WAIC) and Leave-one-out cross-validation (LOO-CV) are preferred over AIC, as outlined by Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2014), Vehtari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' But since the parametric distributions are fit with frequentist and not with Bayesian methods, WAIC and LOO-CV are not considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In this work, the parametric distribution candidates for the MAP priors are normal, normal mixture and t-distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For fitting a mixture distribution, the automixfit function of the RBesT R package (Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021)) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This function uses the EM algorithm to fit a series of normal mixture distributions with increasing number of mixture components and selects the best fit according to the AIC value (which penalizes model complexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For comparison, simple normal and non-centered t-distributions are fit using ML estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Prior predictive checks, as described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4, are perfomred to ensure the priors are reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The approach to formulate a predictive distribution as prior distribution in the new experiment is termed meta-analytic predictive (MAP) approach (Neuenschwander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2010)) and requires the explicit formulation of a prior distribution as predictive distribution, given the historical data (MAP prior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' An alternative to this sequential approach for the analysis of the historical and new data is the meta-analytic-combined (MAC) approach where both historic an new data are analyzed in a common analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Schmidli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2014) show that the MAP approach is theoretically equivalent to the MAC approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In practice, the MAC approach has the advantage to be more direct and easier since it requires only to fit one single Bayesian model instead of one model for the historical data, one for the derivation of a MAP prior from the historic data and one for the analysis of the new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In the context of design analysis and sample size determination, however, the MAP approach has the advantage of allowing better judgment and control over the influence of the historic data on the estimation results in the new analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, an effective sample size (ESS) of the MAP prior neff can be calculated that quantifies the influence and information content of the historical data as ratio of the variance of the MAP prior in a heterogeneous sample to the variance of the MAP prior in a homogeneous, pooled sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' neff gives an estimate of the number of animals that can be saved in the new experiment by using the information in the historical data (given that the new and historical data are exchangeable) (see Neuenschwander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2010, 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 9 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT 3 Application example As an application the simulation based design analysis and sample size determination is carried out for an animal experiment from translational preclinical research that aims to examine the role of the C5aR1 receptor on bone quality under postmenopausal osteoporosis in mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The pathological condition of postmenopausal osteoporosis is realized by ovariectomy in mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Bone quality is measured (among other indicators) by the (unit-less) relative bone volume (bone volume to tissue volume (BV/TV)) in a µCT scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This experiment is referred to as C5aR1 experiment in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The aim of the planned animal experiment is to test whether or not there is evidence for an association between the C5aR1 knockout and the relative bone volume in mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' If there exists such an associations, then a C5aR1 knockout may serve as basis for a potential treatment that can be examined in humans in clinical trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The planned experiment shall consist of data from twelve week old female mice of the C57BL/6J strain, which is the standard mouse strain from the Jackson laboratory research institution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 Data and models in the application example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 Historical data Internal historical data from a previous experiment is available from the proposer for planning the new experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This internal data comprises data from twelve ovariectomized mice and ten mice with Sham operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Yet, the internal historic data from the proposer represents only a fraction of the information that has been collected so far regarding bone quality in mice since bone quality is an active research topic in preclinical research (see for example Ignatius, Schoengraf, Kreja, Liedert, Recknagel, Kandert, Brenner, Schneider, Lambris & Huber-Lang (2011), Ignatius, Ehrnthaller, Brenner, Kreja, Schoengraf, Lisson, Blakytny, Recknagel, Claes, Gebhard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2011), Mödinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2018) and the citations therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The inclusion of additional data has the potential of providing more information about the distribution of the relative bone volume in mice, the variability among and between different experimental groups and upon which effect sizes are realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However most of the available literature cannot be used as further historical data, since the animals characteristics (such as age and the animal strain) of the external data and the experimental conditions under which external data have been collected, deviate from the internal historic data or the outcomes are measured with different methods, have different definitions or scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' An easy and comprehensive option for retrieving control data is the Mouse Phenotype Database (MPD) (Grubb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2004), Bogue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' It is an open-access database that collects phenotype data for the characterization of inbred mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The data can be used for characterizing the correlations of complex traits and as control data or for characterization of mutation effects (see Consortium (2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The MPD data comes in a tidy, standardized format and also the experiment protocols and tools for data analysis are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The data is made available by worldwide researchers and managed by employees of the MPD, who also endow the data with a common public ontology like the Mammalian Phenotype Ontology, that was introduced by Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2005), what makes it easier identify and compare relevant data for the outcome of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The search term “BV/TV" on the MPD web-page leads data from 31 strains of Collaborative Cross (CC) wildtyp mice from an experiment of Levy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' CC mice are recombinant inbred mice from eight genetically divergent strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' They are characterized by a high degree of genetic diversity that represent on average 90% of the allelic diversity in the whole mouse genome Chesler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In the planned experiment the mice shall undergo ovariectomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Since ovariectomy is expected to have an impact on the relative bone volume, the MPD mice cannot be used on its own to estimate a posterior predictive distribution for the mean relative bone volume in the new experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Still, the estimation of the distribution for the mean in the wildtyp mice can give an idea for the range of plausible effect sizes that are examined in the design analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For example, one might expect that the best one can expect from the C5aR1 knockout in the experimental group in the new experiment is to completely reverse the negative effect of the ovariectomy and bring the relative bone volume back to the basic level in wildtyp mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Moreover, the additional consideration of the external data can help to quantify heterogeneity in the outcome in different strains and ideally, it could also give an impression of how representative the internal mice strains are for the mouse genome in general with respect to the outcome relative bone volume by comparing internal wildtyp mice to the MPD wildtyp mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, int his case there is no internal data from wildtyp mice but only ovariectomized and Sham mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The external data and internal data are represented in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Since the hypothesis refers only to female animals, all male animals from the external MPD data set are excluded from the meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The relative bone volume can by definition only take positive values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In the historical data most of the values were close to zero with a couple of very big values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A logarithmic transformation was applied to the data to transform the scale of the outcome to the real numbers and to make it resemble more a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 10 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT Experiment strain OP n � E(BV/TV) � SD(EI) � E(log(EI)) � SD(log(EI)) Internal C57BL/6 Ovx 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='93 Internal C57BL/6 Sham 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='36 MPD PreCC1061/Tau None 2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='08 MPD PreCC111/Tau None 9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='22 MPD PreCC1156/Tau None 2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='01 MPD PreCC1513/Tau None 7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='21 MPD PreCC188/Tau None 9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='29 MPD PreCC1912/Tau None 6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='10 MPD PreCC2126/Tau None 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='76 MPD PreCC2156/Tau None 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='34 MPD PreCC2391/Tau None 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='66 MPD PreCC2573/Tau None 3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='70 MPD PreCC2680/Tau None 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='53 MPD PreCC2689/Tau None 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='33 MPD PreCC2750/Tau None 5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='61 MPD PreCC3348/Tau None 7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='91 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='24 MPD PreCC3438/Tau None 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='49 MPD PreCC3480/Tau None 3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='40 MPD PreCC3912/Tau None 10 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='20 MPD PreCC4052/Tau None 12 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='47 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='81 MPD PreCC4141/Tau None 6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='47 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='46 MPD PreCC4438/Tau None 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='67 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='08 MPD PreCC4457/Tau None 7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='30 MPD PreCC519/Tau None 6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='48 MPD PreCC521/Tau None 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='23 MPD PreCC557/Tau None 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='35 MPD PreCC611/Tau None 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='02 MPD PreCC670/Tau None 3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='23 MPD PreCC711/Tau None 3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='08 MPD PreCC72/Tau None 6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='23 Table 2: Sample size (n), mean (ˆE) and empirical standard deviation ( ˆ SD) of the relative bone volume (BV/TV) in the historical data on the original and the logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The estimates are calculated by the group factors experiment (internal, MPD), mouse strain and operation group (OP as ovariectomy (Ovx), Sham operation and no operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 Meta-analysis model A hierarchical or meta-analysis model with individual data (IP-MA, individual patient meta-analysis) (see for example Lyman & Kuderer (2005), Michiels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2005), van Walraven (2010)) is fit in a Bayesian framework by using the brm function in the brms package (Bürkner (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For comparison, a frequentist model is fit in R with the lme of the nlme package (Pinheiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2022)) to accommodate the random strain effects, where restricted maximum-likelihood (REML) estimates are derived by maximization of the log-restricted maximum-likelihood method (see Pinheiro & Bates (2006)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' An operation group variable OP is included as predictor to indicate if the mice had underdone ovariectomy, a Sham operation or no operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The ovariectomy and Sham operation may not have the same impact on all animals in the respective operation group whereas the data in the animals without an operation is expected to have smaller variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' To allow the mice in the ovariectomy group to have a different residual variance than the residual variance in the Sham operation and allow the residual variance to be yet different than the residual variance in the group without operation, a distributional model (like explained in Bürkner (2020)) is fit, in which the residual variance is modeled by its own operation group specific predictor term, similar to the one in equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Ideally, since bone quality is known to depend on age, this variable should be included as predictor term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' But since each operation group is from a different age interval (the youngest are the animals without operation, and the animals with Sham operation and ovariectomy are all of an older age), the effects of age and the operation group cannot be untangled by the inclusion of the age variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The same problematic applies to the experiment where the data came from (internal, MPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Also this variable should ideally be modeled as fixed or random effects parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' But since both Sham and ovariectomized animals are internal data and the animals without operation are external data from the MPD also this effect cannot be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 11 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT In summary, the the following meta-analysis model is fit: yijk = α + β11(i = 1) + β21(i = 2) + νj + ϵijk, with operation group specific residuals ϵijk ∼ N(0, σi), σi = exp(ησi), ησi = ψ + λ11(i = 1) + λ21(i = 2) and additional independent variance components νj ∼ N(0, τ 2 ν ) (10) where k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , nij indices the mouse in operation group i = 0, 1, 2 (none, ovariectomy (Ovx),Sham) and strain j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The choice of the prior distributions is explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 Prior distributions for the historical data In the brms package non-informative or weakly informative prior distributions are specified as default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Different default priors are chosen for the intercept (intercept of the mean, α, and intercept of the logarithmic residual standard deviation, ψ), than for the population parameters that apply to the whole data (fixed effects in a frequentist framework, here βi, λi, i = 1, 2 and γ), and for group-specific variance parameters that model the heterogeneity between groups (random effects in a frequentist framework, here νj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In table 3 the manually chosen prior distributions for parameters from model (10) are summarized and contrasted with the default priors in the brm function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For the intercept, an unit information prior (UIP) is considered as reference as explained in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The intercept α represents the mean relative bone volume in the animals without operation and at age of eight weeks on the logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' An UIP is set up with information content corresponding to a single observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Therefore the historical animals without operation are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Using ML estimations to fit a normal distribution to the group without operation, the residual standard deviation is estimated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' To make this a bit less informative, the standard deviation for the prior of α is increased to the value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The mean of α’s prior, is set according to the estimated mean in the group without operation which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' These choices result in a 95% interval of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1, 4] on the logarithmic scale and [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1, 53] on the original scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The operation-group specific residual standard deviations σi, i = 0, 1, 2 (no operation, Ovx, Sham), in model (9) are modelled as exponent of a normally distributed linear predictor ησi, that is centered at mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' With a scale parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5, σ0 has a 95% quantile of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3, which is considered as more realistic than the default t3(0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5) distribution in brms that leads to a 95% quantile of 360 of σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For the heterogeneity parameter τν of the variance parameters νj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , 22 (representing the variance due to the different strains) a half-normal prior is chosen as discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The scale parameter of τν is set to 1 2, which corresponds to large heterogeneity when classified with table 1 and the mean of the prior σ0 as reference scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A priori, large heterogeneity is expected to exist in the mice strains since they include CC mice from genetically diverse backgrounds as explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The population effects βi, and λi, i = 1, 2 are kept at their default values in brms and only modified if there are indications of convergence problems or if the prior predictive checks indicate that they are unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Neither is the case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Parameter Default Manual α t3(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='7, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5) N(2, 12) ψ t3(0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5) N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='52) βi U(−∞, ∞) U(−∞, ∞) λi U(−∞, ∞) U(−∞, ∞) τν U(−∞, ∞) HN(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='52) Table 3: Prior distributions (default in brms and manual choices) for the Bayesian meta-analysis model of the historical data on logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4 Prior predictive checks The model including the candidates for the prior distributions are tested in prior predictive checks, introduced by Good (1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In prior predictive checks a large amount of replication data is simulated from the given model and prior distributions with the aim to decide if the model and parameter distributions are (biologically) plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Although plausibility statements can already be made by reviewing the model and parameter distribution definitions, the interplay of all model components is most easily viewed and judged in such prior predictive checks where the generated data 12 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT reflects all these model components and can be compared to a researchers prior expectation of how typical data in this context should look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' To perform the prior predictive checks data sets are generated from model (10) with prior distributions as specified in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For the population effects βi, γ, λi, i = 1, 2, that are necessary to construct the data in the ovariectomy and Sham operation groups, improper, non-informative priors U(−∞, ∞) were chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Since no sampling is possible from this distribution and to reduce the scope of generated plots, the prior predictive checks are only presented for the group without operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Hypothetical data for the other groups could be generated with priors that are approximating the U(−∞, ∞) distribution, for example with a normal distribution N(0, σ2) with very large σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 1000 prior predictive data sets are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For each data set the model parameters are simulated with random number generating functions in R and then the hypothetical data are constructed by the model described in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For the animals in the group without operation this corresponds to ˜yr1jk = ˜αr + ˜νr,j[l] + ˜ϵr1jk (11) for mouse k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , K from mouse strain j[k] = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , 22 in the hypothetical data set r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' , 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The number of mice per simulated data set, K, is chosen to correspond to the number of mice in the historical data which is K = 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, the aim is not to choose prior distributions that exactly reflect the distribution of the historical data but rather to select weakly-informative prior distributions that lead to a prior predictive distribution that is less informative (flatter) compared to the actual observed historical data distribution but that excludes values that seem implausibly high in context of the historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The theory is that, for a big number of generated data sets, the empirical distribution of the simulated data approximates the prior predictive distribution of the data (see Good (1950)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 Results from the application example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 Fitting the Bayesian Normal-normal hierarchical model to the historic data The results of the prior predictive checks for the prior distributions of the meta-analysis model in the historical data are presented in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The distribution of the histograms indicates that the default prior distributions in brms lead to very big values of the logarithmic relative bone volume in the group without ovariectomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The 10% and 90% interval boundaries are almost at −25 and 25, what corresponds to values about zero and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2·1010 on the original scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' With the weakly-informative prior instead the 10% and 90% interval boundaries are at −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5, corresponding still to quite low and high values 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 · 10−4 and 1808 on the original scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' From a biological perspective, the weakly-informative prior distributions seem more reasonable but still allow the actually observed data to have a major impact on the posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Since no convergence problems in the model fit occur and since the posterior estimates seem reasonable and since the meta-analysis model is not the main model but rather is intended mainly for building a prior itself for the analysis of the new experiment, no further fine tuning of the prior distributions for the meta-analysis model is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, additional prior distributions that result in overall smaller relative bone volumes could be examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The plausibility of the estimated meta-analysis model is examined in posterior predictive checks as presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, MCMC diagnostics are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The diagnostics showed no signs of divergence or high auto-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The estimated population effects (fixed effects) and standard deviations of the variance components (random effects) and residuals in the meta-analysis model of the historical data are presented in table 4 as means with 95% quantile intervals and compared with the estimates from a frequentist analysis with REML estimators and approximate confidence intervals under normal assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Overall, the estimations from the Bayesian MCMC and frequentist REML method are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Slightly bigger differences exist in the estimation of the residual standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Notably larger differences exist for the residuals standard deviations of the Sham and Ovx group that have only very few observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A forest plot of the strain effects is represented in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The hierarchical (random effects) model leads to group- specific estimates that are shrunken towards the common mean (pooled effect) and have smaller variance than in an independent estimation as fixed effects model, as discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The point estimates of the Bayesian and frequentist analysis are again quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As part of a Bayesian workflow, posterior predictive checks are performed to ensure that the model generates reasonable data in light of the original data (for details see Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The results are presented in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The posterior distributions seam reasonable in light of the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Compared to the respective prior distributions (here only shown for the group without operation in figure 3), the posterior distributions have smaller tails due to the updated information by the historic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='Designing translational animal experiments by Bayesian MAP approaches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='A PREPRINT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='Weakly informativ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='100 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='Figure 1: Graphical prior predictive checks adapted from Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021) for the relative bone volume in animals without operation on a logarithmic scale with different prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Left column: Default prior distribution in the R package brms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Right column: weakly-informative prior distribution from table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The predictive distributions were calculated over 1000 simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' a) Distribution of histograms calculated per simulated data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The colored areas correspond in the order of increasing intensity to 10-90, 20-80, 30-70 and 40-60 percent intervals over all histogram frequencies of the simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The dark curve in the middle of the intervals represents the distribution of the median over all simulate data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' b) Distribution of arithmetic means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' c) Distribution of standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Extreme log(BV/TV) values < −50 or > 50 are represented as −50 and 50 for representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 Approximation of the MCMC draws and definition of prior predictive distributions The results of the approximations of normal distributions by the ML method and of (according to AIC best) normal mixture distributions by the EM method and selection by the AIC (as desribed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3) are presented in table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The approximations with both methods look quite similar and hence the approximation by a simple normal distribution with the ML method is selected as prior in place of a more complicated mixture distribution to avoid overfitting and for simplification, since it requires less parameter than the mixture distribution with more than one component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' An effective sample size of the normally distributed prior p(θC) for the control (Ovx) group is calculated with the ess function of the RBesT package by Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The calculation requires the specification a reference scale as an estimate of the (within-group) residual standard deviation in the historic and new control group animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This residual standard deviation estimate is set to the mean of the estimated posterior distribution of the residual standard deviation in the historical ovariectomized animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The resulting estimate of the effective sample size of the prior is quite low with neff = 2 indicating that, in this example, the benefit in using the information of the historical data in the analysis of the 14 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT Variable Bayes Frequ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Intercept 2 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='7,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2] 2 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='7,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3] Ovx 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 [-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='46] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 [-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='51] Sham 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='53 [-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='78] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='57 [-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='74] Strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='64 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='47,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='87] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='63 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='45,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='88] SD(Sham) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='41 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='254,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='689] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='36 SD(Ovx) 1 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='666,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='62] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='93 SD(None) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='37 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='301,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='451] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='35 Table 4: Estimated population effects (fixed effects) and standard deviations of the variance components (random effects) and group-specific residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Bayes: arithmetic means with 95% quantile intervals of the MCMC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Frequ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' : frequentist REML estimators where the confidence intervals of the random effects are approximate confidence intervals under normal assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Variable Distribution Method Component Weight E SD Median (95% interval) µC Normal-Mix EM 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 [-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5] µC Normal ML 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 [-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5] σC Normal-Mix EM 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='71,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='7] σC Normal-Mix EM 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='65,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2] σC Normal ML 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='24 1 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='64,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6] −β1 Normal-Mix EM 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='49,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2] −β1 Normal ML 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='53,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3] Table 5: Results from the approximation of the MCMC posterior distribution in the ovariectomized animals by parametric distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' EM: (according to the AIC best) fit normal-mixture approximation of a series of models fitted by the expectation-maximization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' ML: normal distribution fit by the maximum-likelihood method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' E and SD: mean and standard deviation of the respective parametric distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 95% interval: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='025 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='975 quantiles of the parametric distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' new experiment is only small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' More informative prior distributions and models for the design analysis could be derived if there was more historical data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Methods to promote the availability of historical data are described in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 Design analysis and sample size determination The candidates for the true δ in the simulated new data are taken from a range that spans from the minimum zero (corresponding to no effect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' the mean in the control and experimental group are equal on average) to a maximum that corresponds to the mean of the parametric distribution that was fit to the negative difference in predicted means of the ovariectomized animals and the animals without operation (β1 in table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The mean of the parametric distribution was estimated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 and its standard deviation to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' With respect to the estimated mean standard deviation in the ovariectomized group ˆσC = exp( ˆ log(σC)) = 1, this estimate corresponds to a Cohen effect by Cohen (1988) size of d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 (very large to huge according to the classification heuristics by Ferguson (2016) and Sawilowsky (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As further options, mean effect sizes of 1 3 · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 and 2 3 · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 are modelled that correspond to medium and large effect sizes with respect to ˆσC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Additional designs with treatment effect δ = 0 (no effect) are evaluated for investigating type I errors and false discovery rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, heteroscedastic designs with larger residual standard deviations in the experimental group than the control group (that might occur due to varying effects of the treatment) are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Therefore, the coefficient λ is set to log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5) to simulate a standard deviation in the experimental group that is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 times the standard deviation in the control group (σE = exp(ψ + λ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5σC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As sample size candidates typical sample sizes from translational animal experiments are chosen as five and ten animals in either control our experimental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' If in the experimental designs five animals in either group seem to be to few for achieving a certain statistic goal and ten animals seem too much, a more finely-tuned set of candidate sample sizes in an in-between range of five and ten can be examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The designs are summarized in table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The parameters for the prior and data distribution of the mean θC and the residual standard deviation σC in the new experiment’s control group are set according to the estimated parameters from the ML fit of the normal distributions in table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The prior for θE is chosen to be weakly informative unit information prior (UIP) with a mean that equals the mean of θC and a standard deviation that corresponds to the estimated mean of the standard deviation in the historical ovariectomized animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Also the mean of the prior for ησE = log(σE) is set equal to the mean of ησC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The standard deviation of the prior for ησE is set higher than that of ησC, to the value one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' With these parameters, the prior for σC is centered around the same 15 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT value as the prior for σE, but has larger tails that allows also for more extreme values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In prior predictive checks these priors seem reasonable and weakly-informative enough to not overrule the new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Model nE nC µ δ log(σC) σE σC 1 5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 Table 6: Designs (parameter for the simulated data and sample size candidates) for the design analysis for the outcome relative bone volume on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 10000 data sets are simulated with the chosen designs under model (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The results of the design analysis are presented in figures 4, 5, 6, 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The simulations were run on the High Performance Computing cluster of Baden-Wuerrtemberg (bwHPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Using parallel computation, array jobs and the update functionality in brms, the computations took about 7 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In figure 4 a) the number of p-values of the Welch test that are smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5, 95% frequentist confidence intervals and 95% Bayesian quantile credible intervals that don’t include the null effect δ = 0 are shown, for the different experimental designs, as a function of the total sample size (number of animals in both the new experimental and control group, nE + nC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The results with the frequentist regression based decision and the p-values are rather similar with slightly higher rejection percentages with the Welch test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Comparing the frequentist and the Bayesian curves in those designs that were simulated with unequal group means (θC ̸= θE), the additional information in the prior distribution leads to a higher percentage of rejections for the Bayesian model than the frequentist model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Power in this context can be defined as percentage of 95% confidence or rather credible interval that exclude the value null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In those designs, that were simulated with equal standard deviations (σC = σE), such a power of at least 80% is reached with nE = 10 and nC = 5 or nC = 10 in those cases that were the effect was large with δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 and also in the Bayesian model with δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In those designs, that were simulated with unequal standard deviations, a power of at least 80% is only reached with nE = 10 and nC = 10 for δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 and for nE = 10 and nC = 5 also for the Bayesian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Figure 4 b) compares the curves of the Bayesian quantile intervals from a) to those derived from Bayesian highest posterior density intervals (HDI) (for details on HDI and quantile intervals see for example Held & Sabanés Bové (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Figure 5 illustrates the precision for the designs as alternative goal for experimental planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' It shows the widths of a random sample of confidence or credible intervals as suggested by Kruschke (2015) and Elsey (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Using the type HC3 sandwich estimator to account for possibly different standard deviations in the experimental and the control group, many of the frequentist confidence intervals are much wider than the Bayesian ones from the distributional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This can be observed especially for the smaller sample sizes with five animals in the control or experimental group and for actually unequal residual standard deviations σE σC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In contrast, for ten animals in both groups and for actually equal residual standard deviations, the width of the frequentist intervals is more similar to that of the Bayesian intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' If the designs are analyzed with regard to the goal to reach a certain precision instead of power, then a target precision has to be defined in terms of a threshold for the desired interval widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For example, if the goal was that with a high 16 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT probability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 95%) all intervals in the designs with equal standard deviations are not wider than a threshold of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1, then this would be achieved for the larger sample sizes nE = 10 and nC = 5 or nC = 10 in the Bayesian model and for none of the sample sizes in the frequentist model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Figure 6 shows that, for those designs with no to moderate effect δ, the mean squared error (MSE) of the frequentist estimate is on average bigger than the MSE in the Bayesian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For bigger sample sizes, the MSE in the frequentist model gets constantly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In contrast, in the Bayesian model, the MSE only decreases slightly with sample size in the designs with the larger effect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Figure 7 shows the average type M and type S error rate for the different designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In most designs, the type S error rates are quite similar in the Bayesian models compared to the frequentist models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Large differences exist however in the type M error rate of the designs with larger δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 and δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 where the type M error rate is notably larger in the frequentist models than in the Bayesian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For the frequentist model, the type M error is very large in all designs, whereas for the Bayesian model it gets smaller with the larger effects since the choice of prior distribution results in posterior distributions for δ that are pulled towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This indicates that, if the new experiment is conducted with a frequentist analysis and either of these designs (and if the new data is actually reflected by the model used for the fake data generation), then orienting the design choice and statements on statistical significance (in terms of whether or not the confidence or credible intervals didn’t include the null value) almost always leads to an overestimation of the treatment effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The type S error is small in all designs, but around 10% with the models with the small effect δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The analysis of the estimated Bayes factors gives an impression of how much evidence there is for the null and the alternative hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The distributions of the estimated Bayes factors in the different designs are presented in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In the designs with no effect (δ = 0) or small effect (δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3) the distribution of the Bayes factor has a clear peak and the majority of its probability mass below the value one, suggesting that based on the conservative choice of equal priors for the group means and the evidence from the data, H0 is more likely than H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' While for the smaller sample size (nC = nE = 5) the peak of the distribution is closer to the value one, the peak moves further towards zero for bigger sample sizes since then there is more evidence for H0 as compared to the case of smaller sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As compared to the null effect, the distribution of Bayes factors gets flatter for larger values of δ since then the information in the data starts to rule out the tendency of the prior evidence ratio to support the null hypothesis that states equality in the group means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For the very large effect δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9, the distribution of the Bayes factors has its peak above the value one, indicating that there is more evidence for H1, while for the effects that are only of size δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 the peak of the distribution is still very close to the value one, especially for the smaller sample sizes and the case of unequal standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For bigger sample sizes with nE = 10 the distribution of the Bayes factors becomes very flat with very extreme values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Table 7 shows that in those designs, where the data was simulated under the null hypothesis with δ = 0, there is on average more evidence for H0 and the posterior median model probability of H1 is smaller than 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A goal for design analysis could be to find a sample size where the 95& quantile interval of posterior model probability for H1 does exceed the value 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This goal would be achieved in those designs with the very large effect of δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 and with equal standard deviations in both groups, for sample sizes of ten animals in the experimental group and five or ten animals in the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Another goal for design analysis could be to find a sample size that leads to a probability of the Bayes factor indicating at least moderate evidence for H1 of at least 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For this goal a sample size of ten animals would be enough in those designs with δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 that correspond to reversing the negative ovariectomy effect in the knockout animals on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Further designs were evaluated that are not represented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' More specifically, the effect of decreasing the standard deviation in the prior for θC to the half of its previous size was examined with the aim to make it more informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, this did not have a noticeable effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Additionally, the effect of increasing the standard deviation for the prior of δ to two times its previous value and twenty times its previous value was examined, with the aim to make it less informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This lead to a higher FDR since the prior had less effect on the posterior and extreme observations in the small data set could lead to false positive claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, the prior distribution with a standard deviation of twenty times its previous value (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='7 · 20 = 14) lead to a very flat distribution of Bayes factors even for those designs that were simulated with a truly large or very large effect size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Of note, non-informative priors are not recommended to be used in the context with Bayes factors Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Moreover, fake-data was simulated under a Bayesian design with pobability distributions for all parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' There, the curves of the designs that where simulated with a null effect on average (E(δ) = 0) showed, that, if the effect has a large standard deviation (like SD(δ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2), the percentage of intervals that doesn’t include the null effect gets much larger than 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Hence, one would make many more type I error as usually intended in the frequentist framework in the cases with total sample sizes 15 and 20 and in groups that have on average the same relative bone volume but with the experimental group having larger standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Since in the Bayesian simulation framework neither the frequentist nor the Bayesian interval based decision rule was designed for having (asymptotically) such a type I error rate smaller than 5%, this error rate may get much larger than 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 17 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT σE/σC δ nC nE p(M1|y) BF10 BF10 > 3 (%) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='99 [0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='76] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='43 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 Table 7: Alternative quantification of evidence for the model under H1 in the different designs (as represented by the four columns to the left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' p(M1|y): posterior probability for model M1 as model under H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' BF10: Median Bayes factor BF10 with 95% quantile interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' BF10 > 3: percentage with at least moderate evidence for H1 as categorized by Jeffreys (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The distributions are calculated over 10000 simulated data sets- For comparison, classical sample size calculation by solving power equalities is carried out with the frequentist Welch test and the power_t_test() function in the R MESS package (Ekstrøm (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As candidates for the effect sizes, for the group-specific standard deviations and for allocation ratio to the control and experimental group ( nE nC ) the same design settings as in table 6 are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The significance level and target power are set to the conventional values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The results are presented in table 8 According to this calculation, a sample size of six animals in both experimental and control group would be sufficient to detect an effect (as difference in the means) of at least the size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 with a power of at least 80% in this test, if the residual standard deviations in both groups are equal (σE/σC = 1) and the allocation ratio to both groups is also equal (nE/nC = 1) (setting 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' If about twice the animals shall be assigned to the treatment group, then nC = 5 animals for the control group and nE = 9 animals for the experimental group are calculated for detecting at least this effect size and equal standard deviations (setting 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For the same δrel = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9, but a greater standard deviation in the experimental group (σE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5), nine animals are calculated for both groups for an equal allocation ratio (setting 15) and six and eleven animals for an allocation ratio of twice the amount to the experimental group (setting 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 4 Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='1 Summary In this work aspects of sample size determination and analysis of translational animal experiments in a Bayesian framework were discussed and compared to the classical frequentist procedure in a null hypothesis significance testing (NHST) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The considerations where illustrated on a real-world animal experiment examining the knockout effect of the C5aR1 receptor in osteoclasts and osteoblasts on the relative bone volume (C5aR1 example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The determination of a sample size depends on the model assumptions on the new experiment and on the statistical goal of the analysis and model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In the Bayesian framework these assumptions include prior distributions for the model 18 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT Setting δrel σC σE Alloc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' ( nE nC ) nC nE 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1 176 176 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 2 132 264 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 1 285 285 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 2 187 373 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1 45 45 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 2 34 68 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 2 5 9 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 1 9 9 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 2 6 11 Table 8: Results from a classical sample size calculation with a two-sided Welsh test, a power of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8 and a significance level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='05 calculated with the power_t_test() function in the R MESS package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' δrel: Minimal clinically relevant effect size that shall be detected by the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' σC, σE: (estimated) standard deviations in the new experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Alloc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' ( nE nC ): allocation ratio for the mice in the new experiment to the experimental and control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' nC, nE: resulting sample sizes for the new experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As basis for setting up the prior distributions, a Bayesian meta-analysis model was estimated to available historical data, consisting of internal data from the applicant of the new animal experiment and from external data from the Mouse Phenome Database (MPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' For comparison, also a frequentist model was fit that gave quite similar point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Design analysis was performed with prior distributions and fake-data that was based on the fitted meta-analysis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The estimate of the effective sample size of the meta-analytic predictive prior for the control group in the new experiment indicated that the historical control data was only worth two animals, which corresponds to the general impression that sample size planning in translational animal experiments often comes with large uncertainties (see Mayer & Muche (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As sample size candidates for the design analysis, the minimum and maximum of the range of typical sample sizes for preclinical translational animal experiments were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The range of the candidates for the treatment effect was chosen based on what seemed realistic according to the knowledge from the meta-analysis model that was fitted to the historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The simulations required large computational resources, especially when Bayes factors are evaluated (here the High Performance Computing cluster of Baden-Wuerrtemberg (bwHPC) was used and computations in the 30 designs with each 10000 simulated data sets took about 7 hours using parallel computation, array jobs and the update functionality in the brms package).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In this example the power-based sample size calculation (here done with a Welch test) suggested that eleven or less animals in both groups would be enough to detect differences in the means of at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 for an equal allocation ratio of the animals to both groups and residual standard deviations of one in both groups and nine or less for differences of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 or greater or rather six and eleven animals for an unequal allocation ration and larger standard deviations in the experimental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, the analysis the type M error rates showed that the design and analysis with classical frequentist can lead to a high percentage of overestimated effect sizes in those cases where the analysis of the data in the new experiment results in a test decision against the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Using as Bayes factors oriented goal that 95% of the posterior probability of the model under H1 is above the value 50% (representing equal model probability for both the model under H0 and H1), only in those designs with equal standard deviations of one and an effect of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9, ten animals in the experimental group (and five animals or then in the control group) would be enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Using goal that the Bayes factor exceeds with 80% probability the heuristic threshold value for moderate evidence for H1 defined by Jeffreys (1961), Lee & Wagenmakers (2014), then a sample size of ten animals in both groups would also be enough for unequal standard deviations with a standard deviation in the experimental group of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 in the new experiment and a standard deviation in the control group of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Several chances and challenges were identified in this Bayesian meta-analytic predictive framework as compared to the classical frequentist framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' If the goal of planning the new experiment is to achieve a certain statistical power, the use of the historical data did not lead to a lower sample size in a Bayesian analysis with prior predictive distributions from the historical data than with classical frequentist power calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, the use of fake-data design analysis based on the historical data and the evaluation of additional model characteristics and statistical allowed a better representation and estimation of present uncertainties in the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' More specifically, the Bayesian 19 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT model framework allows to formally incorporate a priori knowledge (deduced from historical data) as prior distribution in the analysis of a new experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Secondly, uncertainties, that are almost always present in research stage as early as preclinical translational research, are better represented by modeling all of the model parameters as random variables instead of fixed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Thirdly, the estimation with MCMC methods allows also for more complex models, that might represent some data more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' As an example, different residual standard deviations were modeled for different experimental groups in the historic and the new data of the C5aR1 example and the estimates seemed more precise than frequentist sandwich estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Fitting a meta-analysis model to the historic data provides a quantitative summary and can be used to define the prior distributions for the new experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' With Bayesian methods heterogeneity can be reflected in the means of different historic experiments, also in the case of only few experiments or groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In contrast, frequentist models can deal less well with fitting meta-analysis or hierarchical models in heterogeneous data in the situation of with few, small experiments Gelman (2006), Friede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2017b,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Meta-analysis of similar historical experiments not only provides an initial guess, that can be used for prior specification, but also a tool for quality control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' More specifically, flaws in the experimental design or analysis may stick out or statements may have to be relativized when some measurements originating from a common mean hierarchical model differ significantly from supposedly related measures Walley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Planning and analyzing the new experiment’s in the bigger of the estimated meta-analysis model of the historical data may help to make more appropriate statements and may lead to more reproducible results as a step out of the reproducibility crisis in animal research Ioannidis (2005), Begley & Ellis (2012), Begley & Ioannidis (2015), Loken & Gelman (2017), Goodman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2016), Jilka (2016), Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2015), Macleod & Mohan (2019), Voelkl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The consideration of not only internal but also external data like from the MPD gives a broader picture of the natural variation of the outcome of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This helps to make more generalizable statements that are potentially more likely to being translated to the application in humans or to the reproduction of experiment results in other animals as suggested by Voelkl & Würbel (2016), Voelkl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The point estimates of the Bayesian and frequentist meta-analysis model in this application example were quite similar, but the Bayesian approach allowed an easier estimation of the confidence intervals as quantiles of the posterior draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The frequentist and Bayesian estimates might differ more if heterogeneity was model for a grouping variable with smaller number of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This could be the case if also the data laboratory (MPD, internal) was modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In this case the Bayesian approach has proven to be superior Friede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2017a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, determine sample size by design analysis using fake-data simulation instead of the classical determination by power inequalities, addresses several problems that are common in translational research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In particular, the variation in an experimental design and data may be better represented by a continuous value as the Bayes factor instead of the outcome of a binary decision rule and hence it may be of more value to just report the Bayes factors associated with the different designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In particular, Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2022) show in the context of a standard cognitive experiment that many standard designs don’t have sufficient evidence for making conclusive decisions and support the idea of increasing the sample sizes by sharing data across different researchers and laboratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Concerning the challenges, fitting Bayesian models with MCMC methods requires at least a basic understanding of the additional convergence diagnostics to ensure a proper model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Other recommended steps are prior and posterior predictive checks to make sure that the specified prior and posterior distributions are reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The steps and decisions that are commonly made in a Bayesian framework are described as a Bayesian workflow Gelman, Vehtari, Simpson, Margossian, Carpenter, Yao, Kennedy, Gabry, Bürkner & Modrák (2020) and may become quite complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In particular, Bayesian inference has typically many more determining factors than frequentist inference through the specification of all model parameters’ prior distributions and MCMC parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The problem is even worse when Bayesian methods are considered in an design analysis framework where additional determining factors come with different design options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This abundance of determining factors makes it hard to understand the effect of changing single determining factors for the posterior inference and makes the investigation of all combinations of determining factors becomes soon incomprehensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, there is so far no consensus in literature about which procedure to use for sample size determination in Bayesian framework (if and what decision functions and thresholds should be used etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In the case where only few, small previous experiments are available, special attention has to be paid to the assumptions on the prior model and its hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Especially, setting up a reasonable model for the heterogeneity parameter is important to properly reflect the variation in the background population under focus and prevents from driving overly-confident claims that only apply to standardized animals in a single experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Concerning the use of Bayes factors for design analysis and sample size determination, challenges are firstly the definition of a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Secondly, Bayes factors are highly sensitive to the choice of prior distributions as shown by Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021) and the usual estimation method by bridge sampling or the Savage-Dickey method requires a large number of MCMC iterations to be stable Gronau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2020) and preferably several repeated estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This makes the estimation of Bayes factors also computationally challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Finally, possible bias in the Bayes factors estimate should be examined in simulation based calibrations (SBC) Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' These Bayes factor workflow procedures again increase the already high manual and computational burden of the simulation based design analysis which might also constitute an obstacle for the application in translational research where the resources of the researchers are often quite limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' With 20 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT regard to meta-analysis, challenges are that it is difficult to find the relevant historic information since the rate of annual publications in preclinical research is very high Bannach-Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021) and the published estimates are often subject to bias like publication bias Sena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2010), ter Riet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2012), Conradi & Joffe (2017), of Health at Charité (BIH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, the relevant literature is often unorganized and outcomes do not follow a unique terminology what makes it hard to compare results from different experiments Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' These facts make it currently challenging for researchers of translational animal experiments to understand and correctly apply Bayesian methods and make sample size determination too extensive for practical applications in this context without the development of routines and applications that facilitate their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='2 Extensions There are several extensions to the here presented methodology for planning and analysing translational animal experiments using Bayesian meta-analytic predictive approaches and fake-data design analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In this work the distribution of the Bayes factors were used to visualize the evidence ratio for the null and alternative hypothesis under different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' To transform the distribution into a decision function, a heuristic threshold was defined based on a classification scheme of Jeffreys Jeffreys (1961) or by checking if the 95% posterior model probability for the model under H1 (p(M1|y)) did exceed the value 50% (representing equal probability for model the model undeer H1 and under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A more systematic approach to setting a threshold for the Bayes factors is by the definition of utility functions as illustrated by Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021) and Schönbrodt & Wagenmakers (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Bayes factors compare the “out-of-sample" predictive performance of the two contrasting models (here the model under H0 and under H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A further approach to making a decision whether or not there is evidence in the data that the effect δ differs from that stated by the null hypothesis is to compare the out-of-sample predictive performance by the investigation of posterior predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' A common utility function that measures the out-of-sample predictive performance of a model is the expected log pointwise predictive density (ELPD) (for details see Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2014) and for practical estimation in a Bayesian framework see Vehtari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In this work a Bayesian meta-analysis model was fit to the historical data with the purpose to get prior distributions and to define a reasonable design analysis setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Instead of the Bayesian meta-analysis model, also the estimates from a frequentist meta-analysis model can be used to set up parametric distributions for definition prior distributions and sampling distributions for the fake-data design analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This was done for example by Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2022) in the context of simulations for the examination of the behavior of Bayes factors under different hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In this work age and the historical data’s experiment/ laboratory effects could not be incorporated since necessary data was missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' If only few historical data is available, it is difficult to check assumptions corresponding to a normal distribution and non-parametric models bay be more appropriate Konietschke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Burr and Doss Burr & Doss (2005) suggest a Bayesian semi-parametric meta-analysis model that models the experiment-specific effects through a version of the Dirichlet process prior and implement it in the R package bspmma Burr (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' This model could be used if the normal assumption is in doubt or it can be compared to a parametric model using empirical Bayes to decide which model seems more appropriate Burr (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' There are general efforts for facilitating the use of systematic reviews and meta-analysis in animal research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Examples of such efforts include CAMARADES (Collaborative Approach to Meta-Analysis and Review of Animal Data from Experimental Studies) of Ediburgh (2021) which offers methodological advise and tools for conducting systematic reviews and meta-analysis in animal trials or the online review platform SyRF (Systematic Review Facility) Bahor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Moreover, there are efforts for collecting animal data in common big databases Eppig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2005), Blake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2006), Consortium (2007), Hancock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2008), Hannover (2022), Pognan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, there are advancements towards a mandatory (pre)registration for animal trials which could increase the amount and quality of public available data, reduce bias Chamuleau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2018), Bert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2019), Heinl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2019), Baker (2019), van der Naald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' With these advancements it is realistic, that in future better meta-analysis models of historical evidence can be fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The simulated fake-data in the design analysis represent assumptions on the data in the new experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The assumptions are based on historical data that is summarized in a meta-analysis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In the ideal case of prior distributions that match the true distribution of the future data, the prior represents just additional information that makes the posterior estimates more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' If, however, the prior predictive distribution of the data places major parts of its probability mass to regions that are unlikely according to the empirical distribution of the observed data, there is so-called prior-data conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The effects of prior-data conflict can be a problem when they invalidate the inference based on the posterior draws (see for example Box (1980), Evans & Moshonov (2006)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' To examine the effects of prior-data conflict for the chosen prior distributions on the posterior inference, data that deviates from the prior predictive distribution can be simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Further steps are necessary to get a better understanding and better visualizations of all determining factors that affect the posterior inference in the design analysis with Bayesian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Recently, the R package priorsense Kallioinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021) has been developed which intends to help investigate the impact of the prior with respect to observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Packages like this might help to get a better feeling of how the chosen prior distributions affect the necessary sample size and test decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' If prior-data 21 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT conflict seems to be a problem with the chosen prior distributions, the prior distributions can for example be robustified by adding less informative mixture components to the respective distributions, as suggested by Schmidli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Instead of using data-based priors for the design and analysis of the the new experiment, the priors can also be chosen by prior elicitation Garthwaite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2005), O’Hagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In a prior elicitation approach, the analyst does not specify the prior directly (like here based on historical data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Instead, there is an subject expert that describes properties of the outcome of interest and the task of the analyst is to formalize this description as a prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Prior elicitation could be an appealing alternative for the here used data-based priors, especially When there is no useful historical data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' However, at the current state, technical, practical and societal challenges hinder the use of prior elicitation Mikkola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Acknowledgments The author acknowledges support by Melanie Haffner-Luntzer from the Institute of Orthopedic Research and Biome- chanics in Ulm, Germany, for providing animal data and for discussing the use of animal phenotype databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Furthermore, the author acknowledges support by the state of Baden-Württemberg through bwHPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Conflicts of interest All author declares that they have no potential conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Funding This work was funded by the German Federal Ministry of Education and Research (BMBF), grant number 031L0233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' References Bahor, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=', Liao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=', Currie, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=', Ayder, C.' metadata={'source': 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+page_content='PreCC557/Tau ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='PreCC711/Tau ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='PreCC72/Tau ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='log(BV/TV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='Strain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='Bayes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='Frequ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Observed Figure 2: Forest plot with strain specific means of the historical animals and their common mean (pooled effect) stratified by the operation group (none, ovariectomy (Ovx), Sham).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Bayes: Bayesian estimates as means of the posterior MCMC draws (points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Frequ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' : frequentist REML estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Observed: strain-specific means in the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The intervals are presented as 80% (thick lines) and 85% (thin lines) quantile intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 28 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT None Ovx Sham −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 0 10 20 30 40 0 2 4 6 0 2 4 6 log(BV/TV) Count a None Ovx Sham 0 1 2 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 0 5 10 15 20 Mean(log(BV/TV)) Count b None Ovx Sham 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 0 10 20 30 40 0 5 10 0 5 10 15 20 25 SD(log(BV/TV)) Count c None Ovx Sham −2 0 2 4−2 0 2 4−2 0 2 4 0 5 10 log(BV/TV) Count Data Original Replication Figure 3: Graphical posterior predictive checks adapted from Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (2021) for the relative bone volume in animals without operation on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Grey: original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Blue: replicates from the MCMC fit in 1000 simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' a) Distribution of histograms calculated per simulated data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The colored areas correspond in the order of increasing intensity to 10-90, 20-80, 30-70 and 40-60 percent intervals over all histogram frequencies of the simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The dark curve in the middle of the intervals represents the distribution of the median over all simulate data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' b) Distribution of arithmetic means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' c) Distribution of standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Extreme log(BV/TV) values < −50 or > 50 are represented as −50 and 50 for representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 29 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT σE / σC = 1 σE / σC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content="00 Total sample size Percentage of intervals that don't include 0 Model Frequ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (Welsh−Test) Frequ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (Sandwich) Bayes (HDI) δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 (a) σE / σC = 1 σE / σC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content="00 Total sample size Percentage of intervals that don't include 0 δ 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 Interval Bayes (Quantil) Bayes (HDI) (b) Figure 4: Proportion of the simulated data set in which the decision criteria for "success” is met that the 95% confidence or credible interval does not include the null value 0 or that the p-value of the Welch test is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The distributions are calculated over 10000 simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (a): Dotted line: proportion of simulated data sets with a 2-sided Welch test p-value< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Dashed line: proportion of simulated data set where the frequentist confidence interval with the HC3 sandwich estimator doesn’t include the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Solid line: proportion of simulated data set where the Bayesian highest density intervals (HDI) doesn’t include the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (b): Comparison of the Bayesian quantile interval and HDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Dotted lines: Conventional boundaries for the type I error rate or bower of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 30 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT δ = 0 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 5 10 10 5 5 10 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 nE nC Width of the 95% interval σE σC = 1 δ = 0 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 5 10 10 5 5 10 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 nE nC Width of the 95% interval σE σC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 Model Bayes (HDI) Frequ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (Sandwich) Figure 5: Widths of the 95% Bayesian highest density intervals (HDI)s and the 95% frequentist confidence intervals with the type HC3 sandwich estimator for a sub-sample of 500 of the simulated data sets in different design setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 31 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT σE / σC = 1 σE / σC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 δ = 0 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 10 15 20 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 Total sample size RMSE Approach Bayes (Mean) Bayes (Median) Frequ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (REML) Figure 6: Root mean squared error (RMSE) calculated as root of the average squared difference of the point estimate of the treatment effect and the true mean of the treatment effect (E(δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The RMSEs are represented as average over all simulated data sets together with 95% quantile intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' In the Bayesian model the point estimates of δ are arithmetic means and medians of the posterior MCMC draws of δ and in the frequentist model the estimates of δ are calculated as difference of the means in the treatment and control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 32 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 nC Type M error rate Type M error δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 20.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='00 nC Type S error rate Type S error σE σC$ 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 Model Bayes (HDI) Bayes (Quantil) Frequ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' (Sandwich) Figure 7: Type M (magnitude) error rate: Percentage of the simulated data sets where the effect estimate is bigger than the true treatment δ in absolute value, calculated in those data sets where the confidence/ credible interval did not include the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Type S (sign) error rate: Percentage of the simulated data sets where the effect estimate had a different sign than the true treatment δ in absolute value, calculated in those data sets where the confidence/ credible interval did not include the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The distributions are calculated over 10000 simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 33 Designing translational animal experiments by Bayesian MAP approaches A PREPRINT σE σC=1 σE σC=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='5 δ = 0 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='6 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='3 δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content='9 1 10 100 1 10 100 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 BF10_print Frequency (%) nC + nE 10 15 20 Figure 8: Distribution of the estimated Bayes factors BF10 for evidence for the alternative model that the treatment effect δ is different from zero over the null model where it is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' The distributions are calculated over 10000 simulated data sets for different designs regarding the simulated true distribution in the data where for each design 10000 data sets are simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' Extreme values of ˆ BF10 > 10000 are presented as 10000 for representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} +page_content=' 34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E5T4oBgHgl3EQfYQ9F/content/2301.05572v1.pdf'} diff --git a/LNAyT4oBgHgl3EQfgPhD/vector_store/index.pkl b/LNAyT4oBgHgl3EQfgPhD/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a29d529e1b937815e2491a965069b75c3a97fd99 --- /dev/null +++ b/LNAyT4oBgHgl3EQfgPhD/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fc1ee6431154cd9755410808ba31021d33546b8d3e22d05bad4abc6e8ea0527f +size 165953 diff --git a/NNE4T4oBgHgl3EQfjQ2i/content/tmp_files/2301.05141v1.pdf.txt b/NNE4T4oBgHgl3EQfjQ2i/content/tmp_files/2301.05141v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b66407e8023fcb4cfccd85c1b095246b37fadb64 --- /dev/null +++ b/NNE4T4oBgHgl3EQfjQ2i/content/tmp_files/2301.05141v1.pdf.txt @@ -0,0 +1,1567 @@ +Spin-orbital order and excitons in magnetoresistive HoBi +J. Gaudet,1, 2, 3, ∗ H.-Y. Yang,4 E. M. Smith,5 T. Halloran,1 J. P. Clancy,5 J. A. Rodriguez-Rivera,2, 3 Guangyong +Xu,2 Y. Zhao,2, 3 W. C. Chen,2 G. Sala,6 A. A. Aczel,7 B. D. Gaulin,5, 8, 9 F. Tafti,4 and C. Broholm1, 2, 7 +1Institute for Quantum Matter and Department of Physics and Astronomy, +Johns Hopkins University, Baltimore, MD 21218, USA +2Center for Neutron Research, National Institute of Standards and Technology, MS 6100 Gaithersburg, Maryland 20899, USA +3Department of Materials Science and Eng., University of Maryland, College Park, MD 20742-2115 +4Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA +5Department of Physics and Astronomy, McMaster University, Hamilton, ON L8S 4M1, Canada +6Spallation Neutron Source, Second Target Station, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA +7Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA +8Canadian Institute for Advanced Research, 661 University Avenue, Toronto, Ontario M5G 1M1, Canada. +9Brockhouse Institute for Materials Research, Hamilton, ON L8S 4M1 Canada +(Dated: January 13, 2023) +The magnetism of the rock-salt fcc rare-earth monopnictide HoBi, a candidate topological material with +extreme magnetoresistance, is investigated. From the Ho3+ non-Kramers J=8 spin-orbital multiplet, the cubic +crystal electric field yields six nearly degenerate low-energy levels. These constitute an anisotropic magnetic +moment with a Jahn-Teller-like coupling to the lattice. In the cubic phase for T > TN += +5.72(1) K, the +paramagnetic neutron scattering is centered at k = ( 1 +2 +1 +2 +1 +2) and was fit to dominant antiferromagnetic interactions +between Ho spins separated by {100} and ferromagnetic interactions between spins displaced by { 1 +2 +1 +20}. For +T < TN, a type-II AFM long-range order with k = ( 1 +2 +1 +2 +1 +2) develops along with a tetragonal lattice distortion. +While neutron diffraction from a multi-domain sample cannot unambiguously determine the spin orientation +within a domain, the bulk magnetization, structural distortion, and our measurements of the magnetic excitations +all show the easy axis coincides with the tetragonal axis. The weakly dispersive excitons for T < TN can be +accounted for by a spin Hamiltonian that includes the crystal electric field and exchange interactions within the +Random Phase Approximation. +I. +INTRODUCTION +In spite of their structural simplicity, the fcc rare-earth +monopnictides (see Fig. 1), RX (R=Ce to Yb and X=N, As, +P, Sb, and Bi1,2), display a wide variety of anisotropic mag- +netism and electronic transport properties. The lattice param- +eter varies by 30% across the pnictide series and this provides +opportunities to tune the relative strength of crystal field and +exchange interactions. In the 1960s to 1980s, the rare-earth +monopnictides were studied to understand magnetic phases +driven by oscillatory and highly anisotropic Ruderman-Kittel- +Kasuya-Yosida (RKKY) exchange interactions 3–8. Work on +CeSb for example gave rise to an extensive literature on the +anisotropic nearest and next nearest neighbor Ising model +(ANNNI)9. +This work also resulted in progress towards +a quantitative understanding of their anisotropic exchange +interactions10. +A recent resurgence of interest in these rare-earth monop- +nictides is driven by their extreme magnetoresistance (XMR) +and resistivity plateaus, and the possible connection to the +3D topological state of the non-magnetic lanthanum monop- +nictides LaX11,12. LaAs, LaSb, and LaBi have unsaturated +XMR arising from near perfect electron-hole compensation +and there is a topological transition from a trivial electronic +band structure in LaAs to a topologically non-trivial band +structure in LaBi13–17. +Several studies have confirmed the +presence of protected surface states in LaBi18–21. Since then, +extensive works have been devoted to characterizing the XMR +and topological states of various RX including for example +CeX, HoX, and PrX. XMR has been found in each reported +magnetic RX with characteristics that depend on the rare-earth +ion22–30. The stabilization of topological non-trivial electronic +bands generating protected surface states was proposed for +several of the magnetic monopnictides29,31–33. +FIG. 1. The rock-salt structure of the rare-earth monopnictide HoBi. +Yellow and blue spheres respectively correspond to Ho and Bi. Spins +interacting through the J1 and J2 exchange interaction are shown by +the dashed black arrows. The k = ( 1 +2 +1 +2 +1 +2) magnetic order of the Ho3+ +spins is represented by the red arrows. The local spin orientations of +the Ho3+ spins that are consistent with neutron diffraction are indi- +cated for the Ho ion at (0,0,0). The magnetization, structural distor- +tion, and inelastic neutron scattering however, provide clear evidence +for easy [001] axis anisotropy. +arXiv:2301.05141v1 [cond-mat.str-el] 12 Jan 2023 + +2 +Here we study the magnetism of HoBi using modern neu- +tron scattering techniques to gain insights into its unique +magneto-transport properties29,34. +Consistent with previ- +ous works35–37, we confirm the antiferromagnetic (AFM) +k += +( 1 +2 +1 +2 +1 +2) structure and the associated tetragonal lattice +distortion. Due to multi-domain averaging, our single-crystal +neutron diffraction cannot unambiguously determine the local +spin anisotropy of the k += +( 1 +2 +1 +2 +1 +2) AFM structure. How- +ever, we could resolve this ambiguity by measuring and mod- +eling the magnetic excitations of HoBi, which take the form of +weakly propagating spin-orbital excitons whose energies and +intensities are sensitive to the local orientation of the Ho3+ +moments. Using this method, we found the k += +( 1 +2 +1 +2 +1 +2) +AFM structure has an Ising local spin anisotropy, which is +consistent with the Ising easy-axis bulk magnetization and the +tetragonal distortion. Through analysis of the paramagnetic +diffuse scattering of HoBi and the crystal field excitons in the +low T ordered state, we obtain a spin Hamiltonian with com- +parable crystal field (CEF) and exchange energy scales. +II. +EXPERIMENTAL METHODS +HoBi single crystals with mass of 10-50 mg were grown +following a previously published procedure34. Single crys- +tal low-temperature X-ray diffraction was performed using +a Huber four-circle diffractometer with a Rigaku Rotaflex +18 kW rotating copper anode X-ray generator and a Bicron +point detector. +We used a Ge (111) monochromator with +d111 = 3.266 Å. The sample was aligned for diffraction in the +(HHL) plane and mounted in a closed cycle cryogenic system +with a base temperature of 2.17 K. +We performed thermal neutron diffraction using the HB-1A +triple-axis instrument at Oak Ridge National Laboratory. We +used PG filtered 14.5 meV neutrons, and collected rocking +scans at all accessible magnetic and nuclear Bragg positions +in the (HHL) plane. Polarized neutron diffraction measure- +ments were conducted with the triple-axis instrument BT-7 +at the Center for Neutron Research (NCNR), NIST. Nuclear +spin-polarized 3He gas was used to polarize the incident neu- +tron beam and to analyze the polarization of scattered neu- +trons38,39. Horizontal guide fields were present throughout +the beam path to allow measurements of the spin-flip (SF) and +non-spin-flip (NSF) scattering cross-sections for incident neu- +tron spins polarized parallel to momentum transfer Q. The +flipping ratio measured at nuclear Bragg peaks was greater +than 30. +Cold neutron triple-axis experiments were performed using +the SPINS and the MACS spectrometers at the NCNR. On +both instruments we employed a fixed final neutron energy +E f = 3.7 meV or 5 meV and measured the elastic and inelas- +tic scattering for a single crystal of HoBi aligned for scattering +within the (HHL) and the (HK0) plane in two different exper- +iments. For the E f = 3.7 meV configuration, we used poly- +crystalline cooled Be and BeO filters before and after the sam- +ple, respectively. For the 5 meV configuration we only used +a Be filter after the sample while the incident beam from the +cold neutron source was unfiltered. For both experiments, we +co-mounted 11 HoBi single crystals on an aluminum mount. +We acquired background data using an identical mount with- +out HoBi crystals. We used an ”orange” 4He flow cryostat to +reach a base temperature of 1.6 K for these experiments. +For the highest energy resolution and energy transfer, we +performed time-of-flight neutron scattering experiments us- +ing the CNCS spectrometer at Oak Ridge National Labora- +tory. There we co-aligned two HoBi single crystals on an alu- +minum mount and collected inelastic neutron scattering data +with fixed incident energy Ei = 25 meV at T += 13 K with +a total proton charge of 47 C. We used the high flux mode of +operation of CNCS with a Fermi Chopper, Chopper 2, Chop- +per 3, and a Double Disk frequency of 60, 60, 60, 300, and +300 Hz respectively. The energy resolution (FWHM) at the +elastic line for this configuration is 2.0(1) meV. Finally, we +note that the error bars associated with the neutron scattering +experiments represent one standard deviation. +Both the magnetization and heat capacity measurements +presented here were performed in a Quantum Design physical +properties measurement system (PPMS). We used a PPMS di- +lution refrigerator option for the low-temperature heat capac- +ity. +III. +RESULTS AND ANALYSIS +A. +1st order phase transition +FIG. 2. Low temperature heat capacity of HoBi collected using the +long-pulse method. The red and blue curves respectively correspond +to the warming and cooling protocol and shows a thermal hysteresis +of 13(2) mK. The observation of a plateau at TN in the heating profile +for both warming and cooling protocol (top inset panel) suggests a +1st order phase transition in HoBi. +The thermodynamic properties of HoBi were previously +reported and a long-range k += +( 1 +2 +1 +2 +1 +2) antiferromagnetic +(AFM) order is known to occur concomitantly with a struc- +tural distortion around TN += +5.7 K14,35,36. The order of +the transition, however, remains unknown. To determine the + +AT. = 13(2) mK +HoBi +5.6 +C +K +1000 +5.8 +Cp (J/mol K) +6 +200 +400 +0 +Time (s) +Warming +500 +Cooling +5.6 +5.7 +5.8 +5.9 +T(K)3 +order of the phase transition, we measured the temperature +dependent specific heat capacity using the long-pulse heat +method40. +The resulting Cp data for HoBi is reported in +Fig. 2 for both warming and cooling protocols. +A sharp +peak with a thermal hysteresis of 13(2) mK is observed in +Cp. Correspondingly the inset shows a distinct plateau in the +temperature versus time curves during heating and cooling. +These observations indicate a 1st order phase transition at TN +in HoBi. +B. +Paramagnetic phase +To determine the magnetic interactions leading to this phase +transition, we mapped the neutron elastic scattering for mo- +mentum transfer Q covering the (HHL) plane and for temper- +atures between 150 K and 1.6 K. Representative data sets are +shown in Fig. 3. +In +the +cubic +paramagnetic +phase +for +T += +12 K +> +TN += +5.72(1) K, the scattering is +broad in Q and is centered at k = ( 1 +2 +1 +2 +1 +2) positions ((Fig. 3(a)). +This indicates short-range AFM correlations preceding the +long-range order. +The ”butterfly” pattern of paramagnetic +diffuse scattering is consistent with the equal time structure +factor S(Q) of an fcc Heisenberg paramagnet with FM inter- +actions between the first nearest-neighbor (n.n.) Ho3+ ions +(J1), and AFM interactions between the 2nd n.n. (J2). Dashed +lines in Fig. 1 indicate the lattice geometry associated with +these interactions. The scattered intensity was modeled using +I(Q) = 2 +3N| f(Q)|2 � +ij⟨Si ·Sj⟩ cos(Q·rij) where N is the num- +ber of spins, ri j is the displacement vector from Ho3+ site j to +i, and f(Q) is the Ho3+ atomic form factor41. Including only +self-correlations and correlations between spins separated by +{100} and { 1 +2 +1 +20}, a ratio of ⟨Si ·Sj⟩{100}/⟨Si ·S j⟩{ 1 +2 +1 +2 0} = −2.2(2) +was obtained at T = 12 K. The calculated magnetic diffuse +scattering corresponding to the best fit shown in Fig. 3(d) +accounts for all major features in the data (Fig. 3(a)) and the +introduction of third n.n. correlations does not improve the +fit significantly. A high temperature expansion allows us to +associate the ratio of correlations to the ratio of the corre- +sponding exchange interactions42,43 so that we may infer that +J2/J1 ≈ −2.2(2). Even if some of the J1 bond interactions are +frustrated, this resulting fitted ratio of exchange parameters +stabilize a k = ( 1 +2 +1 +2 +1 +2) order, which is driven by the dominant +AFM J2 interactions44–46. +Upon cooling, the elastic magnetic scattering gets stronger +(T = 5.5 K ≈ TN in Fig. 3(b)) and eventually forms mag- +netic Bragg peaks (T = 1.6 K << TN in Fig. 3(c)) indi- +cating long range magnetic order. To quantify the tempera- +ture dependence of the diffuse and Bragg scattering, as shown +in Fig. 3(e), we fitted the integrated intensity obtained from +one-dimensional (HHH) scans acquired through the magnetic +Bragg peak at Q = ( 1 +2 +1 +2 +1 +2). Each scan was fit to the sum of +a Gaussian function and a Lorentzian function to describe the +long and short range components of the spin correlations, and +a linear background (needed to describe the temperature in- +dependent nuclear and temperature-dependent magnetic inco- +FIG. 3. The elastic diffuse neutron scattering from HoBi measured +in the (HHL) reciprocal lattice plane at (a) 12 K, (b) 5.5 K, and (c) +1.7 K with an incident neutrons energy of 3.7 meV . The scattering +for panels (a,b,c) have been symmetrized to increase statistics. (d) +Calculated paramagnetic diffuse scattering with J1/J2 = -2.17 on an +fcc lattice where J1 is the first n.n. ferromagnetic interaction and J2 +is the 2nd n.n. antiferromagnetic interaction. Panel (e) is the elastic +neutron scattering near the Q = ( 1 +2 +1 +2 +1 +2) Bragg peak acquired through +scans along the the (HHH) direction. The data in panel (e) were fitted +using a Lorentzian function for the diffuse scattering and a Gaussian +function for the resolution limited Bragg component. The inferred +integrated intensity for each component of the scattering are plotted +in panel (f) as a function of temperature. The temperature depen- +dence of the magnetic correlation length is plotted in the inset panel +of (f). +herent elastic scattering). The fits included as dashed curves +in Fig. 3(e) provide a good account of the data. +The temperature dependence of the integrated intensity of +both the Bragg and the diffuse components of the scattering +are reported in Fig. 3(f). The integrated intensity of the diffuse +scattering (red markers) is peaked at TN where the appearance +of Bragg scattering (blue markers) reveals the onset of long +range order and translation symmetry breaking. The tempera- +ture variation of the correlation length ξ, as inferred from the +Lorentzian after correcting for resolution effects, is reported +in the inset of Fig. 3(f). As expected, ξ increases dramatically +at TN. + +do/dQ(b/sr/f.u. +(a) +(d) +a.u. +4 +0 1 +0 +4 +HoBi +Calc. +1.5 +1.5 +1 +1 +0.5 +0.5 +(T00) +(T00) +0 +0 +-0.5 +0.5 +-1 +-1.5 +-1.5 +12 K +-0.5 +0 +0.5 +-1 +-0.5 +0 +0.5 +(b) +(0HH) +(HHO) +(e +●150K +1.5 +·30K +1 +10 +15 K +10 K +0.5 +● 6.7 K +5.7 K +[00 +0 +-0.5 +.6 +-1 +-1.5 +5.5 K +-1 +-0.5 +0 +0.5 +1 +0.2 +0.4 +0.6 +0.8 +(c) +(HHO) +() +(HHH) +1.5 +1.5 +200 +6 +1 +wS 100 +(n'j/q)o +0.5 +D +4 +100) +0 +0 +20 + 40 +60. +T(K) +-0.5 +0.5 +2 +Elastic +-1 +Diffuse +-1.5 +1.7 K +0 +-1 +-0.5 +0 +0.5 +1 +10 +100 +(HHO) +T(K4 +C. +Structural distortion +A previous X-ray diffraction study revealed that a tetrago- +nal distortion accompanies magnetic ordering in HoBi35. We +confirmed the occurrence of this distortion in HoBi with a +four-circle X-ray diffractometer experiment. The θ-2θ scans +of various nuclear Bragg peaks were collected above and be- +low TN with a base temperature of 5 K. Consistent with previ- +ous work35, we observed a splitting of the (H00), (0K0), and +(00L) nuclear Bragg peaks whereas the (HHH) Bragg peaks +do not split. This indicates a tetragonal distortion and specifi- +cally precludes a rhombohedral distortion. +The temperature dependence of a longitudinal θ-2θ scan +through the Q = (006) peak is plotted in Fig. 4(a). This is +an unfiltered copper source with Kα1 and Kα2 radiation. Both +components yield a split (006) peak below TN. The distortion +was quantified by fitting the θ-2θ scans to Lorentzian func- +tions while constraining the ratio of the Kα1 / Kα2t integrated +intensity to be temperature independent and set by its fitted +value obtained at high temperatures. Examples of these fits +are included in Fig. 4(a). The temperature dependent lattice +parameters inferred from this analysis are shown in Fig. 4(b). +The order parameter-like temperature dependence is similar +for both warming and cooling with no hysteresis detected +down to the 100 mK temperature scale. For comparison the +hysteresis detected through heat capacity measurements was +13 mK (Fig. 2). A single (006) Bragg peak with a lattice pa- +rameter of 6.2095(1) Å above TN, splits into two peaks with +lattice parameters 6.2143(1) Å and 6.2075(1) Å below TN. +Assuming an approximately volume conserving phase transi- +tion implies that the lattice parameter that changes most is the +c-axis. This indicates the structural unit cell elongates along +the c-axis in the AFM state with c/a = 1.0011(1) at 5 K. We +note that an orthorhombic distortion with the a and b axis dif- +fering by less than 0.002 Å is not excluded by these data. +A possible space group for HoBi below TN is the maximal +tetragonal subgroup of the paramagnetic space group Fm3m, +which is I4/mmm. The structural parameters in the tetragonal +phase are aT = bT = 6.2075(1)/ +√ +2Å and cT = 6.2143(1) Å +where the aT and bT axes are rotated by 45° relative to the +a and b axes of the paramagnetic simple cubic cell. In this +space group Ho3+ ions occupy a single 2a Wyckoff site and +the magnetic ordering vector is k = ( 3 +20 3 +2). While we must +use the tetragonal space group below TN, we continue to use +the cubic unit cell to index wave vector transfer in the neu- +tron scattering experiments, which do not resolve the multi- +domain tetragonal distortion. +D. +Spin structure +As described in the previous sections, the magnetic order +has a characteristic wavevector k = ( 1 +2 +1 +2 +1 +2). In addition to the +corresponding low T magnetic Bragg peaks, the intensities of +all nuclear Bragg peaks are observed to increase below TN. +The increase of intensity is approximately proportional to the +intensity in the paramagnetic phase, which indicates it arises +from secondary extinction release47. To check this hypothesis, +FIG. 4. A series of θ-2θ X-ray diffraction scans through the Q = (006) +Bragg peak. The inferred temperature dependence of the lattice pa- +rameters is shown in panel (b). The neutron magnetic and nuclear +refinement of HoBi are presented in (c) where the observed cross- +sections for various Bragg peaks are plotted as a function of the cal- +culated cross-sections. The inset in (c) reports the variation of the +χ2 goodness of fit for the magnetic refinement of HoBi assuming a +multi-domain k = ( 1 +2 +1 +2 +1 +2) spin structure with an easy axis defined +by spherical coordinates θ and φ (φ = 0 corresponds to the [110] di- +rection). Panel (d) shows the low-temperature magnetization versus +field for fields applied parallel to the [001] and [110] directions. The +data show that [001] is the easy axis. +we performed polarized neutron diffraction on the (002) and +(220) Bragg peaks below TN and found them to be exclusively +nuclear in origin. +We note that weak k = (001) Bragg peaks also onset at TN. +Examples of these peaks include the (001) and (111) Bragg +peaks (see Fig. 3(c)), which are forbidden within the Fm3m +space group. These Bragg peaks are attributed to multiple +magnetic scattering as their presence depends on both the em- +ployed incident neutron wavelength and the scattering plane, +and they are absent in powder neutron diffraction measure- +ments36. The multiple scattering processes involve magnetic +k = ( 1 +2 +1 +2 +1 +2) Bragg reflections so they occur only for T < TN. +Referring to fcc close packing, the AFM k = ( 1 +2 +1 +2 +1 +2) spin +structure can be described as an AFM stacking of FM trian- +gular lattices. As the magnetic order and structural distortion +in HoBi occur in a single 1st order phase transition, the direc- +tion of the spins in each FM sheet is not constrained by the +usual Landau argument for second order phase transitions. To +determine the local spin orientation of the Ho3+ ions, we col- +lected 18 rocking scans at different magnetic Bragg positions +for a sample presumed to be in an unbiased multi-domain +state. The data were compared to a cubic domain average +of the calculated magnetic Bragg diffraction for a general spin +orientation within one domain given by spherical angles θ, φ +and k = ( 1 +2 +1 +2 +1 +2). Here θ = 0 corresponds to the tetragonal +c-direction and θ = π/2 and φ = 0 corresponds to the [110] +direction. Minimizing with respect to the moment size at each + +(a) +(b) +Kα +Warming + 6K +。 c (Tetragonal) +HoBi +6.214 +Cooling +5.8 K +600 +Q = (006) +5.6 K +(cts/s) +. Par. +6.212 +5 K +400 +Ka2 +Latt. +6.210 +a (Cubic) +200 +6.208 +0 +a (Tetragonal) +96 +96.4 +96.6 +5 +96.2 +5.5 +6 +6.5 +20 +T(K) +(c) +20 +(d) +12 +●H[001] +O H I [110] +CCCCCCCCCCCCCCCCC +15 +Magnetic +Oobs(b/f.u.) +Nuclear +M(μB/Ho) +8 +2 +X +Xmin +10 +180 +4 +90 +5 +45 +90 +0 +d +0 +0 +5 +10 +15 +20 +0 +2 +4 +6 +Ocalc(b/f.u.) +H(T)5 +point, the χ2 measure of fit quality is shown versus θ and φ +in the inset panel of Fig. 4(c). The manifold of states rep- +resented by the red arrows in Fig. 1 are indistinguishable by +neutron diffraction. This degeneracy arises because the mag- +netic diffraction intensity for a multi-domain sample only de- +pends on the smallest angle between the spin and a ⟨111⟩ axis. +From our refinement, we find this angle is 47(10)°. This is ex- +perimentally indistinguishable from the angle between [001] +and [111], which is 55°. This means the magnetic diffraction +data are consistent with spins pointing along the [001] direc- +tions, but also with many other directions including close to +the [110] direction. +Fortunately the spin anisotropy of the Ho3+ ions can be +deduced from other pieces of information. +First, the low- +temperature magnetization of HoBi shown in Fig. 4(d) reveals +the saturation magnetization is larger for fields along the [001] +direction than along [110]. Second, the structural distortion +also occurs along the [001] direction. Both of these measure- +ments are consistent with spins oriented along the tetragonal +cT-axis in the AFM ordered state. Additionally, in Sec. III F +we show that a [001] easy axis anisotropy is needed to ac- +curately model the inelastic neutron scattering spectrum be- +low TN. +We thus conclude the spins in the AFM type II +order of HoBi are oriented along the cT direction, which is +the direction of the structural elongation. +The comparison +between measured and calculated magnetic Bragg intensities +is shown in Fig. 4(c). The corresponding spin structure is +shown in Fig. 1. An ordered moment of 10.3(6) µB was de- +termined, which is experimentally indistinguishable from the +gJµB = 5 +4 · 8 µB = 10 µB saturation magnetization of Ho3+. +E. +Crystal electrical field interaction +For Ho3+ ions, the J = 8 spin-orbit ground state manifold +is (2J+1) = 17 fold degenerate under full rotation symmetry. +This degeneracy is, however, lifted by the symmetry break- +ing crystal electric fields (CEF). Using the Stevens operator +formalism, the CEF Hamiltonian appropriate for Ho3+ in the +high-temperature cubic phase of HoBi can be expressed as +follows: +ˆHcubic +ce f += B4( ˆO0 +4 + 5 ˆO4 +4) + B6( ˆO0 +6 − 21 ˆO4 +6). +(1) +Here ˆOm +n are Stevens operators48 that can be written in terms +of the spin-orbital angular momentum operators ˆJ+, ˆJ− and +ˆJz where ˆz ∥ c. The CEF parameters Bn are scalars of dimen- +sion energy that dictate the strength of the different CEF terms +and can be determined by fitting spectroscopic or thermo- +magnetic data sensitive to the crystal field level scheme. Bn +can also be estimated through the point-charge model49. +Following Hutching’s formalism49 the point charge model +yields +B4 = 7|e||qBi|βJ⟨r4⟩ +64πϵ0d5 +Bi +(2) +and +B6 = 3|e||qBi|γJ⟨r6⟩ +256πϵ0d7 +Bi +. +(3) +Here e is the electron charge, qBi is the charge of the Bi ligand +and ϵ0 is the vacuum permitivity. βJ and γJ are reduced matrix +elements calculated in ref48 whereas the radial integrals for the +4f state ⟨rn⟩ are tabulated in ref50. We used qBi = +− 3e and +the distance between a holmium ion and its first n.n. bismuth +ion dBi = a/2 = 6.2093(1)/2 Å. Introducing these values in +Eqs. 2 and 3 we obtain B4 = −2.2709(2) × 10−4 meV and +B6 = −1.0468(1) × 10−7 meV. +FIG. 5. Determination of the crystal electric field (CEF) level scheme +for the J=8 Ho3+ ion in HoBi. +(a) shows the results of a point +charge (PC) calculation for the cubic and tetragonal phases. The cu- +bic CEF scheme may be compared to the level scheme for the fitted +CEF Hamiltonian of HoBi. Panel (b) and (c) respectively show the +temperature dependence of the magnetic heat capacity (Cp) and the +inverse magnetic susceptibility of HoBi compared to corresponding +properties based on the fitted CEF Hamiltonian. The magnetic en- +tropy obtained from integrating the Cp of HoBi is shown in the inset +of (b). The measured (d) and calculated (e) inelastic neutron scatter- +ing spectra of HoBi are shown for T = 12 K. The neutron inelastic +scattering data were acquired using a 25 meV incident neutron beam. +The corresponding CEF level scheme for Ho3+ in the cubic +phase of HoBi is shown in Fig. 5(a). The Ho3+ J−multiplet +is split into 4 triplets, 2 doublets, and 1 singlet that form three +groups. Group I includes one doublet, one triplet, and one +singlet between 0 and 0.2 meV. Group II is formed by two + +(a)HoBi +P.C. cubic +Fit cubic +P.C. Tetragonal +10 +888888888888888888 T +D +S +8 +D +D +6 +4 +E +2 +S +D +S +0 +D +D +S +(b) +(c) +60 +(J/mol/K) +25 +R ln(17) +20 +/emu) +R ln(6) +20 +15 +H=10 0e +(J/mol/K) +40 +10 + (mol Oe/ +H II [001] +mag +S +0 +10 +100 +10 +20 +T(K) +%/ 1 +CEF fit +CEF +0 +0 +10 +100 +0 +100 +200 +300 +T(K) +(e) +T(K) +(d) +1 +12 K +Data +12 K +Calc. +12 +Ei=25 meV +12 +hw (meV) +I (a.u.) +8 +8 +4 +0 +0 +0 +1 +3 +4 +2 +3 +4 +IQ(A) +IQ(A)6 +triplets between 6 meV and 7 meV, and group III consists of a +doublet and a triplet between 9 meV and 10 meV. +The CEF Hamiltonian estimated from our point-charge cal- +culation can reproduce the temperature dependence of the +magnetic heat capacity Cp (Fig. 5(b)) and magnetic suscepti- +bility χ (Fig. 5(c)). Obtained by integrating Cp/T, the temper- +ature dependence of the entropy shown in the inset of Fig. 5(b) +is informative. A first entropy plateau near 10 K is associ- +ated with the sharp Cp anomaly at the phase transition to long +range magnetic order. The corresponding change in entropy +of ∆S = R ln 6 is that associated with the group I CEF states. +The second plateau at S = R ln 17 is reached at room temper- +ature and encompasses all of the entropy associated with the +three groups of crystal field levels. +For a more stringent test of the point charge model, we turn +to inelastic neutron scattering. Fig. 5(d) shows the 12 K in- +elastic neutron scattering spectrum with energy transfer rang- +ing from 0 to 15 meV. At this temperature, the group II and +III of CEF states are so scarcely populated that only CEF +excitations originating from group I should be visible. No +significant intrinsic broadening of the CEF excitations is ob- +served and we note, also, that the experimental resolution is +too coarse to resolve CEF levels within a group. The mag- +netic neutron scattering cross section associated with CEF +transition from group I to II and from group I to III can +be computed based on the point charge CEF Hamiltonian +(Imn ∝ +� +i |⟨m|Ji|n⟩|2). This calculation predicts the cross +section for transitions from group I to group II is 250 times +stronger than for transitions from group I to group III. The in- +tensity of the transition from I to III is thus predicted to be too +weak to be detected. This explains why Fig. 5(d) shows just a +single peak that we associate with transitions from group I to +group II crystal field levels. +While the measured 7.2 meV gap between group I and +group II CEF levels is just 0.4 meV off from the point charge +prediction of 6.8 meV, we can improve our estimate of the +CEF Hamiltonian by simultaneously fitting B4 and B6 for +the best possible account of the neutron scattering spectra +(Fig. 5(d)), the specific heat data (Fig. 5(b)), and the mag- +netic susceptibility data (Fig. 5(c)). +The best fit parame- +ters thus obtained are B4 += +− 2.24(1) × 10−4 meV and +B6 = − 2.4(1) × 10−7 meV and with them the CEF Hamilto- +nian provides an excellent account of all single ion properties +that we’ve measured, as shown in Fig. 5. +The CEF scheme obtained from our fit (Fig. 5(a)) is remark- +ably similar to the point-charge calculation. Also a re-scaling +of our CEF Hamiltonian for HoBi using Eq. 2 and Eq. 3 con- +sidering only the different ligand spacing successfully predicts +the level scheme for HoN ref51. This is in contrast with the +praseodymium case where a pnictide ligand charge of q = −2e +is needed to bring the point charge model into agreement with +experimental data5. This indicates that holmium monopnic- +tides are more ionic than praseodymium monopnictides. +Finally, we estimated the effect of the tetragonal distortion +on the CEF interaction in HoBi. We performed a point-charge +calculation assuming that the first n.n. Ho-Bi bond is shorter +along the a and b direction (da) as compared to the c direction +(dc). The calculated CEF Hamiltonian can be written as: +Htet +ce f = |e||qBi| +4πϵ0 +[αJ⟨r2⟩( 1 +d3c +− 1 +d3a +) ˆO0 +2+ +(4) +βJ⟨r4⟩(( 1 +4d5c ++ +3 +16d5a +) ˆO0 +4 + +35 +16d5a +ˆO4 +4)+ +γJ⟨r6⟩(( 1 +8d7c +− +5 +64d7a +) ˆO0 +6 − +63 +64d7a +ˆO4 +6)]. +The corresponding level scheme is shown in Fig. 5(a). For +this calculation, we used the lattice parameters determined +in our high-resolution X-ray scattering experiment. The de- +generacy of all the triplets and doublets associated with cubic +symmetry is lifted. This results in four doublets and nine sin- +glets and a significant broadening of each of the three groups +of crystal field levels. +F. +Low energy spin dynamics +We now turn our attention to the collective physics of HoBi, +which we explore using inelastic magnetic neutron scatter- +ing. Fig. 6(a) shows the temperature dependence of the in- +elastic scattering for Q = ( 1 +2 +1 +2 +1 +2). Just above TN, the scat- +tering is quasi-elastic with a physical (resolution corrected) +FWHM of 0.30(5) meV. No inelastic intensity is observed up +to 2 meV. This is consistent with the CEF energy scheme +shown in Fig. 5(c). Below TN, the quasi-elastic scattering +splits into an elastic and an inelastic component. +To probe any dispersion of the low energy spin excitations, +we acquired low energy spectra at momentum transfer Q cor- +responding to high symmetry points in the Brillouin zone. +Fig. 6(b) shows the spectrum consists of a peak that is broader +than the experimental resolution (FWHM indicated by hor- +izontal bar) and that shifts by less than the peak width be- +tween the different values of Q. A gaussian fit finds the peak +centered at 1.7(2) meV with a FWHM of 0.48(4) meV that +exceeds the instrumental resolution (FWHM of 0.22 meV). +The limited resolution and statistical accuracy of the data does +not rule out the possibility of multiple dispersive components +within the approximately Gaussian envelope of the peak. +We also examined the higher energy excitations for T < TN +by acquiring momentum resolved inelastic scattering data up +to 11.5 meV. A representative slice through the data is dis- +played as a color image versus Q along the (HH0) direction +and energy transfer in Fig. 6(c). No dispersion is resolved. +The data are similar to the high-temperature plot of intensity +versus |Q| and ℏω in Fig. 5(d) though with additional inelastic +features at 9.0(3) meV and 1.7(2) meV. +Fig. 6(e) shows the momentum dependence of the inte- +grated intensity of the 1.7 meV mode throughout the (HHL) +zone. +The Q dependence of the intensity is subtle albeit +peaked at the magnetic ( 1 +2 +1 +2 +1 +2) zone center and smoothly de- +creases with |Q| in accordance with the Ho3+ magnetic form +factor41. We note that the 1.7 meV gap is about an order of +magnitude greater than the predicted CEF gap arising from +the tetragonal distortion. This indicates the phase transition is +driven by the magnetic interactions, which we model below. + +7 +FIG. 6. The temperature dependence of the low energy inelastic neu- +tron spectrum of HoBi at Q = ( 1 +2 +1 +2 +1 +2) is shown in panel (a). The spec- +trum of neutron scattering at some high symmetry positions within +the first Brillouin zone of HoBi are shown in (b).The horizontal +black bar indicates the FWHM energy resolution of the spectrom- +eter while the black dashed lines show the predicted spectrum based +on the spin Hamiltonian presented in this work. The energies asso- +ciated with each exciton are indicated by vertical black dashed lines. +The observed and calculated inelastic neutron scattering spectrum +up to 11.5 meV are respectively plotted in (c) and (d) for momentum +transfer Q along the [HH0] direction. The observed and calculated +momentum dependence of the 1.75 meV exciton scattering inten- +sity is shown in (e) and (f). The energy integration for panel (e) is +±0.25 meV. +IV. +MODELING SPIN DYNAMICS OF SPIN-ORBITAL +EXCITONS +The low-temperature excitations in HoBi are similar to +other rare-earth metallic compounds where exchange interac- +tions are strong enough to mix crystal field levels4,52,53. Be- +cause components that are longitudinal with respect to the or- +dered moment are involved, these are not conventional trans- +verse spin wave excitations. They may be described as crystal +field excitations that can propagate through the lattice due to +inter-site interactions. We shall adopt the practice of calling +these “crystal field exciton” or simply “exciton”54–56. +A common theoretical approach to describing excitons in +rare-earth magnets is to use a pseudo-boson theory where the +exciton creation operator is a linear combination of single-ion +operators53,57,58. In this theory, the Q = 0 single-ion opera- +tors are obtained by diagonalizing the mean-field spin Hamil- +tonian and the dispersion at finite Q is produced by the ex- +change terms. We use this pseudo-boson theory to describe +the magnetic excitation spectrum of HoBi below TN. +The +Hamiltonian Hs includes the single-ion tetragonal crystal field +terms and isotropic exchange interactions. Hs is decomposed +into a mean-field term (H0,k) and an interacting part (Hint) so +Hs = � +k H0,k + Hint where: +H0,k = Htet +ce f,k + (−1)kHzJk +jz +(5) +and +Hint = +� +j, j′,k,k′ +Jk,k′ +j, j′ Jk +j · Jk′ +j′ − +� +j,k +(−1)kHzJk +jz. +(6) +Here j indexes the unit cell while k = 1, 2 specifies the +anti-parallel sub-lattices of the AFM order (Fig. 1). We define +Hz = 2 � +r ZrJr⟨Jz⟩ where Jr and Zr are respectively the ex- +change constant and coordination number associated with the +rth neighbor. ⟨Jz⟩ is the thermal average of Jz on each site, +which we found to be ⟨Jz⟩ = 8 in our diffraction and CEF +analysis. By definition, Hint carries no mean value and so can +be written in terms of creation (ˆa† +n,k = |n, k⟩⟨0, k|) and annihila- +tion (ˆan,k = |0, k⟩⟨n, k|) operators that connect the ground state +|0, k⟩ and the excited eigenstates |n, k⟩ of ˆH0,k. In this case, +ˆH0,k = � +n Enˆa† +n,kˆan,k where En,k is the eigenvalue of the |n, k⟩ +eigenstate of ˆHo,k. After writing ˆHs in terms of these operators +and Fourier transforming it, we obtain: +ˆHs = 1 +2 +� +Q +� +ˆa†(Q)A(Q)ˆa(Q) + ˆa†(−Q)A(−Q)ˆa(−Q) +(7) ++ˆa†(Q)B(Q)ˆa†(−Q) + ˆa(−Q)B(Q)ˆa(−Q) +� +with +ˆA = +ˆ∆ + 2ˆhzz + ˆh+− + ˆh−+ and +ˆB = 2ˆhzz ++ +ˆh++ ++ +ˆh−− +where +ˆ∆ += +En,kδk,k′δn,n′ +and +ˆhαβ(k, k′, n, n′, Q) = J(Q)⟨k, n| ˆJα|0, k⟩⟨k′, 0| ˆJβ|n′, k′⟩. +The procedure to compute the spin dynamics first consist of +diagonalizing ˆH0,k to obtain the eigenvalues En,k and eigenvec- +tors |n, k⟩ for Q = 0. At finite Q, the matrix ˆHs = +� ˆA +ˆB +− ˆB − ˆA +� +is +then computed and diagonalized to obtain the perturbed ener- +gies (E˜n(Q)) and eigenstates |˜n(Q)⟩ for each exciton. We con- +sider all the excited CEF states belonging to the (2J+1) spin- +orbit manifold of HoBi so there are 32 creation and annihla- +tion operators for each of the 2 Ho3+ spins within the magnetic +unit cell. This give a Hilbert space of 64 states for ˆHs. The +associated inelastic magnetic neutron scattering cross-section +for a single magnetic domain is then53,57: +d2σ +dEdΩ = N(γr0)2 k f +ki +|g +2 f(Q)|2 +(8) +× +� +˜n,q,τm +|⟨˜n(q)| ˆJQ|GS ⟩|2δ(E − E˜n(q))∆(Q − q − τm) +Here N is the number of primitive magnetic unit cells, γ = - +1.91 is the gyromagnetic ratio of the neutron, r0 = 2.818 × + +(a) +I (a.u.) +0 +3 +HoBi +AE +(Aaw) m +100 +7 +(a.u.) +50 +0 +0 +2 +4 +6 +8 +10 +12 +1.5 +2 +2.5 +T(K) +hw (meV) +(c) +(d) +4 +Data +Calc. +10 +10 +(meV) +8 +8 +(a.u.) +6 +6 +hw +4 +2 +2 +1.7 K +0 +0 +0 +2 +3 +0 +2 +3 +0 +[HH0] +[HHO] +(f) +(e) +5 +1.75 meV +1.75 meV +Data +Calc. +2 +2 +1.7 K +[00L] +L +1001 +(a.u.) +I +L +UX +U +X +0 +0 +0 +0.5 +1 +1.5 +0 +0.5 +1 +1.5 +2 +[HH0] +[HH0]8 +10−15 m is the classical electron radius, τm is the magnetic +zone center, q is the reduced momentum transfer within the +first magnetic Brillouin zone, while k f and ki respectively are +the scattered and incoming neutron wave vector. The mea- +sured spectrum is subject to the finite resolution of the instru- +ment which we account for by replacing the delta functions by +a united normalized Gaussian functions with the Q-integrated +energy resolution width. The final calculated spectrum was +averaged over all possible magnetic domains. +V. +MICROSCOPIC SPIN HAMILTONIAN FOR HOLMIUM +BISMUTH +We determined the microscopic parameters of ˆHs for HoBi +by fitting the Q = 0 spectrum consisting of three excitons +at E1 = 1.7(2) meV, E2 = 7.4(2) meV and E3 = 9.0(3) meV +with relative intensities I2/I1 = 5.5(3) and I2/I3 = 37(7). Em- +ploying the ratio ∥J2/J1∥ = 2.17 obtained by analyzing the +magnetic diffuse scattering (section III B) leaves just one free +parameter. The tetragonal CEF Hamiltonian has six free pa- +rameters that were initially estimated from the point-charge +model. To reproduce the exact energies of the excitons at E2 +and E3, we allowed the CEF parameters to relax away from +their point-charge values which results in many combinations +of parameters consistent with the data. We estimated the ex- +change constants by varying the CEF parameters away from +their point-charge calculation values and keeping all solutions +that have a χ2 within 20% (1/Nobs) of the global minimum. +The exchange parameters refined to J1 = − 1.4(2) µeV and +J2 += +3.0(5) µeV. +A mean-field critical temperature of +20(7) K is obtained from these parameters. For comparison, +the actual ordering temperature is only TN += +5.72(1) K. +We hypothesize that fluctuations arising from competition be- +tween the ferromagnetic J1 and the antiferromagnetic J2 in- +teractions lead to the reduced critical temperature. +The right column of Fig. 6 compares the optimized model +for a multi-domain sample to the experimental data. Fig. 6(d) +shows the full intensity versus ℏω and Q ∥ (HH0) for com- +parison with Fig. 6(c). The position and relative intensity of +the three modes are well reproduced. Looking more closely +at the 1.75 meV mode, Fig. 6(b) compares the intensity ver- +sus energy transfer at select high symmetry points in the Bril- +louin zone. The vertical dashed lines show that multiple ex- +citons contribute at each Q. This is generally consistent with +the featured spectrum observed though there is more broad- +ening/dispersion observed than reproduced by the model. In- +clusion of anisotropic or longer range interactions might be +needed to remedy this discrepancy though data with higher +energy resolution is needed to justify the greater model com- +plexity. Fig. 6(f) shows the calculated Q-dependent integrated +intensity of the 1.75 meV mode. The dominant features of +the experimental result in Fig. 6(e) are reproduced, includ- +ing mainly the increase of scattered intensity at the magnetic +zone centers. We note the presence of phonon scattering near +Q += +(002) that may account for the discrepancy between +the calculation and the experimental data at that momentum +point. +VI. +DISCUSSION AND CONCLUSION +In this manuscript, we have characterized an antiferro- +magnetic order and the associated crystal field excitons +that develop below TN += +5.72(1) K in the rare-earth +monopnictide HoBi. This magnetic state is driven by strong +2nd n.n. antiferromagnetic and weaker 1st n.n. ferromag- +netic interactions, which we quantified via modeling of the +diffuse paramagnetic and low temperature inelastic neutron +scattering. The excitation spectrum is sensitive to the local +orientation of the Ho3+ ordered spins, which allowed us +to establish the Ising nature of the antiferromagnetic order +in HoBi that cannot be deduced from neutron diffraction +of a multi-domain sample. +We used X-ray diffraction to +provide evidence for a tetragonal structural distortion that +accompanies magnetic ordering. +Our CEF analysis and +modelling of inelastic scattering data indicates the elongated +c-axis coincides with the easy magnetic axis within a domain. +The magnetic excitations that we have documented here +surely have significant impacts on the magneto-transport +properties of HoBi34. For example, we found strong quasi- +elastic neutron scattering in the paramagnetic state. +The +associated short range correlated spin fluctuations, which +may be accompanied by short range tetragonal lattice dis- +tortions too given the non-Kramers nature of the Ho3+, are +expected to enhance the electrical resistivity above TN. Below +TN, these gapless fluctuations are replaced by a coherent +exciton at 1.7(2) meV and correspondingly the electrical +resistivity is reduced by an order of magnitude upon cooling +below TN34. The field-dependence of spin-orbital excitons +may be responsible for various features observed in the +magnetoresistance of HoBi and more broadly in the rare-earth +monopnictides23–29. +VII. +ACKNOWLEDGEMENTS +This work was supported as part of the Institute for Quan- +tum Matter, an Energy Frontier Research Center funded by the +U.S. Department of Energy, Office of Science, Basic Energy +Sciences Under Award No.DE-SC0019331. CB was further +supported by the Gordon and Betty Moore foundation EPIQS +program under GBMF9456. The work at Boston College was +supported by the U.S. Department of Energy, Office of Basic +Energy Sciences, Division of Physical Behavior of Materials +under Award DE-SC0023124. This work was supported in +part by the Natural Sciences and Engineering Research Coun- +cil of Canada (NSERC). We acknowledge the support of the +National Institute of Standards and Technology, U.S. Depart- +ment of Commerce. Access to MACS was provided by the +Center for High Resolution Neutron Scattering, a partnership +between the National Institute of Standards and Technology +and the National Science Foundation under Agreement No. +DMR-1508249. The identification of any commercial prod- +uct or trade name does not imply endorsement or recommen- +dation by the National Institute of Standards and Technology. + +9 +A portion of this research used resources at the High Flux Iso- +tope Reactor, a DOE Office of Science User Facility operated +by the Oak Ridge National Laboratory. +∗ Correspondence email address: Jonathan.Gaudet@nist.gov +1 C.-G. Duan, R. F. Sabirianov, W. N. Mei, P. A. Dowben, S. S. +Jaswal, and E.Y. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Oak Ridge National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Oak Ridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' TN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 37831,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' USA 7Neutron Scattering Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Oak Ridge National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Oak Ridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Tennessee 37831,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' USA 8Canadian Institute for Advanced Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 661 University Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Ontario M5G 1M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 9Brockhouse Institute for Materials Research, Hamilton, ON L8S 4M1 Canada (Dated: January 13, 2023) The magnetism of the rock-salt fcc rare-earth monopnictide HoBi, a candidate topological material with extreme magnetoresistance, is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' From the Ho3+ non-Kramers J=8 spin-orbital multiplet, the cubic crystal electric field yields six nearly degenerate low-energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' These constitute an anisotropic magnetic moment with a Jahn-Teller-like coupling to the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' In the cubic phase for T > TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='72(1) K, the paramagnetic neutron scattering is centered at k = ( 1 2 1 2 1 2) and was fit to dominant antiferromagnetic interactions between Ho spins separated by {100} and ferromagnetic interactions between spins displaced by { 1 2 1 20}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' For T < TN, a type-II AFM long-range order with k = ( 1 2 1 2 1 2) develops along with a tetragonal lattice distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' While neutron diffraction from a multi-domain sample cannot unambiguously determine the spin orientation within a domain, the bulk magnetization, structural distortion, and our measurements of the magnetic excitations all show the easy axis coincides with the tetragonal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The weakly dispersive excitons for T < TN can be accounted for by a spin Hamiltonian that includes the crystal electric field and exchange interactions within the Random Phase Approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' INTRODUCTION In spite of their structural simplicity, the fcc rare-earth monopnictides (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 1), RX (R=Ce to Yb and X=N, As, P, Sb, and Bi1,2), display a wide variety of anisotropic mag- netism and electronic transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The lattice param- eter varies by 30% across the pnictide series and this provides opportunities to tune the relative strength of crystal field and exchange interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' In the 1960s to 1980s, the rare-earth monopnictides were studied to understand magnetic phases driven by oscillatory and highly anisotropic Ruderman-Kittel- Kasuya-Yosida (RKKY) exchange interactions 3–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Work on CeSb for example gave rise to an extensive literature on the anisotropic nearest and next nearest neighbor Ising model (ANNNI)9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This work also resulted in progress towards a quantitative understanding of their anisotropic exchange interactions10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' A recent resurgence of interest in these rare-earth monop- nictides is driven by their extreme magnetoresistance (XMR) and resistivity plateaus, and the possible connection to the 3D topological state of the non-magnetic lanthanum monop- nictides LaX11,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' LaAs, LaSb, and LaBi have unsaturated XMR arising from near perfect electron-hole compensation and there is a topological transition from a trivial electronic band structure in LaAs to a topologically non-trivial band structure in LaBi13–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Several studies have confirmed the presence of protected surface states in LaBi18–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Since then, extensive works have been devoted to characterizing the XMR and topological states of various RX including for example CeX, HoX, and PrX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' XMR has been found in each reported magnetic RX with characteristics that depend on the rare-earth ion22–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The stabilization of topological non-trivial electronic bands generating protected surface states was proposed for several of the magnetic monopnictides29,31–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The rock-salt structure of the rare-earth monopnictide HoBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Yellow and blue spheres respectively correspond to Ho and Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Spins interacting through the J1 and J2 exchange interaction are shown by the dashed black arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The k = ( 1 2 1 2 1 2) magnetic order of the Ho3+ spins is represented by the red arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The local spin orientations of the Ho3+ spins that are consistent with neutron diffraction are indi- cated for the Ho ion at (0,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The magnetization, structural distor- tion, and inelastic neutron scattering however, provide clear evidence for easy [001] axis anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='05141v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='str-el] 12 Jan 2023 2 Here we study the magnetism of HoBi using modern neu- tron scattering techniques to gain insights into its unique magneto-transport properties29,34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Consistent with previ- ous works35–37, we confirm the antiferromagnetic (AFM) k = ( 1 2 1 2 1 2) structure and the associated tetragonal lattice distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Due to multi-domain averaging, our single-crystal neutron diffraction cannot unambiguously determine the local spin anisotropy of the k = ( 1 2 1 2 1 2) AFM structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' How- ever, we could resolve this ambiguity by measuring and mod- eling the magnetic excitations of HoBi, which take the form of weakly propagating spin-orbital excitons whose energies and intensities are sensitive to the local orientation of the Ho3+ moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Using this method, we found the k = ( 1 2 1 2 1 2) AFM structure has an Ising local spin anisotropy, which is consistent with the Ising easy-axis bulk magnetization and the tetragonal distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Through analysis of the paramagnetic diffuse scattering of HoBi and the crystal field excitons in the low T ordered state, we obtain a spin Hamiltonian with com- parable crystal field (CEF) and exchange energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' EXPERIMENTAL METHODS HoBi single crystals with mass of 10-50 mg were grown following a previously published procedure34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Single crys- tal low-temperature X-ray diffraction was performed using a Huber four-circle diffractometer with a Rigaku Rotaflex 18 kW rotating copper anode X-ray generator and a Bicron point detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We used a Ge (111) monochromator with d111 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='266 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The sample was aligned for diffraction in the (HHL) plane and mounted in a closed cycle cryogenic system with a base temperature of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='17 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We performed thermal neutron diffraction using the HB-1A triple-axis instrument at Oak Ridge National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We used PG filtered 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 meV neutrons, and collected rocking scans at all accessible magnetic and nuclear Bragg positions in the (HHL) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Polarized neutron diffraction measure- ments were conducted with the triple-axis instrument BT-7 at the Center for Neutron Research (NCNR), NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Nuclear spin-polarized 3He gas was used to polarize the incident neu- tron beam and to analyze the polarization of scattered neu- trons38,39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Horizontal guide fields were present throughout the beam path to allow measurements of the spin-flip (SF) and non-spin-flip (NSF) scattering cross-sections for incident neu- tron spins polarized parallel to momentum transfer Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The flipping ratio measured at nuclear Bragg peaks was greater than 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Cold neutron triple-axis experiments were performed using the SPINS and the MACS spectrometers at the NCNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' On both instruments we employed a fixed final neutron energy E f = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 meV or 5 meV and measured the elastic and inelas- tic scattering for a single crystal of HoBi aligned for scattering within the (HHL) and the (HK0) plane in two different exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' For the E f = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 meV configuration, we used poly- crystalline cooled Be and BeO filters before and after the sam- ple, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' For the 5 meV configuration we only used a Be filter after the sample while the incident beam from the cold neutron source was unfiltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' For both experiments, we co-mounted 11 HoBi single crystals on an aluminum mount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We acquired background data using an identical mount with- out HoBi crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We used an ”orange” 4He flow cryostat to reach a base temperature of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='6 K for these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' For the highest energy resolution and energy transfer, we performed time-of-flight neutron scattering experiments us- ing the CNCS spectrometer at Oak Ridge National Labora- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' There we co-aligned two HoBi single crystals on an alu- minum mount and collected inelastic neutron scattering data with fixed incident energy Ei = 25 meV at T = 13 K with a total proton charge of 47 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We used the high flux mode of operation of CNCS with a Fermi Chopper, Chopper 2, Chop- per 3, and a Double Disk frequency of 60, 60, 60, 300, and 300 Hz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The energy resolution (FWHM) at the elastic line for this configuration is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='0(1) meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Finally, we note that the error bars associated with the neutron scattering experiments represent one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Both the magnetization and heat capacity measurements presented here were performed in a Quantum Design physical properties measurement system (PPMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We used a PPMS di- lution refrigerator option for the low-temperature heat capac- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' RESULTS AND ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 1st order phase transition FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Low temperature heat capacity of HoBi collected using the long-pulse method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The red and blue curves respectively correspond to the warming and cooling protocol and shows a thermal hysteresis of 13(2) mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The observation of a plateau at TN in the heating profile for both warming and cooling protocol (top inset panel) suggests a 1st order phase transition in HoBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The thermodynamic properties of HoBi were previously reported and a long-range k = ( 1 2 1 2 1 2) antiferromagnetic (AFM) order is known to occur concomitantly with a struc- tural distortion around TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 K14,35,36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The order of the transition, however, remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' To determine the AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' = 13(2) mK HoBi 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='6 C K 1000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='8 Cp (J/mol K) 6 200 400 0 Time (s) Warming 500 Cooling 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='9 T(K)3 order of the phase transition, we measured the temperature dependent specific heat capacity using the long-pulse heat method40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The resulting Cp data for HoBi is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 2 for both warming and cooling protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' A sharp peak with a thermal hysteresis of 13(2) mK is observed in Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Correspondingly the inset shows a distinct plateau in the temperature versus time curves during heating and cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' These observations indicate a 1st order phase transition at TN in HoBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Paramagnetic phase To determine the magnetic interactions leading to this phase transition, we mapped the neutron elastic scattering for mo- mentum transfer Q covering the (HHL) plane and for temper- atures between 150 K and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Representative data sets are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' In the cubic paramagnetic phase for T = 12 K > TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='72(1) K, the scattering is broad in Q and is centered at k = ( 1 2 1 2 1 2) positions ((Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This indicates short-range AFM correlations preceding the long-range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The ”butterfly” pattern of paramagnetic diffuse scattering is consistent with the equal time structure factor S(Q) of an fcc Heisenberg paramagnet with FM inter- actions between the first nearest-neighbor (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=') Ho3+ ions (J1), and AFM interactions between the 2nd n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' (J2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 1 indicate the lattice geometry associated with these interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The scattered intensity was modeled using I(Q) = 2 3N| f(Q)|2 � ij⟨Si ·Sj⟩ cos(Q·rij) where N is the num- ber of spins, ri j is the displacement vector from Ho3+ site j to i, and f(Q) is the Ho3+ atomic form factor41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Including only self-correlations and correlations between spins separated by {100} and { 1 2 1 20}, a ratio of ⟨Si ·Sj⟩{100}/⟨Si ·S j⟩{ 1 2 1 2 0} = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2(2) was obtained at T = 12 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The calculated magnetic diffuse scattering corresponding to the best fit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3(d) accounts for all major features in the data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3(a)) and the introduction of third n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' correlations does not improve the fit significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' A high temperature expansion allows us to associate the ratio of correlations to the ratio of the corre- sponding exchange interactions42,43 so that we may infer that J2/J1 ≈ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Even if some of the J1 bond interactions are frustrated, this resulting fitted ratio of exchange parameters stabilize a k = ( 1 2 1 2 1 2) order, which is driven by the dominant AFM J2 interactions44–46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Upon cooling, the elastic magnetic scattering gets stronger (T = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 K ≈ TN in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3(b)) and eventually forms mag- netic Bragg peaks (T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='6 K << TN in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3(c)) indi- cating long range magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' To quantify the tempera- ture dependence of the diffuse and Bragg scattering, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3(e), we fitted the integrated intensity obtained from one-dimensional (HHH) scans acquired through the magnetic Bragg peak at Q = ( 1 2 1 2 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Each scan was fit to the sum of a Gaussian function and a Lorentzian function to describe the long and short range components of the spin correlations, and a linear background (needed to describe the temperature in- dependent nuclear and temperature-dependent magnetic inco- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The elastic diffuse neutron scattering from HoBi measured in the (HHL) reciprocal lattice plane at (a) 12 K, (b) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 K, and (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 K with an incident neutrons energy of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 meV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The scattering for panels (a,b,c) have been symmetrized to increase statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' (d) Calculated paramagnetic diffuse scattering with J1/J2 = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='17 on an fcc lattice where J1 is the first n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' ferromagnetic interaction and J2 is the 2nd n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' antiferromagnetic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Panel (e) is the elastic neutron scattering near the Q = ( 1 2 1 2 1 2) Bragg peak acquired through scans along the the (HHH) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The data in panel (e) were fitted using a Lorentzian function for the diffuse scattering and a Gaussian function for the resolution limited Bragg component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The inferred integrated intensity for each component of the scattering are plotted in panel (f) as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The temperature depen- dence of the magnetic correlation length is plotted in the inset panel of (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' herent elastic scattering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The fits included as dashed curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3(e) provide a good account of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The temperature dependence of the integrated intensity of both the Bragg and the diffuse components of the scattering are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The integrated intensity of the diffuse scattering (red markers) is peaked at TN where the appearance of Bragg scattering (blue markers) reveals the onset of long range order and translation symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The tempera- ture variation of the correlation length ξ, as inferred from the Lorentzian after correcting for resolution effects, is reported in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' As expected, ξ increases dramatically at TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' do/dQ(b/sr/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' (a) (d) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 4 0 1 0 4 HoBi Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 (T00) (T00) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 12 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 (b) (0HH) (HHO) (e 150K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 30K 1 10 15 K 10 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 K [00 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='6 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 K 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='8 (c) (HHO) () (HHH) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content="5 200 6 1 wS 100 (n'j/q)o 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 D 4 100) 0 0 20 40 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' T(K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 2 Elastic 1 Diffuse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 K 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 1 10 100 (HHO) T(K4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Structural distortion A previous X-ray diffraction study revealed that a tetrago- nal distortion accompanies magnetic ordering in HoBi35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We confirmed the occurrence of this distortion in HoBi with a four-circle X-ray diffractometer experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The θ-2θ scans of various nuclear Bragg peaks were collected above and be- low TN with a base temperature of 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Consistent with previ- ous work35, we observed a splitting of the (H00), (0K0), and (00L) nuclear Bragg peaks whereas the (HHH) Bragg peaks do not split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This indicates a tetragonal distortion and specifi- cally precludes a rhombohedral distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The temperature dependence of a longitudinal θ-2θ scan through the Q = (006) peak is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This is an unfiltered copper source with Kα1 and Kα2 radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Both components yield a split (006) peak below TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The distortion was quantified by fitting the θ-2θ scans to Lorentzian func- tions while constraining the ratio of the Kα1 / Kα2t integrated intensity to be temperature independent and set by its fitted value obtained at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Examples of these fits are included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The temperature dependent lattice parameters inferred from this analysis are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The order parameter-like temperature dependence is similar for both warming and cooling with no hysteresis detected down to the 100 mK temperature scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' For comparison the hysteresis detected through heat capacity measurements was 13 mK (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' A single (006) Bragg peak with a lattice pa- rameter of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2095(1) Å above TN, splits into two peaks with lattice parameters 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2143(1) Å and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2075(1) Å below TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Assuming an approximately volume conserving phase transi- tion implies that the lattice parameter that changes most is the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This indicates the structural unit cell elongates along the c-axis in the AFM state with c/a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='0011(1) at 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We note that an orthorhombic distortion with the a and b axis dif- fering by less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='002 Å is not excluded by these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' A possible space group for HoBi below TN is the maximal tetragonal subgroup of the paramagnetic space group Fm3m, which is I4/mmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The structural parameters in the tetragonal phase are aT = bT = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2075(1)/ √ 2Å and cT = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2143(1) Å where the aT and bT axes are rotated by 45° relative to the a and b axes of the paramagnetic simple cubic cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' In this space group Ho3+ ions occupy a single 2a Wyckoff site and the magnetic ordering vector is k = ( 3 20 3 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' While we must use the tetragonal space group below TN, we continue to use the cubic unit cell to index wave vector transfer in the neu- tron scattering experiments, which do not resolve the multi- domain tetragonal distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Spin structure As described in the previous sections, the magnetic order has a characteristic wavevector k = ( 1 2 1 2 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' In addition to the corresponding low T magnetic Bragg peaks, the intensities of all nuclear Bragg peaks are observed to increase below TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The increase of intensity is approximately proportional to the intensity in the paramagnetic phase, which indicates it arises from secondary extinction release47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' To check this hypothesis, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' A series of θ-2θ X-ray diffraction scans through the Q = (006) Bragg peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The inferred temperature dependence of the lattice pa- rameters is shown in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The neutron magnetic and nuclear refinement of HoBi are presented in (c) where the observed cross- sections for various Bragg peaks are plotted as a function of the cal- culated cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The inset in (c) reports the variation of the χ2 goodness of fit for the magnetic refinement of HoBi assuming a multi-domain k = ( 1 2 1 2 1 2) spin structure with an easy axis defined by spherical coordinates θ and φ (φ = 0 corresponds to the [110] di- rection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Panel (d) shows the low-temperature magnetization versus field for fields applied parallel to the [001] and [110] directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The data show that [001] is the easy axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' we performed polarized neutron diffraction on the (002) and (220) Bragg peaks below TN and found them to be exclusively nuclear in origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We note that weak k = (001) Bragg peaks also onset at TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Examples of these peaks include the (001) and (111) Bragg peaks (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3(c)), which are forbidden within the Fm3m space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' These Bragg peaks are attributed to multiple magnetic scattering as their presence depends on both the em- ployed incident neutron wavelength and the scattering plane, and they are absent in powder neutron diffraction measure- ments36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The multiple scattering processes involve magnetic k = ( 1 2 1 2 1 2) Bragg reflections so they occur only for T < TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Referring to fcc close packing, the AFM k = ( 1 2 1 2 1 2) spin structure can be described as an AFM stacking of FM trian- gular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' As the magnetic order and structural distortion in HoBi occur in a single 1st order phase transition, the direc- tion of the spins in each FM sheet is not constrained by the usual Landau argument for second order phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' To determine the local spin orientation of the Ho3+ ions, we col- lected 18 rocking scans at different magnetic Bragg positions for a sample presumed to be in an unbiased multi-domain state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The data were compared to a cubic domain average of the calculated magnetic Bragg diffraction for a general spin orientation within one domain given by spherical angles θ, φ and k = ( 1 2 1 2 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Here θ = 0 corresponds to the tetragonal c-direction and θ = π/2 and φ = 0 corresponds to the [110] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Minimizing with respect to the moment size at each (a) (b) Kα Warming 6K 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' c (Tetragonal) HoBi 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='214 Cooling 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='8 K 600 Q = (006) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='6 K (cts/s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='212 5 K 400 Ka2 Latt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='210 a (Cubic) 200 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='208 0 a (Tetragonal) 96 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='6 5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 20 T(K) (c) 20 (d) 12 H[001] O H I [110] CCCCCCCCCCCCCCCCC 15 Magnetic Oobs(b/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=') Nuclear M(μB/Ho) 8 2 X Xmin 10 180 4 90 5 45 90 0 d 0 0 5 10 15 20 0 2 4 6 Ocalc(b/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=') H(T)5 point, the χ2 measure of fit quality is shown versus θ and φ in the inset panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The manifold of states rep- resented by the red arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 1 are indistinguishable by neutron diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This degeneracy arises because the mag- netic diffraction intensity for a multi-domain sample only de- pends on the smallest angle between the spin and a ⟨111⟩ axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' From our refinement, we find this angle is 47(10)°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This is ex- perimentally indistinguishable from the angle between [001] and [111], which is 55°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This means the magnetic diffraction data are consistent with spins pointing along the [001] direc- tions, but also with many other directions including close to the [110] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Fortunately the spin anisotropy of the Ho3+ ions can be deduced from other pieces of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' First, the low- temperature magnetization of HoBi shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 4(d) reveals the saturation magnetization is larger for fields along the [001] direction than along [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Second, the structural distortion also occurs along the [001] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Both of these measure- ments are consistent with spins oriented along the tetragonal cT-axis in the AFM ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Additionally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' III F we show that a [001] easy axis anisotropy is needed to ac- curately model the inelastic neutron scattering spectrum be- low TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We thus conclude the spins in the AFM type II order of HoBi are oriented along the cT direction, which is the direction of the structural elongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The comparison between measured and calculated magnetic Bragg intensities is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The corresponding spin structure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' An ordered moment of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='3(6) µB was de- termined, which is experimentally indistinguishable from the gJµB = 5 4 · 8 µB = 10 µB saturation magnetization of Ho3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Crystal electrical field interaction For Ho3+ ions, the J = 8 spin-orbit ground state manifold is (2J+1) = 17 fold degenerate under full rotation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This degeneracy is, however, lifted by the symmetry break- ing crystal electric fields (CEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Using the Stevens operator formalism, the CEF Hamiltonian appropriate for Ho3+ in the high-temperature cubic phase of HoBi can be expressed as follows: ˆHcubic ce f = B4( ˆO0 4 + 5 ˆO4 4) + B6( ˆO0 6 − 21 ˆO4 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' (1) Here ˆOm n are Stevens operators48 that can be written in terms of the spin-orbital angular momentum operators ˆJ+, ˆJ− and ˆJz where ˆz ∥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The CEF parameters Bn are scalars of dimen- sion energy that dictate the strength of the different CEF terms and can be determined by fitting spectroscopic or thermo- magnetic data sensitive to the crystal field level scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Bn can also be estimated through the point-charge model49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Following Hutching’s formalism49 the point charge model yields B4 = 7|e||qBi|βJ⟨r4⟩ 64πϵ0d5 Bi (2) and B6 = 3|e||qBi|γJ⟨r6⟩ 256πϵ0d7 Bi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' (3) Here e is the electron charge, qBi is the charge of the Bi ligand and ϵ0 is the vacuum permitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' βJ and γJ are reduced matrix elements calculated in ref48 whereas the radial integrals for the 4f state ⟨rn⟩ are tabulated in ref50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We used qBi = − 3e and the distance between a holmium ion and its first n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' bismuth ion dBi = a/2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2093(1)/2 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Introducing these values in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 2 and 3 we obtain B4 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2709(2) × 10−4 meV and B6 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='0468(1) × 10−7 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Determination of the crystal electric field (CEF) level scheme for the J=8 Ho3+ ion in HoBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' (a) shows the results of a point charge (PC) calculation for the cubic and tetragonal phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The cu- bic CEF scheme may be compared to the level scheme for the fitted CEF Hamiltonian of HoBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Panel (b) and (c) respectively show the temperature dependence of the magnetic heat capacity (Cp) and the inverse magnetic susceptibility of HoBi compared to corresponding properties based on the fitted CEF Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The magnetic en- tropy obtained from integrating the Cp of HoBi is shown in the inset of (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The measured (d) and calculated (e) inelastic neutron scatter- ing spectra of HoBi are shown for T = 12 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The neutron inelastic scattering data were acquired using a 25 meV incident neutron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The corresponding CEF level scheme for Ho3+ in the cubic phase of HoBi is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The Ho3+ J−multiplet is split into 4 triplets, 2 doublets, and 1 singlet that form three groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Group I includes one doublet, one triplet, and one singlet between 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Group II is formed by two (a)HoBi P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' cubic Fit cubic P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Tetragonal 10 888888888888888888 T D S 8 D D 6 4 E 2 S D S 0 D D S (b) (c) 60 (J/mol/K) 25 R ln(17) 20 /emu) R ln(6) 20 15 H=10 0e (J/mol/K) 40 10 (mol Oe/ H II [001] mag S 0 10 100 10 20 T(K) %/ 1 CEF fit CEF 0 0 10 100 0 100 200 300 T(K) (e) T(K) (d) 1 12 K Data 12 K Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 12 Ei=25 meV 12 hw (meV) I (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=') 8 8 4 0 0 0 1 3 4 2 3 4 IQ(A) IQ(A)6 triplets between 6 meV and 7 meV, and group III consists of a doublet and a triplet between 9 meV and 10 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The CEF Hamiltonian estimated from our point-charge cal- culation can reproduce the temperature dependence of the magnetic heat capacity Cp (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(b)) and magnetic suscepti- bility χ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Obtained by integrating Cp/T, the temper- ature dependence of the entropy shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(b) is informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' A first entropy plateau near 10 K is associ- ated with the sharp Cp anomaly at the phase transition to long range magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The corresponding change in entropy of ∆S = R ln 6 is that associated with the group I CEF states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The second plateau at S = R ln 17 is reached at room temper- ature and encompasses all of the entropy associated with the three groups of crystal field levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' For a more stringent test of the point charge model, we turn to inelastic neutron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(d) shows the 12 K in- elastic neutron scattering spectrum with energy transfer rang- ing from 0 to 15 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' At this temperature, the group II and III of CEF states are so scarcely populated that only CEF excitations originating from group I should be visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' No significant intrinsic broadening of the CEF excitations is ob- served and we note, also, that the experimental resolution is too coarse to resolve CEF levels within a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The mag- netic neutron scattering cross section associated with CEF transition from group I to II and from group I to III can be computed based on the point charge CEF Hamiltonian (Imn ∝ � i |⟨m|Ji|n⟩|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This calculation predicts the cross section for transitions from group I to group II is 250 times stronger than for transitions from group I to group III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The in- tensity of the transition from I to III is thus predicted to be too weak to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This explains why Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(d) shows just a single peak that we associate with transitions from group I to group II crystal field levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' While the measured 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='2 meV gap between group I and group II CEF levels is just 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='4 meV off from the point charge prediction of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='8 meV, we can improve our estimate of the CEF Hamiltonian by simultaneously fitting B4 and B6 for the best possible account of the neutron scattering spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(d)), the specific heat data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(b)), and the mag- netic susceptibility data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The best fit parame- ters thus obtained are B4 = − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='24(1) × 10−4 meV and B6 = − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='4(1) × 10−7 meV and with them the CEF Hamilto- nian provides an excellent account of all single ion properties that we’ve measured, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The CEF scheme obtained from our fit (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(a)) is remark- ably similar to the point-charge calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Also a re-scaling of our CEF Hamiltonian for HoBi using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 2 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 3 con- sidering only the different ligand spacing successfully predicts the level scheme for HoN ref51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This is in contrast with the praseodymium case where a pnictide ligand charge of q = −2e is needed to bring the point charge model into agreement with experimental data5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This indicates that holmium monopnic- tides are more ionic than praseodymium monopnictides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Finally, we estimated the effect of the tetragonal distortion on the CEF interaction in HoBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We performed a point-charge calculation assuming that the first n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Ho-Bi bond is shorter along the a and b direction (da) as compared to the c direction (dc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The calculated CEF Hamiltonian can be written as: Htet ce f = |e||qBi| 4πϵ0 [αJ⟨r2⟩( 1 d3c − 1 d3a ) ˆO0 2+ (4) βJ⟨r4⟩(( 1 4d5c + 3 16d5a ) ˆO0 4 + 35 16d5a ˆO4 4)+ γJ⟨r6⟩(( 1 8d7c − 5 64d7a ) ˆO0 6 − 63 64d7a ˆO4 6)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The corresponding level scheme is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' For this calculation, we used the lattice parameters determined in our high-resolution X-ray scattering experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The de- generacy of all the triplets and doublets associated with cubic symmetry is lifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This results in four doublets and nine sin- glets and a significant broadening of each of the three groups of crystal field levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Low energy spin dynamics We now turn our attention to the collective physics of HoBi, which we explore using inelastic magnetic neutron scatter- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6(a) shows the temperature dependence of the in- elastic scattering for Q = ( 1 2 1 2 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Just above TN, the scat- tering is quasi-elastic with a physical (resolution corrected) FWHM of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='30(5) meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' No inelastic intensity is observed up to 2 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This is consistent with the CEF energy scheme shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Below TN, the quasi-elastic scattering splits into an elastic and an inelastic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' To probe any dispersion of the low energy spin excitations, we acquired low energy spectra at momentum transfer Q cor- responding to high symmetry points in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6(b) shows the spectrum consists of a peak that is broader than the experimental resolution (FWHM indicated by hor- izontal bar) and that shifts by less than the peak width be- tween the different values of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' A gaussian fit finds the peak centered at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7(2) meV with a FWHM of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='48(4) meV that exceeds the instrumental resolution (FWHM of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='22 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The limited resolution and statistical accuracy of the data does not rule out the possibility of multiple dispersive components within the approximately Gaussian envelope of the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We also examined the higher energy excitations for T < TN by acquiring momentum resolved inelastic scattering data up to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' A representative slice through the data is dis- played as a color image versus Q along the (HH0) direction and energy transfer in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' No dispersion is resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The data are similar to the high-temperature plot of intensity versus |Q| and ℏω in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 5(d) though with additional inelastic features at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='0(3) meV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7(2) meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6(e) shows the momentum dependence of the inte- grated intensity of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 meV mode throughout the (HHL) zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The Q dependence of the intensity is subtle albeit peaked at the magnetic ( 1 2 1 2 1 2) zone center and smoothly de- creases with |Q| in accordance with the Ho3+ magnetic form factor41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We note that the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 meV gap is about an order of magnitude greater than the predicted CEF gap arising from the tetragonal distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This indicates the phase transition is driven by the magnetic interactions, which we model below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The temperature dependence of the low energy inelastic neu- tron spectrum of HoBi at Q = ( 1 2 1 2 1 2) is shown in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The spec- trum of neutron scattering at some high symmetry positions within the first Brillouin zone of HoBi are shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='The horizontal black bar indicates the FWHM energy resolution of the spectrom- eter while the black dashed lines show the predicted spectrum based on the spin Hamiltonian presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The energies asso- ciated with each exciton are indicated by vertical black dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The observed and calculated inelastic neutron scattering spectrum up to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 meV are respectively plotted in (c) and (d) for momentum transfer Q along the [HH0] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The observed and calculated momentum dependence of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='75 meV exciton scattering inten- sity is shown in (e) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The energy integration for panel (e) is ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='25 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' MODELING SPIN DYNAMICS OF SPIN-ORBITAL EXCITONS The low-temperature excitations in HoBi are similar to other rare-earth metallic compounds where exchange interac- tions are strong enough to mix crystal field levels4,52,53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Be- cause components that are longitudinal with respect to the or- dered moment are involved, these are not conventional trans- verse spin wave excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' They may be described as crystal field excitations that can propagate through the lattice due to inter-site interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We shall adopt the practice of calling these “crystal field exciton” or simply “exciton”54–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' A common theoretical approach to describing excitons in rare-earth magnets is to use a pseudo-boson theory where the exciton creation operator is a linear combination of single-ion operators53,57,58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' In this theory, the Q = 0 single-ion opera- tors are obtained by diagonalizing the mean-field spin Hamil- tonian and the dispersion at finite Q is produced by the ex- change terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We use this pseudo-boson theory to describe the magnetic excitation spectrum of HoBi below TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The Hamiltonian Hs includes the single-ion tetragonal crystal field terms and isotropic exchange interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Hs is decomposed into a mean-field term (H0,k) and an interacting part (Hint) so Hs = � k H0,k + Hint where: H0,k = Htet ce f,k + (−1)kHzJk jz (5) and Hint = � j, j′,k,k′ Jk,k′ j, j′ Jk j · Jk′ j′ − � j,k (−1)kHzJk jz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' (6) Here j indexes the unit cell while k = 1, 2 specifies the anti-parallel sub-lattices of the AFM order (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We define Hz = 2 � r ZrJr⟨Jz⟩ where Jr and Zr are respectively the ex- change constant and coordination number associated with the rth neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' ⟨Jz⟩ is the thermal average of Jz on each site, which we found to be ⟨Jz⟩ = 8 in our diffraction and CEF analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' By definition, Hint carries no mean value and so can be written in terms of creation (ˆa† n,k = |n, k⟩⟨0, k|) and annihila- tion (ˆan,k = |0, k⟩⟨n, k|) operators that connect the ground state |0, k⟩ and the excited eigenstates |n, k⟩ of ˆH0,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' In this case, ˆH0,k = � n Enˆa† n,kˆan,k where En,k is the eigenvalue of the |n, k⟩ eigenstate of ˆHo,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' After writing ˆHs in terms of these operators and Fourier transforming it, we obtain: ˆHs = 1 2 � Q � ˆa†(Q)A(Q)ˆa(Q) + ˆa†(−Q)A(−Q)ˆa(−Q) (7) +ˆa†(Q)B(Q)ˆa†(−Q) + ˆa(−Q)B(Q)ˆa(−Q) � with ˆA = ˆ∆ + 2ˆhzz + ˆh+− + ˆh−+ and ˆB = 2ˆhzz + ˆh++ + ˆh−− where ˆ∆ = En,kδk,k′δn,n′ and ˆhαβ(k, k′, n, n′, Q) = J(Q)⟨k, n| ˆJα|0, k⟩⟨k′, 0| ˆJβ|n′, k′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The procedure to compute the spin dynamics first consist of diagonalizing ˆH0,k to obtain the eigenvalues En,k and eigenvec- tors |n, k⟩ for Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' At finite Q, the matrix ˆHs = � ˆA ˆB − ˆB − ˆA � is then computed and diagonalized to obtain the perturbed ener- gies (E˜n(Q)) and eigenstates |˜n(Q)⟩ for each exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We con- sider all the excited CEF states belonging to the (2J+1) spin- orbit manifold of HoBi so there are 32 creation and annihla- tion operators for each of the 2 Ho3+ spins within the magnetic unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This give a Hilbert space of 64 states for ˆHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The associated inelastic magnetic neutron scattering cross-section for a single magnetic domain is then53,57: d2σ dEdΩ = N(γr0)2 k f ki |g 2 f(Q)|2 (8) × � ˜n,q,τm |⟨˜n(q)| ˆJQ|GS ⟩|2δ(E − E˜n(q))∆(Q − q − τm) Here N is the number of primitive magnetic unit cells, γ = - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='91 is the gyromagnetic ratio of the neutron, r0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='818 × (a) I (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=') 0 3 HoBi AE (Aaw) m 100 7 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=') 50 0 0 2 4 6 8 10 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 T(K) hw (meV) (c) (d) 4 Data Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 10 10 (meV) 8 8 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=') 6 6 hw 4 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 K 0 0 0 2 3 0 2 3 0 [HH0] [HHO] (f) (e) 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='75 meV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='75 meV Data Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7 K [00L] L 1001 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=') I L UX U X 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5 2 [HH0] [HH0]8 10−15 m is the classical electron radius, τm is the magnetic zone center, q is the reduced momentum transfer within the first magnetic Brillouin zone, while k f and ki respectively are the scattered and incoming neutron wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The mea- sured spectrum is subject to the finite resolution of the instru- ment which we account for by replacing the delta functions by a united normalized Gaussian functions with the Q-integrated energy resolution width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The final calculated spectrum was averaged over all possible magnetic domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' MICROSCOPIC SPIN HAMILTONIAN FOR HOLMIUM BISMUTH We determined the microscopic parameters of ˆHs for HoBi by fitting the Q = 0 spectrum consisting of three excitons at E1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7(2) meV, E2 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='4(2) meV and E3 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='0(3) meV with relative intensities I2/I1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='5(3) and I2/I3 = 37(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Em- ploying the ratio ∥J2/J1∥ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='17 obtained by analyzing the magnetic diffuse scattering (section III B) leaves just one free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The tetragonal CEF Hamiltonian has six free pa- rameters that were initially estimated from the point-charge model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' To reproduce the exact energies of the excitons at E2 and E3, we allowed the CEF parameters to relax away from their point-charge values which results in many combinations of parameters consistent with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We estimated the ex- change constants by varying the CEF parameters away from their point-charge calculation values and keeping all solutions that have a χ2 within 20% (1/Nobs) of the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The exchange parameters refined to J1 = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='4(2) µeV and J2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='0(5) µeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' A mean-field critical temperature of 20(7) K is obtained from these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' For comparison, the actual ordering temperature is only TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='72(1) K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We hypothesize that fluctuations arising from competition be- tween the ferromagnetic J1 and the antiferromagnetic J2 in- teractions lead to the reduced critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The right column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6 compares the optimized model for a multi-domain sample to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6(d) shows the full intensity versus ℏω and Q ∥ (HH0) for com- parison with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The position and relative intensity of the three modes are well reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Looking more closely at the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='75 meV mode, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6(b) compares the intensity ver- sus energy transfer at select high symmetry points in the Bril- louin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The vertical dashed lines show that multiple ex- citons contribute at each Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This is generally consistent with the featured spectrum observed though there is more broad- ening/dispersion observed than reproduced by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' In- clusion of anisotropic or longer range interactions might be needed to remedy this discrepancy though data with higher energy resolution is needed to justify the greater model com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6(f) shows the calculated Q-dependent integrated intensity of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='75 meV mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The dominant features of the experimental result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 6(e) are reproduced, includ- ing mainly the increase of scattered intensity at the magnetic zone centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We note the presence of phonon scattering near Q = (002) that may account for the discrepancy between the calculation and the experimental data at that momentum point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION In this manuscript, we have characterized an antiferro- magnetic order and the associated crystal field excitons that develop below TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='72(1) K in the rare-earth monopnictide HoBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This magnetic state is driven by strong 2nd n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' antiferromagnetic and weaker 1st n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' ferromag- netic interactions, which we quantified via modeling of the diffuse paramagnetic and low temperature inelastic neutron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The excitation spectrum is sensitive to the local orientation of the Ho3+ ordered spins, which allowed us to establish the Ising nature of the antiferromagnetic order in HoBi that cannot be deduced from neutron diffraction of a multi-domain sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We used X-ray diffraction to provide evidence for a tetragonal structural distortion that accompanies magnetic ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Our CEF analysis and modelling of inelastic scattering data indicates the elongated c-axis coincides with the easy magnetic axis within a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The magnetic excitations that we have documented here surely have significant impacts on the magneto-transport properties of HoBi34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' For example, we found strong quasi- elastic neutron scattering in the paramagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The associated short range correlated spin fluctuations, which may be accompanied by short range tetragonal lattice dis- tortions too given the non-Kramers nature of the Ho3+, are expected to enhance the electrical resistivity above TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Below TN, these gapless fluctuations are replaced by a coherent exciton at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='7(2) meV and correspondingly the electrical resistivity is reduced by an order of magnitude upon cooling below TN34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The field-dependence of spin-orbital excitons may be responsible for various features observed in the magnetoresistance of HoBi and more broadly in the rare-earth monopnictides23–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work was supported as part of the Institute for Quan- tum Matter, an Energy Frontier Research Center funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Department of Energy, Office of Science, Basic Energy Sciences Under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='DE-SC0019331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' CB was further supported by the Gordon and Betty Moore foundation EPIQS program under GBMF9456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The work at Boston College was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Department of Energy, Office of Basic Energy Sciences, Division of Physical Behavior of Materials under Award DE-SC0023124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' This work was supported in part by the Natural Sciences and Engineering Research Coun- cil of Canada (NSERC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' We acknowledge the support of the National Institute of Standards and Technology, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Depart- ment of Commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Access to MACS was provided by the Center for High Resolution Neutron Scattering, a partnership between the National Institute of Standards and Technology and the National Science Foundation under Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' DMR-1508249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' The identification of any commercial prod- uct or trade name does not imply endorsement or recommen- dation by the National Institute of Standards and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' 9 A portion of this research used resources at the High Flux Iso- tope Reactor, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' ∗ Correspondence email address: Jonathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='Gaudet@nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='gov 1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Duan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' Sabirianov, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQfjQ2i/content/2301.05141v1.pdf'} +page_content=' N.' metadata={'source': 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a/ONFOT4oBgHgl3EQf3DT4/content/tmp_files/2301.12945v1.pdf.txt b/ONFOT4oBgHgl3EQf3DT4/content/tmp_files/2301.12945v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8fb3979900d3c234c6e5eb1821d580722a967cc3 --- /dev/null +++ b/ONFOT4oBgHgl3EQf3DT4/content/tmp_files/2301.12945v1.pdf.txt @@ -0,0 +1,1307 @@ +arXiv:2301.12945v1 [math.CO] 30 Jan 2023 +CONTINUED FRACTIONS FOR PARTITION GENERATING +FUNCTIONS +GEOFFREY B CAMPBELL +Dedicated to Professor Rodney J Baxter on his 83rd birthday. +Abstract. We derive continued fractions for partition generating functions, uti- +lizing both Euler’s techniques and Ramanujan’s techniques. Although our results +are for integer partitions there is scope to extend this work to vector partitions, +including for binary and n-ary partitions. +1. Euler’s Continued Fraction +Almost 290 years ago in 1737, Leonhard Euler wrote De fractionibus continuis dis- +sertatio, which gave mathematics a first ever comprehensive account of the properties +of continued fractions, and included the first proof that the number e is irrational. +(See Sandifer [50]) Later, but still 275 years ago in 1748, Euler, in his Introductio in +analysin infinitorum Vol. I, Chapter 18 [28], proved: +(a) the equivalence of his continued fraction to a generalized infinite series, +(b) every rational number can be written as a finite continued fraction, and +(c) the continued fraction of an irrational number is infinite. +2010 Mathematics Subject Classification. Primary: 11J70; Secondary: 05A15, 05E40, 11Y11, +11P21. +Key words and phrases. Continued fractions and generalizations. Exact enumeration problems, +generating functions. +Partitions of integers. +Elementary theory of partitions. +Combinatorial +identities, bijective combinatorics. Lattice points in specified regions. +Thanks are due to Professor Dr Henk Koppelaar, whose discussions and suggestions have been +very helpful for the book for which this paper is essentially a chapter. +1 + +2 +GEOFFREY B CAMPBELL +Euler’s continued fraction is the very nice identity, whose first few cases are: +a0 + a0a1 += +a0/(1 − a1/(1 + a1)) += +a0 +1 − +a1 +1 + a1 +; +a0 + a0a1 + a0a1a2 += +a0/(1 − a1/(1 + a1 − a2/(1 + a2))) += +a0 +1 − +a1 +1 + a1 − +a2 +1 + a2 +; +a0 + a0a1 + a0a1a2 + a0a1a2a3 += +a0/(1 − a1/(1 + a1 − a2/(1 + a2 − a3/(1 + a3)))) += +a0 +1 − +a1 +1 + a1 − +a2 +1 + a2 − +a3 +1 + a3 +. +Hence, we can state Euler’s Continued Fraction in the following +Theorem 1.1. If a0, a1, a3, ... an are defined functions such that no denominator +is zero in the following equations then +(1.1) +n +� +k=0 +k +� +j=0 +aj = a0 + a0a1 + a0a1a2 + ... + a0a1...an += a0/(1 − a1/(1 + a1 − a2/(1 + a2 − a3/(1 + ... an−1/(1 + an−1 − an/(1 + an))))). += +a0 +1 − +a1 +1 + a1 − +a2 +1 + a2 − +a3 +1 + a3 − +... +... +an−1 +1 + an−1 − +an +1 + an +. +Obviously, this lends itself to many of the elementary series that arise in school +and university analysis. +However, we shall put this to good use in applying it +to partition generating functions. The fact of this theorem involving a finite sum +allows us to incrementally extend the number of terms until we can infer the infinite +versions of the theorem. +Example 1: The exponential function is +(1.2) exp(z) = 1 + z +1! + z2 +2! + z3 +3! + ... = 1 + +�z +1 +� ++ +�z +1 +� �z +2 +� ++ +�z +1 +� �z +2 +� �z +3 +� ++ ... += 1/ +� +1 − z/ +� +1 + z − +�z +2 +� +/ +� +1 + +�z +2 +� +− +�z +3 +� +/ +� +1 + +�z +3 +� +− +�z +4 +� +/ +� +1 + +�z +4 +� +− ... +����� +. + +CONTINUED FRACTION PARTITION IDENTITIES +3 +Applying an “equivalence transformation” that consists of clearing the fractions, +this example is simplified to +exp(z) = 1/(1 − z/(1 + z − z/(2 + z − 2z/(3 + z − 3z/(4 + z − . . .))))), +or the equivalent statement +exp(z) = +1 +1 − +z +1 + z − +z +2 + z − +2z +3 + z − +3z +4 + z − . . . +and we know this continued fraction converges uniformly on every bounded domain +in the complex plane because it is equivalent to the power series for exp(z). +Example 2: There is the well-known logarithmic function series +(1.3) +log +�1 + z +1 − z +� += 2z(1 +1 + z2 +3 + z4 +5 + ...) += 2z(1 + (z2 +3 ) + (z2 +3 )(3z2 +5 ) + (z2 +3 )(3z2 +5 )(5z2 +7 ) + ...). +Applying Euler’s continued fraction formula to this expression shows that: +log +�1 + z +1 − z +� += 2z/(1−(z2 +3 )/(1+(z2 +3 )−(3z2 +5 )/(1+(3z2 +5 )−(5z2 +7 )/(1+(5z2 +7 )−(7z2 +9 )/(1+(7z2 +9 )−...))))). +Applying the “equivalence transformation” this example is simplified to +log +�1 + z +1 − z +� += 2z/(1−z2/(z2+3−(3z)2/(3z2+5−(5z)2/(5z2+7−(7z)2/(7z2+9−...))))) += +2z +1 − +z2 +z2 + 3 − +(3z)2 +3z2 + 5 − +(5z)2 +5z2 + 7 − +(7z)2 +7z2 + 9 − . . . +Example 3: A continued fraction for π. We can use the previous example involving +the principal branch of the natural logarithm function to construct a continued +fraction representation of π. First we note that +(i + 1)/(i − 1) = i, +so +then +log((i + 1)/(i − 1)) = iπ/2. + +4 +GEOFFREY B CAMPBELL +Setting z = i in the previous result, and remembering that i2 = −1, we obtain +immediately +π = +4 +1 + +12 +2 + +32 +2 + +52 +2 + +72 +2 + . . . +2. Euler’s continued fraction applied to partitions +In this section we will technically do no more than apply the previous section. +However, the theory of partitions is full of generating functions that are emenable to +the Euler continued fraction. In a subsequent section we will examine Ramanujan +type continued fractions, but firstly we will gather some ”low hanging fruit” from +some elementary series-product identities. +We begin with the well-known telescoping identities: +If a1, a2, a3, ... , an, are functions chosen for nonzero denominators, then +(2.1) +1 + +a1 +1 − a1 ++ +a2 +(1 − a1)(1 − a2) + ... + +an +(1 − a1)(1 − a2)...(1 − an) += +1 +(1 − a1)(1 − a2)(1 − a3)...(1 − an); +and +(2.2) 1 + a1 + a2(1 + a1) + a3(1 + a1)(1 + a2) + ... + an(1 + a1)(1 + a2)...(1 + an−1) += (1 + a1)(1 + a2)(1 + a3)...(1 + an). +The series in (2.1) and (2.2) are already close to being in the required form to +apply the Euler continued fraction since +(2.3) +1 + +a1 +1 − a1 ++ +a2 +(1 − a1)(1 − a2) + ... + +an +(1 − a1)(1 − a2)...(1 − an) += 1 + +a1 +1 − a1 ++ +a1 +1 − a1 +a2(1 − a1) +a1(1 − a2) + ... + +a1 +1 − a1 +a2(1 − a1) +a1(1 − a2)...an(1 − an−1) +an−1(1 − an); +and +(2.4) 1 + a1 + a2(1 + a1) + a3(1 + a1)(1 + a2) + ... + an(1 + a1)(1 + a2)...(1 + an−1) += 1+a1+a1 +a2(1 + a1) +a1 ++a1 +a2(1 + a1) +a1 +a3(1 + a2) +a2 ++...+a1 +a2(1 + a1) +a1 +a3(1 + a2) +a2 +...an(1 + an−1) +an−1 +. +Hence combining (2.1) with (2.3) and then (2.2) with (2.4) respectively, we obtain +(2.5) +1 +(1 − a1)(1 − a2)(1 − a3)...(1 − an) + +CONTINUED FRACTION PARTITION IDENTITIES +5 += +1 +1 − +a1 +1−a1 +1 + +a1 +1−a1 − +a2(1−a1) +a1(1−a2) +1 + a2(1−a1) +a1(1−a2) − +a3(1−a2) +a2(1−a3) +1 + a3(1−a2) +a2(1−a3) − +... +... +an−1(1−an−2) +an−2(1−an−1) +1 + an−1(1−an−2) +an−2(1−an−1) − +an(1−an−1) +an−1(1−an) +1 + an(1−an−1) +an−1(1−an) +; +and +(2.6) +(1 + a1)(1 + a2)(1 + a3)...(1 + an) += +1 +1 − +a1 +1 + a1 − +a2(1+a1) +a1 +1 + a2(1+a1) +a1 +− +a3(1+a2) +a2 +1 + a3(1+a2) +a2 +− +... +... +an−1(1+an−2) +an−2 +1 + an−1(1+an−2) +an−2 +− +an(1+an−1) +an−1 +1 + an(1+an−1) +an−1 +. +After applying the “equivalence transformation” to both of (2.5) and then (2.6) +to eliminate denominator terms, each continued fraction is simplified giving us the +following two theorems. +Theorem 2.1. If a1, a2, a3, ... , an, are functions chosen for nonzero denominators, +then +(2.7) +1 +(1 − a1)(1 − a2)(1 − a3)...(1 − an) += +1 +1 − +a1 +1 − +a2 +a1 + a2 − 2a1a2 − +a1a3 +a2 + a3 − 2a2a3 − +... +... +an−2an +an−1 + an − 2an−1an +. +At first glance we can see this theorem as being applicable to generating functions +for unrestricted partitions of various kinds. Similarly the next theorem applies for +partitions of various sorts into distinct parts. + +6 +GEOFFREY B CAMPBELL +Theorem 2.2. If a1, a2, a3, ... , an, are functions chosen for nonzero denominators, +then +(2.8) +(1 + a1)(1 + a2)(1 + a3)...(1 + an) += +1 +1 − +a1 +1 + a1 − +(1 + a1)a2 +a1 + a2 + a1a2 − +(1 + a2)a3 +a2 + a3 + a2a3 − +... +... +(1 + an−1)an +an−1 + an + an−1an +. +There are many examples we could choose for substitution into theorems 2.2 and +2.2. So, let’s start with the generating functions for unrestricted partitions, and for +distinct partitions as follows. +Corollary 2.1. If pn(k), is the number of unrestricted partitions of k into integers +no greater than n, then +(2.9) +1 +(1 − q1)(1 − q2)(1 − q3)...(1 − qn) = +∞ +� +k=0 +pn(k)qk += +1 +1 − +q1 +1 − +q2 +q1 + q2 − 2q1q2 − +q1q3 +q2 + q3 − 2q2q3 − +... +... +qn−2qn +qn−1 + qn − 2qn−1qn +. +Corollary 2.2. If pn(D, k), is the number of distinct partitions of k into integers +no greater than n, then +(2.10) +(1 + q1)(1 + q2)(1 + q3)...(1 + qn) = +∞ +� +k=0 +pn(D, k)qk += +1 +1 − +q1 +1 + q1 − +(1 + q1)q2 +q1 + q2 + q1q2 − +(1 + q2)q3 +q2 + q3 + q2q3 − +... +... +(1 + qn−1)qn +qn−1 + qn + qn−1qn +. +Next we choose the odd integer powers substituted into the two theorems. + +CONTINUED FRACTION PARTITION IDENTITIES +7 +Corollary 2.3. If pn(O, k), is the number of unrestricted partitions of k into odd +integers no greater than 2n − 1, then +(2.11) +1 +(1 − q1)(1 − q3)(1 − q5)...(1 − q2n−1) = +∞ +� +k=0 +pn(O, k)qk += +1 +1 − +q1 +1 − +q3 +q1 + q3 − 2q1q3 − +q1q5 +q3 + q5 − 2q3q5 − +... +... +qn−2qn +q2n−3 + q2n−1 − 2q2n−3q2n−1 +. +Corollary 2.4. If pn(DO, k), is the number of distinct partitions of k into odd +integers no greater than 2n − 1, then +(2.12) +(1 + q1)(1 + q3)(1 + q5)...(1 + q2n−1) = +∞ +� +k=0 +pn(DO, k)qk += +1 +1 − +q1 +1 + q1 − +(1 + q1)q3 +q1 + q3 + q1q3 − +(1 + q3)q5 +q3 + q5 + q3q5 − +... +... +(1 + q2n−3)q2n−1 +q2n−3 + q2n−1 + q2n−3q2n−1 +. +It is a well-known result due to Euler that p∞(DO, k) = p∞(O, k). Explicitly, as +n → ∞ equations (2.12) and (2.11) are equal to each other. +Next, let us give the cases covering binary partitions. +Corollary 2.5. If bn(2, k), is the number of unrestricted binary partitions of k into +non-negative powers of two no greater than 2n, then +(2.13) +1 +(1 − q1)(1 − q2)(1 − q4)...(1 − q2n) = +∞ +� +k=0 +bn(2, k)qk += +1 +1 − +q1 +1 − +q2 +q1 + q2 − 2q1q2 − +q1q4 +q2 + q4 − 2q2q4 − +... +... +q2n−2q2n +q2n−1 + q2n − 2q2n−1q2n +. +The following distinct binary partitions example is completely solvable. + +8 +GEOFFREY B CAMPBELL +Corollary 2.6. If pn(2D, k), is the number of binary partitions of k into distinct +non-negative powers of two no greater than 2n, then +(2.14) +(1 + q1)(1 + q2)(1 + q4)...(1 + q2n) = 1 − q2n+1 +1 − q += +2n+1−1 +� +k=0 +pn(2D, k)qk += +1 +1 − +q1 +1 + q1 − +(1 + q1)q2 +q1 + q2 + q1q2 − +(1 + q2)q4 +q2 + q4 + q2q4 − +... +... +(1 + q2n−1)q2n +q2n−1 + q2n + q2n−1q2n +. +Note that from (2.14) we have directly that +pn(2D, k) = +� +1, +when 0 ≤ k < 2n+1; +0, +when k ≥ 2n+1. +The following distinct ternary partitions example is easily stated. +Corollary 2.7. If pn(3D, k), is the number of ternary partitions of k into distinct +non-negative powers of three no greater than 3n, then +(2.15) +(1 + q1)(1 + q3)(1 + q9)...(1 + q3n) = +3n−1 +� +k=0 +pn(3D, k)qk += +1 +1 − +q1 +1 + q1 − +(1 + q1)q3 +q1 + q3 + q1q3 − +(1 + q3)q9 +q3 + q9 + q3q9 − +... +... +(1 + q3n−1)q3n +q3n−1 + q3n + q3n−1q3n +. +Note that from (2.15) we have directly that +pn(3D, k) = + + + +1, +for 0 ≤ k < 3n+1; k is a sum of distinct powers of 3. +0, +for 0 ≤ k < 3n+1; k not a sum of distinct powers of 3. +0, +for k ≥ 3n+1. +Clearly this topic of Euler Continued Fractions applied to partition generating +functions is an interesting elementary study for students, and a possible tool for +researchers. The above results are old, and have probably been well-worked over +time. + +CONTINUED FRACTION PARTITION IDENTITIES +9 +3. Rogers-Ramanujan Continued Fractions for partition functions +The fraction given here was mentioned by Ramanujan in his second letter to +Hardy (see Adiga et al. [2, p. xxviii]); namely +(3.1) +R(a, b) = 1 + +bq +1 + aq + +bq2 +1 + aq2 + +bq3 +1 + aq3 + bq4 +... +. +However, these now famous continued fractions, as with the Rogers-Ramanujan +identities, were first discovered in 1894 by Rogers (see [49]). We define the functions +G(q) and H(q) in the context of the Rogers–Ramanujan identities, +(3.2) +G(q) = +∞ +� +n=0 +qn2 +(1 − q)(1 − q2) · · · (1 − qn) = +∞ +� +n=0 +qn2 +(q : q)n += +1 +(q; q5)(q4; q5) = +∞ +� +n=1 +1 +(1 − q5n−4)(1 − q5n−1), +and +(3.3) +H(q) = +∞ +� +n=0 +qn2+n +(1 − q)(1 − q2) · · ·(1 − qn) = +∞ +� +n=0 +qn2+n +(q : q)n += +1 +(q2; q5)(q3; q5) = +∞ +� +n=1 +1 +(1 − q5n−3)(1 − q5n−2). +The Rogers–Ramanujan continued fraction is then, +(3.4) +R(q) = q +11 +60 H(q) +q +−1 +60 G(q) = q +1 +5 +∞ +� +n=1 +(1 − q5n−4)(1 − q5n−1) +(1 − q5n−3)(1 − q5n−2) += 1 + +q +1 +5 +1 + +q +1 + +q2 +1 + q3 +... +. +So, we note that R(0, 1) leads us to the celebrated Rogers-Ramanujan contin- +ued fraction, which has been researched by many (see Andrews [4, Chapter 7], for +example). In the course of analyzing identities from Ramanujan’s Lost Notebook +[7], Andrews and Berndt have discussed the fraction R(a, b), but mainly from the +viewpoint of transformation formulas. +Our emphasis here is on using (3.1) in a +generalized approach to several partition identities, but there is a whole adjacent +theory on particular values of these continued fractions determined from applying +the theory of modular forms. +Hence the examples, using ϕ as the golden ratio +( +√ +5 + 1)/2, + +10 +GEOFFREY B CAMPBELL +(3.5) +e− −π +5 +1 + +e−π +1 + +e−2π +1 + e−3π +... += 1 +2ϕ( +√ +5 − ϕ3/2)( +4√ +5 + ϕ3/2), +(3.6) +e− −2π +5 +1 + +e−2π +1 + +e−4π +1 + e−6π +... += +4√ +5ϕ1/2 − ϕ, +(3.7) +e− −4π +5 +1 + +e−4π +1 + +e−8π +1 + e−12π +... += 1 +2ϕ( +√ +5 − ϕ3/2)(− +4√ +5 + ϕ3/2). +So next we examine the continued fraction R(a, b) of Ramanujan and consider +various restricted partition functions. For further reading, a good reference is Alladi +and Gordon [3]. We use the continued fraction to give results for several partition +identities, some of which generalize results of Bressoud [12] and G¨ollnitz [34]. We also +give a combinatorial interpretation for the coefficients in the power series expansion +of the reciprocal +1 +R(−a,−b), extending a result of Odlyzko and Wilf [42]. The full +description of this approach would add several more pages to our work, but [3] +covers all of this very nicely. +It turns out that Lebesgue’s identity plays a major role in our analysis with +respect to the numerators and denominators of the finite continued fractions we +consider. +(3.8) +� +k≥0 +qk(k+1)/2 �k +j=1(1 + bqj) +(1 − q)(1 − q2)...(1 − qk) = +� +m≥1 +(1 + bq2m)(1 + qm). +It is known that Lebesgue’s identity implies Ramanujan’s fraction R(a, b) has a +product representation when a = 1. More precisely (3.14) and (3.15) (see below) +yield +(3.9) +1 + +bq +1 + q + +bq2 +1 + q2 + +bq3 +1 + q3 + bq4 +... += +∞ +� +m=1 +(1 + bq2m−1) +(1 + bq2m) . + +CONTINUED FRACTION PARTITION IDENTITIES +11 +A neat case of (3.9) is obtained from q �→ q2 and b �→ bq−1 so then +(3.10) +1 + +bq +1 + q2 + +bq3 +1 + q4 + +bq5 +1 + q6 + bq7 +... += +∞ +� +m=1 +(1 + bq4m−3) +(1 + bq4m−1). +For a continued fraction F, let Pn/Qn denote its nth convergent, and suppose +that limn→∞ Pn = P, limn→∞ Qn = Q in a suitable topology. We then say that F +has numerator P and denominator Q, and write P = F N, Q = F D. Consider the +fraction +F(a, c) = 1 + a + +acq +1 + aq + +acq2 +1 + aq2 + +acq3 +1 + aq3 + acq4 +... +. +This can be written in the form +F(a, c) = f(a, c) +f(aq, c), +where +f(a, c) = +� +k≥0 +Akqk. +We now compute the coefficients Ak = Ak(c, q), observing that f(a, c) satisfies +the recurrence +f(a, c) = (1 + a)f(aq, c) + acq f(aq2, c). +Therefore the coefficients Ak satisfy +Ak = qk Ak + qk−1Ak−1 q − cq2k−1 Ak−1, +which is the same as +Ak = qk−1(1 + cqk) +(1 − qk) +Ak−1. +By iteration this yields +F(a, c) = +� +k≥0 +akq +k(k−1) +2 +(−cq)k +(q)k +. +Let c = a−1b. Then +R(a, b) = f(a, a−1b) +f(aq, a−1b) − a +is Ramanujan’s fraction (3.1). +Lemma 3.1. For the fraction R(a, b), the numerator is +(3.11) +RN(a, b) = +� +k≥0 +akqk(k+1)/2(−a−1b)k +(q)k +, +and the denominator is +(3.12) +RD(a, b) = +� +k≥0 +akqk(k+1)/2(−a−1bq)k +(q)k +. + +12 +GEOFFREY B CAMPBELL +Proof : The expansion (3.12) is an immediate consequence of +(3.13) +RD(a, b) = f(aq, a−1b). +The expansion (3.11) is more complicated. To obtain it, observe that +RN(a, b) += +f(a, a−1b) − a f(aq, a−1b) += +� +k≥0 +akqk(k−1)/2(−a−1bq)k +(q)k +− +� +k≥0 +ak+1qk(k+1)/2(−a−1bq)k +(q)k += +1 + +� +k≥0 +ak+1qk(k+1)/2(−a−1bq)k +(q)k +�1 + a−1bqk+1 +1 − qk+1 +− 1 +� += +1 + +� +k≥0 +ak+1q(k+1)(k+2)/2(−a−1bq)k(1 − a−1b) +(q)k+1 += +� +k≥0 +akqk(k+1)/2(−a−1b)k +(q)k +as required. +■ +Andrews (see [5] and [6]) considered the expansions in lemma 3.1 while discussing +a transformation formula of Ramanujan [47] for R(a, b). Our emphasis here is on +the partition theorems that can be derived using R(a, b), and for this the following +lemma is crucial. +Lemma 3.2. For the fraction R(a, b), we also have the expansions +(3.14) +RN(a, b) = +� +i,j≥0 +aibjq(i2+i)/2+ij+j2 +(q)i(q)j +, +and the denominator is +(3.15) +RD(a, b) = +� +i,j≥0 +aibjq(i2+i)/2+ij+j2+j +(q)i(q)j +. +Proof : To obtain (3.14) and (3.15) from (3.12) and (3.13) we use the q-binomial +theorem, +(−z)k = +k +� +j=0 +zjqj(j−1)/2 +�k +j +� +q +with z = a−1b and z = a−1bq. +(See Campbell [22] for the n-space q-binomial +theorem.) Therefore +RN(a, b) += +� +k≥0 +akqk(k+1)/2 +(q)k +k +� +j=0 +a−jbjqj(j−1)/2(q)k +(q)j(q)j−k += +� +i,j≥0 +aibjq(i+j)(i+j+1)/2 +(q)i(q)j +, +where i = k − j; this is equivalent to (3.12). To obtain (3.13), observe that +(3.16) +RD(a, b) = RN(a, bq) +by comparing (3.14) and (3.15). + +CONTINUED FRACTION PARTITION IDENTITIES +13 +The following two theorems relate successively to the numerator and the denom- +inator of the fraction (3.1), so then to (3.14) and (3.15). For a proof of these see +Alladi and Gordon [3]. +Theorem 3.1. (Numerator) +Let AN(n; i, j) be the number of partitions of n into i + j distinct red parts and j +distinct blue parts such that one of the blue parts may be zero and every blue part is +≤ i + j − 1. +Let BN(n; i, j) be the number of partitions of n into i distinct red parts and j +distinct non-consecutive blue parts such that every red part is > j. +Let CN(n; i, j) be the number of partitions of n into i red parts and j blue parts +such that all parts are distinct and after each blue part there is a gap of at least 2. +Then +AN(n; i, j) = BN(n; i, j) = CN(n; i, j). +Theorem 3.2. (Denominator) +Let AD(n; i, j) be as in AN(n; i, j) except that every blue part is > 0 and ≤ i + j. +Let BD(n; i, j) be as in BN(n; i, j) except that part 1 cannot be blue. +Let CD(n; i, j) be as in CN(n; i, j) except that part 1 cannot be blue. Then +AD(n; i, j) = BD(n; i, j) = CD(n; i, j). +So reprising (3.10) namely +1 + +bq +1 + q2 + +bq3 +1 + q4 + +bq5 +1 + q6 + bq7 +... += +∞ +� +m=1 +(1 + bq4m−3) +(1 + bq4m−1)), +we have interesting cancellations in numerator-denominator equations. That is, +the numerator is given by +� +k≥0 +qk(k+1)(−bq−1; q2)k +(q2; q2)k += +∞ +� +m=1 +(1 + bq4m−3)(1 + q2m) += +∞ +� +m=1 +(1 + bq4m−3)(1 + q4m−2)(1 + q4m) +and the denominator is given by +� +k≥0 +qk(k+1)(−bq; q2)k +(q2; q2)k += +∞ +� +m=1 +(1 + bq4m−1)(1 + q4m−2)(1 + q4m) +with right sides having common factors that eliminate. +This leads in particular to the continued fraction identity +(3.17) +1 + +q +1 + q2 + +q3 +1 + q4 + +q5 +1 + q6 + q7 +... += +� +j≡2,3,7 (mod8)(1 − qj) +� +j≡1,5,6 (mod8)(1 − qj). + +14 +GEOFFREY B CAMPBELL +G¨o11nitz [34] states similar results, but (3.1) seems to have escaped attention. There +is a continued fraction identity due to Gordon [33] and G¨o11nitz [34] which looks +very similar to (3.17), namely +(3.18) +1 + q + +q2 +1 + q3 + +q4 +1 + q5 + +q4 +1 + q7 + bq6 +... += +� +j≡3,4,5 (mod8)(1 − qj) +� +j≡1,4,7 (mod8)(1 − qj). +However, this result first appears in Alladi and Gordon [3] almost 30 years after +(3.1). +4. Ramanujan’s three parameter continued fraction +Ramanujan [45] obtained in addition to (3.1), the following continued fraction +with three parameters a, b, q which has also a product representation +(4.1) +1 − ab + +(a − bq)(b − aq) +(1 − ab)(1 + q2) + +(a − bq3)(b − aq3) +(1 − ab)(1 + q4) + +(a − bq5)(b − aq5) +(1 − ab)(1 + q6) + (a − bq7)(b − aq7) +... += +∞ +� +m=1 +(1 + a2q4m−3)(1 + b2q4m−3) +(1 + a2q4m−1)(1 + b2q4m−1). +This was proved only in 1985 by the reviewers of Chapter 16 of Ramanujan’s Second +Notebook [2], 65 years after Ramanujan’s death. If we put a = 0 and replace b2 by +−b in (4.1), we get (3.10). It seems there is still scope to study the combinatorial +properties of the coefficients in the power series expansion of this fraction. +References +[1] ABRAMOWITZ, M., and STEGUN, I. Handbook of Mathematical Functions, Dover Publi- +cations Inc., New York, 1972. +[2] ADIGA,C. BERNDT,B. C.BHARGAVA,S. AND WATSON,G. N. ”Chapter 16 of Ramanu- +jan’s Second Notebook: Theta Functions and q-Series”, Memoirs of the American Mathemat- +ical Society, Vol. 315, Amer. Math. Soc., Providence, RI, 1985. +[3] ALLADI, K. and GORDON H., Partition Identities and a Continued Fraction of Ramanujan, +Journal of Combinatorial Theory, Series A 63, 275-300 (1993) +[4] ANDREWS, G.E. The Theory of Partitions, Addison-Wesley Publishing Company, Advanced +Book Program, Reading, Massachusetts, 1976. +[5] ANDREWS,G. E. An introduction to Ramanujan’s ”lost” notebook, Amer. Math. Monthly 86 +(1979), 89-108. +[6] ANDREWS,G. E. Ramanujan’s ”Lost” Notebbook. III. The Rogers-Ramanujan continued frac- +tion, Adv. 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Math., Contempo- +rary Mathematics, 67: 279–293, doi:10.1090/conm/067/902599, ISBN 9780821850749, Zbl +0634.14012 +[56] WRIGHT, E. M. Partitions of multipartite numbers, Proc. Amer. Math. Soc. 28, 1956, 880- +890. +Mathematical Sciences Institute, The Australian National University, Canberra, +ACT, 0200, Australia +Email address: Geoffrey.Campbell@anu.edu.au + diff --git a/ONFOT4oBgHgl3EQf3DT4/content/tmp_files/load_file.txt b/ONFOT4oBgHgl3EQf3DT4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d2c26791a36f41727a15a1ba8839752527ee3c5 --- /dev/null +++ b/ONFOT4oBgHgl3EQf3DT4/content/tmp_files/load_file.txt @@ -0,0 +1,782 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf,len=781 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='12945v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='CO] 30 Jan 2023 CONTINUED FRACTIONS FOR PARTITION GENERATING FUNCTIONS GEOFFREY B CAMPBELL Dedicated to Professor Rodney J Baxter on his 83rd birthday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' We derive continued fractions for partition generating functions, uti- lizing both Euler’s techniques and Ramanujan’s techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Although our results are for integer partitions there is scope to extend this work to vector partitions, including for binary and n-ary partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Euler’s Continued Fraction Almost 290 years ago in 1737, Leonhard Euler wrote De fractionibus continuis dis- sertatio, which gave mathematics a first ever comprehensive account of the properties of continued fractions, and included the first proof that the number e is irrational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' (See Sandifer [50]) Later, but still 275 years ago in 1748, Euler, in his Introductio in analysin infinitorum Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' I, Chapter 18 [28], proved: (a) the equivalence of his continued fraction to a generalized infinite series, (b) every rational number can be written as a finite continued fraction, and (c) the continued fraction of an irrational number is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Primary: 11J70;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Secondary: 05A15, 05E40, 11Y11, 11P21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Continued fractions and generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Exact enumeration problems, generating functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Partitions of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Elementary theory of partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Combinatorial identities, bijective combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Lattice points in specified regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Thanks are due to Professor Dr Henk Koppelaar, whose discussions and suggestions have been very helpful for the book for which this paper is essentially a chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 1 2 GEOFFREY B CAMPBELL Euler’s continued fraction is the very nice identity, whose first few cases are: a0 + a0a1 = a0/(1 − a1/(1 + a1)) = a0 1 − a1 1 + a1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' a0 + a0a1 + a0a1a2 = a0/(1 − a1/(1 + a1 − a2/(1 + a2))) = a0 1 − a1 1 + a1 − a2 1 + a2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' a0 + a0a1 + a0a1a2 + a0a1a2a3 = a0/(1 − a1/(1 + a1 − a2/(1 + a2 − a3/(1 + a3)))) = a0 1 − a1 1 + a1 − a2 1 + a2 − a3 1 + a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Hence, we can state Euler’s Continued Fraction in the following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' If a0, a1, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' an are defined functions such that no denominator is zero in the following equations then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1) n � k=0 k � j=0 aj = a0 + a0a1 + a0a1a2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' + a0a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='an = a0/(1 − a1/(1 + a1 − a2/(1 + a2 − a3/(1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' an−1/(1 + an−1 − an/(1 + an))))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = a0 1 − a1 1 + a1 − a2 1 + a2 − a3 1 + a3 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' an−1 1 + an−1 − an 1 + an .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Obviously, this lends itself to many of the elementary series that arise in school and university analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' However, we shall put this to good use in applying it to partition generating functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' The fact of this theorem involving a finite sum allows us to incrementally extend the number of terms until we can infer the infinite versions of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Example 1: The exponential function is (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='2) exp(z) = 1 + z 1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' + z2 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' + z3 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = 1 + �z 1 � + �z 1 � �z 2 � + �z 1 � �z 2 � �z 3 � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = 1/ � 1 − z/ � 1 + z − �z 2 � / � 1 + �z 2 � − �z 3 � / � 1 + �z 3 � − �z 4 � / � 1 + �z 4 � − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' CONTINUED FRACTION PARTITION IDENTITIES 3 Applying an “equivalence transformation” that consists of clearing the fractions, this example is simplified to exp(z) = 1/(1 − z/(1 + z − z/(2 + z − 2z/(3 + z − 3z/(4 + z − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' ))))), or the equivalent statement exp(z) = 1 1 − z 1 + z − z 2 + z − 2z 3 + z − 3z 4 + z − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' and we know this continued fraction converges uniformly on every bounded domain in the complex plane because it is equivalent to the power series for exp(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Example 2: There is the well-known logarithmic function series (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='3) log �1 + z 1 − z � = 2z(1 1 + z2 3 + z4 5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=') = 2z(1 + (z2 3 ) + (z2 3 )(3z2 5 ) + (z2 3 )(3z2 5 )(5z2 7 ) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Applying Euler’s continued fraction formula to this expression shows that: log �1 + z 1 − z � = 2z/(1−(z2 3 )/(1+(z2 3 )−(3z2 5 )/(1+(3z2 5 )−(5z2 7 )/(1+(5z2 7 )−(7z2 9 )/(1+(7z2 9 )−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='))))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Applying the “equivalence transformation” this example is simplified to log �1 + z 1 − z � = 2z/(1−z2/(z2+3−(3z)2/(3z2+5−(5z)2/(5z2+7−(7z)2/(7z2+9−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='))))) = 2z 1 − z2 z2 + 3 − (3z)2 3z2 + 5 − (5z)2 5z2 + 7 − (7z)2 7z2 + 9 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Example 3: A continued fraction for π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' We can use the previous example involving the principal branch of the natural logarithm function to construct a continued fraction representation of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' First we note that (i + 1)/(i − 1) = i, so then log((i + 1)/(i − 1)) = iπ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 4 GEOFFREY B CAMPBELL Setting z = i in the previous result, and remembering that i2 = −1, we obtain immediately π = 4 1 + 12 2 + 32 2 + 52 2 + 72 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Euler’s continued fraction applied to partitions In this section we will technically do no more than apply the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' However, the theory of partitions is full of generating functions that are emenable to the Euler continued fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' In a subsequent section we will examine Ramanujan type continued fractions, but firstly we will gather some ”low hanging fruit” from some elementary series-product identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' We begin with the well-known telescoping identities: If a1, a2, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' , an, are functions chosen for nonzero denominators, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1) 1 + a1 1 − a1 + a2 (1 − a1)(1 − a2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' + an (1 − a1)(1 − a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 − an) = 1 (1 − a1)(1 − a2)(1 − a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 − an);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='2) 1 + a1 + a2(1 + a1) + a3(1 + a1)(1 + a2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' + an(1 + a1)(1 + a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 + an−1) = (1 + a1)(1 + a2)(1 + a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 + an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' The series in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='2) are already close to being in the required form to apply the Euler continued fraction since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='3) 1 + a1 1 − a1 + a2 (1 − a1)(1 − a2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' + an (1 − a1)(1 − a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 − an) = 1 + a1 1 − a1 + a1 1 − a1 a2(1 − a1) a1(1 − a2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' + a1 1 − a1 a2(1 − a1) a1(1 − a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='an(1 − an−1) an−1(1 − an);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='4) 1 + a1 + a2(1 + a1) + a3(1 + a1)(1 + a2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' + an(1 + a1)(1 + a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 + an−1) = 1+a1+a1 a2(1 + a1) a1 +a1 a2(1 + a1) a1 a3(1 + a2) a2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='+a1 a2(1 + a1) a1 a3(1 + a2) a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='an(1 + an−1) an−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Hence combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='3) and then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='2) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='4) respectively, we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='5) 1 (1 − a1)(1 − a2)(1 − a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 − an) CONTINUED FRACTION PARTITION IDENTITIES 5 = 1 1 − a1 1−a1 1 + a1 1−a1 − a2(1−a1) a1(1−a2) 1 + a2(1−a1) a1(1−a2) − a3(1−a2) a2(1−a3) 1 + a3(1−a2) a2(1−a3) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' an−1(1−an−2) an−2(1−an−1) 1 + an−1(1−an−2) an−2(1−an−1) − an(1−an−1) an−1(1−an) 1 + an(1−an−1) an−1(1−an) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='6) (1 + a1)(1 + a2)(1 + a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 + an) = 1 1 − a1 1 + a1 − a2(1+a1) a1 1 + a2(1+a1) a1 − a3(1+a2) a2 1 + a3(1+a2) a2 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' an−1(1+an−2) an−2 1 + an−1(1+an−2) an−2 − an(1+an−1) an−1 1 + an(1+an−1) an−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' After applying the “equivalence transformation” to both of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='5) and then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='6) to eliminate denominator terms, each continued fraction is simplified giving us the following two theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' If a1, a2, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' , an, are functions chosen for nonzero denominators, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='7) 1 (1 − a1)(1 − a2)(1 − a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 − an) = 1 1 − a1 1 − a2 a1 + a2 − 2a1a2 − a1a3 a2 + a3 − 2a2a3 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' an−2an an−1 + an − 2an−1an .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' At first glance we can see this theorem as being applicable to generating functions for unrestricted partitions of various kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Similarly the next theorem applies for partitions of various sorts into distinct parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 6 GEOFFREY B CAMPBELL Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' If a1, a2, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' , an, are functions chosen for nonzero denominators, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='8) (1 + a1)(1 + a2)(1 + a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 + an) = 1 1 − a1 1 + a1 − (1 + a1)a2 a1 + a2 + a1a2 − (1 + a2)a3 a2 + a3 + a2a3 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' (1 + an−1)an an−1 + an + an−1an .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' There are many examples we could choose for substitution into theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' So, let’s start with the generating functions for unrestricted partitions, and for distinct partitions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' If pn(k), is the number of unrestricted partitions of k into integers no greater than n, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='9) 1 (1 − q1)(1 − q2)(1 − q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 − qn) = ∞ � k=0 pn(k)qk = 1 1 − q1 1 − q2 q1 + q2 − 2q1q2 − q1q3 q2 + q3 − 2q2q3 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' qn−2qn qn−1 + qn − 2qn−1qn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' If pn(D, k), is the number of distinct partitions of k into integers no greater than n, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='10) (1 + q1)(1 + q2)(1 + q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 + qn) = ∞ � k=0 pn(D, k)qk = 1 1 − q1 1 + q1 − (1 + q1)q2 q1 + q2 + q1q2 − (1 + q2)q3 q2 + q3 + q2q3 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' (1 + qn−1)qn qn−1 + qn + qn−1qn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Next we choose the odd integer powers substituted into the two theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' CONTINUED FRACTION PARTITION IDENTITIES 7 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' If pn(O, k), is the number of unrestricted partitions of k into odd integers no greater than 2n − 1, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='11) 1 (1 − q1)(1 − q3)(1 − q5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 − q2n−1) = ∞ � k=0 pn(O, k)qk = 1 1 − q1 1 − q3 q1 + q3 − 2q1q3 − q1q5 q3 + q5 − 2q3q5 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' qn−2qn q2n−3 + q2n−1 − 2q2n−3q2n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' If pn(DO, k), is the number of distinct partitions of k into odd integers no greater than 2n − 1, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='12) (1 + q1)(1 + q3)(1 + q5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 + q2n−1) = ∞ � k=0 pn(DO, k)qk = 1 1 − q1 1 + q1 − (1 + q1)q3 q1 + q3 + q1q3 − (1 + q3)q5 q3 + q5 + q3q5 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' (1 + q2n−3)q2n−1 q2n−3 + q2n−1 + q2n−3q2n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' It is a well-known result due to Euler that p∞(DO, k) = p∞(O, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Explicitly, as n → ∞ equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='11) are equal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Next, let us give the cases covering binary partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' If bn(2, k), is the number of unrestricted binary partitions of k into non-negative powers of two no greater than 2n, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='13) 1 (1 − q1)(1 − q2)(1 − q4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 − q2n) = ∞ � k=0 bn(2, k)qk = 1 1 − q1 1 − q2 q1 + q2 − 2q1q2 − q1q4 q2 + q4 − 2q2q4 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' q2n−2q2n q2n−1 + q2n − 2q2n−1q2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' The following distinct binary partitions example is completely solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 8 GEOFFREY B CAMPBELL Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' If pn(2D, k), is the number of binary partitions of k into distinct non-negative powers of two no greater than 2n, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='14) (1 + q1)(1 + q2)(1 + q4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 + q2n) = 1 − q2n+1 1 − q = 2n+1−1 � k=0 pn(2D, k)qk = 1 1 − q1 1 + q1 − (1 + q1)q2 q1 + q2 + q1q2 − (1 + q2)q4 q2 + q4 + q2q4 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' (1 + q2n−1)q2n q2n−1 + q2n + q2n−1q2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Note that from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='14) we have directly that pn(2D, k) = � 1, when 0 ≤ k < 2n+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 0, when k ≥ 2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' The following distinct ternary partitions example is easily stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' If pn(3D, k), is the number of ternary partitions of k into distinct non-negative powers of three no greater than 3n, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='15) (1 + q1)(1 + q3)(1 + q9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 + q3n) = 3n−1 � k=0 pn(3D, k)qk = 1 1 − q1 1 + q1 − (1 + q1)q3 q1 + q3 + q1q3 − (1 + q3)q9 q3 + q9 + q3q9 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' (1 + q3n−1)q3n q3n−1 + q3n + q3n−1q3n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Note that from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='15) we have directly that pn(3D, k) = \uf8f1 \uf8f2 \uf8f3 1, for 0 ≤ k < 3n+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' k is a sum of distinct powers of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 0, for 0 ≤ k < 3n+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' k not a sum of distinct powers of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 0, for k ≥ 3n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Clearly this topic of Euler Continued Fractions applied to partition generating functions is an interesting elementary study for students, and a possible tool for researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' The above results are old, and have probably been well-worked over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' CONTINUED FRACTION PARTITION IDENTITIES 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Rogers-Ramanujan Continued Fractions for partition functions The fraction given here was mentioned by Ramanujan in his second letter to Hardy (see Adiga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' xxviii]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' namely (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1) R(a, b) = 1 + bq 1 + aq + bq2 1 + aq2 + bq3 1 + aq3 + bq4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' However, these now famous continued fractions, as with the Rogers-Ramanujan identities, were first discovered in 1894 by Rogers (see [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' We define the functions G(q) and H(q) in the context of the Rogers–Ramanujan identities, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='2) G(q) = ∞ � n=0 qn2 (1 − q)(1 − q2) · · · (1 − qn) = ∞ � n=0 qn2 (q : q)n = 1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' q5)(q4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' q5) = ∞ � n=1 1 (1 − q5n−4)(1 − q5n−1), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='3) H(q) = ∞ � n=0 qn2+n (1 − q)(1 − q2) · · ·(1 − qn) = ∞ � n=0 qn2+n (q : q)n = 1 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' q5)(q3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' q5) = ∞ � n=1 1 (1 − q5n−3)(1 − q5n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' The Rogers–Ramanujan continued fraction is then, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='4) R(q) = q 11 60 H(q) q −1 60 G(q) = q 1 5 ∞ � n=1 (1 − q5n−4)(1 − q5n−1) (1 − q5n−3)(1 − q5n−2) = 1 + q 1 5 1 + q 1 + q2 1 + q3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' So, we note that R(0, 1) leads us to the celebrated Rogers-Ramanujan contin- ued fraction, which has been researched by many (see Andrews [4, Chapter 7], for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' In the course of analyzing identities from Ramanujan’s Lost Notebook [7], Andrews and Berndt have discussed the fraction R(a, b), but mainly from the viewpoint of transformation formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Our emphasis here is on using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1) in a generalized approach to several partition identities, but there is a whole adjacent theory on particular values of these continued fractions determined from applying the theory of modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Hence the examples, using ϕ as the golden ratio ( √ 5 + 1)/2, 10 GEOFFREY B CAMPBELL (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='5) e− −π 5 1 + e−π 1 + e−2π 1 + e−3π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = 1 2ϕ( √ 5 − ϕ3/2)( 4√ 5 + ϕ3/2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='6) e− −2π 5 1 + e−2π 1 + e−4π 1 + e−6π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = 4√ 5ϕ1/2 − ϕ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='7) e− −4π 5 1 + e−4π 1 + e−8π 1 + e−12π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = 1 2ϕ( √ 5 − ϕ3/2)(− 4√ 5 + ϕ3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' So next we examine the continued fraction R(a, b) of Ramanujan and consider various restricted partition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' For further reading, a good reference is Alladi and Gordon [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' We use the continued fraction to give results for several partition identities, some of which generalize results of Bressoud [12] and G¨ollnitz [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' We also give a combinatorial interpretation for the coefficients in the power series expansion of the reciprocal 1 R(−a,−b), extending a result of Odlyzko and Wilf [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' The full description of this approach would add several more pages to our work, but [3] covers all of this very nicely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' It turns out that Lebesgue’s identity plays a major role in our analysis with respect to the numerators and denominators of the finite continued fractions we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='8) � k≥0 qk(k+1)/2 �k j=1(1 + bqj) (1 − q)(1 − q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='(1 − qk) = � m≥1 (1 + bq2m)(1 + qm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' It is known that Lebesgue’s identity implies Ramanujan’s fraction R(a, b) has a product representation when a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' More precisely (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='15) (see below) yield (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='9) 1 + bq 1 + q + bq2 1 + q2 + bq3 1 + q3 + bq4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = ∞ � m=1 (1 + bq2m−1) (1 + bq2m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' CONTINUED FRACTION PARTITION IDENTITIES 11 A neat case of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='9) is obtained from q �→ q2 and b �→ bq−1 so then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='10) 1 + bq 1 + q2 + bq3 1 + q4 + bq5 1 + q6 + bq7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = ∞ � m=1 (1 + bq4m−3) (1 + bq4m−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' For a continued fraction F, let Pn/Qn denote its nth convergent, and suppose that limn→∞ Pn = P, limn→∞ Qn = Q in a suitable topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' We then say that F has numerator P and denominator Q, and write P = F N, Q = F D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Consider the fraction F(a, c) = 1 + a + acq 1 + aq + acq2 1 + aq2 + acq3 1 + aq3 + acq4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' This can be written in the form F(a, c) = f(a, c) f(aq, c), where f(a, c) = � k≥0 Akqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' We now compute the coefficients Ak = Ak(c, q), observing that f(a, c) satisfies the recurrence f(a, c) = (1 + a)f(aq, c) + acq f(aq2, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Therefore the coefficients Ak satisfy Ak = qk Ak + qk−1Ak−1 q − cq2k−1 Ak−1, which is the same as Ak = qk−1(1 + cqk) (1 − qk) Ak−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' By iteration this yields F(a, c) = � k≥0 akq k(k−1) 2 (−cq)k (q)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Let c = a−1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Then R(a, b) = f(a, a−1b) f(aq, a−1b) − a is Ramanujan’s fraction (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' For the fraction R(a, b), the numerator is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='11) RN(a, b) = � k≥0 akqk(k+1)/2(−a−1b)k (q)k , and the denominator is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='12) RD(a, b) = � k≥0 akqk(k+1)/2(−a−1bq)k (q)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 12 GEOFFREY B CAMPBELL Proof : The expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='12) is an immediate consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='13) RD(a, b) = f(aq, a−1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' The expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='11) is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' To obtain it, observe that RN(a, b) = f(a, a−1b) − a f(aq, a−1b) = � k≥0 akqk(k−1)/2(−a−1bq)k (q)k − � k≥0 ak+1qk(k+1)/2(−a−1bq)k (q)k = 1 + � k≥0 ak+1qk(k+1)/2(−a−1bq)k (q)k �1 + a−1bqk+1 1 − qk+1 − 1 � = 1 + � k≥0 ak+1q(k+1)(k+2)/2(−a−1bq)k(1 − a−1b) (q)k+1 = � k≥0 akqk(k+1)/2(−a−1b)k (q)k as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' ■ Andrews (see [5] and [6]) considered the expansions in lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1 while discussing a transformation formula of Ramanujan [47] for R(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Our emphasis here is on the partition theorems that can be derived using R(a, b), and for this the following lemma is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' For the fraction R(a, b), we also have the expansions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='14) RN(a, b) = � i,j≥0 aibjq(i2+i)/2+ij+j2 (q)i(q)j , and the denominator is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='15) RD(a, b) = � i,j≥0 aibjq(i2+i)/2+ij+j2+j (q)i(q)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Proof : To obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='15) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='13) we use the q-binomial theorem, (−z)k = k � j=0 zjqj(j−1)/2 �k j � q with z = a−1b and z = a−1bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' (See Campbell [22] for the n-space q-binomial theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=') Therefore RN(a, b) = � k≥0 akqk(k+1)/2 (q)k k � j=0 a−jbjqj(j−1)/2(q)k (q)j(q)j−k = � i,j≥0 aibjq(i+j)(i+j+1)/2 (q)i(q)j , where i = k − j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' this is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' To obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='13), observe that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='16) RD(a, b) = RN(a, bq) by comparing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' CONTINUED FRACTION PARTITION IDENTITIES 13 The following two theorems relate successively to the numerator and the denom- inator of the fraction (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1), so then to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' For a proof of these see Alladi and Gordon [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' (Numerator) Let AN(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) be the number of partitions of n into i + j distinct red parts and j distinct blue parts such that one of the blue parts may be zero and every blue part is ≤ i + j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Let BN(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) be the number of partitions of n into i distinct red parts and j distinct non-consecutive blue parts such that every red part is > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Let CN(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) be the number of partitions of n into i red parts and j blue parts such that all parts are distinct and after each blue part there is a gap of at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Then AN(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) = BN(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) = CN(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' (Denominator) Let AD(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) be as in AN(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) except that every blue part is > 0 and ≤ i + j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Let BD(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) be as in BN(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) except that part 1 cannot be blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Let CD(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) be as in CN(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) except that part 1 cannot be blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Then AD(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) = BD(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j) = CD(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' So reprising (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='10) namely 1 + bq 1 + q2 + bq3 1 + q4 + bq5 1 + q6 + bq7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = ∞ � m=1 (1 + bq4m−3) (1 + bq4m−1)), we have interesting cancellations in numerator-denominator equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' That is, the numerator is given by � k≥0 qk(k+1)(−bq−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' q2)k (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' q2)k = ∞ � m=1 (1 + bq4m−3)(1 + q2m) = ∞ � m=1 (1 + bq4m−3)(1 + q4m−2)(1 + q4m) and the denominator is given by � k≥0 qk(k+1)(−bq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' q2)k (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' q2)k = ∞ � m=1 (1 + bq4m−1)(1 + q4m−2)(1 + q4m) with right sides having common factors that eliminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' This leads in particular to the continued fraction identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='17) 1 + q 1 + q2 + q3 1 + q4 + q5 1 + q6 + q7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = � j≡2,3,7 (mod8)(1 − qj) � j≡1,5,6 (mod8)(1 − qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 14 GEOFFREY B CAMPBELL G¨o11nitz [34] states similar results, but (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1) seems to have escaped attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' There is a continued fraction identity due to Gordon [33] and G¨o11nitz [34] which looks very similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='17), namely (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='18) 1 + q + q2 1 + q3 + q4 1 + q5 + q4 1 + q7 + bq6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = � j≡3,4,5 (mod8)(1 − qj) � j≡1,4,7 (mod8)(1 − qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' However, this result first appears in Alladi and Gordon [3] almost 30 years after (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Ramanujan’s three parameter continued fraction Ramanujan [45] obtained in addition to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1), the following continued fraction with three parameters a, b, q which has also a product representation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1) 1 − ab + (a − bq)(b − aq) (1 − ab)(1 + q2) + (a − bq3)(b − aq3) (1 − ab)(1 + q4) + (a − bq5)(b − aq5) (1 − ab)(1 + q6) + (a − bq7)(b − aq7) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' = ∞ � m=1 (1 + a2q4m−3)(1 + b2q4m−3) (1 + a2q4m−1)(1 + b2q4m−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' This was proved only in 1985 by the reviewers of Chapter 16 of Ramanujan’s Second Notebook [2], 65 years after Ramanujan’s death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' If we put a = 0 and replace b2 by −b in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='1), we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' It seems there is still scope to study the combinatorial properties of the coefficients in the power series expansion of this fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' References [1] ABRAMOWITZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=', and STEGUN, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Handbook of Mathematical Functions, Dover Publi- cations Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=', New York, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [2] ADIGA,C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' BERNDT,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='BHARGAVA,S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' AND WATSON,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' ”Chapter 16 of Ramanu- jan’s Second Notebook: Theta Functions and q-Series”, Memoirs of the American Mathemat- ical Society, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 315, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=', Providence, RI, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [3] ALLADI, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' and GORDON H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=', Partition Identities and a Continued Fraction of Ramanujan, Journal of Combinatorial Theory, Series A 63, 275-300 (1993) [4] ANDREWS, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' The Theory of Partitions, Addison-Wesley Publishing Company, Advanced Book Program, Reading, Massachusetts, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [5] ANDREWS,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' An introduction to Ramanujan’s ”lost” notebook, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Monthly 86 (1979), 89-108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [6] ANDREWS,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Ramanujan’s ”Lost” Notebbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' The Rogers-Ramanujan continued frac- tion, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 41 (1981), 186-208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [7] ANDREWS, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=', and BERNDT, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Ramanujan’s Lost Notebook: Part V Paperback (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Springer-Verlag, New York, ISBN-13: 978-3030085506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [8] ANDREWS, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' and ERIKSSON, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Integer Partitions, Cambridge University Press, Cam- bridge, UK, New York, USA, Port Melbourne, Australia, Madrid, Spain, Cape Town, South Africa, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' CONTINUED FRACTION PARTITION IDENTITIES 15 [9] APOSTOL, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [10] BAXTER, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [11] BIRKHOFF, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' and MACLAINE, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' A survey of modern algebra, fourth ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=', Macmillan, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [12] BRESSOUD, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' On a partition theorem of G¨ollnitz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=', Multipartitions and multipermutations, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 19, 1971, 37-39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [28] EULER, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} 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+page_content=' reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' G¨ott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=', Vol II;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' reprinted in Werke 3 (1876), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 123–162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [31] GOLDFELD, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Beyond the last theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Math Horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 4 (September): 26–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 16 GEOFFREY B CAMPBELL [34] G¨OLLNITZ, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Partitionen mit Differenzenbedingungen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 225 (1967), 154-190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' [35] HARDY, G.' metadata={'source': 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numbers, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' 28, 1956, 880- 890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content=' Mathematical Sciences Institute, The Australian National University, Canberra, ACT, 0200, Australia Email address: Geoffrey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='Campbell@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} +page_content='au' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFOT4oBgHgl3EQf3DT4/content/2301.12945v1.pdf'} diff --git a/PNE0T4oBgHgl3EQfjwHm/content/tmp_files/2301.02465v1.pdf.txt b/PNE0T4oBgHgl3EQfjwHm/content/tmp_files/2301.02465v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3681f59ce698605a67ca904fc2b52a5bedce97e --- /dev/null +++ b/PNE0T4oBgHgl3EQfjwHm/content/tmp_files/2301.02465v1.pdf.txt @@ -0,0 +1,711 @@ +Direct electrical probing of anomalous Nernst conductivity +Weinan Zhou,1, ∗ Asuka Miura,2, † Yuya Sakuraba,2 and Ken-ichi Uchida2, 3, ‡ +1International Center for Young Scientists, National Institute for Materials Science, Tsukuba 305-0047, Japan +2Research Center for Magnetic and Spintronic Materials, +National Institute for Materials Science, Tsukuba 305-0047, Japan +3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan +Despite the usefulness of the anomalous Nernst conductivity (αA +xy) for studying electronic band +structures and exploring magnetic materials with large transverse thermopower, there has not been a +straightforward way to obtain αA +xy in the experiment. Here, we propose a simple and versatile method +enabling direct electrical probing of αA +xy, which is realized by creating a closed circuit consisting +of a target magnetic material and a non-magnetic conductor. +This method was experimentally +demonstrated on a thin film of magnetic Weyl semimetal Co2MnGa, where the closed circuit was +formed simply by connecting both ends of the Co2MnGa film with a Au wire. A good approximation +of αA +xy was obtained, validating the proposed method and exhibiting its potential for aiding the +further development of topological materials science and transverse thermoelectrics. +The anomalous Nernst conductivity, i.e., the off- +diagonal component of the thermoelectric conductivity +tensor (αA +xy) stemming from magnetic moments, de- +scribes an intrinsic material property that directly con- +verts a longitudinal temperature gradient into a trans- +verse electric field in a magnetic material. It has been +shown that αA +xy is closely linked to the Berry curvature +of the electronic bands; in comparison with the anoma- +lous Hall conductivity, which is determined by all oc- +cupied bands, αA +xy can be more sensitive to the elec- +tronic band structures close to the Fermi level, rendering +it a valuable tool to study the topological features of +magnetic materials through transport measurements [1– +18]. In addition to this rapidly increasing interest from +the viewpoint of fundamental physics, αA +xy is regarded +as a crucial parameter to explain unconventionally large +transverse thermoelectric output in some magnetic mate- +rials where intrinsic contribution plays a dominant role. +Therefore, exploring magnetic materials with large val- +ues of αA +xy has become a major strategy for thermoelec- +tric applications [19–23]. Due to the orthogonal relation- +ship between the applied temperature gradient and gen- +erated electric field, the transverse thermoelectric gen- +eration module can be a simple slab or sheet, where no +complicated three-dimensional structures are necessary +unlike conventional Seebeck-effect-based modules. Thus, +transverse thermoelectric modules could potentially cir- +cumvent the problems of durability, flexibility, and cost +that the Seebeck modules encounter [22–26], as well as +be exploited for additional functionalities, such as heat +flux sensing [23, 25, 27, 28]. Despite the significant role of +αA +xy in topological materials science and transverse ther- +moelectrics, there has not been a straightforward way +to experimentally obtain αA +xy, and establishing such a +method is of great importance. +The conventional experimental method for estimat- +ing αA +xy consists of the measurements of the anomalous +Nernst effect (ANE), anomalous Hall effect (AHE), See- +beck effect (SE), and electrical resistivity of a magnetic +material. The anomalous Nernst coefficient (SANE), i.e., +the transverse thermopower due to ANE, is expressed as +SANE = ρxxαA +xy − ρAHEαxx, +(1) +where ρxx, ρAHE, and αxx are the longitudinal resistivity, +anomalous Hall resistivity, and diagonal component of +the thermoelectric conductivity tensor, respectively. The +first term on the right-hand side of Eq. (1) (SI = ρxxαA +xy) +is regarded as an intrinsic component of ANE, while the +second term appears as a consequence of AHE acting on +the longitudinal electric field induced by SE, which can +be rewritten as SII = −SSEρAHE/ρxx [Fig. 1(a)] with SSE +being the Seebeck coefficient. As a result, αA +xy is obtained +by experimentally measuring all four parameters of ρxx, +ρAHE, SSE, and SANE, then calculating using Eq. (1). +Many studies have exploited this conventional method +to obtain αA +xy of a variety of magnetic materials [2–5, 7– +FIG. 1. +(a) Schematic illustration of ANE in a magnetic +material. The orange and green arrows represent the contri- +bution from the SI and SII terms of SANE, while the black +arrow represents the direction of magnetization (M). The + +and − symbols indicate the accumulated electric charges due +to SE and ANE. (b) Schematic illustration of the closed circuit +in which a magnetic material (cyan) is electrically connected +to a non-magnetic conductor (gray) at both ends along the +direction of the applied temperature gradient (∇T). +arXiv:2301.02465v1 [cond-mat.mtrl-sci] 6 Jan 2023 + +(a) +(b) +S = -SsE PAHE IPxx +e +M +e +VT +++++++ +S +e +e2 +19, 22, 23, 25, 28]. However, such a task could be cum- +bersome, and sometimes challenging to complete, since +it requires various experimental techniques and measure- +ment systems. +In this study, we propose a method to directly mea- +sure the intrinsic component of ANE of a magnetic ma- +terial and probe its αA +xy with ease. This method is real- +ized simply by creating a closed circuit consisting of the +target magnetic material and a non-magnetic conductor, +and then measuring transverse thermopower, as shown +in Fig. 1(b). The formation of the closed circuit tunes +the boundary conditions for electron transport, resulting +in the direct emergence of αA +xy reflecting the Berry curva- +ture in the transverse thermopower. We experimentally +demonstrated this method using a Co2MnGa thin film, +and compared the result with the value of αA +xy obtained +using the conventional method. The proposed method +grants easy access to αA +xy, and could be a useful tool in +studying topological features and transverse thermoelec- +tric conversion properties of magnetic materials. +When a magnetic material is electrically connected to +a non-magnetic conductor at both ends along the direc- +tion of the applied temperature gradient (∇T), a closed +circuit is formed, and its total transverse thermopower +measured at the magnetic material (Sy +tot) is derived to +be [29, 30] +Sy +tot = SANE − +ρAHE +ρC/r + ρM +(SC − SM). +(2) +Here, ρC(M) and SC(M) are the longitudinal resistivity +and Seebeck coefficient of the non-magnetic conductor +(magnetic material), respectively. The size ratio r is de- +termined by the geometry of the closed circuit, and in +this case, can be expressed as r = (LM/LC) × (AC/AM), +where LC(M) is the length of the non-magnetic conduc- +tor (magnetic material) along the closed circuit [x axis +in Fig. 1(b)] and AC(M) is the cross-section area of the +non-magnetic conductor (magnetic material) perpendic- +ular to the LC(M) direction [yz plane in Fig. 1(b)]. Pre- +viously, thermoelectric materials have been connected to +magnetic materials to create closed circuits in order to +generate large transverse thermopower [29, 31], which is +referred to as the Seebeck-driven transverse thermoelec- +tric generation. However, Eq. (2) is still valid when a +non-magnetic conductor having negligible SE is used in- +stead of thermoelectric materials. If |SC| ≪ |SM| and we +make ρC/r ≪ ρM through small ρC, large r, or both, the +second term on the right-hand side of Eq. (2) is reduced +to SMρAHE/ρM. By substituting Eq. (1) into Eq. (2), the +SII term in SANE is canceled out, leaving only the SI term +in Sy +tot [Fig. 1(b)]. In other words, SE of the magnetic +material is shunted by connecting to the non-magnetic +conductor, leading to the disappearance of the SII term. +Then, αA +xy can be easily obtained as +αA +xy ≈ Sy +tot +ρM +. +(3) +FIG. 2. +(a) Schematic illustration of the sample structure +and measurement setup for the experimental demonstration +of the proposed method to directly probe αA +xy. V1, V2, V3, and +V4 represent four nanovoltmeters measuring the longitudinal +thermoelectric signal, transverse thermoelectric signal, and +resistance of two Pt wires, respectively. (b), (c) H dependence +of the transverse electric field (Ey) divided by ∇T for the +closed-circuit sample (b) and the reference sample (c). (d) H +dependence of the transverse resistivity (ρyx) of the reference +sample, showing AHE of Co2MnGa. +(e) H dependence of +the voltage from V1 of the closed-circuit (blue diamond) and +reference (red square) samples. The magneto-Seebeck effect +[32] in Co2MnGa was found to be negligibly small. +In comparison with the conventional method based on +Eq. (1), the method proposed here reduces the required +parameters for obtaining αA +xy from four to two. If ρM is +known, a simple measurement of Sy +tot in the closed circuit +enables the direct probing of αA +xy. +We experimentally demonstrated the proposed method +using a Co2MnGa thin film. +We chose Co2MnGa be- +cause it is known as a magnetic Weyl semimetal hav- +ing substantial SI and SII terms contributing to its large +SANE [7, 11, 14, 15]. The 26-nm-thick Co2MnGa thin +film was epitaxially deposited on a single crystal MgO +(100) substrate at room temperature by magnetron sput- + +(a) +H +V +Au bonding wire +Co2MnGa +Au electrode +MgO substrate +Pt wire +b +E*/VT(μVK-1) +K-1 +2 +2 +(μV +0 +0 +2 +-2 +3 +2 +1 +0 +2 +3 +-3 +-2 +-1 +0 +1 +2 +μoH (T) +HoH (T) +20 +-135 +(d) +(e) +-130 +10 +Pyx (μQ cm) +(μV) +-125 ++ Reference +0 ++ Closed circuit +V +-10 +-10 +-5 +-20 +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +-3 +3 +μoH (T) +μoH (T)3 +tering, followed by post annealing at 500◦C. After the +sample was cooled down to room temperature, a 2-nm- +thick Al capping layer was deposited to prevent oxidiza- +tion. The composition of Co2MnGa was determined to be +Co45.7Mn25.4Ga28.9 by X-ray fluorescence spectroscopy. +The 111 superlattice peak of Co2MnGa was confirmed +in the X-ray diffraction pattern, indicating the forma- +tion of L21 atomic ordering. +Then, we patterned the +Co2MnGa film into a 2-mm-wide and 8-mm-long Hall +bar structure using photolithography and Ar ion milling, +followed by the formation of Au electrodes through a lift- +off process. On-chip thermometers made of Pt wires were +subsequently formed through a lift-off process at the po- +sitions corresponding to the electrodes of the Hall bar +along the x axis, as shown in Fig. 2(a). In order to cre- +ate the closed circuit, we simply connected both ends of +the the Co2MnGa film along the x axis with a 30-µm- +diameter Au bonding wire. Here, the Co2MnGa is the +magnetic material under study, while the Au wire serves +as the non-magnetic conductor. The electrical resistivity +of Au wire is 2.3 µΩ cm at room temperature, two orders +of magnitude smaller than that of the Co2MnGa film, +which was measured to be ρM = 222.589±0.001 µΩ cm. +Meanwhile, we assumed a 30-µm-diameter circle as AC, +and estimated LC = 12 mm for the Au wire, leading to +estimation of r = 7. Together with SC = 2.0 µV K−1 of +Au [33] and experimentally measured SM = −32.7 ± 0.2 +µV K−1 for Co2MnGa, the close circuit satisfies the as- +sumptions of |SC| ≪ |SM| and ρC/r ≪ ρM for Eq. (3). +To measure the transverse thermopower, we set the sam- +ple on a home-made holder, where one side of the sample +was thermally connected to a Cu block then to a heat +sink while the other side was thermally connected to a +heater and insulated from the heat sink by a bakelite +plate, similar to the one used in Ref. 34. When a charge +current is applied to the heater, ∇T along the x axis +is generated in the sample. To evaluate ∇T, we placed +the holder in a physical property measurement system +(PPMS; Quantum Design), and first calibrated the on- +chip thermometers by measuring the resistance of the Pt +wires as a function of temperature using the four-terminal +method under zero magnetic field (H). Then, we set the +temperature of PPMS at 295 K, applied the current to +the heater, and swept H along the z axis while monitor- +ing the longitudinal and transverse thermoelectric signals +from the closed circuit with two nanovoltmeters, V1 and +V2, respectively. The measured resistance of the Pt wires +during the sweep of H was used to obtain ∇T. As a ref- +erence, the same measuring process was carried out with- +out the Au wire connecting both ends of the Co2MnGa +film; this is the conventional ANE measurement. +The +average temperature and ∇T of the closed-circuit (ref- +erence) sample were 302.56±0.02 (302.01±0.02) K and +0.977±0.005 (0.937±0.004) K mm−1, respectively. For +the reference sample, the ρM and ρAHE were separately +measured at room temperature. +FIG. 3. +(a) SANE and SI of the reference sample in compar- +ison with Sy +tot of the closed-circuit sample. (b) αA +xy obtained +using the conventional method and Sy +tot/ρM, which approxi- +mately corresponds to αA +xy through Eq. (3). +Figures 2(b) and 2(c) show the H dependence of the +transverse electric field (Ey) divided by ∇T for the +closed-circuit and reference samples, respectively. +The +observed signal of the reference sample showed the H- +odd dependence and saturation at |µ0H| ∼ 1 T, which +is attributed to ANE of Co2MnGa in the open circuit +condition. By contrast, the signal of the closed-circuit +sample is smaller than that of the reference sample, al- +though the shapes of the H dependence of the signals +are similar to each other. The curve in Fig. 2(b) also +saturates at |µ0H| ∼ 1 T along the z axis, suggesting +the transverse thermopower of the closed-circuit sample +is determined by the magnetization (M) of Co2MnGa +as well. Figure 2(d) shows the H dependence of ρyx of +Co2MnGa measured using the reference sample, where +the signal is mostly due to AHE of Co2MnGa. The Sy +tot, +SANE, and ρAHE values were evaluated by extrapolating +the curves in Figs. 2(b)-2(d) at high H after the sat- +uration of M down to zero H. Figure 2(e) shows the +longitudinal thermopower from V1 measured at the same +time when the results in Figs. 2(b) and 2(c) were ob- +tained. In case of the reference sample, this voltage was +due to SE of the Co2MnGa-Au thermocouple (note that +similar Au bonding wires were used to connect the elec- +trodes of the sample to the home-made holder), and SM +can be calculated by dividing the voltage at zero H with +the corresponding temperature difference then adding SC +of Au. On the other hand, the magnitude of the longitu- +dinal thermopower of the closed-circuit sample was dra- +matically reduced, indicating that SE of Co2MnGa was +indeed shunted by the connection to the Au wire at both +ends. +By applying Eq. (3) to the experimental results of +the closed-circuit sample, we were able to probe αA +xy +of Co2MnGa with ease. The values obtained using the +proposed method and the conventional method are com- +pared in Fig. 3. +SANE of Co2MnGa was estimated to +be 4.09±0.02 µV K−1, consistent with the previously re- +ported result of the sample having similar composition + +5 +1.4 +a +(b) +1.2 +1.0 +3 +0.8 +0.6 +2 +0.4 +0.2 +0 +0 +S, +PANE +xy4 +FIG. 4. +Size ratio r dependence of Sy +tot calculated using +Eq. (2) (cyan line) in comparison with SI of Co2MnGa ob- +tained in the experiment (black dashed line). +Sy +tot of the +closed-circuit sample (blue circle) is also plotted at the corre- +sponding r. +[15]. Meanwhile, Sy +tot = 1.89±0.01 µV K−1 of the closed +circuit is smaller than SANE, but comparable to its SI = +2.01±0.02 µV K−1 [Fig. 3(a)]. For αA +xy, the value based +on Eq. (3) was calculated to be 0.848±0.005 A m−1 K−1, +while 0.905±0.010 A m−1 K−1 was obtained using Eq. (1) +of the conventional method [Fig. 3(b)]. As one can see, +the proposed method exhibits a close approximation of +αA +xy, although the value is slightly smaller than that ob- +tained from the conventional method: the difference is +∼6%. To understand this difference, we calculated Sy +tot of +the closed circuit as a function of r using Eq. (2) and ma- +terial parameters of Co2MnGa and Au, then compared +it with the SI term from the conventional method, as +shown in Fig. 4. The experimentally measured Sy +tot is +also plotted at its corresponding r = 7. One can see a +quantitative agreement in Sy +tot between the experiment +and calculation. As r increases, the calculated Sy +tot de- +creases from the initial value ∼SANE of Co2MnGa down +to ∼Sy +tot measured in the experiment. The tendency of +the curve suggests that the r of the closed circuit used +for the demonstration is large enough to neglect the in- +fluence of ρC. On the other hand, the difference between +the calculated Sy +tot and SI at large r is attributed to finite +SC of Au. The Sy +tot value being slightly smaller than SI is +consistent with the fact that SC of Au is positive and op- +posite to SM of Co2MnGa in sign. These results indicate +that we should be mindful to the Seebeck coefficient of +the magnetic material and non-magnetic conductor while +using the proposed method, as SC being much smaller +in magnitude than SM is important to achieve a better +approximation. A non-magnetic conductor having zero +SC would be an ideal material for the proposed method, +which could further reduce the difference in αA +xy. +As shown above, the proposed method can be eas- +ily implemented in the experiment to directly measure +the SI term of a magnetic thin film and probe its αA +xy. +While multiple measurement setups are required to use +the conventional method and evaluate the material pa- +rameters in Eq. (1), the proposed method can be carried +out mostly on one setup. This would lead to better relia- +bility and reproductivity of the results as well as consid- +erable time and effort saving for the experiment, which +is especially beneficial for high-throughput materials re- +search. In addition, using the first-principles calculations +to obtain the Berry curvature and derive αA +xy has been +popularized in recent years and plays an important role in +exploiting and predicting materials with valuable prop- +erties. The proposed method could make αA +xy a direct +observable in the experiment, thereby enabling fast and +straightforward comparison with the theory and promot- +ing further understanding of the matter. It is worth men- +tioning that although the experimental demonstration +was done on a magnetic thin film, the proposed method +should also be applicable to study bulk materials, as long +as the assumptions of |SC| ≪ |SM| and ρC/r ≪ ρM for +Eq. (3) are satisfied. +In summary, we have proposed a method to directly +probe αA +xy of a magnetic material, which is realized sim- +ply by connecting both ends of the magnetic material +along the direction of ∇T with a non-magnetic conduc- +tor to create a closed circuit. Sy +tot of the closed circuit +approximates the SI term of the magnetic material, and +αA +xy can be easily obtained from Sy +tot and ρM, in con- +trast to four different parameters required in the conven- +tional method. The proposed method was experimentally +demonstrated to probe αA +xy of a Co2MnGa thin film. The +closed circuit was easily realized using a Au wire, and a +good approximation was obtained for both SI and αA +xy, +validating this method. Further analysis of the results +revealed that the small difference was due to finite SC, +and provided guides for the utilization of the proposed +method. As the popularity of using αA +xy is growing, our +finding could become a powerful tool propelling studies +of topological materials science and application of trans- +verse thermoelectric phenomena. +The authors thank R. Toyama and T. Hirai for their +support in sample preparation and measurement. This +work was supported by JST CREST “Creation of In- +novative Core Technologies for Nano-enabled Thermal +Management” (Grant No. JPMJCR17I1), JST ERATO +“Magnetic Thermal Management Materials” (Grant No. +JPMJER2201), JSPS KAKENHI Grant-in-Aid for Sci- +entific Research (B) (Grant No. +JP21H01608) and +Grant-in-Aid for Research Activity Start-up (Grant No. +JP22K20494), and NEC Corporation. +∗ ZHOU.Weinan@nims.go.jp +† Present address: +Integrated Research for Energy and +Environment Advanced Technology, Kyushu Institute of +Technology, Fukuoka 804-8550, Japan +‡ UCHIDA.Kenichi@nims.go.jp +[1] D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Berry-phase effect + +5 +4 +t (μV K-1) +3 +S +2 +0 +102 +100 +102 +10° +Size ratio r5 +in anomalous thermoelectric transport, Phys. Rev. Lett. +97, 026603 (2006). +[2] T. Miyasato, N. Abe, T. Fujii, A. Asamitsu, S. Onoda, +Y. Onose, N. Nagaosa, and Y. 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Seki, Y. Sakuraba, +T. +Kubota, +K. +Ito, +and +K. +Takanashi, +Strain- +induced large anomalous Nernst effect in polycrystalline +Co2MnGa/AlN multilayers, Adv. Electron. Mater. 8, +2101380 (2022). + diff --git a/PNE0T4oBgHgl3EQfjwHm/content/tmp_files/load_file.txt b/PNE0T4oBgHgl3EQfjwHm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e4a35048c0cf5616b891ac0b911eff7fd1f4adc --- /dev/null +++ b/PNE0T4oBgHgl3EQfjwHm/content/tmp_files/load_file.txt @@ -0,0 +1,602 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf,len=601 +page_content='Direct electrical probing of anomalous Nernst conductivity Weinan Zhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' ∗ Asuka Miura,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' † Yuya Sakuraba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='2 and Ken-ichi Uchida2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' ‡ 1International Center for Young Scientists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Tsukuba 305-0047,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Japan 2Research Center for Magnetic and Spintronic Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Tsukuba 305-0047,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Japan 3Institute for Materials Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Sendai 980-8577,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Japan Despite the usefulness of the anomalous Nernst conductivity (αA xy) for studying electronic band structures and exploring magnetic materials with large transverse thermopower,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' there has not been a straightforward way to obtain αA xy in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Here, we propose a simple and versatile method enabling direct electrical probing of αA xy, which is realized by creating a closed circuit consisting of a target magnetic material and a non-magnetic conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' This method was experimentally demonstrated on a thin film of magnetic Weyl semimetal Co2MnGa, where the closed circuit was formed simply by connecting both ends of the Co2MnGa film with a Au wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' A good approximation of αA xy was obtained, validating the proposed method and exhibiting its potential for aiding the further development of topological materials science and transverse thermoelectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The anomalous Nernst conductivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=', the off- diagonal component of the thermoelectric conductivity tensor (αA xy) stemming from magnetic moments, de- scribes an intrinsic material property that directly con- verts a longitudinal temperature gradient into a trans- verse electric field in a magnetic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' It has been shown that αA xy is closely linked to the Berry curvature of the electronic bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' in comparison with the anoma- lous Hall conductivity, which is determined by all oc- cupied bands, αA xy can be more sensitive to the elec- tronic band structures close to the Fermi level, rendering it a valuable tool to study the topological features of magnetic materials through transport measurements [1– 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' In addition to this rapidly increasing interest from the viewpoint of fundamental physics, αA xy is regarded as a crucial parameter to explain unconventionally large transverse thermoelectric output in some magnetic mate- rials where intrinsic contribution plays a dominant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Therefore, exploring magnetic materials with large val- ues of αA xy has become a major strategy for thermoelec- tric applications [19–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Due to the orthogonal relation- ship between the applied temperature gradient and gen- erated electric field, the transverse thermoelectric gen- eration module can be a simple slab or sheet, where no complicated three-dimensional structures are necessary unlike conventional Seebeck-effect-based modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Thus, transverse thermoelectric modules could potentially cir- cumvent the problems of durability, flexibility, and cost that the Seebeck modules encounter [22–26], as well as be exploited for additional functionalities, such as heat flux sensing [23, 25, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Despite the significant role of αA xy in topological materials science and transverse ther- moelectrics, there has not been a straightforward way to experimentally obtain αA xy, and establishing such a method is of great importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The conventional experimental method for estimat- ing αA xy consists of the measurements of the anomalous Nernst effect (ANE), anomalous Hall effect (AHE), See- beck effect (SE), and electrical resistivity of a magnetic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The anomalous Nernst coefficient (SANE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=', the transverse thermopower due to ANE, is expressed as SANE = ρxxαA xy − ρAHEαxx, (1) where ρxx, ρAHE, and αxx are the longitudinal resistivity, anomalous Hall resistivity, and diagonal component of the thermoelectric conductivity tensor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The first term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (1) (SI = ρxxαA xy) is regarded as an intrinsic component of ANE, while the second term appears as a consequence of AHE acting on the longitudinal electric field induced by SE, which can be rewritten as SII = −SSEρAHE/ρxx [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 1(a)] with SSE being the Seebeck coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' As a result, αA xy is obtained by experimentally measuring all four parameters of ρxx, ρAHE, SSE, and SANE, then calculating using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Many studies have exploited this conventional method to obtain αA xy of a variety of magnetic materials [2–5, 7– FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (a) Schematic illustration of ANE in a magnetic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The orange and green arrows represent the contri- bution from the SI and SII terms of SANE, while the black arrow represents the direction of magnetization (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The + and − symbols indicate the accumulated electric charges due to SE and ANE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (b) Schematic illustration of the closed circuit in which a magnetic material (cyan) is electrically connected to a non-magnetic conductor (gray) at both ends along the direction of the applied temperature gradient (∇T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='02465v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='mtrl-sci] 6 Jan 2023 (a) (b) S = -SsE PAHE IPxx e M e VT ++++++ S e e2 19, 22, 23, 25, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' However, such a task could be cum- bersome, and sometimes challenging to complete, since it requires various experimental techniques and measure- ment systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' In this study, we propose a method to directly mea- sure the intrinsic component of ANE of a magnetic ma- terial and probe its αA xy with ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' This method is real- ized simply by creating a closed circuit consisting of the target magnetic material and a non-magnetic conductor, and then measuring transverse thermopower, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The formation of the closed circuit tunes the boundary conditions for electron transport, resulting in the direct emergence of αA xy reflecting the Berry curva- ture in the transverse thermopower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' We experimentally demonstrated this method using a Co2MnGa thin film, and compared the result with the value of αA xy obtained using the conventional method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The proposed method grants easy access to αA xy, and could be a useful tool in studying topological features and transverse thermoelec- tric conversion properties of magnetic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' When a magnetic material is electrically connected to a non-magnetic conductor at both ends along the direc- tion of the applied temperature gradient (∇T), a closed circuit is formed, and its total transverse thermopower measured at the magnetic material (Sy tot) is derived to be [29, 30] Sy tot = SANE − ρAHE ρC/r + ρM (SC − SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (2) Here, ρC(M) and SC(M) are the longitudinal resistivity and Seebeck coefficient of the non-magnetic conductor (magnetic material), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The size ratio r is de- termined by the geometry of the closed circuit, and in this case, can be expressed as r = (LM/LC) × (AC/AM), where LC(M) is the length of the non-magnetic conduc- tor (magnetic material) along the closed circuit [x axis in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 1(b)] and AC(M) is the cross-section area of the non-magnetic conductor (magnetic material) perpendic- ular to the LC(M) direction [yz plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Pre- viously, thermoelectric materials have been connected to magnetic materials to create closed circuits in order to generate large transverse thermopower [29, 31], which is referred to as the Seebeck-driven transverse thermoelec- tric generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' However, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (2) is still valid when a non-magnetic conductor having negligible SE is used in- stead of thermoelectric materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' If |SC| ≪ |SM| and we make ρC/r ≪ ρM through small ρC, large r, or both, the second term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (2) is reduced to SMρAHE/ρM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' By substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (1) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (2), the SII term in SANE is canceled out, leaving only the SI term in Sy tot [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' In other words, SE of the magnetic material is shunted by connecting to the non-magnetic conductor, leading to the disappearance of the SII term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Then, αA xy can be easily obtained as αA xy ≈ Sy tot ρM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (3) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (a) Schematic illustration of the sample structure and measurement setup for the experimental demonstration of the proposed method to directly probe αA xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' V1, V2, V3, and V4 represent four nanovoltmeters measuring the longitudinal thermoelectric signal, transverse thermoelectric signal, and resistance of two Pt wires, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (b), (c) H dependence of the transverse electric field (Ey) divided by ∇T for the closed-circuit sample (b) and the reference sample (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (d) H dependence of the transverse resistivity (ρyx) of the reference sample, showing AHE of Co2MnGa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (e) H dependence of the voltage from V1 of the closed-circuit (blue diamond) and reference (red square) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The magneto-Seebeck effect [32] in Co2MnGa was found to be negligibly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' In comparison with the conventional method based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (1), the method proposed here reduces the required parameters for obtaining αA xy from four to two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' If ρM is known, a simple measurement of Sy tot in the closed circuit enables the direct probing of αA xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' We experimentally demonstrated the proposed method using a Co2MnGa thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' We chose Co2MnGa be- cause it is known as a magnetic Weyl semimetal hav- ing substantial SI and SII terms contributing to its large SANE [7, 11, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The 26-nm-thick Co2MnGa thin film was epitaxially deposited on a single crystal MgO (100) substrate at room temperature by magnetron sput- (a) H V Au bonding wire Co2MnGa Au electrode MgO substrate Pt wire b E*/VT(μVK-1) K-1 2 2 (μV 0 0 2 2 3 2 1 0 2 3 3 2 1 0 1 2 μoH (T) HoH (T) 20 135 (d) (e) 130 10 Pyx (μQ cm) (μV) 125 + Reference 0 + Closed circuit V 10 10 5 20 2 1 0 1 2 3 3 2 1 0 1 2 3 3 μoH (T) μoH (T)3 tering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' followed by post annealing at 500◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' After the sample was cooled down to room temperature, a 2-nm- thick Al capping layer was deposited to prevent oxidiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The composition of Co2MnGa was determined to be Co45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='7Mn25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='4Ga28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='9 by X-ray fluorescence spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The 111 superlattice peak of Co2MnGa was confirmed in the X-ray diffraction pattern, indicating the forma- tion of L21 atomic ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Then, we patterned the Co2MnGa film into a 2-mm-wide and 8-mm-long Hall bar structure using photolithography and Ar ion milling, followed by the formation of Au electrodes through a lift- off process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' On-chip thermometers made of Pt wires were subsequently formed through a lift-off process at the po- sitions corresponding to the electrodes of the Hall bar along the x axis, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' In order to cre- ate the closed circuit, we simply connected both ends of the the Co2MnGa film along the x axis with a 30-µm- diameter Au bonding wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Here, the Co2MnGa is the magnetic material under study, while the Au wire serves as the non-magnetic conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The electrical resistivity of Au wire is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='3 µΩ cm at room temperature, two orders of magnitude smaller than that of the Co2MnGa film, which was measured to be ρM = 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='589±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='001 µΩ cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Meanwhile, we assumed a 30-µm-diameter circle as AC, and estimated LC = 12 mm for the Au wire, leading to estimation of r = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Together with SC = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='0 µV K−1 of Au [33] and experimentally measured SM = −32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='2 µV K−1 for Co2MnGa, the close circuit satisfies the as- sumptions of |SC| ≪ |SM| and ρC/r ≪ ρM for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' To measure the transverse thermopower, we set the sam- ple on a home-made holder, where one side of the sample was thermally connected to a Cu block then to a heat sink while the other side was thermally connected to a heater and insulated from the heat sink by a bakelite plate, similar to the one used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' When a charge current is applied to the heater, ∇T along the x axis is generated in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' To evaluate ∇T, we placed the holder in a physical property measurement system (PPMS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Quantum Design), and first calibrated the on- chip thermometers by measuring the resistance of the Pt wires as a function of temperature using the four-terminal method under zero magnetic field (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Then, we set the temperature of PPMS at 295 K, applied the current to the heater, and swept H along the z axis while monitor- ing the longitudinal and transverse thermoelectric signals from the closed circuit with two nanovoltmeters, V1 and V2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The measured resistance of the Pt wires during the sweep of H was used to obtain ∇T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' As a ref- erence, the same measuring process was carried out with- out the Au wire connecting both ends of the Co2MnGa film;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' this is the conventional ANE measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The average temperature and ∇T of the closed-circuit (ref- erence) sample were 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='56±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='02 (302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='02) K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='977±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='937±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='004) K mm−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' For the reference sample, the ρM and ρAHE were separately measured at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (a) SANE and SI of the reference sample in compar- ison with Sy tot of the closed-circuit sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (b) αA xy obtained using the conventional method and Sy tot/ρM, which approxi- mately corresponds to αA xy through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Figures 2(b) and 2(c) show the H dependence of the transverse electric field (Ey) divided by ∇T for the closed-circuit and reference samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The observed signal of the reference sample showed the H- odd dependence and saturation at |µ0H| ∼ 1 T, which is attributed to ANE of Co2MnGa in the open circuit condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' By contrast, the signal of the closed-circuit sample is smaller than that of the reference sample, al- though the shapes of the H dependence of the signals are similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 2(b) also saturates at |µ0H| ∼ 1 T along the z axis, suggesting the transverse thermopower of the closed-circuit sample is determined by the magnetization (M) of Co2MnGa as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Figure 2(d) shows the H dependence of ρyx of Co2MnGa measured using the reference sample, where the signal is mostly due to AHE of Co2MnGa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The Sy tot, SANE, and ρAHE values were evaluated by extrapolating the curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 2(b)-2(d) at high H after the sat- uration of M down to zero H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Figure 2(e) shows the longitudinal thermopower from V1 measured at the same time when the results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 2(b) and 2(c) were ob- tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' In case of the reference sample, this voltage was due to SE of the Co2MnGa-Au thermocouple (note that similar Au bonding wires were used to connect the elec- trodes of the sample to the home-made holder), and SM can be calculated by dividing the voltage at zero H with the corresponding temperature difference then adding SC of Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' On the other hand, the magnitude of the longitu- dinal thermopower of the closed-circuit sample was dra- matically reduced, indicating that SE of Co2MnGa was indeed shunted by the connection to the Au wire at both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' By applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (3) to the experimental results of the closed-circuit sample, we were able to probe αA xy of Co2MnGa with ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The values obtained using the proposed method and the conventional method are com- pared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' SANE of Co2MnGa was estimated to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='02 µV K−1, consistent with the previously re- ported result of the sample having similar composition 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='4 a (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='0 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='6 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='2 0 0 S, PANE xy4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Size ratio r dependence of Sy tot calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (2) (cyan line) in comparison with SI of Co2MnGa ob- tained in the experiment (black dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Sy tot of the closed-circuit sample (blue circle) is also plotted at the corre- sponding r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Meanwhile, Sy tot = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='01 µV K−1 of the closed circuit is smaller than SANE, but comparable to its SI = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='02 µV K−1 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' For αA xy, the value based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (3) was calculated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='848±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='005 A m−1 K−1, while 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='905±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='010 A m−1 K−1 was obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (1) of the conventional method [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 3(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' As one can see, the proposed method exhibits a close approximation of αA xy, although the value is slightly smaller than that ob- tained from the conventional method: the difference is ∼6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' To understand this difference, we calculated Sy tot of the closed circuit as a function of r using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (2) and ma- terial parameters of Co2MnGa and Au, then compared it with the SI term from the conventional method, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The experimentally measured Sy tot is also plotted at its corresponding r = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' One can see a quantitative agreement in Sy tot between the experiment and calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' As r increases, the calculated Sy tot de- creases from the initial value ∼SANE of Co2MnGa down to ∼Sy tot measured in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The tendency of the curve suggests that the r of the closed circuit used for the demonstration is large enough to neglect the in- fluence of ρC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' On the other hand, the difference between the calculated Sy tot and SI at large r is attributed to finite SC of Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The Sy tot value being slightly smaller than SI is consistent with the fact that SC of Au is positive and op- posite to SM of Co2MnGa in sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' These results indicate that we should be mindful to the Seebeck coefficient of the magnetic material and non-magnetic conductor while using the proposed method, as SC being much smaller in magnitude than SM is important to achieve a better approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' A non-magnetic conductor having zero SC would be an ideal material for the proposed method, which could further reduce the difference in αA xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' As shown above, the proposed method can be eas- ily implemented in the experiment to directly measure the SI term of a magnetic thin film and probe its αA xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' While multiple measurement setups are required to use the conventional method and evaluate the material pa- rameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (1), the proposed method can be carried out mostly on one setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' This would lead to better relia- bility and reproductivity of the results as well as consid- erable time and effort saving for the experiment, which is especially beneficial for high-throughput materials re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' In addition, using the first-principles calculations to obtain the Berry curvature and derive αA xy has been popularized in recent years and plays an important role in exploiting and predicting materials with valuable prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The proposed method could make αA xy a direct observable in the experiment, thereby enabling fast and straightforward comparison with the theory and promot- ing further understanding of the matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' It is worth men- tioning that although the experimental demonstration was done on a magnetic thin film, the proposed method should also be applicable to study bulk materials, as long as the assumptions of |SC| ≪ |SM| and ρC/r ≪ ρM for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' (3) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' In summary, we have proposed a method to directly probe αA xy of a magnetic material, which is realized sim- ply by connecting both ends of the magnetic material along the direction of ∇T with a non-magnetic conduc- tor to create a closed circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Sy tot of the closed circuit approximates the SI term of the magnetic material, and αA xy can be easily obtained from Sy tot and ρM, in con- trast to four different parameters required in the conven- tional method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The proposed method was experimentally demonstrated to probe αA xy of a Co2MnGa thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The closed circuit was easily realized using a Au wire, and a good approximation was obtained for both SI and αA xy, validating this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Further analysis of the results revealed that the small difference was due to finite SC, and provided guides for the utilization of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' As the popularity of using αA xy is growing, our finding could become a powerful tool propelling studies of topological materials science and application of trans- verse thermoelectric phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' The authors thank R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Toyama and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' Hirai for their support in sample preparation and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' This work was supported by JST CREST “Creation of In- novative Core Technologies for Nano-enabled Thermal Management” (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' JPMJCR17I1), JST ERATO “Magnetic Thermal Management Materials” (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' JPMJER2201), JSPS KAKENHI Grant-in-Aid for Sci- entific Research (B) (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' JP21H01608) and Grant-in-Aid for Research Activity Start-up (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' JP22K20494), and NEC Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content=' ∗ ZHOU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='Weinan@nims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} +page_content='jp † Present address: Integrated Research for Energy and Environment Advanced Technology, Kyushu Institute of Technology, Fukuoka 804-8550, Japan ‡ UCHIDA.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQfjwHm/content/2301.02465v1.pdf'} diff --git a/R9FRT4oBgHgl3EQfLjfi/content/tmp_files/2301.13503v1.pdf.txt b/R9FRT4oBgHgl3EQfLjfi/content/tmp_files/2301.13503v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..11aee151849618182afc3df83cfde1c253b39953 --- /dev/null +++ b/R9FRT4oBgHgl3EQfLjfi/content/tmp_files/2301.13503v1.pdf.txt @@ -0,0 +1,2011 @@ +Identical Bands Around the Isobaric Rare Earth Even-Even Nuclei +with the Mass Number A = 164 +M. A. Abdelsalam⋆, H. A. Ghanim⋆, M. Kotb⋆, and A. M. Khalaf⋆ +⋆Physics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt +Corresponding author: mahmoudkotb@azhar.edu.eg +Abstract +Eight pairs of rare-earth normally - deformed (ND) nuclei around the isobaric nuclei with A = 164 +and have identical values of F-spin, ± F0 and Np Nn (Np and Nn are the number of valence protons and +valence neutrons respectively ) have been studied. These pairs of identical bands (IB’s) cover 16 mass +units and are classified as (i) 3 pairs of nuclei separated by (2p,2n) :(162Y b −166 Hf), (162Er −166 Y b), +(162Dy −166 Er) (ii) 2 pairs of nuclei separated by (4p,4n): (160Dy −168 Y b), (160Er −168 Hf) (iii) 2 pairs +of nuclei separated by (6p,6n): (158Er −170 W) (158Dy −170 Hf) and (iv) one pair of nuclei separated +by (8p,8n): (156Dy −172 W). +We suggested a theoretical collective rotational formula containing three parameters (CRF3) as an +extended version of Bohr-Mottelson model to calculate the ground state positive parity excitation en- +ergies. Also, the sd-version of the interacting boson model (IBM) has been used to describe the nuclear +shapes by using the intrinsic coherent-state. The optimized models parameters for each nucleus are +adjusted by using a simulation search program to minimize the root mean square deviation between +the theoretical calculation and experimental excitation energies. The best adopted model parameters +of the CRF3 are used to calculate the rotational frequencies ¯hω, the kinematic J(1) and dynamic J(2) +moments of inertia and the evolution of J(1) and J(2) with increasing ¯hω are systematically analyzed. +A smooth gradual increase in both moments of inertia was seen. +The calculated results agree excellently with the experimental ones which give strong support to +the suggested CRF3. +The adopted IBM parameters are used to calculate the potential energy surfaces (PES’s) which +describe the nuclear deformation. The PES’s for our nuclei shows two wells corresponding to prolate +and oblate sides which indicate that these nuclei are deformed and have rotational behaviors. +The correlation quantities which identify the IB’s are extracted. It is found that the nuclei having +NpNn/△ where △ is the average pairing gap, exhibit identical excitation energies and energy ratios in +their ground state rotational bands. +Keywords : Interacting Boson model (IBM) - Identical Bands - Potential Energy Surface +1 +Introduction +The discovery of rotational bands in adjacent even-even and odd-mass superdeformed (SD) nuclei in +which the γ-ray transition energies are nearly identical to within a few KeV was an exotic and unex- +pected phenomenon in nuclear structure physics [1–5]. Since the identical bands (IB’s) have essentially +identical transition energies, then the associated dynamical moment of inertia are thus identical. Sev- +eral explanations were put forward [4–12] to understand the origin of IB’s phenomenon assuming the +occurrence of such IB’s to be a specific property of the SD states in nuclei. The explanations of these IB’s +includes: the Coriolis force, the particle alignment and pairing [13], the roles of special high-N orbitals of +intruder configuration and band crossing [14–17], the pseudo-spin in supersymmetry [7, 18, 19] and the +supersymmetry with many-body interactions [20]. +Soon the phenomenon of low-spin identical bands was found in pairs of even-even normal deformed +(ND) nuclei [21], and in neighboring even-even and odd-mass nuclei in rare-earth region where they have +similar moments of inertia [22,23]. If was noted that low spin IB’s are not limited to nearby nuclei but are +widespread and found in pairs of even-even nucleoside as separated by 24 mass unit (like 156Dy,180 Os) +1 +arXiv:2301.13503v1 [nucl-th] 31 Jan 2023 + +[24]. Attempts were made to understand the low-spin IB’s in terms of some simple systematics of the +moments of inertia in the rare-earth region [25–30] or from several types of consideration [31]. +For the description of normally deformed (ND) bands, some useful models were proposed. Bohr and +Mottelson [32] pointed out that, under the adiabatic approximation, the rotational energy of an axially +symmetric nucleus may be expanded for K = 0 band as a power series in the I(I+1) term. The expansion +for the K ̸= 0 band takes the same form, but includes a band head energy and the I(I+1) is replaced by +� +I(I + 1) − K2� +. Another useful models for nuclear rotational spectra are the particle-rotor model (PRM) +[33], the variable moment of inertia (VMI) model [34, 35], the soft rotor model [36] and the interacting +boson model [37]. +In the concept of F-spin and its projection [38] any pairs of conjugate nuclei with the same F-spin and +F0 values in any F-multiplet will have the same NpNn [24, 39, 40] where Np and Nn are respectively the +number of valence protons and valence neutrons. The product NpNn was used in the classification of the +changes that occur in nuclear structure [41,42]. It was assumed that [25,43] the moment and the P-factor +depends also on the product NpNn. +The purpose of the present paper is (i) to analyse the excitation energies for even-even normally de- +formed nuclei in rare earth region in framework of suggested new collective rotational formula (CRF3). +(ii) to exhibit the occurrence of IB’s in eight pairs of nuclei in rare earth region. (iii) to present the parame- +ters which characterize the appearance of IB’s. (iv) use the sd version of interacting boson model (sdIBM) +to calculate the potential energy surfaces (PES’s). +2 +Outline of the Suggested Collective Rotational Formula with Three Pa- +rameters (CRF3) +Rotational states in normal deformed (ND) nuclei can be characterized by their excitation energies E(I) +as a function of spin I, which generally lie low as compared to the single-particle excitation. In the strong +coupling limit, the rotational ground state energy for an axially symmetric even-even nucleus obeys the +I(I+1) rule, i.e form bands of levels that fulfill the relation +E(I) = ¯h2 +2J I(I + 1) = α Î +2 +(1) +where α = ¯h2/2J and Î = I(I+1) +The relation (1) defines in addition the nuclear moment of inertia J as a constant for an ideal rotor. +This simple rotational formula gives deviations from experimental data, So Bohr and Mottelson pointed +out that agreement was improved by adding to it a second team to yield +E(I) = αI(I + 1) + β[I(I + 1)]2 += α Î +2 + β Î +4 +E(I) = α Î +2(1 + γ Î +2) +(2) +where γ = β/α +Since the moment of inertia J increases on rotation of the nucleus, the observed deviations from the +experiment were still more evident. +According to the variable moment of inertia(VMI) model [34, 35], there is a gradual increase in mo- +ment of inertia J with increasing the spin I, so we suggest that the moment inertia J can be written as +J = J(I) = J (1 + σ Î +2) +(3) +Substituting in equation (2), yield +E(I) = α Î +2 +� +1 + γ Î +2 +1 + σ Î +2 +� +(4) +Therefore, the two-term Bohr-Mottelson formula becomes an extended new formula with three pa- +rameters. We denote formula (4) as the collective rotational formula with three parameters (CRF3). The +parameters are α, β, γ. +2 + +The suggested CRF3 is more general because it leads to the following three predictions: +a) when σ = γ it gives pure rigid rotor equation(1) +b) when σ = 0 it gives the two parameters Bohr-Mottelson equation (2) +c) when γ = 0 it gives soft rotor model [36] +E(I) = ¯h2 +2J +I(I + 1) +1 + σ(I + I2) +(5) +Two types of moments of inertia were suggested by Bohr-Mottelson which reflect two different as- +pects of nuclear dynamics. The first moment of inertia is the kinematic J(1), it is equal to the inverse of +the slope of the curve of energy E versus Î +2 (or I(I+1)) times ¯h2/2, while the second moment of inertia is +the dynamic J(2), it is related to the curvature in the curve of E versus Î (or +� +I(I + 1) ). +The kinematic J(1)) and dynamic J(2) moments of inertia are defined as: +J(1) = ¯h2 +2 +� +dE +dI(I + 1) +�−1 += ¯h +� +I(I + 1) +ω += ¯h2 +2 +�dE +dÎ +2 +�−1 += ¯h Î +ω +(6) +J(2) = ¯h2 +� +d2E +d( +� +I(I + 1))2 +�−1 += ¯hd +� +I(I + 1) +dω += ¯h2 +�d2E +dÎ +2 +�−1 += ¯h dÎ +dω +(7) +In the case of our CRF3, the two moments of inertia becomes +J(1)(I) = ¯h2 +2α +(1 + σÎ +2)2 +[1 + γÎ +2(2 + σÎ +2)] +(8) +J(2)(I) = ¯h2 +2α +(1 + σÎ +2)3 +[(1 + 6γÎ +2) + σÎ +2(3γÎ +2 + αγÎ +4 − 3)] +(9) +Experimentally ¯hω, J(1)and J(2) are extracted in terms of the transition energy Eγ(I) = E(I)−E(I−2) +as: +¯hω(I) = 1 +4[Eγ(I + 2) + Eγ(I)] +(MeV ) +(10) +J(1)(I) = 2I − 1 +Eγ(I) +(¯h2MeV −1) +(11) +J(2)(I) = +4 +Eγ(I + 2) − Eγ(I) +(¯h2MeV −1) +(12) +As a special case, the lowest dynamical moment of inertia reads +J(2) +lowest = +4 +Eγ(4+ +1 → 2+ +1 ) − Eγ(2+ +1 → 0+ +1 ) +(13) +3 +Determination of Ground State Band Properties of Even-Even Nuclei and +the Physical Identical Parameters +In order to understand the behavior of low lying states of an axially symmetric normally deformed nuclei, +it is insightful to examine some physical observables which exist in a pair of IB’s, the observables include: +1. The P- Factor, Structure Factor (SF), and Saturation Parameter (SP) +Casten [43] introduced the P-Factor +P = +NpNn +Np + Nn +(14) +3 + +where Np and Nn are the numbers of valence protons and valence neutrons respectively which are +counted as particles or holes from the nearest closed shell +Np = min[(Z − 50), (82 − Z)] +(15) +Nn = min[(N − 82), (126 − N)] +(16) +The P- Factor represents the average number of interactions of each valence nucleon with those of the +other type. It can be viewed as the ratio of the number of valences p-n residual interactions to the number +of valence like-nucleon pairing interactions, or if the p-n and pairing interactions are orbit independent, +then P is proportional to the ratio of the integrated p-n interaction strength to the integrated pairing +interaction strength. The nuclear collectivity and deformation depend sensitively on the P- Factor. +The structure factor (SF) and the saturation parameter (SP) are given by +SF = NpNn(Np + Nn) +(17) +SP = +� +1 + +SF +SFmax +�−1 +(18) +It is found that the lowest dynamical moment of inertia J(2) +lowest is proportional to +√ +SF. +2. The Concept of F-Spin +A nucleus with Np valence protons and Nn valence neutrons has a total boson number +NB = Np + Nn +2 += Nπ + Nν +(19) +The Nπ proton bosons and neutron bosons are assigned F-Spin, F = +1 +2 with projection F0 = + 1 +2 +for proton bosons and F0 = − 1 +2 for neutron bosons. A given nucleus is characterized by two quantum +numbers [38]: +F = Nπ + Nν +2 +and its projection F0 = Nπ − Nν +2 +Squaring and subtracting, yield +4(F 2 − F 2 +0 ) = 4NπNν = NpNn +(20) +That is any pair of conjugate nuclei with the same F-spin and F0 values in any F-spin multiplet have +identical NpNn values. +In our chosen nuclei, the F-spin multiplet is given by: (A+4, Z+2), (A+8, Z+4), (A+12, Z+6) and (A+16, +Z+8) for Dy, Er, Yb, Hf, and W isotopes. +Any pair of nuclei which show identical excitation energies have nearly equal value of the product of +their valence nucleon numbers Np and Nn [41]. However, the analysis of experimental data shows that +the converse is not true. The simple quantity NpNn helps also in the evolution of nuclear deformation +and collectivity in nuclei [40]. On the other hand, the product NpNn or the P- Factor plays an important +role in studying the orbit dependence, shell gaps, and intruder orbitals. +3. Pairing Interaction Energy +The pairing interaction energy △ in an even-even nucleus is the average pairing gap ((△p + △n)/2 +where △p and △n are respectively the proton and neutron pairing gaps which are determined from the +difference in binding energies of the neighboring odd and even nuclei +△p = 1 +4[B(N, Z − 2) − 3B(N, Z − 1) + 3B(N, Z) − B(N, Z + 1)] +(21) +△n = 1 +4[B(N − 2, Z) − 3B(N − 1, Z) + 3B(N, Z) − B(N + 1, Z)] +(22) +The pairing gaps △p and △n are determined empirically from the relation +△p ≃ △n = 12 +√ +A +(MeV ) +(23) +The average pairing gap of the nucleus is then +4 + +△ = △p + △n +2 += 12 +√ +A +MeV +(24) +It is observed that [39, 43] the even-even nuclei belong to different mass number having identical +(NpNn/△) values exhibit identical excitation energies and identical energy ratios. +4. Quadrupole Transition Probabilities and Deformation Parameters +The quadrupole transition probability per unit time for the transition Ii → If is given by +T(E2) = 4π +75 +�5 +¯h +� �E2+ +1 +¯hc +�5 +B(E2; Ii → If) +(25) +where B(E2) is the reduced transition probability and E2+ +1 is the energy of the 2+ +1 state. +Experimentally T(E2) for transition 2+ +1 → 0+ +1 is obtained by +T(E2, 2+ +1 → 0+ +1 ) = +ln2 +(1 + α)T1/2 += +0.693 +(1 + α)T1/2 +(26) +where α is the total conversion coefficient taken from the tabulated values given by Rose [44] and T1/2 +is the lifetime of the rotational level. +The B(E2, 2+ +1 → 0+ +1 ) values carry important information about the collectivity of nuclear rotation and +can be extracted from the equations (25,26). +The relation between the intrinsic nuclear quadrupole moment Q0 and B(E2) is given by +Q2 +0 = 16π +e B(E2, 2+ +1 → 0+ +1 ) +(27) +Practically the most reliable method of determining the quadrupole deformation parameter β2 in +framework of geometric collective model (GCM) is to extract β2 from Q0 according to the formula +β2(exp) = +√ +5π +3ZR2 +0 +Q0 +(28) +assuming a uniformly charged nucleus of spheroidal shape, where the nuclear radius has the value +R0 = 1.2A1/3(fm) and Z is the nuclear charge number. +The expression (28) for β2 is widely used to compare the quadrupole deformation of different nuclei. +It is noticed that the B(E2, 2+ +1 → 0+ +1 ) values increase when going from the closed shell at N=82 toward +midshell where maximum values are occur, while from midshell toward the shell closure at N= 126 its +values are decreases. +In a second way , specially where the B(E2, 2+ +1 → 0+ +1 ) value is not known, we estimate β by using the +approximate empirical Grodzins relation [45]: +E2+ +1 B(E2, 2+ +1 → 0+ +1 ) = 2.5 × 10−3 Z2 +A +(29) +where +B(E2, 2+ +1 → 0+ +1 ) = +1 +16πe2Q2 +0 = +9 +80π2 e2Z2R4 +0β2 +(in units of e2b2) +(30) +We can relate β and E2+ +1 as: +β2 +G = +1224 +E2+ +1 A7/3 +(31) +where E2+ +1 is in MeV. +Also β2 can be determined by using the SU(3) rotational limit of interacting boson model(IBM) [37], +the square of the deformation parameter β2 in a state of angular momentum I is given by [46]: +⟨β2⟩I = +α2 +6(2N − 1)[I(I + 1) + 8N2 +B + 22NB − 15] +(32) +5 + +where NB is the total number of valence bosons and α is a normalization constant (α = 0.101 for rare- +earth nuclei). The expectation value of β2 in the ground state becomes +⟨β2⟩0 = α2 8N2 +B + 22NB − 15 +6(2N − 1) +(33) +which is an almost linearly increasing function of the boson number NB and has the same value for +nuclei having the same number of valence nucleons +N = [Np + Nn], N = [(Np − 1) + (Nn − 1)] +(34) +It is evident that βIBM extracted from IBM is much larger than βGCM extracted from GCM because +βGCM refer to the deformation of all A nucleons while βIBM describe only 2N valence bosons, the ap- +proximate relation between them is given by: +βGCM = 1.18 +�2N +A +� +βIBM +(35) +The deformation parameter β reflects the equilibrium shape and structure of the nucleus such as the +energy ratio R4/2 = E(4+ +1 )/E(2+ +1 ) and the reduced transition probability B(E2, 2+ +1 → 0+ +1 ) which are the +best indicators to exhibit the collective properties of the even-even nuclei. +5. Energy Ratios and Percentage Difference in Transition Energies +The energy ratios and the percentage difference in transition energies give the characteristic of the +evolution of the collectivity in the even-even nuclei. Only deformed nuclei show rotational levels and +particularly the even-even nuclei display a simple structure energies proportional to I(I+1) with only +even values of the spin I considering that the moment of inertia is constant (rigid rotator), therefore +the energy ratio R4/2 = 3.333. The observed moment of inertia extracted from the experiment is only +one-quarter to one-half of what one would expect from a rigid rotator which means that not the whole +nucleons are participating in the collective motion. +On the other hand for an ideal harmonic quadrupole spectrum for spherical nuclei a system of +equidistant states is formed by the composition of vibrational quanta. The first excited state is 2+ +1 fol- +lowed by the degenerate 0+ +2 , 2+ +2 , 4+ +1 , and so forth. Therefore energy ratioR4/2 = 2. +To compare level spacing in two nuclei with masses A1, and A2 where A2 > A1, we define the per- +centage differences ratios in transition energies as : +δ = △Eγ(I) +Eγ2(I) +(36) +where +Eγ = E(I) − E(I − 2) +(37) +△Eγ(I) = Eγ1(I) − Eγ2(I) +(38) +So that +Eγ1 = (1 + δ)Eγ2 +(39) +For rigid rotor the ratio +δR = +�A2 +A1 +�5/3 +− 1 +(40) +define the fractional change in A5/3. +The fractional change in transition energies δ divided by the rigid rotor ratio δR is denoted by δγ. If +the spacings are identical, then δ = 0, δγ = 0 and if they scale as A5/3 then δγ=1. +Similarly, the percentage difference in kinematic moment of inertia J(1) is given by +K = −△J(1)(I) +J(1) +2 (I) +(41) +6 + +where +J(1)(I) = 2I − 1 +Eγ(I) +(42) +△J(1)(I) = J(1) +1 (I) − J(1) +2 (I) +(43) +So that +J(2) +2 += (1 + K)J(1) +1 +(44) +Substituting for J(1), yield K = δ. +4 +The Interacting Boson Model to Calculate the Potential Energy Surfaces +and Electric Quadrupole Transition Probability +We consider the Hamiltonian of the first order U(5)- SU(3) quantum shape phase transition in the form +H = ϵdˆnd + a2 ˆQ(x) ˆQ(x) +(45) +where ˆnd and ˆQ(x) are respectively the d-boson number operator and quadrupole operator defined as +ˆnd = +� +µ +d† +µ +∼ +dµ +(46) +ˆQ(x) = +� +d†s + s† ∼ +d +�(2) ++ x +� +d†× +∼ +d +�(2) +(47) +where +� +s†, d†� +and +� +s, +∼ +d +� +are the boson creation and annihilation operators respectively, and x is +the structure parameter of the quadrupole operator of IBM (x for pure rotational SU(3) limit is equal to +− +√ +7/2). Here dµ = (−1)µd−µ and standard notation of angular momentum coupling is used. +To get the potential energy surface (PES) of the Hamiltonian, we introduce the intrinsic coherent +frame in which the ground state of a nucleus with N bosons can be expressed as a boson condensate. The +bosonic intrinsic coherent state for the ground state band of a given even-even nucleus can be written in +the form [47–49] +|Nβγ⟩ = +1 +√ +N! +[b†(β, γ)]N|0⟩ +(48) +where |0⟩ is the boson vacuum and b† is the boson creation operator which acts in the intrinsic system +and is given by: +b† = +1 +� +1 + β2 [s† + βcosγ(d† +0) + 1 +√ +2βsinγ(d† +2 + d† +−2)] +(49) +where β is the quadrupole deformation parameter which measures the axial deviation from spherical +symmetry and the parameter γ controls the departure from axial symmetries. +The ground state PES is the expectation value of the Hamiltonian in the intrinsic coherent state +PES = ⟨Nβγ|H|Nβγ⟩ +(50) +The associated PES of the Hamiltonian (45) for x = − +√ +7/2 reads +E(N, β, γ) = ϵd +Nβ2 +1 + β2 + a2 +� +N +1 + β2 (5 + 11 +4 β2) + N(N − 1) +(1 + β2)2 (4β2 − 2 +√ +2β3cos3γ + 1 +2β4) +� +(51) +Equation (51) can be written in another form as +E(N, β, γ) = g1 +Nβ2 +1 + β2 + N(N − 1) +(1 + β2)2 [g2β2 + g3β3cos3γ + g4β4] + c +(52) +7 + +where the coefficients involve linear combination of the Hamiltonian parameters +g1 = ϵd − 9 +4a2, +g2 = 4a2 +g3 = 2 +√ +2a2, +g4 = 1 +2a2, +c = 5Na2 +Also, equation (51) can be rewritten in general form as +E(N, β, γ) = A2β2 + A3β3cos3γ + A4β4 +(1 + β2)2 ++ A0 +(53) +where the coefficients read +A2 = +� +ϵ + +� +4N − 25 +4 +� +a2 +� +N, +A3 = 2 +√ +2a2(N − 1)N +A4 = +� +ϵ + +�2N + 5 +4 +− 4 +� +a2 +� +N, +A0 = 5a2N +For a2 = 0, we get the pure spherical vibrator U(5) limit and for ϵd = 0, we get the pure deformed +rotational Su(3) limit. +Another important quantity that tests the nature of the shape phase transition of low lying collective +states the reduced electric quadrupole transition probabilities B(E2). +In IBM, the general form of the electric quadrupole operator is written in the form [50] +T(E2) = eQ(sdIBM) +(54) +The coefficient e is the boson’s effective charge. +The reduced electric quadrupole transition probabilities are given by +B[E2, Ii → If] = +1 +2Ii + 1|⟨If||T(E2)||Ii⟩|2 +(55) +For rotational SU(3), yield +B(E2, I + 2 → I) = e2 3 +4 +(I + 2)(I + 1) +(2I + 3)(2I + 5)(2N − 1)(2N + I + 3) +(56) +Q(I) = −e +� +16π +40 +I +2I + 3(4N + 3) +(57) +For the special case for I=0, we have +B(E2, 2+ +1 → 0+ +1 ) = e2 1 +5N(2N + 3) +(58) +5 +Numerical Calculations and Discussion +In this section, we applied our formalism to eight pairs of nuclei having identical bands (IB’s) in rare- +earth region namely: (162Y b−166 Hf), (162Er−166 Y b), (162Dy −166 Er), (160Dy −168 Y b), (160Er−168 Hf), +(158Er −170 W), (158Dy −170 Hf) and (156Dy −172 W). +To calculate the ground state positive parity excitation energy E(I) for each nucleus, we suggested the +CRF3. +The parameters α, γ, σ of CRF3 have been determined by a fitting procedure using a computer- +simulated search program to minimize the root mean square deviation of the calculated excitation ener- +gies from the experimental ones. The quality of the fitting is indicated by the standard common definition +of x +x = +� +1 +N Σi +�Eexp(Ii) − Ecal(Ii) +δEexp(Ii) +�2 +8 + +where N is the number of experimental data points entering the fitting procedure and δEexp(Ii) is the +experimental error in the excitation energies - The experimental excitation energies are taken from [51]. +The optimized best adopted values of parameters for each nucleus of our studied nuclei are listed in +Table (1). +Figure 1: Systematic of the calculated (solid curves) ground state energies for our selected even-even rare earth Dy, +Er, YB, Hf, W isotopes versus neutron number N and comparison with the experimental ones (dashed curves). The +spin-parity are labeled by Iπ. +9 + +68Er Exp +6Dy Exp +68Er Cal +68Er Exp +2500F +2500F +2500 +2500 +12+ +12* +12t +2000 +12+ +2000 +2000 +2000 +1500 +10* +10+ +1500* +1500Q +10 +1500 +10 + (KeV) +Energies (KeV) +01 +KeV +KeV +Energies +Energies ( +Energies +8+ +1000Q +8+ +1000 +1000 +1000 +10 +6 +6 +500 +500 +500 +4+ +2 +G +2 +2+ +92 +94 +96 +92 +t6 +96 +90 +92 +t6 +96 +98 +90 +92 +t6 +96 +98 +N +N +70 Yb Cal +70Yb Exp +72Hf Cal +72Hf Exp +2500 +2500F +2500 +2500 +12 +12 +12 +12 +2000* +2000Q +2000 +2000 +10* +10* +10+ +10 +1500 +1500 +1500 +1500 +Energies (KeV) +Energies (KeV) +Energies (KeV) +8 +8 +1000 +1000 +1000 +1000 +6 +6 +6 +500* +500G +500 +4 +4 +21 +2 +10 +2 +G +o2 +oL +96 +94 +98 +92 +94 +96 +98 +95 +96 +97 +98 +94 +95 +86 +N +N +N +74 W Cal +74W Exp +2500 +2500 +12 +12t +2000 +2000 +10° +10* +500 +1500 +(KeV) +(KeV) +Energies +8t +1000 +1000 +6 +G +61 +500 +4t +G +96 +96.5 +97 +97.5 +98 +96 +96.5 +97 +97.5 + 98 +NTable 1: Values of optimized best parameters α, γ, σ of the collective rotational formula(CRF3) for ground state +bands in our selected even-even rare-earth nuclei. Np and Nn are the number of valance protons and the number of +valance neutrons respectively. +Nuclide +α (KeV) +γ (10−3) +σ (10−3) +Np +Nn +Dy 156 +22.96 +6.964 +14.54 +16 +8 +158 +16.48 +2.163 +4.339 +16 +10 +160 +14.49 +0.8683 +2.021 +16 +12 +162 +13.49 +1.398 +2.233 +16 +14 +Er 158 +32.76 +9.699 +23.52 +14 +8 +160 +20.73 +3.017 +6.641 +14 +10 +162 +17.01 +1.440 +3.212 +14 +12 +166 +13.49 +0.2573 +1.188 +14 +16 +Yb 162 +27.87 +6.334 +14.27 +12 +10 +166 +17.08 +2.053 +3.95 +12 +14 +168 +14.72 +1.039 +2.425 +12 +16 +Hf 166 +26.60 +5.565 +12.67 +10 +12 +168 +20.58 +3.116 +6.849 +10 +14 +170 +15.92 +-0.00749 +1.391 +10 +16 +W 170 +26.44 +5.714 +13.55 +8 +14 +172 +20.68 +3.944 +9.279 +8 +16 +Figure 2: The calculated energy ratio R4/2 = E(4+ +1 )/E(2+ +1 ) versus neutron number N characterizes the low lying +spectrum in Dy, Er, Yb, Hf, and W isotopes. The symbols o, ∗, �, △, and x denote 66Dy,68 Er,70 Y b,72 Hf, and +74W respectively. +The systematic of the excitation energies of the low spin states as a function of neutron number N +in the considered even-even Dy, Er, Yb, Hf, W isotopes in the mass region A= 156 - 172 in the normally +deformed nuclear are shown in Figure(1) and compared with the experimental ones. Only the ground +state of positive parity and spin Iπ = 2+, 4+, 6+, 8+, 10+, 12+ has been indicated. We can see that the +excitation energies decrease with increasing the neutron number. Also, Figure(2) illustrate the calculated +10 + +o Dy +162 +Dy +3.3 +* Er +166 +3* + Yb +168. +Yb +△ Hf +162 +Er* +166. +Yb +× W +158 +Dyo +170. +3.2 +ZHf +168 +160 +AHf +3.1 +Er* +172, +W +3 +R4/2 +166 +156 +aHf +170. +Dy +162 +Ybo +2.9 +2.8 +158 +Er +2.7 +90 +92 +94 +96 +98 +Nenergy ratio R4/2 as a function of neutron number N for our studied nuclei. We observe that for each +isotopic chain the value of R4/2 increases with increasing N (that is the deformation increased), and the +difference in R4/2 for all pairs of IB’s is ranging from 0.4 % to 2.5 % except the two pairs including the +two isotopes 170,172W (the difference is about 5%). +Figure 3: The calculated results of kinematic J(1) (dashed curves) and dynamic J(2) (solid curves) moments of +inertia plotted as a function of rotational frequency ¯hω for the studied eight pairs of identical bands in the rare-earth +region. The ∗ and o correspond to the lighter and heavier nucleus respectively. +For the eight pairs of IB’S, the kinematic J(1) and the dynamic J(2) moments of inertia derived from +the transition energies are plotted versus the rotational frequency ¯hω as shown in Figure(3). It can be +seen that for all bands J(1) is smaller than J(2) and a smooth gradual increase in both J(1) and J(2) with +increasing ¯hω are seen and the similarities between each pair of IB’S are observed. +11 + +170W +162Yb -_ 166Hf +70 +70 +60 +60 +J(), J(2) (h? MeV-1) +50 +40 +40 +30 +30 +20 +G +10 +0 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +0.26 +0.28 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +0.26 +0.28 +ho(MeV) +ho(MeV) +156Dy _ 172W +160Er -_ 168Hf +90 +80 +J(M), J(2) (h? MeV-l) +70 +J(I), J(2) (h? MeV-l) +50 +50 +40 +40 +30 +20 +20 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +0.26 +0.28 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +0.26 +0.28 +ho(MeV) +ho(MeV) +158Dy _ 170Hf +162Er - 166Yb +100 +70 +06 +65 +(h? MeV-l) +J(), J(2) (h? MeV-1) +80 +60 +70 +50 +J(I), J(2) ( +60 +45 +50 +40 +40 +30 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +0.26 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +0.26 +0.28 +ho(MeV) +ho(MeV) +160Dy +168Yb +162Dy +166Er +一 +70 +70 +65 +65 +J(I), J2) (h? MeV-1) +60 +J(), J(2) (h? MeV-1) +60 +55 +5 +50 +50 +45 +45 +40 +35 +40 +30 +35 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +0.26 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +0.26 +ho(MeV) +ho(MeV)The IB’s correlation quantities exist between the considered pairs of nuclei which exhibit the same +identical excitation energies in their ground state bands are listed in Table (2). These quantities include +the P. Factor, structure Factor SF, Saturation parameter SP, the F-Spin and its projection F0, pairing gaps +△, and the deformation parameter β. The maximum structure factor for our region of nuclei is SF= 6720. +It is seen that the ratio NpNn/△ rather than the product NpNn may be a better parameter for studying +the IB’s. Note that nuclei with symmetric ±F0 values have identical NpNn values. For example the pair +(160Er and 168Hf) have (Np, Nn) = (14, 10) and (10, 14) respectively, so that NpNn = 140 and F0 = ±1. +Therefore if any F-spin multiplet has F0 =|Np − Nn|/4, those indicate that the pair of nuclei are similar in +structure if they have identical (|F0|, NpNn). +Table 2: The identical band quantities of our eight pairs of nuclei. +NpNn +P +SF +SP +|δ|% +|k|% +(158Er − 170W ) +112 +5.090 +2464 +0.7317 +1.28 +1.27 +(162Y b − 166Hf) +120 +5.4545 +2640 +0.7179 +2.94 +2.45 +(156Dy − 172W ) +128 +5.333 +3072 +0.6862 +6.73 +6.28 +(160Er − 168Hf) +140 +5.833 +3360 +0.6666 +1.35 +1.22 +(158Dy − 170Hf) +160 +6.1538 +4160 +0.6176 +1.28 +1.27 +(162Er − 166Y b) +168 +6.6461 +4368 +0.6060 +0.22 +0.20 +(160Dy − 168Y b) +192 +6.6857 +5376 +0.5555 +0.10 +0.30 +(162Dy − 166Er) +224 +7.466 +6720 +0.5 +1.29 +1.26 +(Nπ, Nν) +N +Nν +Nπ +(F, F0) +△ (MeV) +NpNn +△ +(MeV−1) +βG +158Er +(7,4) +11 +0.571 +(5.5,1.5) +0.954 +117.4 +0.2173 +170W +(4,7) +11 +1.750 +(5.5,-1.5) +0.920 +121.739 +0.2206 +162Y b +(6,5) +11 +0.833 +(5.5,0.5) +0.942 +127.388 +0.2270 +166Hf +(5,6) +11 +1.2 +(5.5,-0.5) +0.931 +128.893 +0.2254 +156Dy +(8,4) +12 +0.5 +(6,2) +0.960 +133.333 +0.2601 +172W +(4,8) +12 +2.0 +(6,-2) +0.914 +140.043 +0.2459 +160Er +(7,5) +12 +0.714 +(6,1) +0.948 +147.679 +0.2643 +168Hf +(5,7) +12 +1.4 +(6,-1) +0.925 +151.351 +0.2517 +158Dy +(8,5) +13 +0.625 +(6.5,1.5) +0.954 +167.714 +0.3026 +170Hf +(5,8) +13 +1.6 +(6.5,-1.5) +0.920 +173.913 +0.2754 +162Er +(7,6) +13 +0.857 +(6.5,0.5) +0.942 +178.343 +0.2896 +166Y b +(6,7) +13 +1.166 +(6.5,-0.5) +0.931 +180.451 +0.2814 +160Dy +(8,6) +14 +0.75 +(7,1) +0.948 +202.531 +0.3181 +168Y b +(6,8) +14 +1.333 +(7,-1) +0.925 +207.567 +0.2993 +162Dy +(8,7) +15 +0.875 +(7.5,0.5) +0.942 +237.791 +0.3256 +166Er +(7,8) +15 +1.142 +(7.5,-0.5) +0.931 +240.601 +0.3167 +The percentage differences ratios in transition energy δ and the rigid rotor ratio δR between pairs +of levels in two nuclei are calculated and listed in Table(3) for our eight pairs of IB’s. In spite of the +parameters NpNn, P, SF and SP are the same for the pairs (156Dy,172 W), this pair is not really identical +according to their high average percentage differences in transition energies (approximately 6.7%). +For each nucleus in isotopic chains of 66Dy,68 Er,70 Y b,72 Hf and 74W, the values of lowest dynamical +moments of inertia J(2) +lowest were calculated and displayed against the neutron number N in Figure(4) - It +can be seen that J(2) +lowest increases with increasing the neutron number N and the difference inJ(2) +lowest for +each pair of IB’s is very small ( approximately a horizontal line). As an example of two nuclei that exhibit +good IB’s, the pair 162 +68 Er(J(2) +lowest = 31.525¯h2MeV −1) and 166 +70 Y b(J(2) +lowest = 31.519¯h2MeV −1), that is nearly +the same J(2) +lowest. +12 + +Table 3: The percentage differences ratios in transition energies δ, the fractional change in transition energies +divided by the rigid rotor ratio δR and the ratio R = δ/δR for the eight pairs of identical bands. +Identical pairs +|δ| = △Eγ +Eγ2 +% +δR +⟨Rδ⟩ +(162Y b − 166Hf) +2.964 +4.149 +0.714 +(162Er − 166Y b) +0.415 +4.149 +0.100 +(162Dy − 166Er) +1.297 +4.149 +0.312 +(160Er − 168Hf) +1.352 +8.471 +0.159 +(160Dy − 168Y b) +1.131 +8.471 +0.133 +(158Er − 170W ) +10.826 +12.976 +0.834 +(158Dy − 170Hf) +1.765 +12.976 +0.136 +(156Dy − 172W ) +7.410 +17.671 +0.419 +Figure 4: The lowest dynamical moment of inertia J(2) +lowest against the neutron number N for the eight pairs of +identical bands. The solid line connects each pair and symbols o, ∗, △, �, and ♦ denotes 66Dy,68 Er,70 Y b,72 Hf, +and 74W respectively. +We classified our selected pairs of IB’s into four multiplets = (A+4), Z+2), (A+B,Z+4), (A+12,Z+6), and +(A+16,Z+8) and the percentage differences in transition energies δ = △Eγ/Eγ2 as a function of spin I (up +to I=10) have been calculated and illustrated Figure (5). It is seen that the pairs of IB’s have approximately +similar δ ( less than 2.5 %) except the two pairs which include the tungsten isotopes 170,172W where the +value of δ reaches ∼ 6 − 10% in spite of they have the same NpNn value (NpNn = 112 for 158Er,170 W and +NpNn = 128 for 156Dy,172 W). +To further investigation for IB’s we used the SU(3) rotational limit of the IBM to extract the quadrupole +deformation βIBM for each nucleus. The calculated βIBM is plotted against the ratio Nν/Nπ (where Nν +and Nπ are the number of valence neutron and valence proton bosons respectively) in Figure(6). It is seen +that βIBM is the same for each pair of IB’s (horizontal line). +13 + +o Dy +162 +166 +米 +Er +Dy +Er +38 +△Yb +Hf +160 +168 +Yb +36 +170 +34 +158 +Hf +Dy +32 +162 +166 +Er +Yb +2 MeV-l) +172 +30 +W +156 +Dy +168 +28 +160 +west +Er* +JHO +26 +170 +M +158 +Er +24 +米 +162. +166. +JH. +Yb +90 +92 +94 +96 +98 +NFigure 5: Percentage difference in transition energies δ = △Eγ/Eγ2 for the eight pairs of multiplet (A+4,Z+2), +(A+8,Z+4), (A+12,Z+6), and (A+16,Z+8) for Dy, Er, Yb, Hf, and W isotopes. The dashed curve represents the +ratio of the rigid rotor. +Figure 6: The quadrupole deformation parameter βIBM was calculated from SU(3) limit of IBM as a function of +Nν/Nπ for our eight pairs of identical bands. +14 + +162Yb - 166Hf +162Er - 166Yb +162Dy _ 166Er +0.07 +0.07 +0.07 +8| = 2.94 % +8/ = 0.22 % +[8| = 1.29 % +0.06 +0.06 +0.06 +0.05 +0.05 +900 +0.04 + 0.04 +0.04 +8 +8 0.03 +0.03 +0.03 +0.02 +0.02 +0.02 +0.01 +0.01 +0.01 +0.01 +10 +10 +160Er_168Hf +60Dy +168Yb +0.14 +[8|= 1.35 % +0.12 +=.1 % +d' +0.1 +0.08 +0.08 +8 0.06 +0.06 +0.04 +0.04 +0.02 +0.02 +６ +10 +6 +10 +160Dy _ 168Yb +158Dy +_ 170Hf +0.14 +0.25 +0.12 +[8/ = .1 % +0.2 +[8/ = 1.28 % +0.1 +0.15 +0.08 +8 0.06 +8 0.1 +0.04 +0.05 +0.02 +0.05 +6 +10 +6 +156Dy - 172W +[8| =6.73 % +.25 +0.2 +8 0.150.355 +162Dy +166Er +N=15 +0.35 +0.345 +160Dy +168Yb +N-14 +0.34 +0.335 +162Er +166Yb +170Hf +N=13 +βIBM +0.33 +0.325 +156Dy +160Er +168Hf +N=12172W +0.32 +0.315 +158Er +162Yb +166Hf +G +0.31 +0.5 +A +1.5 +2 +Nv/N元Figure 7: Sketch of the potential energy surface PES calculated from the U(5)-SU(3) shape phase transitions of +IBM with intrinsic coherent state versus the deformation parameters β for the eight pairs of even-even nuclei +having identical bands. +For each nucleus, by using the IBM Hamiltonian equation (45) and its eigenvalues equation (53), the +PES’s have been calculated as a function of deformation parameter β along the axial trajectory γ = 0°, 60°. +The results are illustrated in Figure(7) and the corresponding calculated parameter of the PES’s A2, A3, A4 +and Ao which are linear combinations of the original parameters ϵ0 and a2 are listed in Table(4). From +the graphs presented in Figure(7), we observe the similarity in PES’s for each pair of IB’s. All studied +nuclei are deformed and have rotational characters, the prolate deformation is deeper than the oblate +deformation. +15 + +162Dy 166Er +162Er-166Yb +2.5 +1.5k +2 +1.5 +0.5 + (KeV) +(KeV) +0 +PES +0.5 +0 +-1 +-0.5 +-1.5 +-2 +-1 +-2 +-1 +0 +-1.5 +-0.50 +0.5 +1 +1.5 +β +β +162Yb _ 166Hf +168Yb +1.5 +1.5 +0.5 + (KeV) +0.5 +0 +0 +-1 +-1 +-1.5 +-1.5 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +-2 +-1 +0 +β +β +160Er - 168Hf +158Dy - 170Hf +2 +2.5 +1.5 +2 +1.5 +PES (KeV) +(KeV) +0.5 +0.5 +PES( +0 +0 +-0.5 +0.5 +-1 +-1.5 +-1.5 +-2 +-1 +0 +1 +2 +-1.5 +-1 +0 +0.5 +1.5 +2 +β +158Er-170W +156Dy 172W +2.5 +0.6 +2 +0.4 +(KeV) +1.5 +0.2 +1 +0 +-0.4 +-0.5 +0.6 +-1 +-2-1.5 +-1 +-0.5 +0 +0.5 +1.5 +-1.5 +-1 +-0.5 +0.5 +1.5 +β +βTable 4: Values of the adopted best (PES) parameters A2, A3, A4, A0 ( in KeV ) for the studied eight pairs of +identical bands. NB is the total number of bosons. +NB +A2 +A3 +A4 +A0 +162Dy +15 +-2.4667 +-0.5863 +1.6665 +-0.3265 +166Er +15 +-1.6586 +-2.0341 +4.4739 +-0.7875 +162Er +13 +-5.0526 +-2.5496 +3.7667 +-0.9375 +166Y b +13 +-5.3088 +-3.1366 +4.0554 +-0.925 +162Y b +11 +-4.84 +-1.6163 +3.6775 +-0.9 +166Hf +11 +-2.8484 +-1.9547 +3.9131 +-0.8625 +160Dy +14 +-1.9568 +-0.8838 +1.1005 +-0.3 +168Y b +14 +-5.3088 +-3.1366 +4.0554 +-0.925 +160Er +12 +-3.0403 +-2.3636 +4.1401 +-0.8625 +168Hf +12 +-3.463 +-2.4694 +4.039 +-0.875 +158Dy +13 +-1.6288 +-1.1822 +1.0095 +-0.288 +170Hf +13 +-3.1845 +-3.395 +4.497 +-0.8375 +158Er +11 +-1.6586 +-2.0541 +4.4739 +-0.7875 +170W +11 +-0.9761 +-2.4841 +4.7606 +-0.7546 +156Dy +12 +-1.5043 +-1.2135 +0.9961 +-0.3 +172W +12 +-0.8852 +-1.4675 +1.0599 +-0.313 +6 +Conclusion +By using a novel three parameters collective rotational formula (CRF3), the positive parity ground state +excitation energies are calculated for sixteen nuclei in rare-earth region. The optimized three parameters +are deduced by using a computer simulated search program in order to obtain a minimum root mean +square deviation of the calculated excitation energies from the measured ones. The potential energy +surfaces are calculated by using the sd-version of the interacting boson model. +The problem of low-spin identical bands in normal deformed nuclei in rare-earth region is treated. We +have exhibited identical bands in eight pairs of conjugate even-even nuclei of widely dispersed spanning +as much as sixteen mass unit. Each pair with the same F-spin and projections ±F0 values have identical +product of valence proton and neutron numbers NpNn values. Also, the values of dynamical moments +of inertia for each identical band pair are approximately the same. We extracted all the identical band +symmetry parameters like P-factor, saturation parameter, and structure factor which all depend on Np +and Nn. 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Physical Review C, 28(4):1798, 1983. +[51] Evaluated Nuclear Structure Data File National Nuclear Data Center. https://www.nndc.bnl.gov/. +19 + diff --git a/R9FRT4oBgHgl3EQfLjfi/content/tmp_files/load_file.txt b/R9FRT4oBgHgl3EQfLjfi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d80bfe8f0fd15af45bc3d028221822651f2fb49 --- /dev/null +++ b/R9FRT4oBgHgl3EQfLjfi/content/tmp_files/load_file.txt @@ -0,0 +1,1106 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf,len=1105 +page_content='Identical Bands Around the Isobaric Rare Earth Even-Even Nuclei with the Mass Number A = 164 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Abdelsalam⋆, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Ghanim⋆, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Kotb⋆, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Khalaf⋆ ⋆Physics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt Corresponding author: mahmoudkotb@azhar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='eg Abstract Eight pairs of rare-earth normally - deformed (ND) nuclei around the isobaric nuclei with A = 164 and have identical values of F-spin, ± F0 and Np Nn (Np and Nn are the number of valence protons and valence neutrons respectively ) have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' These pairs of identical bands (IB’s) cover 16 mass units and are classified as (i) 3 pairs of nuclei separated by (2p,2n) :(162Y b −166 Hf), (162Er −166 Y b), (162Dy −166 Er) (ii) 2 pairs of nuclei separated by (4p,4n): (160Dy −168 Y b), (160Er −168 Hf) (iii) 2 pairs of nuclei separated by (6p,6n): (158Er −170 W) (158Dy −170 Hf) and (iv) one pair of nuclei separated by (8p,8n): (156Dy −172 W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' We suggested a theoretical collective rotational formula containing three parameters (CRF3) as an extended version of Bohr-Mottelson model to calculate the ground state positive parity excitation en- ergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Also, the sd-version of the interacting boson model (IBM) has been used to describe the nuclear shapes by using the intrinsic coherent-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The optimized models parameters for each nucleus are adjusted by using a simulation search program to minimize the root mean square deviation between the theoretical calculation and experimental excitation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The best adopted model parameters of the CRF3 are used to calculate the rotational frequencies ¯hω, the kinematic J(1) and dynamic J(2) moments of inertia and the evolution of J(1) and J(2) with increasing ¯hω are systematically analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' A smooth gradual increase in both moments of inertia was seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The calculated results agree excellently with the experimental ones which give strong support to the suggested CRF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The adopted IBM parameters are used to calculate the potential energy surfaces (PES’s) which describe the nuclear deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The PES’s for our nuclei shows two wells corresponding to prolate and oblate sides which indicate that these nuclei are deformed and have rotational behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The correlation quantities which identify the IB’s are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' It is found that the nuclei having NpNn/△ where △ is the average pairing gap, exhibit identical excitation energies and energy ratios in their ground state rotational bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Keywords : Interacting Boson model (IBM) - Identical Bands - Potential Energy Surface 1 Introduction The discovery of rotational bands in adjacent even-even and odd-mass superdeformed (SD) nuclei in which the γ-ray transition energies are nearly identical to within a few KeV was an exotic and unex- pected phenomenon in nuclear structure physics [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Since the identical bands (IB’s) have essentially identical transition energies, then the associated dynamical moment of inertia are thus identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Sev- eral explanations were put forward [4–12] to understand the origin of IB’s phenomenon assuming the occurrence of such IB’s to be a specific property of the SD states in nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The explanations of these IB’s includes: the Coriolis force, the particle alignment and pairing [13], the roles of special high-N orbitals of intruder configuration and band crossing [14–17], the pseudo-spin in supersymmetry [7, 18, 19] and the supersymmetry with many-body interactions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Soon the phenomenon of low-spin identical bands was found in pairs of even-even normal deformed (ND) nuclei [21], and in neighboring even-even and odd-mass nuclei in rare-earth region where they have similar moments of inertia [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' If was noted that low spin IB’s are not limited to nearby nuclei but are widespread and found in pairs of even-even nucleoside as separated by 24 mass unit (like 156Dy,180 Os) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='13503v1 [nucl-th] 31 Jan 2023 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Attempts were made to understand the low-spin IB’s in terms of some simple systematics of the moments of inertia in the rare-earth region [25–30] or from several types of consideration [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' For the description of normally deformed (ND) bands, some useful models were proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Bohr and Mottelson [32] pointed out that, under the adiabatic approximation, the rotational energy of an axially symmetric nucleus may be expanded for K = 0 band as a power series in the I(I+1) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The expansion for the K ̸= 0 band takes the same form, but includes a band head energy and the I(I+1) is replaced by � I(I + 1) − K2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Another useful models for nuclear rotational spectra are the particle-rotor model (PRM) [33], the variable moment of inertia (VMI) model [34, 35], the soft rotor model [36] and the interacting boson model [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' In the concept of F-spin and its projection [38] any pairs of conjugate nuclei with the same F-spin and F0 values in any F-multiplet will have the same NpNn [24, 39, 40] where Np and Nn are respectively the number of valence protons and valence neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The product NpNn was used in the classification of the changes that occur in nuclear structure [41,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' It was assumed that [25,43] the moment and the P-factor depends also on the product NpNn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The purpose of the present paper is (i) to analyse the excitation energies for even-even normally de- formed nuclei in rare earth region in framework of suggested new collective rotational formula (CRF3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' (ii) to exhibit the occurrence of IB’s in eight pairs of nuclei in rare earth region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' (iii) to present the parame- ters which characterize the appearance of IB’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' (iv) use the sd version of interacting boson model (sdIBM) to calculate the potential energy surfaces (PES’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 2 Outline of the Suggested Collective Rotational Formula with Three Pa- rameters (CRF3) Rotational states in normal deformed (ND) nuclei can be characterized by their excitation energies E(I) as a function of spin I, which generally lie low as compared to the single-particle excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' In the strong coupling limit, the rotational ground state energy for an axially symmetric even-even nucleus obeys the I(I+1) rule, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='e form bands of levels that fulfill the relation E(I) = ¯h2 2J I(I + 1) = α Î 2 (1) where α = ¯h2/2J and Î = I(I+1) The relation (1) defines in addition the nuclear moment of inertia J as a constant for an ideal rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' This simple rotational formula gives deviations from experimental data, So Bohr and Mottelson pointed out that agreement was improved by adding to it a second team to yield E(I) = αI(I + 1) + β[I(I + 1)]2 = α Î 2 + β Î 4 E(I) = α Î 2(1 + γ Î 2) (2) where γ = β/α Since the moment of inertia J increases on rotation of the nucleus, the observed deviations from the experiment were still more evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' According to the variable moment of inertia(VMI) model [34, 35], there is a gradual increase in mo- ment of inertia J with increasing the spin I, so we suggest that the moment inertia J can be written as J = J(I) = J (1 + σ Î 2) (3) Substituting in equation (2), yield E(I) = α Î 2 � 1 + γ Î 2 1 + σ Î 2 � (4) Therefore, the two-term Bohr-Mottelson formula becomes an extended new formula with three pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' We denote formula (4) as the collective rotational formula with three parameters (CRF3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The parameters are α, β, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 2 The suggested CRF3 is more general because it leads to the following three predictions: a) when σ = γ it gives pure rigid rotor equation(1) b) when σ = 0 it gives the two parameters Bohr-Mottelson equation (2) c) when γ = 0 it gives soft rotor model [36] E(I) = ¯h2 2J I(I + 1) 1 + σ(I + I2) (5) Two types of moments of inertia were suggested by Bohr-Mottelson which reflect two different as- pects of nuclear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The first moment of inertia is the kinematic J(1), it is equal to the inverse of the slope of the curve of energy E versus Î 2 (or I(I+1)) times ¯h2/2, while the second moment of inertia is the dynamic J(2), it is related to the curvature in the curve of E versus Î (or � I(I + 1) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The kinematic J(1)) and dynamic J(2) moments of inertia are defined as: J(1) = ¯h2 2 � dE dI(I + 1) �−1 = ¯h � I(I + 1) ω = ¯h2 2 �dE dÎ 2 �−1 = ¯h Î ω (6) J(2) = ¯h2 � d2E d( � I(I + 1))2 �−1 = ¯hd � I(I + 1) dω = ¯h2 �d2E dÎ 2 �−1 = ¯h dÎ dω (7) In the case of our CRF3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' the two moments of inertia becomes J(1)(I) = ¯h2 2α (1 + σÎ 2)2 [1 + γÎ 2(2 + σÎ 2)] (8) J(2)(I) = ¯h2 2α (1 + σÎ 2)3 [(1 + 6γÎ 2) + σÎ 2(3γÎ 2 + αγÎ 4 − 3)] (9) Experimentally ¯hω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' J(1)and J(2) are extracted in terms of the transition energy Eγ(I) = E(I)−E(I−2) as: ¯hω(I) = 1 4[Eγ(I + 2) + Eγ(I)] (MeV ) (10) J(1)(I) = 2I − 1 Eγ(I) (¯h2MeV −1) (11) J(2)(I) = 4 Eγ(I + 2) − Eγ(I) (¯h2MeV −1) (12) As a special case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' the lowest dynamical moment of inertia reads J(2) lowest = 4 Eγ(4+ 1 → 2+ 1 ) − Eγ(2+ 1 → 0+ 1 ) (13) 3 Determination of Ground State Band Properties of Even-Even Nuclei and the Physical Identical Parameters In order to understand the behavior of low lying states of an axially symmetric normally deformed nuclei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' it is insightful to examine some physical observables which exist in a pair of IB’s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' the observables include: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The P- Factor, Structure Factor (SF), and Saturation Parameter (SP) Casten [43] introduced the P-Factor P = NpNn Np + Nn (14) 3 where Np and Nn are the numbers of valence protons and valence neutrons respectively which are counted as particles or holes from the nearest closed shell Np = min[(Z − 50), (82 − Z)] (15) Nn = min[(N − 82), (126 − N)] (16) The P- Factor represents the average number of interactions of each valence nucleon with those of the other type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' It can be viewed as the ratio of the number of valences p-n residual interactions to the number of valence like-nucleon pairing interactions, or if the p-n and pairing interactions are orbit independent, then P is proportional to the ratio of the integrated p-n interaction strength to the integrated pairing interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The nuclear collectivity and deformation depend sensitively on the P- Factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The structure factor (SF) and the saturation parameter (SP) are given by SF = NpNn(Np + Nn) (17) SP = � 1 + SF SFmax �−1 (18) It is found that the lowest dynamical moment of inertia J(2) lowest is proportional to √ SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The Concept of F-Spin A nucleus with Np valence protons and Nn valence neutrons has a total boson number NB = Np + Nn 2 = Nπ + Nν (19) The Nπ proton bosons and neutron bosons are assigned F-Spin, F = 1 2 with projection F0 = + 1 2 for proton bosons and F0 = − 1 2 for neutron bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' A given nucleus is characterized by two quantum numbers [38]: F = Nπ + Nν 2 and its projection F0 = Nπ − Nν 2 Squaring and subtracting, yield 4(F 2 − F 2 0 ) = 4NπNν = NpNn (20) That is any pair of conjugate nuclei with the same F-spin and F0 values in any F-spin multiplet have identical NpNn values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' In our chosen nuclei, the F-spin multiplet is given by: (A+4, Z+2), (A+8, Z+4), (A+12, Z+6) and (A+16, Z+8) for Dy, Er, Yb, Hf, and W isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Any pair of nuclei which show identical excitation energies have nearly equal value of the product of their valence nucleon numbers Np and Nn [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' However, the analysis of experimental data shows that the converse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The simple quantity NpNn helps also in the evolution of nuclear deformation and collectivity in nuclei [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' On the other hand, the product NpNn or the P- Factor plays an important role in studying the orbit dependence, shell gaps, and intruder orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Pairing Interaction Energy The pairing interaction energy △ in an even-even nucleus is the average pairing gap ((△p + △n)/2 where △p and △n are respectively the proton and neutron pairing gaps which are determined from the difference in binding energies of the neighboring odd and even nuclei △p = 1 4[B(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Z − 2) − 3B(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Z − 1) + 3B(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Z) − B(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Z + 1)] (21) △n = 1 4[B(N − 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Z) − 3B(N − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Z) + 3B(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Z) − B(N + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Z)] (22) The pairing gaps △p and △n are determined empirically from the relation △p ≃ △n = 12 √ A (MeV ) (23) The average pairing gap of the nucleus is then 4 △ = △p + △n 2 = 12 √ A MeV (24) It is observed that [39,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 43] the even-even nuclei belong to different mass number having identical (NpNn/△) values exhibit identical excitation energies and identical energy ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Quadrupole Transition Probabilities and Deformation Parameters The quadrupole transition probability per unit time for the transition Ii → If is given by T(E2) = 4π 75 �5 ¯h � �E2+ 1 ¯hc �5 B(E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Ii → If) (25) where B(E2) is the reduced transition probability and E2+ 1 is the energy of the 2+ 1 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Experimentally T(E2) for transition 2+ 1 → 0+ 1 is obtained by T(E2, 2+ 1 → 0+ 1 ) = ln2 (1 + α)T1/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='693 (1 + α)T1/2 (26) where α is the total conversion coefficient taken from the tabulated values given by Rose [44] and T1/2 is the lifetime of the rotational level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The B(E2, 2+ 1 → 0+ 1 ) values carry important information about the collectivity of nuclear rotation and can be extracted from the equations (25,26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The relation between the intrinsic nuclear quadrupole moment Q0 and B(E2) is given by Q2 0 = 16π e B(E2, 2+ 1 → 0+ 1 ) (27) Practically the most reliable method of determining the quadrupole deformation parameter β2 in framework of geometric collective model (GCM) is to extract β2 from Q0 according to the formula β2(exp) = √ 5π 3ZR2 0 Q0 (28) assuming a uniformly charged nucleus of spheroidal shape, where the nuclear radius has the value R0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2A1/3(fm) and Z is the nuclear charge number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The expression (28) for β2 is widely used to compare the quadrupole deformation of different nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' It is noticed that the B(E2, 2+ 1 → 0+ 1 ) values increase when going from the closed shell at N=82 toward midshell where maximum values are occur, while from midshell toward the shell closure at N= 126 its values are decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' In a second way , specially where the B(E2, 2+ 1 → 0+ 1 ) value is not known, we estimate β by using the approximate empirical Grodzins relation [45]: E2+ 1 B(E2, 2+ 1 → 0+ 1 ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 × 10−3 Z2 A (29) where B(E2, 2+ 1 → 0+ 1 ) = 1 16πe2Q2 0 = 9 80π2 e2Z2R4 0β2 (in units of e2b2) (30) We can relate β and E2+ 1 as: β2 G = 1224 E2+ 1 A7/3 (31) where E2+ 1 is in MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Also β2 can be determined by using the SU(3) rotational limit of interacting boson model(IBM) [37], the square of the deformation parameter β2 in a state of angular momentum I is given by [46]: ⟨β2⟩I = α2 6(2N − 1)[I(I + 1) + 8N2 B + 22NB − 15] (32) 5 where NB is the total number of valence bosons and α is a normalization constant (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='101 for rare- earth nuclei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The expectation value of β2 in the ground state becomes ⟨β2⟩0 = α2 8N2 B + 22NB − 15 6(2N − 1) (33) which is an almost linearly increasing function of the boson number NB and has the same value for nuclei having the same number of valence nucleons N = [Np + Nn],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' N = [(Np − 1) + (Nn − 1)] (34) It is evident that βIBM extracted from IBM is much larger than βGCM extracted from GCM because βGCM refer to the deformation of all A nucleons while βIBM describe only 2N valence bosons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' the ap- proximate relation between them is given by: βGCM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='18 �2N A � βIBM (35) The deformation parameter β reflects the equilibrium shape and structure of the nucleus such as the energy ratio R4/2 = E(4+ 1 )/E(2+ 1 ) and the reduced transition probability B(E2, 2+ 1 → 0+ 1 ) which are the best indicators to exhibit the collective properties of the even-even nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Energy Ratios and Percentage Difference in Transition Energies The energy ratios and the percentage difference in transition energies give the characteristic of the evolution of the collectivity in the even-even nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Only deformed nuclei show rotational levels and particularly the even-even nuclei display a simple structure energies proportional to I(I+1) with only even values of the spin I considering that the moment of inertia is constant (rigid rotator), therefore the energy ratio R4/2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The observed moment of inertia extracted from the experiment is only one-quarter to one-half of what one would expect from a rigid rotator which means that not the whole nucleons are participating in the collective motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' On the other hand for an ideal harmonic quadrupole spectrum for spherical nuclei a system of equidistant states is formed by the composition of vibrational quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The first excited state is 2+ 1 fol- lowed by the degenerate 0+ 2 , 2+ 2 , 4+ 1 , and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Therefore energy ratioR4/2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' To compare level spacing in two nuclei with masses A1, and A2 where A2 > A1, we define the per- centage differences ratios in transition energies as : δ = △Eγ(I) Eγ2(I) (36) where Eγ = E(I) − E(I − 2) (37) △Eγ(I) = Eγ1(I) − Eγ2(I) (38) So that Eγ1 = (1 + δ)Eγ2 (39) For rigid rotor the ratio δR = �A2 A1 �5/3 − 1 (40) define the fractional change in A5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The fractional change in transition energies δ divided by the rigid rotor ratio δR is denoted by δγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' If the spacings are identical, then δ = 0, δγ = 0 and if they scale as A5/3 then δγ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Similarly, the percentage difference in kinematic moment of inertia J(1) is given by K = −△J(1)(I) J(1) 2 (I) (41) 6 where J(1)(I) = 2I − 1 Eγ(I) (42) △J(1)(I) = J(1) 1 (I) − J(1) 2 (I) (43) So that J(2) 2 = (1 + K)J(1) 1 (44) Substituting for J(1), yield K = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 4 The Interacting Boson Model to Calculate the Potential Energy Surfaces and Electric Quadrupole Transition Probability We consider the Hamiltonian of the first order U(5)- SU(3) quantum shape phase transition in the form H = ϵdˆnd + a2 ˆQ(x) ˆQ(x) (45) where ˆnd and ˆQ(x) are respectively the d-boson number operator and quadrupole operator defined as ˆnd = � µ d† µ ∼ dµ (46) ˆQ(x) = � d†s + s† ∼ d �(2) + x � d†× ∼ d �(2) (47) where � s†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' d†� and � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' ∼ d � are the boson creation and annihilation operators respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' and x is the structure parameter of the quadrupole operator of IBM (x for pure rotational SU(3) limit is equal to − √ 7/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Here dµ = (−1)µd−µ and standard notation of angular momentum coupling is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' To get the potential energy surface (PES) of the Hamiltonian, we introduce the intrinsic coherent frame in which the ground state of a nucleus with N bosons can be expressed as a boson condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The bosonic intrinsic coherent state for the ground state band of a given even-even nucleus can be written in the form [47–49] |Nβγ⟩ = 1 √ N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' [b†(β, γ)]N|0⟩ (48) where |0⟩ is the boson vacuum and b† is the boson creation operator which acts in the intrinsic system and is given by: b† = 1 � 1 + β2 [s† + βcosγ(d† 0) + 1 √ 2βsinγ(d† 2 + d† −2)] (49) where β is the quadrupole deformation parameter which measures the axial deviation from spherical symmetry and the parameter γ controls the departure from axial symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The ground state PES is the expectation value of the Hamiltonian in the intrinsic coherent state PES = ⟨Nβγ|H|Nβγ⟩ (50) The associated PES of the Hamiltonian (45) for x = − √ 7/2 reads E(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' γ) = ϵd Nβ2 1 + β2 + a2 � N 1 + β2 (5 + 11 4 β2) + N(N − 1) (1 + β2)2 (4β2 − 2 √ 2β3cos3γ + 1 2β4) � (51) Equation (51) can be written in another form as E(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' γ) = g1 Nβ2 1 + β2 + N(N − 1) (1 + β2)2 [g2β2 + g3β3cos3γ + g4β4] + c (52) 7 where the coefficients involve linear combination of the Hamiltonian parameters g1 = ϵd − 9 4a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' g2 = 4a2 g3 = 2 √ 2a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' g4 = 1 2a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' c = 5Na2 Also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' equation (51) can be rewritten in general form as E(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' γ) = A2β2 + A3β3cos3γ + A4β4 (1 + β2)2 + A0 (53) where the coefficients read A2 = � ϵ + � 4N − 25 4 � a2 � N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' A3 = 2 √ 2a2(N − 1)N A4 = � ϵ + �2N + 5 4 − 4 � a2 � N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' A0 = 5a2N For a2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' we get the pure spherical vibrator U(5) limit and for ϵd = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' we get the pure deformed rotational Su(3) limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Another important quantity that tests the nature of the shape phase transition of low lying collective states the reduced electric quadrupole transition probabilities B(E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' In IBM, the general form of the electric quadrupole operator is written in the form [50] T(E2) = eQ(sdIBM) (54) The coefficient e is the boson’s effective charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The reduced electric quadrupole transition probabilities are given by B[E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Ii → If] = 1 2Ii + 1|⟨If||T(E2)||Ii⟩|2 (55) For rotational SU(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' yield B(E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' I + 2 → I) = e2 3 4 (I + 2)(I + 1) (2I + 3)(2I + 5)(2N − 1)(2N + I + 3) (56) Q(I) = −e � 16π 40 I 2I + 3(4N + 3) (57) For the special case for I=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' we have B(E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 2+ 1 → 0+ 1 ) = e2 1 5N(2N + 3) (58) 5 Numerical Calculations and Discussion In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' we applied our formalism to eight pairs of nuclei having identical bands (IB’s) in rare- earth region namely: (162Y b−166 Hf),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' (162Er−166 Y b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' (162Dy −166 Er),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' (160Dy −168 Y b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' (160Er−168 Hf),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' (158Er −170 W),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' (158Dy −170 Hf) and (156Dy −172 W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' To calculate the ground state positive parity excitation energy E(I) for each nucleus, we suggested the CRF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The parameters α, γ, σ of CRF3 have been determined by a fitting procedure using a computer- simulated search program to minimize the root mean square deviation of the calculated excitation ener- gies from the experimental ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The quality of the fitting is indicated by the standard common definition of x x = � 1 N Σi �Eexp(Ii) − Ecal(Ii) δEexp(Ii) �2 8 where N is the number of experimental data points entering the fitting procedure and δEexp(Ii) is the experimental error in the excitation energies - The experimental excitation energies are taken from [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The optimized best adopted values of parameters for each nucleus of our studied nuclei are listed in Table (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Figure 1: Systematic of the calculated (solid curves) ground state energies for our selected even-even rare earth Dy, Er, YB, Hf, W isotopes versus neutron number N and comparison with the experimental ones (dashed curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The spin-parity are labeled by Iπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='68Er Exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6Dy Exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='68Er Cal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='68Er Exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='74 W Cal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='74W Exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12t ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='(KeV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='Energies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='8t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='61 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='4t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 97 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 98 96 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 97 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 98 NTable 1: Values of optimized best parameters α, γ, σ of the collective rotational formula(CRF3) for ground state bands in our selected even-even rare-earth nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Np and Nn are the number of valance protons and the number of valance neutrons respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Nuclide α (KeV) γ (10−3) σ (10−3) Np Nn Dy 156 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='96 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='964 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='54 16 8 158 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='163 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='339 16 10 160 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='8683 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='021 16 12 162 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='398 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='233 16 14 Er 158 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='76 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='699 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='52 14 8 160 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='73 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='017 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='641 14 10 162 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='440 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='212 14 12 166 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2573 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='188 14 16 Yb 162 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='87 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='334 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='27 12 10 166 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='053 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='95 12 14 168 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='039 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='425 12 16 Hf 166 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='60 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='565 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='67 10 12 168 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='116 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='849 10 14 170 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='00749 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='391 10 16 W 170 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='44 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='714 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='55 8 14 172 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='944 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='279 8 16 Figure 2: The calculated energy ratio R4/2 = E(4+ 1 )/E(2+ 1 ) versus neutron number N characterizes the low lying spectrum in Dy, Er, Yb, Hf, and W isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The symbols o, ∗, �, △, and x denote 66Dy,68 Er,70 Y b,72 Hf, and 74W respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The systematic of the excitation energies of the low spin states as a function of neutron number N in the considered even-even Dy, Er, Yb, Hf, W isotopes in the mass region A= 156 - 172 in the normally deformed nuclear are shown in Figure(1) and compared with the experimental ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Only the ground state of positive parity and spin Iπ = 2+, 4+, 6+, 8+, 10+, 12+ has been indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' We can see that the excitation energies decrease with increasing the neutron number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Also, Figure(2) illustrate the calculated 10 o Dy 162 Dy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='3 Er 166 3* Yb 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Yb △ Hf 162 Er* 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Yb × W 158 Dyo 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2 ZHf 168 160 AHf 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 Er* 172, W 3 R4/2 166 156 aHf 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Dy 162 Ybo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='8 158 Er 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='7 90 92 94 96 98 Nenergy ratio R4/2 as a function of neutron number N for our studied nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' We observe that for each isotopic chain the value of R4/2 increases with increasing N (that is the deformation increased), and the difference in R4/2 for all pairs of IB’s is ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='4 % to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 % except the two pairs including the two isotopes 170,172W (the difference is about 5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Figure 3: The calculated results of kinematic J(1) (dashed curves) and dynamic J(2) (solid curves) moments of inertia plotted as a function of rotational frequency ¯hω for the studied eight pairs of identical bands in the rare-earth region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The ∗ and o correspond to the lighter and heavier nucleus respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' For the eight pairs of IB’S, the kinematic J(1) and the dynamic J(2) moments of inertia derived from the transition energies are plotted versus the rotational frequency ¯hω as shown in Figure(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' It can be seen that for all bands J(1) is smaller than J(2) and a smooth gradual increase in both J(1) and J(2) with increasing ¯hω are seen and the similarities between each pair of IB’S are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 11 170W 162Yb -_ 166Hf 70 70 60 60 J(), J(2) (h?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' MeV-1) 50 40 40 30 30 20 G 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='28 ho(MeV) ho(MeV) 156Dy _ 172W 160Er -_ 168Hf 90 80 J(M), J(2) (h?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' MeV-l) 70 J(I), J(2) (h?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' MeV-l) 50 50 40 40 30 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='28 ho(MeV) ho(MeV) 158Dy _ 170Hf 162Er - 166Yb 100 70 06 65 (h?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' MeV-l) J(), J(2) (h?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' MeV-1) 80 60 70 50 J(I), J(2) ( 60 45 50 40 40 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='14 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='28 ho(MeV) ho(MeV) 160Dy 168Yb 162Dy 166Er 一 70 70 65 65 J(I), J2) (h?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' MeV-1) 60 J(), J(2) (h?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' MeV-1) 60 55 5 50 50 45 45 40 35 40 30 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='26 ho(MeV) ho(MeV)The IB’s correlation quantities exist between the considered pairs of nuclei which exhibit the same identical excitation energies in their ground state bands are listed in Table (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' These quantities include the P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Factor, structure Factor SF, Saturation parameter SP, the F-Spin and its projection F0, pairing gaps △, and the deformation parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The maximum structure factor for our region of nuclei is SF= 6720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' It is seen that the ratio NpNn/△ rather than the product NpNn may be a better parameter for studying the IB’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Note that nuclei with symmetric ±F0 values have identical NpNn values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' For example the pair (160Er and 168Hf) have (Np, Nn) = (14, 10) and (10, 14) respectively, so that NpNn = 140 and F0 = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Therefore if any F-spin multiplet has F0 =|Np − Nn|/4, those indicate that the pair of nuclei are similar in structure if they have identical (|F0|, NpNn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Table 2: The identical band quantities of our eight pairs of nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' NpNn P SF SP |δ|% |k|% (158Er − 170W ) 112 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='090 2464 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='7317 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='27 (162Y b − 166Hf) 120 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='4545 2640 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='7179 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='94 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='45 (156Dy − 172W ) 128 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='333 3072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6862 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='73 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='28 (160Er − 168Hf) 140 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='833 3360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6666 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='22 (158Dy − 170Hf) 160 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1538 4160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6176 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='27 (162Er − 166Y b) 168 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6461 4368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='20 (160Dy − 168Y b) 192 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6857 5376 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='30 (162Dy − 166Er) 224 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='466 6720 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='26 (Nπ, Nν) N Nν Nπ (F, F0) 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='931 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2814 160Dy (8,6) 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='75 (7,1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='948 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='3181 168Y b (6,8) 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='333 (7,-1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='925 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2993 162Dy (8,7) 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='875 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='942 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='3256 166Er (7,8) 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='142 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='931 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='3167 The percentage differences ratios in transition energy δ and the rigid rotor ratio δR between pairs of levels in two nuclei are calculated and listed in Table(3) for our eight pairs of IB’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' In spite of the parameters NpNn, P, SF and SP are the same for the pairs (156Dy,172 W), this pair is not really identical according to their high average percentage differences in transition energies (approximately 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' For each nucleus in isotopic chains of 66Dy,68 Er,70 Y b,72 Hf and 74W, the values of lowest dynamical moments of inertia J(2) lowest were calculated and displayed against the neutron number N in Figure(4) - It can be seen that J(2) lowest increases with increasing the neutron number N and the difference inJ(2) lowest for each pair of IB’s is very small ( approximately a horizontal line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' As an example of two nuclei that exhibit good IB’s, the pair 162 68 Er(J(2) lowest = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='525¯h2MeV −1) and 166 70 Y b(J(2) lowest = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='519¯h2MeV −1), that is nearly the same J(2) lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 12 Table 3: The percentage differences ratios in transition energies δ, the fractional change in transition energies divided by the rigid rotor ratio δR and the ratio R = δ/δR for the eight pairs of identical bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Identical pairs |δ| = △Eγ Eγ2 % δR ⟨Rδ⟩ (162Y b − 166Hf) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='964 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='714 (162Er − 166Y b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='415 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='100 (162Dy − 166Er) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='297 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='312 (160Er − 168Hf) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='352 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='159 (160Dy − 168Y b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='131 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='133 (158Er − 170W ) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='826 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='834 (158Dy − 170Hf) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='765 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='136 (156Dy − 172W ) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='410 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='671 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='419 Figure 4: The lowest dynamical moment of inertia J(2) lowest against the neutron number N for the eight pairs of identical bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The solid line connects each pair and symbols o, ∗, △, �, and ♦ denotes 66Dy,68 Er,70 Y b,72 Hf, and 74W respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' We classified our selected pairs of IB’s into four multiplets = (A+4), Z+2), (A+B,Z+4), (A+12,Z+6), and (A+16,Z+8) and the percentage differences in transition energies δ = △Eγ/Eγ2 as a function of spin I (up to I=10) have been calculated and illustrated Figure (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' It is seen that the pairs of IB’s have approximately similar δ ( less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 %) except the two pairs which include the tungsten isotopes 170,172W where the value of δ reaches ∼ 6 − 10% in spite of they have the same NpNn value (NpNn = 112 for 158Er,170 W and NpNn = 128 for 156Dy,172 W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' To further investigation for IB’s we used the SU(3) rotational limit of the IBM to extract the quadrupole deformation βIBM for each nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The calculated βIBM is plotted against the ratio Nν/Nπ (where Nν and Nπ are the number of valence neutron and valence proton bosons respectively) in Figure(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' It is seen that βIBM is the same for each pair of IB’s (horizontal line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 13 o Dy 162 166 米 Er Dy Er 38 △Yb Hf 160 168 Yb 36 170 34 158 Hf Dy 32 162 166 Er Yb 2 MeV-l) 172 30 W 156 Dy 168 28 160 west Er* JHO 26 170 M 158 Er 24 米 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' JH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Yb 90 92 94 96 98 NFigure 5: Percentage difference in transition energies δ = △Eγ/Eγ2 for the eight pairs of multiplet (A+4,Z+2), (A+8,Z+4), (A+12,Z+6), and (A+16,Z+8) for Dy, Er, Yb, Hf, and W isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The dashed curve represents the ratio of the rigid rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Figure 6: The quadrupole deformation parameter βIBM was calculated from SU(3) limit of IBM as a function of Nν/Nπ for our eight pairs of identical bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 14 162Yb - 166Hf 162Er - 166Yb 162Dy _ 166Er 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='07 8| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='94 % 8/ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='22 % [8| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='29 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='05 900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='04 8 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='01 10 10 160Er_168Hf 60Dy 168Yb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='14 [8|= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='35 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12 =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content="1 % d' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='08 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='02 ６ 10 6 10 160Dy _ 168Yb 158Dy _ 170Hf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='12 [8/ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2 [8/ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='28 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='08 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='06 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='05 6 10 6 156Dy - 172W [8| =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='73 % .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='355 162Dy 166Er N=15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='345 160Dy 168Yb N-14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='335 162Er 166Yb 170Hf N=13 βIBM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='325 156Dy 160Er 168Hf N=12172W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='315 158Er 162Yb 166Hf G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 2 Nv/N元Figure 7: Sketch of the potential energy surface PES calculated from the U(5)-SU(3) shape phase transitions of IBM with intrinsic coherent state versus the deformation parameters β for the eight pairs of even-even nuclei having identical bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' For each nucleus, by using the IBM Hamiltonian equation (45) and its eigenvalues equation (53), the PES’s have been calculated as a function of deformation parameter β along the axial trajectory γ = 0°, 60°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The results are illustrated in Figure(7) and the corresponding calculated parameter of the PES’s A2, A3, A4 and Ao which are linear combinations of the original parameters ϵ0 and a2 are listed in Table(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' From the graphs presented in Figure(7), we observe the similarity in PES’s for each pair of IB’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' All studied nuclei are deformed and have rotational characters, the prolate deformation is deeper than the oblate deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 15 162Dy 166Er 162Er-166Yb 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5k 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 (KeV) (KeV) 0 PES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 2 1 2 1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 β β 162Yb _ 166Hf 168Yb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 (KeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0 0 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 2 2 1 0 β β 160Er - 168Hf 158Dy - 170Hf 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 PES (KeV) (KeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 PES( 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 2 1 0 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 2 β 158Er-170W 156Dy 172W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='4 (KeV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6 1 2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5 β βTable 4: Values of the adopted best (PES) parameters A2, A3, A4, A0 ( in KeV ) for the studied eight pairs of identical bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' NB is the total number of bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' NB A2 A3 A4 A0 162Dy 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='4667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5863 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='3265 166Er 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6586 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='0341 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='4739 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='7875 162Er 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='0526 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5496 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='7667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='9375 166Y b 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='3088 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='1366 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='0554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='925 162Y b 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='6163 3.' metadata={'source': 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+page_content='9761 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='4841 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='7606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='7546 156Dy 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='5043 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='2135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='9961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='3 172W 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='8852 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='4675 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='0599 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content='313 6 Conclusion By using a novel three parameters collective rotational formula (CRF3), the positive parity ground state excitation energies are calculated for sixteen nuclei in rare-earth region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The optimized three parameters are deduced by using a computer simulated search program in order to obtain a minimum root mean square deviation of the calculated excitation energies from the measured ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The potential energy surfaces are calculated by using the sd-version of the interacting boson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The problem of low-spin identical bands in normal deformed nuclei in rare-earth region is treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' We have exhibited identical bands in eight pairs of conjugate even-even nuclei of widely dispersed spanning as much as sixteen mass unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Each pair with the same F-spin and projections ±F0 values have identical product of valence proton and neutron numbers NpNn values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Also, the values of dynamical moments of inertia for each identical band pair are approximately the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' We extracted all the identical band symmetry parameters like P-factor, saturation parameter, and structure factor which all depend on Np and Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' The pairing interaction energy, the quadrupole transition probabilities, and the energy ratios are also treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' References [1] Th Byrski, FA Beck, D Curien, C Schuck, P Fallon, A Alderson, I Ali, MA Bentley, AM Bruce, PD Forsyth, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Observation of identical superdeformed bands in N = 86 nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Physical review letters, 64(14):1650, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Haas, D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Flibotte, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Galindo-Uribarri, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Janzen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Johansson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Kluge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Kuehner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Omar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Pilotte, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Prevost, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Rodriguez, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Radford, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Taras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Vivien, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Waddington, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Aberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Observation of excited proton and neutron configurations in the superdeformed 149Gd nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' C, 42:R1817–R1821, Nov 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' [3] Cyrus Baktash, Bernard Haas, and Witold Nazarewicz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Identical bands in deformed and superde- formed nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Annual Review of Nuclear and Particle Science, 45(1):485–541, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 16 [4] FS Stephens, MA Deleplanque, JE Draper, RM Diamond, CW Beausang, W Korten, WH Kelly, F Azaiez, JA Becker, EA Henry, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Spin alignment in superdeformed hg nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Physical review letters, 64(22):2623, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' [5] FS Stephens, MA Deleplanque, JE Draper, RM Diamond, AO Macchiavelli, CW Beausang, W Korten, WH Kelly, F Azaiez, JA Becker, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Pseudospin symmetry and quantized alignment in nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Physical review letters, 65(3):301, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' [6] Ingemar Ragnarsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Additivity in superdeformed bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Physics Letters B, 264(1-2):5–10, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' [7] W Nazarewicz, PJ Twin, P Fallon, and JD Garrett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Natural-parity states in superdeformed bands and pseudo su (3) symmetry at extreme conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Physical Review Letters, 64(14):1654, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' [8] Z Szyma´nski and W Nazarewicz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Rotating pseudo-oscillator scheme: pseudo-spin symmetry and identical bands.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Microscopic mechanism of normally deformed identi- cal bands at low spin in the rare-earth nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Physical Review C, 63(2):024305, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' [11] Shu-Xin Liu, Jin-Yan Zeng, and En-Guang Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Microscopic mechanism of identical superde- formed bands in 192,193,194Hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' Physical Review C, 66(2):024320, 2002.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FRT4oBgHgl3EQfLjfi/content/2301.13503v1.pdf'} diff --git a/RtE0T4oBgHgl3EQfUQDk/content/tmp_files/2301.02249v1.pdf.txt b/RtE0T4oBgHgl3EQfUQDk/content/tmp_files/2301.02249v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eab94e9fdeb07aa0007b3db8c2c4c9666dbf74e5 --- /dev/null +++ b/RtE0T4oBgHgl3EQfUQDk/content/tmp_files/2301.02249v1.pdf.txt @@ -0,0 +1,4428 @@ +Generalized Symmetries and Anomalies of 3d N = 4 +SCFTs +Lakshya Bhardwaj1, Mathew Bullimore2, Andrea E. V. Ferrari2, Sakura Sch¨afer-Nameki1 +1 Mathematical Institute, University of Oxford, +Woodstock Road, Oxford, OX2 6GG, United Kingdom +2 Department of Mathematical Sciences, Durham University, +Upper Mountjoy, Stockton Road, Durham, DH1 3LE, United Kingdom +Abstract: +We study generalized global symmetries and their ’t Hooft anomalies in 3d +N = 4 superconformal field theories (SCFTs). Following some general considerations, we +focus on good quiver gauge theories, comprised of balanced unitary nodes and unbalanced +unitary and special unitary nodes. While the global form of the Higgs branch symmetry +group may be determined from the UV Lagrangian, the global form of Coulomb branch +symmetry groups and associated mixed ’t Hooft anomalies are more subtle due to potential +symmetry enhancement in the IR. We describe how Coulomb branch symmetry groups and +their mixed ’t Hooft anomalies can be deduced from the UV Lagrangian by studying center +charges of various types of monopole operators, providing a concrete and unambiguous way +to implement ’t Hooft anomaly matching. The final expression for the symmetry group and ’t +Hooft anomalies has a concise form that can be easily read off from the quiver data, specifically +from the positions of the unbalanced and flavor nodes with respect to the positions of the +balanced nodes. We provide consistency checks by applying our method to compute symmetry +groups of 3d N = 4 theories corresponding to magnetic quivers of 4d Class S theories and +5d SCFTs. We are able to match these results against the flavor symmetry groups of the 4d +and 5d theories computed using independent methods. Another strong consistency check is +provided by comparing symmetry groups and anomalies of two theories related by 3d mirror +symmetry. +arXiv:2301.02249v1 [hep-th] 5 Jan 2023 + +Contents +1 +Introduction +1 +2 +Generalized Symmetries of 3d N = 4 Theories +4 +2.1 +BPS Defects +5 +2.2 +A- and B-type Symmetries +7 +2.2.1 +0-form Symmetry +7 +2.2.2 +1-form Symmetry +11 +2.2.3 +2-group Symmetry +12 +2.3 +Solitonic defects +14 +2.4 +’t Hooft Anomalies +16 +2.5 +Discrete Gauging +18 +3 +Warmup: T[SU(n)] and its Gaugings +19 +3.1 +T[SU(2)] and its Gaugings +19 +3.1.1 +T[SU(2)] +19 +3.1.2 +SU(2)H Gauging +22 +3.1.3 +SO(3)H Gauging +25 +3.1.4 +U(2) Gauging +27 +3.2 +T[SU(n)] and its Gaugings +28 +3.2.1 +T[SU(n)] +28 +3.2.2 +SU(n)H Gauging +30 +3.2.3 +Other su(n)H Gaugings +32 +3.2.4 +U(n) Gauging +33 +4 +General Symmetry and Anomaly Analysis for 3d N = 4 SCFTs +34 +4.1 +Symmetries +35 +4.2 +Anomaly +36 +4.3 +Other Gauge Groups +37 +4.4 +Including Flavors +39 +4.5 +Special Case 1: Single Special Unitary Node +39 +4.6 +Special Case 2: Single Unbalanced Unitary Node +41 +5 +Consistency Checks +41 +5.1 +Class S +42 +5.1.1 +General Matching +42 +5.1.2 +Examples +44 +5.2 +5d SCFTs +49 +5.2.1 +Flavor Symmetry Groups from MQs +49 +– i – + +5.2.2 +Flavor Symmetry Groups from String Theory Constructions +54 +5.3 +3d Mirror Symmetry +57 +6 +Some Generalizations +59 +6.1 +N = 2 Gauging of T[SU(n)] +59 +6.2 +T[SU(2)]/ZC +2 and Its Gaugings +61 +A Geometric Computations for 5d SCFTs +64 +1 +Introduction +Gauge theories in three spacetime dimensions are extremely interesting to study from a the- +oretical viewpoint. Since the gauge coupling has positive mass dimension, any gauge theory +can be given an ultraviolet (UV) complete definition, but in the infrared (IR) the effective +gauge coupling becomes strong, opening up the possibility of interesting strong coupling be- +haviour. The case of 3d gauge theories with eight supercharges, i.e. N = 4 supersymmetry, +has been very well studied in this context, where it is known that with enough matter the +gauge theory flows in the IR to a 3d N = 4 superconformal field theory (SCFT). +There are many interesting non-perturbative phenomena that arise in the context of N = +4 supersymmetry. Of these, the most well-known phenomenon is that of 3d mirror symmetry +[1–4] which relates two different 3d N = 4 gauge theories such that the corresponding IR 3d +N = 4 SCFTs are same, up to the exchange of Coulomb and Higgs branches. +Arguably the most interesting aspect of 3d N = 4 supersymmetric gauge theories and +mirror symmetry is symmetry enhancement in the IR. The flavor symmetries of 3d N = 4 +gauge theories arise from hyper-K¨ahler isometries of the Higgs and Coulomb branch moduli +spaces. While the Higgs branch and associated symmetries may be understood classically, +the Coulomb branch receives 1-loop and non-perturbative corrections and the hyper-K¨ahler +metric depends on the gauge coupling. This allows for the emergence of additional hyper- +K¨ahler isometries and associated symmetry enhancement on the Coulomb branch in the IR +SCFT. This phenomenon plays a fundamental role in mirror symmetry. +This symmetry +enhancement was systematically explained in [5], using techniques developed in [6–8] based +on the study of monopole operators [9, 10]. +In this paper, we extend the discussion of symmetries in 3d N = 4 supersymmetric gauge +theories to include generalized symmetries [11], including global aspects of traditional 0-form +symmetries, 1-form symmetries, 2-groups symmetries and discrete ’t Hooft anomalies. +A key result is to identify the monopole operators in the UV gauge theory that allow a +determination of the global form of the IR Coulomb branch 0-form symmetry group. The +result matches the global form of the Higgs branch 0-form symmetry group of the mirror UV +– 1 – + +gauge theory, which can be computed classically from the matter content pf the mirror gauge +theory without considering its monopole operators. +Once the global form of the IR Coulomb 0-form symmetry group is known, an impor- +tant question is to determine the ’t Hooft anomalies of the Coulomb 0-form symmetry with +the Higgs 0-form symmetry. In three space-time dimensions, such anomalies are necessar- +ily discrete. The general structure of such anomalies for 3d gauge theories was explored in +our previous work [12]. This captured the information about the anomaly in terms of flavor +charges carried by mixed flavor-gauge monopole operators, which are in general non-genuine +local operators that arise at the end points of flavor-gauge vortex line defects. In N = 4 +supersymmetric gauge theories, there exist BPS configurations of flavor-gauge monopole op- +erators sitting at the ends of flavor-gauge vortex lines. These configurations thus descend to +configurations of local operators sitting at the ends of line operators in the corresponding IR +N = 4 SCFTs. During the flow the flavor charges of these non-genuine local operators get +mixed, and the mixing can be deduced using the methods of [11]. Finally, reversing the logic +of [12], one can use the information about the charges of these non-genuine local operators +to deduce the ’t Hooft anomaly of the IR SCFT. This provides a concrete and unambiguous +way of implementing ’t Hooft anomaly matching from UV 3d N = 4 gauge theories to IR 3d +N = 4 SCFTs. +In a similar fashion, we also determine the ’t Hooft anomalies of the IR Coulomb 0- +form symmetry with 1-form symmetries. The requisite operators are now what were dubbed +fractional gauge monopole operators in [12], which are non-genuine local operators living +at the ends of topological line operators generating the 1-form symmetry. For 3d N = 4 +gauge theories, there exist BPS configurations of fractional gauge monopole operators living +at the ends of topological line operators1. These configurations survive in the IR SCFT and +the charges of such non-genuine local operators under IR Coulomb symmetry determine the +precise form of the mixed ’t Hooft anomaly between the Coulomb 0-form symmetry and the +1-form symmetry of the IR 3d N = 4 SCFT. While this work was being written, we received +[13] which also used the analysis of [12] to obtain some of the results appearing in sections 3 +and 6 of this paper. +The analysis of this paper also opens the door for the determination of symmetries and +anomalies of higher-dimensional (i.e. in d > 3) SCFTs with at least eight supercharges. This +can be done by applying the analysis of this paper to 3d N = 4 magnetic quivers (MQs) +associated to these higher-dimensional SCFTs. MQs are 3d N = 4 gauge theories whose IR +behaviour captures information about the Higgs branch of vacua of the corresponding higher- +dimensional SCFTs. +MQs have been a subject of much interest and exploration recently +[14–43]. In particular they have been instrumental in studying Higgs branches for 4d N = 2 +and 5d N = 1 SCFTs, which are otherwise more difficult to access due to quantum corrections. +In some instances, we expect the symmetries of the 4d or 5d SCFT and that of its 3d +MQ theories to agree. This is in particular the case, when the higher-dimensional theory only +1Note that a topological operator automatically preserves all supersymmetry. +– 2 – + +exhibits 0-form symmetries, which then should then agree with the 0-form symmetries of the +MQ theory2. We test our proposal by computing the global 0-form symmetries of MQs, and +compare them to the global form of flavor symmetries of 4d class S theories and 5d SCFTs, +and find agreement. +Outlook. +This paper opens up many interesting avenues to explore in the connection be- +tween generalized symmetries and their ’t Hooft anomalies and SCFTs (in particular with 8 +supercharges). +As mentioned above, 0-form global symmetries act by hyper-K¨ahler isometries of Higgs +and Coulomb branch moduli spaces of vacua. A natural question is therefore whether there +exists such a geometric realization for generalised symmetries such as 1-form symmetries and +2-groups and their ’t Hooft anomalies. In forthcoming work [44], we will show that such +a realization may be found in the algebraic setting by promoting the Higgs and Coulomb +branch moduli spaces to moduli stacks. These stacky enhancements of moduli spaces keep +track of unbroken discrete gauge symmetry when flowing to the IR at points on the underlying +moduli space and carry actions of generalized symmetries. Moreover, the ’t Hooft anomalies +for generalised symmetries considered in this paper may be understood geometrically in terms +of equivariance properties of distinguished line bundles on these moduli stacks associated to +half-BPS line operators. +Another natural generalization in light of the fact that we consider invertible symmetries +in 3d, is the extension to non-invertible symmetries. There is a multitude of realizations +now. Most relevant for the field theoretic approach that was the focus in this paper are the +following constructions, which have direct 3d realizations [45–53]. Non-invertible symmetries +in 3d are of course very well explored in the context of modular tensor categories, however +here the interesting question is related to the interplay between non-invertible symmetries +and superconformal symmetry in 3d. One construction, which relies on the presence of mixed +anomalies has been explored in 3d in [45, 48]. To explore these in full it will be useful to +characterize systematically the symmetry topological field theories for SCFTs in 3d, as started +in [54, 55]. +Finally one can extend the considerations of this paper involving mostly continuous 0- +form symmetries to also include discrete 0-form symmetries and their associated ’t Hooft +anomalies. The two types of symmetries in general combine to form a disconnected 0-form +symmetry (Lie) group, which when combined with 1-form symmetries generally gives rise to +disconnected 2-group symmetries introduced recently in [56]. +The paper is organized as follows. In section 2, we provide a discussion of supersymmetric +defect operators and generalized symmetries of A/B-type (Coulomb/Higgs in the gauge theory +setting) and their ’t Hooft anomalies in 3d N = 4 theories. This is an application of our +general results in [12] to this supersymmetric setting. +2If the higher dimensional theory has higher-form symmetries, then these can in principle contribute to the +0-form symmetry of the 3d MQ theory. +– 3 – + +In section 3, we study the simplest example, in great detail, where one can apply the +considerations of this paper. This concerns T[SU(n)] theories and related theories that can +be obtained by gauging Higgs 0-form symmetry of T[SU(n)]. +In section 4, we generalize the methods employed in the previous section 3 to study a +large class of 3d N = 4 SCFTs that can be obtained in the IR of good 3d N = 4 quiver gauge +theories composed of balanced unitary and unbalanced unitary and special unitary gauge +groups along with matter hypermultiplets transforming in fundamental and bifundamental +representations. We describe how the Coulomb 0-form symmetry groups and its mixed ’t +Hooft anomalies with 1-form and Higgs 0-forms symmetries can be obtained easily by a +visual analysis of the UV quiver and noting the placement of unbalanced and flavor nodes +with respect to the positions of balanced nodes. +In section 5, we present a variety of consistency checks of our general results of section +4. We check that the IR Coulomb 0-form symmetry groups of magnetic quivers of Class S +theories and 5d SCFTs match the flavor symmetry groups of these higher dimensional SCFTs +computed via other methods. We also check that the IR Coulomb 0-form symmetry group +matches the Higgs 0-form symmetry group of the 3d mirror gauge theory. +In the final section 6, we study a few interesting theories that lie outside the general class +of theories studied in section 4. Our methods presented in section 4 can be easily generalized +to include such theories and lead to many interesting phenomena not observed in the class of +theories studied in section 4. These include pure ’t Hooft anomalies for 1-form symmetry, the +existence of 2-group symmetries in 3d N = 4 SCFTs, and mixed ’t Hooft anomalies between +2-group and 0-form symmetries of 3d N = 4 SCFTs. The latter two phenomena are exhibited +by the 3d N = 4 SCFT called T[SU(2)]/ZC +2 that can be obtained from T[SU(2)] by gauging +a Z2 subgroup of SO(3)C Coulomb 0-form symmetry of T[SU(2)], and can be obtained as the +IR SCFT corresponding to 3d N = 4 SQED with U(1) gauge group and 2 hypermultiplets of +charge 2. +Appendix A provides details on the computation of global forms of flavor symmetry +groups of 5d SCFTs from Calabi-Yau threefold singularities. +2 +Generalized Symmetries of 3d N = 4 Theories +In this section, we consider general aspects of invertible generalized symmetries in 3d N = +4 supersymmetric theories, including 0-form symmetries, 1-form symmetries and 2-group +symmetries as well as their ’t Hooft anomalies. For ordinary 0-form symmetries, we distinguish +between R-symmetries and flavor symmetries. We focus on flavor symmetries, which commute +with all of the supercharges, and consider possible 2-group symmetries that combine flavor +symmetries and 1-form symmetries. +We will introduce two types of such symmetries called “A-type” and “B-type” depending +on which class of BPS operators are charged under them. For continuous 0-form symmetries, +this corresponds to the known the classification of supermultiplets that conserved currents or +background gauge fields for continuous symmetries may transform in, or equivalently central +– 4 – + +extensions of the supersymmetry algebra. However, we explain how this classification can be +applied more broadly to both finite and continuous symmetry groups, in addition to 1-form +and 2-group symmetries.3 +These symmetries may have various ’t Hooft anomalies, which we study using the tech- +niques introduced in our previous paper [12]. In particular, we consider “A-type” and “B- +type” BPS solitonic local operators and line defects that source background fields for the above +symmetries and explain how their properties capture different types of ’t Hooft anomaly. We +also explain how gauging discrete symmetries interchanges A-type and B-type symmetries in +a manner compatible with such ’t Hooft anomalies and how this leads to examples of mirror +symmetry involving generalised symmetries. +Our primary example throughout this section will be standard 3d N = 4 supersymmetric +gauge theories built from vectormultiplets and hypermultiplets. In such case, the A-type and +B-type symmetries are associated to Coulomb and Higgs branch geometry respectively. It +is also possible to consider gauge theories with less supersymmetry that flow to N = 4 +supersymmetry in the IR [57–59]. In these cases, the identification of the A-type and B-type +symmetries from a UV perspective is more intricate. +2.1 +BPS Defects +We begin with a discussion of BPS local operators, line defects and junctions that will play +a role in the classification of flavor symmetries. First recall that a theory with 3d N = 4 +supersymmetry has R-symmetry algebra so(4) ∼= su(2) ⊕ su(2) and supercharges QA ˙A +α +where +α is a euclidean space-time spinor index and the indices A, ˙A denote the spinor representation +of the two factors of the R-symmetry group. +Local Operators. +The half-BPS genuine local operators come in two types: +• A-type: annihilated by the four supercharges QA ˙+ +α . +• B-type: annihilated by the four supercharges Q+ ˙A +α . +The A-type operators are constructed from the bottom scalar components of vectormultiplets +and twisted hypermultiplets, while the B-type operators are constructed from the bottom +scalar components of hypermultiplets and twisted vectormultiplets. This classification into +A-type and B-type applies equally well to non-genuine twisted sector local operators attached +to a topological line defect. +The two sets of half-BPS genuine local operators generate two chiral rings CA,CB whose +spectra define complex affine moduli spaces +XA := Spec(CA) +(2.1) +XB := Spec(CB) . +(2.2) +3This does not preclude the existence of additional discrete global symmetries that are not of this type. +Examples of such symmetries include outer automorphisms of gauge groups such as charge conjugation or +automorphisms of quiver diagrams, and anomalous 1-form symmetries that arise when coupling to a 3d TQFT. +– 5 – + +In a standard supersymmetric gauge theory constructed from vectormultiplets and hyper- +multiplets, they coincide with the Coulomb and Higgs branch respectively, in the absence of +resolution or deformation parameters, viewed as complex algebraic varieties. +We might consider the possibility that there are local operators annihilated by all of the +supercharges. Such operators are necessarily topological. We will assume that there is a +unique (upto multiplication by a complex number) such topological local operator, namely +the identity operator. This is tantamount to the statement that the theory is irreducible or +equivalently that there are no 2-form symmetries. +In the opposite direction, we may have occasion to consider more general quarter-BPS +operators annihilated by two supercharges Q+ ˙+ +α . +They are half-BPS for the 3d N = 2 +supersymmetry algebra generated by Q+ ˙+ +α , Q− ˙− +α . +Line Operators. +We consider half-BPS line defects along the x3-axis preserving a 1d N = 4 +supersymmetric quantum mechanics sub-algebra of the 3d N = 4 supersymmetry algebra. +Such line operators were first introduced in supersymmetric gauge theories in [60] and have +been further studied in [61–63]. +There are two classes of half-BPS lines: +• A-type: annihilated by four supercharges QA ˙+ ++ , QA ˙− +− . +• B-type: annihilated by four supercharges Q+ ˙A ++ , Q− ˙A +− . +The line defects can be described uniformly by consistent couplings to 1d N = 4 supersym- +metric quantum mechanics with super-multiplets obtained by dimensional reduction from 2d +N = (2, 2) and N = (0, 4) supersymmetry respectively. Examples include B-type Wilson +lines for dynamical vectormultiplets and A-type Wilson lines for dynamical twisted vector- +multiplets. We note that this classification applies equally well to half-BPS twisted sector +line defects that are attached to a topological surface. +A special case is line defects annihilated by all of the supercharges, which are simultane- +ously A-type and B-type and therefore necessarily topological line defects. Such line defects +are normally considered as generators of 1-form symmetries, but may also be charged under +them in the presence of ’t Hooft anomalies. This situation may arise when coupling to a +general 3d TQFT in a way that preserves N = 4 supersymmetry but does not arise in the- +ories constructed from standard supermultiplets. Incorporating such topological line defects +as charged objects will require a refinement of the classification of symmetries presented here +and some examples are presented in subsequent sections. +In the opposite direction, we may have occasion to consider more general quarter-BPS +line defects preserving the common pair of supercharges Q+ ˙+ ++ , Q− ˙− +− . They can be regarded as +half-BPS line defects for the 3d N = 2 supersymmetry algebra generated by the supercharges +Q+ ˙+ +α , Q− ˙− +α . +Junctions. +Finally we consider various local junction operators between pairs of line de- +fects. We consider two classes of quarter-BPS junctions between pairs of A-type and B-type +– 6 – + +lines and preserve two supercharges lying in the intersections of the two sets of four super- +charges preserved by genuine local operators and line defects: +• A-type: annihilated by two supercharges QA ˙+ ++ . +• B-type: annihilated by two supercharges Q+ ˙A ++ . +It is also possible to consider local junction operators between a half-BPS A-type and a B-type +line defect, or alternatively between a pair of quarter-BPS line defects, which both preserve +the single supercharge Q++ ++ . +Comment on relation to topological twist +The above classification of BPS operators +is related but distinct to the classification of operators in topological twists of 3d N = 4 +supersymmetry, where A-type and B-type operators are defined as those in the cohomology +of the nilpotent supercharges +QA := Q+ ˙+ ++ ++ Q− ˙+ +− +(2.3) +QB := Q+ ˙+ ++ ++ Q+ ˙− +− +. +(2.4) +Correspondingly, we are interested only in genuine symmetries generated by extended opera- +tors that are topological in the full 3d N = 4 theory, not merely after performing a topological +twist. +2.2 +A- and B-type Symmetries +We now consider the classification of invertible flavor symmetries in 3d N = 4 theories. As +mentioned above, we assume that the theory is irreducible and therefore restrict ourselves to +at most 2-group symmetries. In particular, there is a unique genuine local operator that is +simultaneously A-type and B-type, which is the identity operator. +The proposal is then that the most general flavor symmetry is a product of A-type and +B-type 2-group symmetries associated to the above classification of BPS defects. +2.2.1 +0-form Symmetry +For continuous 0-form symmetries, it is well known that the flavor symmetry takes the form +of a product FA × FB for compact Lie groups FA, FB. The two factors are known as A-type +and B-type symmetry groups. +At the level of the associated Lie algebra fA ⊕ fB, this decomposition may be understood +from the allowed central extensions of the 3d N = 4 supersymmetry algebra. This admits +a pair of central charges ZAB, Z ˙A ˙B transforming in the adjoint representations of the two +su(2) R-symmetries. The central charges are proportional to the generators of the A-type and +B-type symmetries respectively with coefficients given by scalar fields σAB, σ ˙A ˙B in vector- +multiplets and twisted vectormultiplets respectively. In summary, A-type symmetries couple +to vectormultiplets and B-type symmetries to twisted vectormultiplets. +– 7 – + +However, in order to provide a definition of the flavor symmetry group FA × FB, which +also applies to discrete symmetries, and in addition to formulate obstruction classes that +appear in ’t Hooft anomalies for these symmetries, it is convenient to define symmetries +starting from the BPS operators on which they act. +Definitions. +The flavor symmetry groups FA, FB are defined as the maximal compact +Lie groups with Lie algebras fA, fB that act faithfully on A-type and B-type genuine half- +BPS local operators respectively. +This definition also applies when fA, fB are trivial, in +which case the flavor symmetry groups are discrete. These symmetry groups (or rather their +complexification in the continuous case) will act by complex isometries on the moduli spaces +XA, XB. +In the construction of 2-group symmetries involving these flavor symmetries and their +’t Hooft anomalies, we will also need to consider non-genuine local operators that sit at the +junctions between half-BPS line defects. +Let us consider A-type or B-type half-BPS line defects that preserve the whole symmetry +group FA or FB respectively. Such line defects may then end on A-type or B-type quarter- +BPS local operators that transform in representations of central extensions of FA, FB by +discrete abelian groups, that are not representations of FA, FB. It is convenient to write +down the short exact sequences +0 −→ ZA −→ FA −→ FA −→ 0 +(2.5) +0 −→ ZB −→ FB −→ FB −→ 0 , +(2.6) +where ZA, ZB are finite abelian groups and FA, FB denotes the extended symmetry groups. +Equivalently, we have the quotients FA = FA/ZA, FB = FB/ZB. In summary, local operators +at the end of line defects may be charged under ZA, ZB. +There are associated obstruction classes +wA +2 ∈ H2(BFA, ZA) +(2.7) +wB +2 ∈ H2(BFB, ZB) , +(2.8) +for lifting FA, FB bundles to FA, FB bundles, which play an important role in the description +of 2-groups and ’t Hooft anomalies involving these symmetries. In particular, introducing +background fields BA +1 , BB +1 : M → BFA, BFB, there are associated obstruction classes on +spacetime via pull-back (BA +1 )∗wA +2 , (BB +1 )∗wB +2 . In what follows, we will often abuse notation +and denote these spacetime obstruction classes also by wA +2 , wB +2 . +Gauge theories. +Let us consider standard supersymmetric gauge theories constructed from +vectormultiplets and hypermultiplets. In such cases, it is appropriate to replace the monikers +A/B by C/H, which refer to Coulomb and Higgs respectively. +The B-type symmetry FH acts faithfully on gauge-invariant combinations of hypermul- +tiplet fields, while the central extension FH is constructed by examining the charges of non- +gauge invariant combinations of hypermultiplet fields attached to B-type half-BPS Wilson +lines for the dynamical vectormultiplet. +– 8 – + +This is conveniently captured by introducing the structure group S, which captures the +combination of gauge and B-type flavor symmetries acting faithfully on all supermultiplets. +In other words, the bundles for S correspond to the most general combination of gauge and +B-type flavor symmetry bundles (transforming in dynamical and background vectormultiplets +respectively) to which the theory may be consistently coupled. It takes the form +S = G × FH +E +, +(2.9) +where G denotes the gauge group, which we assume is connected, and E is a subgroup of the +center Z(G × FH) of G × FH such that pH(E) = ZH where pH : Z(G) × Z(FH) → Z(FH) is +the natural projection. +On the other hand, the A-type symmetry group FC acts faithfully on genuine half-BPS +monopole operators. This is the topological symmetry +FC = � +π1(G) , +(2.10) +which measures the topological class of the G-bundles on a sphere surrounding the monopole +operator. It may be continuous or discrete. Unlike the B-type symmetry group, this may +undergo enhancement at an IR superconformal fixed point. Determining the precise global +form of the enhanced symmetry group is a major goal of this paper. +The gauge theory may couple to bundles for the structure group S. Correspondingly, +there exist A-type half-BPS line defects corresponding to gauge-flavour vortex lines for the +structure group labelled by a co-character φ : U(1) → S. This may involve a fractional gauge +vortex lines, which by definition are vortices associated to co-characters for the quotient group +G = G/Zg , +(2.11) +where Zg = pg(E) where pg : Z(G) × Z(FH) → Z(G) is the natural projection. Note that the +co-character associated to a fractional gauge vortex must be a co-character simultaneously +for G and for S, and a general co-character for G does not satisfy this criterion. Gauge- +flavour vortex line defects may end on monopole operators of fractional magnetic charge, +which results in a short exact sequence +1 −→ ZC −→ FC −→ FC −→ 1 , +(2.12) +where +FC = � +π1(G) +(2.13) +and we identify ZC = � +Zg. +Let us note that in general 3d gauge theories it is necessary to include such topological +symmetries as part of the structure group, thus extending the above discussed structure +group S into an extended structure group �S. Thus is due to the fact that monopole operators +receive charges under gauge and flavor symmetries due to effective Chern-Simons levels, as +discussed in our previous paper [12]. However, with N = 4 supersymmetry and only standard +supermultiplets, this extended structure group factorises as �S = S × FC. +– 9 – + +Example. +A basic example to illustrate these points is supersymmetric QED with G = U(1) +and N hypermultiplets of charge q. We assume without loss of generality that q > 0. +The hypermultiplets contain complex scalar fields Xj, Yj of charge q, −q transforming +in the fundamental and anti-fundamental representations of the flavor symmetry algebra +fH = su(N). The B-type genuine local operators are the gauge-invariant combinations XiYj +transforming in the adjoint representation. The B-type flavor symmetry group is therefore +FH = PSU(N). +There are B-type Wilson lines Wn labelled by an integer charge n. Consider the case where +n > 0. If n is a multiple of the minimal charge q, the Wilson line may end on local operators +consisting of homogeneous polynomials in Xj of degree m = n/q, which transform in the m-th +symmetric power of the fundamental representation of su(N). This includes representations +of the central extension FH = SU(N) that are not representations of FH = PSU(N) forming +a short exact sequence +1 −→ ZN −→ SU(N) −→ PSU(N) −→ 1 +(2.14) +with ZH = ZN. The associated obstruction class may be denoted by wH +2 ∈ H2(X, ZN). This +is reflected in the structure group +S = U(1) × SU(N) +ZqN +, +(2.15) +where the denominator is generated by the central element (e2πi/qN, e2πi/N1N). +The genuine A-type local operators correspond to half-BPS monopole operators labelled +by a co-character m : U(1) → G, which is an integer magnetic charge m ∈ Z. Correspondingly, +the A-type symmetry group is the topological symmetry +FC = � +π1(G) = U(1) , +(2.16) +whose charge measures the topological type of a G-bundle on a two-sphere surrounding a +monopole operator. +However, the theory may in general be coupled to bundles for the structure group S and +therefore there exist A-type gauge-flavor vortex lines labelled by co-characters φ : U(1) → S. +They are conveniently labelled by a pair of co-characters φ = (φg, φH) ∈ Z × Z≥0 with +obstructions +αg = φg mod qN +(2.17) +αH = φH mod N . +(2.18) +For this to define a consistent co-character of S, the obstructions must arise as projections +αg = pg(α) and αH = pH(α) of a common obstruction α ∈ E, where pg, ph are projections +from Z(G) × Z(FH) to Z(G), Z(FH). This requires αg mod N = αH. Let us summarise some +examples that will play a role in what follows: +– 10 – + +• Consider pure fractional gauge vortex lines φ = (φg, 0), which requires φg is a multiple +of N. If φg is a multiple of qN, this lifts to dynamical vortex for the gauge group +and corresponds to a trivial line defect. They end on genuine A-type gauge monopole +operators of magnetic charge m = φ/qN ∈ Z. +• The remaining fractional gauge vortex lines, modulo dynamical vortices, are indexed by +φg = nN with n = 0, 1, . . . , q − 1 and end on A-type monopoles of fractional magnetic +charge n/q. +• More general gauge-flavour vortex lines φ = (φg, φH) may end on A-type gauge-flavour +monopoles of fractional magnetic charge in multiples of 1/qN. +In particular, the final bullet point means that we must introduce the qN-fold cover of the +topological symmetry FC = � +π1(G) ∼= U(1), which is an extension of the topological symmetry +by ZC = ZqN. The associated obstruction class for background fields is wC +2 = cC +1 mod qN, +where cC +1 denotes the first Chern class of a background FC = U(1) bundle. +2.2.2 +1-form Symmetry +Following the same philosophy, we define A/B-type 1-form symmetries by applying the recipe +studied in [12], but restricted to half-BPS A/B-type line defects. +Definition +The construction begins by considering equivalence classes of A-type or B-type +line defects. We say that two line defects L1, L2 are equivalent L1 ∼ L2 if there exists a +non-trivial quarter-BPS junction of the appropriate type connecting them. In other words, +equivalence classes capture the half-BPS line defects that cannot be screened by quarter-BPS +junctions of A-type or B-type. +The equivalence classes of A-type and B-type lines inherit the structure of abelian groups +�ΓA, �ΓB from the OPE of parallel line defects. The A-type and B-type 1-form symmetries are +defined as the Pontryagin dual groups +ΓA := Hom(�ΓA, U(1)) +ΓB := Hom(�ΓB, U(1)) , +(2.19) +such that these 1-form symmetries act on half-BPS lines via the natural pairings Γ×�Γ → U(1). +Correspondingly, we can introduce ΓA, ΓB-valued 2-cochain backgrounds BA +2 , BB +2 +for +these 1-form symmetries. If the 1-form symmetries do not participate in 2-groups, the back- +ground field are closed and define ΓA, ΓB-valued 2-cocycles. +Gauge theories. +In standard gauge theories built from vectormultiplets and hypermulti- +plets, the A-type and B-type 1-form symmetries may be determined from the properties of +vortex lines and dynamical Wilson lines respectively. +The B-type symmetry arises from half-BPS Wilson lines in representations of the gauge +group G. +In the absence of hypermultiplets, quarter-BPS junctions may only arise from +– 11 – + +vectormultiplet fields in the adjoint representation of the gauge group. In this case, Wilson +lines in representations R1, R2 are equivalent if and only if the central characters of the +representations coincide. Therefore +�ΓH = Hom(Z(G), U(1)) +(2.20) +is the abelian group of central characters and the 1-form symmetry coincides with the centre +of the gauge group ΓH = Z(G). +More generally, incorporating hypermultiplet fields, the 1-form symmetry ΓH is the sub- +group of the center of the gauge group that acts trivially on hypermultiplets. This can be +formulated in terms of the structure group +S = G × FH +E +, +(2.21) +where the B-type 1-form symmetry may be identified with the intersection ΓH = Z(G) ∩ E. +This naturally forms a short exact sequence +1 −→ ΓH −→ E −→ ZH −→ 1 , +(2.22) +which will play a role in the construction of 2-group symmetries below. +An A-type 1-form symmetry may arise in gauge theories with discrete or continuous but +disconnected gauge groups, which we will not discuss here. We will instead explain below how +A-type 1-form symmetries arise generally when gauging discrete B-type 0-form symmetries. +Example. +Let us again consider supersymmetric QED with G = U(1) and N hypermulti- +plets of charge q > 0. It is a standard result that this has a B-type 1-form symmetry ΓB = Zq +as B-type Wilson lines Wn cannot be screened unless n is a multiple of q. +2.2.3 +2-group Symmetry +The 0-form and 1-form symmetries defined above may combine to form A-type and B-type +2-group symmetries. +In order to define the 2-group symmetry structure, we will need to +consider a more refined equivalence relation for line defects that takes into account the fact +that junctions may transform in representations of central extensions of symmetry groups. +For further background on this perspective see [12, 64–67]. +Definition. +We first define another equivalence relation such that L1 ∼′ L2 if the two +line operators admit quarter-BPS junctions of the appropriate type transforming in honest +representations of FA, FB that are not charged under ZA, ZB. +These equivalence classes form larger abelian groups � +EA, � +EB sitting in short exact se- +quences +0 −→ � +ZA −→ � +EA −→ � +ΓA −→ 0 +(2.23) +0 −→ � +ZB −→ � +EB −→ � +ΓB −→ 0 . +(2.24) +– 12 – + +The first terms in the sequence can be understood as follows. The quarter-BPS local operators +screening line operators in equivalence classes corresponding to elements �zA ∈ � +ZA ⊂ � +EA, +�zB ∈ � +ZB ⊂ � +EB transform in representations of FA, FB with charges �zA, �zB under ZA, ZB. +The Pontryagin dual exact sequences are +1 −→ ΓA −→ EA −→ ZA −→ 1 +(2.25) +1 −→ ΓB −→ EB −→ ZB −→ 1 . +(2.26) +The 0-form symmetries FA, FB and 1-form symmetries ΓA, ΓB now combine into 2-groups +whose Postnikov classes are given by +ΘA = Bock(wA +2 ) +(2.27) +ΘB = Bock(wB +2 ) , +(2.28) +using the appropriate Bockstein homomorphisms Bock : H2(X, ZA) → H3(X, ΓA) or Bock : +H2(X, ZB) → H3(X, ΓB) associated to the above short exact sequences. If the Postnikov +classes are trivial the 2-group symmetry is a product of a 0-form and a 1-form symmetry. +We may then introduce backgrounds for the 2-group symmetry given by the EA, EB-valued +combinations +BA +w = i(BA +2 ) + �wA +2 +(2.29) +BB +w = i(BB +2 ) + �wB +2 +(2.30) +where i : ΓA, ΓB → EA, EB denotes the relevant inclusion maps and �wA +2 , �wB +2 are co-chain lifts +of wA +2 , wB +2 under the projections p : EA, EB → ZA, ZB in the above short exact sequences. +These combinations are closed by construction and define EA, EB-valued co-cycles. If the +Postnikov class is trivial, we may work independently with closed backgrounds BA +2 , BB +2 and +wA +2 , wB +2 for the 1-form and 0-form symmetries respectively. +Gauge theories. +Consider a standard supersymmetric 3d N = 4 gauge theory built from +ordinary vectormultiplets and hypermultiplets. The data determining the B-type 2-group is +encoded in the structure group +S = G × FH +E +. +(2.31) +In particular, we have already identified ZH = pH(E) and the 1-form symmetry ΓB = Z(G)∩E. +The remaining ingredient is simply the identification EH = E, which forms the appropriate +short exact sequence. +Example. +Let us consider again supersymmetric QED with G = U(1) and N hypermulti- +plets of charge q > 1. Recall that there are B-type symmetry groups FH = PSU(N) and +ΓH = Zq sitting in short exact sequences +1 −→ ZN −→ SU(N) −→ PSU(N) −→ 1 +(2.32) +– 13 – + +and +1 −→ Zq −→ ZqN −→ ZN −→ 1 +(2.33) +respectively. There is therefore a potential Postnikov class Θ = Bock(wH +2 ) where wH +2 is the +obstruction class for the first sequence and Bock : H2(PSU(N), ZN) → H3(PSU(N), Zq) +is the Bockstein homomorphism for the second. The Bockstein homomorphism may or may +not be trivial. In the former case, there is no 2-group symmetry. An example of vanishing +Bockstein is provided if the first sequence splits, which requires gcd(q, N) = 1. +A non- +supersymmetric version of this example was considered already in [12]. +There is also an A-type 0-form symmetry FA = U(1), which does not participate in a +2-group. However, we will show later that it has a mixed ’t Hooft anomaly with the above +B-type 2-group. +2.3 +Solitonic defects +The A-type and B-type local, line and junctions operators may induce background fields +for flavor symmetries and correspond to solitonic defects in the terminology of [12]. More +specifically they induce vortex and monopole configurations for background fields associated +to flavor symmetries. Such defects play a crucial role in determining ’t Hooft anomalies from +the spectrum of BPS charged objects. +The proposal is that A-type defects may source background fields for B-type flavor sym- +metries and vice-versa. We substantiate this claim for vortex and monopole backgrounds in +the remainder of this subsection. +Definitions. +For concreteness, let us first consider A-type line defects. The most general +situation is that they induce a background field configuration for the B-type 2-group symmetry +such that +� +D2 +BB +w = αB , +(2.34) +where D2 denotes a small disk intersecting the line defect transversely and αB ∈ EB. We refer +to this as a background vortex configuration for the 2-group symmetry. Such line defects may +end on A-type quarter-BPS local operators with the property that +� +S2 BB +w = αB , +(2.35) +where S2 is now a small 2-sphere surrounding the local operator and intersecting the line +defect transversely. +We refer to this as a background monopole configuration for the 2- +group symmetry. Entirely analogous statements hold with A-type and B-type symmetries +and defects interchanged. +This reduces to simpler statements in special cases of individual 0-form and 1-form sym- +metry groups. For example, an A-type line defect may induce a vortex background for a +B-type 0-form symmetry such that +� +D2 +wB +2 = αB , +(2.36) +– 14 – + +where now αB ∈ ZB. Similarly, if there is a B-type 1-form symmetry that does not participate +in a 2-group symmetry then an A-type line defect may induce a vortex background for the +1-form symmetry such that +� +D2 +BB +2 = αB , +(2.37) +where now αB ∈ ΓB. Similar comments apply to local operators and monopole backgrounds. +Again, entirely analogous statements fold with A-type and B-type symmetries and defects +interchanged. +Gauge theory. +In a standard supersymmetric gauge theory, the B-type symmetry back- +ground field configurations sourced by A-type line defects can be understood systematically +in terms of the structure group S. +Since the gauge theory may be consistently coupled to S-bundles, A-type line defects +include gauge-flavor vortex defects Vφ labelled by co-characters +φ : U(1) → S . +(2.38) +Such A-type gauge-flavor vortex defects source background field configurations for the B-type +2-group symmetry such that +� +D2 +BH +2 = αH +(2.39) +where αH ∈ E is the obstruction for lifting φ to a co-character for G × F. This reduces to +corresponding simpler statements for individual 0-form and 1-form symmetries. There are +many special cases of interest and further examples are considered in the example below. +In the opposite direction, B-type Wilson lines for a dynamical vectormultiplet source a +background vortex configuration for the dual A-type topological symmetry. This is discussed +for G = U(1) in the example below. +Example. +Consider again supersymmetric QED with N hypermultiplets of charge q > 0. +For simplicity, we assume here that gcd(q, N) = 1 so that the 2-group structure is trivial. +We consider general A-type gauge-flavor vortex lines labelled by a co-character of the +structure group φ : U(1) → S. This can be conveniently labelled by a pair of co-characters +φ = (φg, φH) ∈ Z × Z≥0 as discussed previously. We can then summarise the backgrounds +that they induce as follows: +• Consider pure fractional gauge vortex lines φ = (φg, 0), which requires φg is a multiple +of N. If φg is a multiple of qN, this lifts to dynamical vortex for the gauge group and +corresponds to a trivial line defect. +• The remaining fractional gauge vortex lines, modulo dynamical vortices, are indexed +by φ = nN with n = 0, 1, . . . , q − 1 and source backgrounds for the B-type 1-form +symmetry ΓH = Zq such that +� +D2 +BH +2 = n . +(2.40) +– 15 – + +• General vortex lines φ = (φg, φH) induce combinations of 0-form and 1-form symmetry +backgrounds. Note that a pure flavor vortex φ = (0, φH) cannot induce an obstruction +background wH +2 for the B-type 0-form symmetry since this would require that αH = +φH mod N = 0. +In the opposite direction, the B-type Wilson lines Wn source a background for the A-type +topological symmetry FC = U(1) such that +� +D2 +cC +1 = n . +(2.41) +These statements will be utilised to derive mixed ’t Hooft anomalies below. +2.4 +’t Hooft Anomalies +We now consider the ’t Hooft anomalies captured by BPS operators considered so far. These +are primarily mixed ’t Hooft anomalies between A-type and B-type symmetries. +The most general situation assuming potential A-type and B-type 2-group symmetries +is as follows. Let us consider A-type line defects that source background field configurations +for the B-type 2-group symmetry labelled by elements αB ∈ EB. Such line defects define +equivalence classes in �EA and this provides a homomorphism +�γ : � +EB → EA . +(2.42) +In this situation there is a mixed ’t Hooft anomaly represented by the four-dimensional SPT +phase +A4 = +� +BA +w ∪ γ(BB +w ) . +(2.43) +This construction may be performed exchanging A-type and B-type defects and symmetries +and these constructions must be compatible. +There are various simpler special cases that are worth considering: +• Let us assume there are 1-form symmetries ΓA, ΓB that do not participate in 2-groups. +The A-type line defects may source backgrounds for a B-type 1-form symmetry labelled +by elements αB ∈ ΓB and simultaneously charged under the A-type 1-form symmetry +ΓA. This determined a homomorphism +γ : ΓB → �ΓA +(2.44) +and mixed ’t Hooft anomaly +A4 = +� +BA +2 ∪ γ(BB +2 ) . +(2.45) +• Consider an A-type 0-form symmetry FA and a B-type 1-form symmetry ΓB not par- +ticipating in a 2-group. The A-type line defects may source backgrounds for a B-type +– 16 – + +1-form symmetry labelled by elements αB ∈ ΓB and simultaneously end on A-type local +operators charged under ZA. This determines a homomorphism +γ : ΓB → � +ZA +(2.46) +and mixed ’t Hooft anomaly +A4 = +� +wA +2 ∪ γ(BB +2 ) . +(2.47) +• Consider an A-type 0-form symmetry FA and a B-type 0-form symmetry FB. +The +A-type line defects may source backgrounds wB +2 for a B-type 0-form symmetry labelled +by elements αH ∈ ZB and simultaneously end on A-type local operators charged under +ZA. This determines a homomorphism +γ : ZB → � +ZA +(2.48) +and mixed ’t Hooft anomaly +A4 = +� +wA +2 ∪ γ(wB +2 ) . +(2.49) +Gauge theory. +For a standard supersymmetric gauge theory with connected gauge group, +there may be a mixed ’t Hooft anomaly between the A-type topological symmetry and the +B-type 2-group symmetry. This may be determined, for example, by examining gauge-flavor +vortex line defects that induce backgrounds for the B-type flavor symmetry and the fractional +A-type topological charges of the monopoles on which they end. An example is presented +below. +Example. +Let us consider again supersymmetric QED with N hypermultiplets of charge +q > 0. The symmetries are summarises as follows: +• An A-type topological symmetry FA = U(1) whose background field has an ZqN-valued +obstruction class wC +2 = c1 mod qN. +• A B-type 2-group symmetry with FH = PSU(N) and ΓH = Zq with ZqN-valued +background field BH +w = NB2 + �wH +2 . +The mixed ’t Hooft anomaly between these symmetries is derived from the fractional topolog- +ical charges of the A-type local operators on which gauge-flavour vortex lines end. A marginal +generalisation of the examples presented in [12] shows that this is represented by the 4d SPT +phase +A4 = exp +�2πi +qN +� +(cC +1 mod qN) ∪ BH +w +� +. +(2.50) +When gcd(q, N) = 1 and the 2-group structure is trivial, this simplifies to a sum of mixed +anomalies for the individual B-type 0-form and 1-form symmetries +A4 = exp +�2πi +q +� +(cC +1 mod q) ∪ BH +2 + 2πi +N +� +(cC +1 mod N) ∪ wB +2 +� +. +(2.51) +– 17 – + +2.5 +Discrete Gauging +In three dimensions, gauging a discrete abelian 0/1-form symmetry Γ group results in a +Pontryagin dual 1/0-form symmetry group �Γ := Hom(A, U(1)). In the context of symmetries +in theories with N = 4 supersymmetry these operations interchange A-type and B-type +symmetries. In summary: +• Gauging an A-type discrete abelian 0/1-form symmetry Γ results in a B-type Pontryagin +dual 1/0-form symmetry �Γ := Hom(Γ, U(1)). +• Gauging an B-type discrete abelian 0/1-form symmetry Γ results in a A-type Pontryagin +dual 1/0-form symmetry �Γ := Hom(Γ, U(1)). +This is compatible with our discussion of mixed ’t Hooft anomalies between A-type and B- +type symmetries. A slight generalisation of the above is that gauging a normal subgroup +Γ ⊂ Γ′ of a 0-form symmetry results in a 1-form symmetry �Γ with a mixed anomaly with +the remaining quotient group Γ′/Γ controlled by the extension class [68].4 In theories with +N = 4 supersymmetry and with the above identifications, this is always a mixed anomaly +between A-type and B-type symmetries considered above. +This provides a clean and general method to construct new examples of mirror symmetry +that involved 1-form symmetries and their anomalies can be explicitly matched. Examples +are presented below and in the remainder of the paper. +Example. +An example of this phenomenon arises in U(1) supersymmetric gauge theories, +which have an A-type topological symmetry FC = U(1) under which genuine monopole oper- +ators are charged. Gauging a subgroup Zq ⊂ U(1) of the topological symmetry is equivalent +to multiplying the charges of all hypermultiplet fields by q. This results in a B-type 1-form +symmetry ΓH = Zq due since a subgroup of the gauge group now acts trivially on all hyper- +multiplet fields. This B-type symmetry has a mixed anomaly with the remaining topological +symmetry after gauging. +As an example consider supersymmetric QED with N hypermultiplets of charge 1. This +has A-type topological symmetry FC = U(1) and B-type symmetry FH = PSU(N) with +mixed ’t Hooft anomaly +A4 = exp +�2πi +N +� +(cC +1 mod N) ∪ wH +2 +� +. +(2.52) +We now gauge Zq ⊂ FC assuming gcd(q, N) = 1. This results in a dual B-type 1-form sym- +metry ΓH = Zq and an additional mixed anomaly with the remaining topological symmetry +such that the total anomaly is +A4 = exp +�2πi +q +� +(cC +1 mod q) ∪ BH +2 + 2πi +N +� +(cC +1 mod N) ∪ wH +2 +� +. +(2.53) +4This conclusion holds even when Γ′ is continuous. +– 18 – + +This is indeed the anomaly of supersymmetric QED with N hypermultiplets of charge q. A +slightly more intricate argument is required when gcd(q, N) > 1. +The mirror of supersymmetric QED with N hypermultiplets of charge q can therefore +be obtained from the mirror of supersymmetric QED with N hypermultiplets of charge 1 by +gauging a Zq symmetry. This results in a circular quiver gauge theory with (N − 1) U(1) +nodes and 1 Zq node, which has an A-type 1-form symmetry ΓA = Zq. In particular, when +N = 1, the mirror theory is a Zq-quotient of a free hypermultiplet. +3 +Warmup: T[SU(n)] and its Gaugings +In this section, we study the simplest example, which is provided by 3d N = 4 SCFTs known +as T[SU(n)] theories. We study the global forms of flavor symmetry groups of T[SU(n)] along +with their ’t Hooft anomalies. We also study some other 3d N = 4 SCFTs closely related to +T[SU(n)], in that they can be obtained by gauging (along with the addition of extra flavors +for balancing purposes) the Higgs branch flavor symmetries of the UV unitary quiver theory +whose IR fixed point is T[SU(n)]. +3.1 +T[SU(2)] and its Gaugings +Let us begin with T[SU(2)] and theories related to it by gauging. Some of the results on the +symmetries and anomalies of the T[SU(2)] theory are known already in the literature [69, 70]. +We derive them here using the perspective of our earlier paper [12]. +3.1.1 +T[SU(2)] +The T[SU(2)] theory arises as an IR fixed point of the following 3d N = 4 Lagrangian theory +U(1) +[su(2)H] , +(3.1) +having a U(1) gauge group and 2 hypermultiplets of charge 1 that are rotated by an su(2)H +flavor symmetry algebra5, where the subscript H indicates that the su(2)H symmetry acts +non-trivially (i.e. is spontaneously broken) on the Higgs branch of vacua of the theory. +B-Type 0-Form Symmetry. +The genuine local operators charged under su(2)H arise from +gauge invariant combinations of hypermultiplets. Such gauge invariant combinations all form +representations of the Lie group SO(3)H with Lie algebra su(2)H. +Thus, the Higgs branch 0-form symmetry group is +FH = SO(3)H = SU(2)H/Z2 , +(3.2) +where SU(2)H is the simply connected group with Lie algebra su(2)H. +5Note that the flavor symmetry is not u(2)H = su(2)H ⊕ u(1)H. One way to see it is to note that the u(1)H +part acts in the same way as the u(1) gauge algebra and hence is absorbed into that. +– 19 – + +Another way of deducing the above 0-form symmetry group is as follows. Let us put +the theory (3.1) on a non-trivial compact 3-manifold M3. The charges of the vector and +hypermultiplets allow us to turn on bundles for the structure group +S = U(1) × SU(2)H +Z2 +, +(3.3) +where U(1) is the gauge group. The Z2 in the denominator is the diagonal combination of the +Z2 element in U(1) and the non-trivial element in the Z2 center of SU(2)H. This diagonal +combination leaves all vector and hyper multiplets invariant. Thus the flavor symmetry group +associated to su(2)H is +SO(3)H = SU(2)H/Z2 +(3.4) +because, according to (3.3), we can couple the theory (3.1) to non-trivial background bundles +for SO(3)H, provided we turn on non-trivial bundles for U(1)/Z2. In more detail, let wH +2 +be the obstruction class for lifting the SO(3)H bundle to an SU(2)H bundle, i.e. the second +Stiefel-Whitney class of the SO(3)H bundle. The obstruction class for lifting the U(1)/Z2 +bundle to a U(1) bundle can then be written as c1 (mod 2), where c1 is the first Chern class +for the U(1)/Z2 bundle. The group (3.3) then requires that +wH +2 = c1 (mod 2) . +(3.5) +That is, SO(3)H bundle can be lifted to SU(2)H bundle if and only if U(1)/Z2 bundle can +be lifted to U(1) bundle. Once an SO(3)H bundle is specified, the gauge theory sums over +all U(1)/Z2 bundles satisfying the constraint (3.5). +A-Type 0-Form Symmetry. +In addition of the SO(3)H 0-form symmetry, the Lagrangian +theory (3.1) admits a magnetic +FUV +C += U(1)C +(3.6) +0-form symmetry whose associated topological operators are +exp +� +iα +� +F +� +(3.7) +parametrized by α ∈ [0, 2π), where F is the field strength of the U(1) gauge group. The +subscript C denotes that the U(1)C symmetry acts non-trivially on the Coulomb branch +of vacua of the theory. Indeed, the monopole operators, whose vacuum expectation values +parametrize the Coulomb branch, are charged under this symmetry, with the charges valued +in Z. +It is well-known, from the analysis of [5], that at the level of Lie algebras, the u(1)C 0- +form symmetry enhances in the IR to an su(2)C 0-form symmetry. In particular, the Cartan +of the simply connected Lie group SU(2)C with Lie algebra su(2)C is a double cover of U(1)C, +or in other words we have an inclusion map +U(1)C �→ SO(3)C = SU(2)C/Z2 , +(3.8) +– 20 – + +which embeds U(1)C as the maximal torus of SO(3)C. +Since the gauge monopole operators have integer charges under U(1)C in the UV, they +descend to genuine local operators of the IR SCFT transforming in representations of SO(3)C. +Thus the Coulomb 0-form symmetry group of the IR SCFT is +FIR +C = SO(3)C . +(3.9) +In other words, at the level of Lie groups the U(1)C 0-form symmetry group enhances in the +IR to SO(3)C 0-form symmetry group. +Mixed 0-Form Symmetry Anomaly. +As is well-known, there is a mixed ’t Hooft anomaly +between the SO(3)H and SO(3)C 0-form symmetries of T[SU(2)], see [69, 70]. +Here we +describe how this ’t Hooft anomaly can be derived as a consequence of a mixed ’t Hooft +anomaly between the SO(3)H and U(1)C 0-form symmetries in the UV gauge theory. For this +purpose, we will use the analysis of [12] which described a general way of deducing ’t Hooft +anomalies for any 3d gauge theory by computing center charges of flavor-gauge monopole +operators. The relevant monopole operator is associated to a co-character +U(1) → S , +(3.10) +where S is the structure group of the gauge theory appearing in (3.3), with winding number +half around the U(1) factor in the numerator of S and winding number half around the +U(1) maximal torus of SU(2)H. This is an allowed co-character because of the Z2 quotient +appearing in the denominator of S. +The mixed flavor-gauge monopole operator O associated to such a co-character is neces- +sarily a non-genuine local operator of the gauge theory lying at the end of a vortex line defect. +From the methods of [12], we can compute that O carries a charge 1/2 under U(1)C. A quick +way to see this is to note that a purely gauge monopole operator associated to a co-character +with winding number 1 around U(1) gauge group has charge 1 under U(1)C; a purely flavor +monopole operator associated to a co-character with winding number 1 around U(1) maximal +torus of SU(2)H is uncharged under U(1)C; and twice of the co-character associated to O is +a product of such purely gauge and purely flavor co-characters. +As explained in [12], the half-integral charge under U(1)C of the monopole operator O is +equivalent to a mixed ’t Hooft anomaly between the Coulomb and Higgs 0-form symmetry +groups of the UV gauge theory +AUV +4 += exp +� +πi +� +wH +2 ∪ +� +c1 +� +U(1)C +� +(mod 2) +�� +, +(3.11) +where wH +2 denotes the Stiefel-Whitney class for the background SO(3)H bundle and c1 +� +U(1)C +� +denotes the first Chern class of the background U(1)C bundle. +After flowing to the IR, the monopole operator O descends to a non-genuine local operator +of the T[SU(2)] theory that is now a purely flavor monopole operator (because there is no +– 21 – + +gauge group in the IR SCFT) associated to a co-character having winding number 1 around +the SO(3)H 0-form symmetry group of T[SU(2)]. Any flavor monopole operator is a non- +genuine local operator living at the end of a flavor vortex line defect associated to the same +co-character. The flavor monopole operator O must transform in a representation of su(2)C +which is not an allowed representation of SO(3)C. This is a straightforward consequence of +the embedding (3.8). Again using the analysis of [12], this fact is equivalent to a mixed ’t +Hooft anomaly between the Coulomb and Higgs 0-form symmetry groups of the IR SCFT +T[SU(2)] +AIR +4 = exp +� +πi +� +wH +2 ∪ wC +2 +� +, +(3.12) +where wC +2 is the second Stiefel-Whitney class of the background SO(3)C bundle. +One can think of the IR anomaly (3.12) as being obtained from the UV anomaly (3.11) +by ’t Hooft anomaly matching. The above analysis in terms of charges of flavor-monopole +operators thus provides a precise and unambiguous way of performing such a ’t Hooft anomaly +matching. +3.1.2 +SU(2)H Gauging +Consider now the 3d N = 4 theory +T[SU(2)] +SU(2)H +(3.13) +obtained by gauging the su(2)H flavor symmetry of T[SU(2)] by an SU(2)H gauge group. We +can reach a very closely related cousin of this theory by flowing from the 3d N = 4 quiver +U(1) +SU(2)H +(3.14) +if the gauge coupling for SU(2)H is extremely small compared to the U(1) gauge coupling. +However, in this way we never land precisely on the theory (3.13). +Note that (3.13) is a “bad” theory in the sense of [5]. After a discussion of the symmetries +and anomalies of this theory, we will add flavors for the SU(2)H gauge group converting the +above theory into a “good” theory and study its symmetries. +1-Form Symmetry. +This theory carries a +Γ(1) = Z2 +(3.15) +1-form symmetry, as can be seen by the following argument. After gauging we obtain Wilson +line defects valued in representations of SU(2)H. A genuine local operator of T[SU(2)] trans- +forming in representation R of SU(2)H becomes a non-genuine local operator of the gauged +theory (3.13) that lives at the end of Wilson line defect in representation R. In other words, +the Wilson line in representation R is screened in the gauged theory (3.13). From our previous +discussion, we know that the genuine local operators transform only in those representations +– 22 – + +of SU(2)H that are also representations of SO(3)H. Thus, only Wilson lines in representa- +tions of SO(3)H are screened, but the Wilson lines in representations of SU(2)H that are not +representations of SO(3)H are not screened. The non-screened Wilson lines are non-trivially +charged under the ZH +2 center of SU(2)H, which descends to a Z2 1-form symmetry of the +gauged theory (3.13). +0-Form Symmetry. +We claim that the SO(3)C 0-form symmetry of T[SU(2)] is not im- +pacted by the gauging procedure and the gauged theory (3.13) also has +FC = SO(3)C +(3.16) +0-form symmetry. To see this, we need to show that all genuine local operators of the gauged +theory form representations of SO(3)C. It is sufficient to show this fact for the following two +types of genuine local operators: +1. The genuine local operators of T[SU(2)] that are uncharged under SU(2)H descend to +genuine local operators of the gauged theory (3.13). Since such operators form SO(3)C +representations before gauging, they also form SO(3)C representations after gauging. +2. The flavor monopole operators for su(2)H are non-genuine in T[SU(2)] as they are +attached to vortex line defects. +Some of these vortex line defects become invisible +after gauging and the attached flavor monopole operators thus become genuine local +operators of the gauged theory (3.13). The co-characters associated to such monopoles +are those which have even winding numbers around the maximal torus of SO(3)H. +Such monopole operators form SO(3)C representations before gauging, so only lead to +SO(3)C representations after gauging. +Is There a 2-Group Symmetry? +The question we would now like to address is whether +the above Z2 1-form and SO(3)C 0-form symmetries combine to form a 2-group symmetry +with a non-trivial Postnikov class. We claim that the answer is negative, due to the following +reason. +The existence of such a 2-group symmetry requires the presence of a local operator O in +the gauged theory (3.13) which sits at the end of a Wilson line operator transforming in an +allowed representation of SO(3)H and transforms in a representation of SU(2)C that is not +an allowed representation of SO(3)C. This means that in the ungauged theory T[SU(2)], O +must be a local operator (transforming in same representations of SO(3)H and SU(2)C) of +one of the following two types: +1. O is a genuine local operator in T[SU(2)]. But then O must transform in an SO(3)C +representation because the Coulomb 0-form symmetry group of T[SU(2)] is SO(3)C. +2. O is a flavor monopole operator in T[SU(2)] associated to a co-character with even +winding number around the maximal torus of SO(3)H. But then O must transform in +an SO(3)C representation as already discussed above. +Thus, we conclude that there is no non-trivial 2-group symmetry in the gauged theory (3.13). +– 23 – + +Mixed 0-/1-Form Symmetry ’t Hooft Anomaly. +A flavor monopole operator in T[SU(2)] +with odd winding number around maximal torus of SO(3)H becomes a gauge monopole op- +erator in the gauged theory (3.13). Such a gauge monopole operator is not a standard gauge +monopole operator usually discussed in the literature. The latter monopole operators are +genuine local operators while the former monopole operator must be non-genuine attached to +a non-trivial solitonic line defect which induces a non-trivial flux for the background field B2 +for the Z2 1-form symmetry [12]. For this reason, such monopole operators were referred to +as fractional gauge monopole operators in [12] to distinguish them from the standard (non- +fractional) gauge monopole operators. Some fractional monopole operators lie in the twisted +sector of the Z2 1-form symmetry, that is, they lie at the end of a topological line operator +generating the Z2 1-form symmetry. +Since such a fractional gauge monopole operator forms a representation of SU(2)C that +is not an allowed representation of SO(3)C, using the analysis of [12] we learn that there is a +mixed ’t Hooft anomaly +A4 = exp +� +πi +� +B2 ∪ wC +2 +� +(3.17) +between the Z2 1-form symmetry and SO(3)C 0-form symmetry of the gauged theory (3.13). +A more straightforward way of deriving the above anomaly is to first note that the +background field B2 for the Z2 1-form symmetry can be identified with the obstruction class +wH +2 +B2 = wH +2 +(3.18) +and then the anomaly (3.17) follows simply from the anomaly (3.12) of T[SU(2)]. +Adding Flavors. +We are interested in understanding the global form of the Coulomb +symmetry group of the IR SCFT obtained (for large enough N) from the UV 3d N = 4 +Lagrangian theory +U(1) +SU(2)H +N F , +(3.19) +where we have N hypermultiplets transforming in fundamental representation of SU(2)H +along with the bifundamental hypermultiplet between U(1) and SU(2)H. +Note that the +corresponding IR SCFT can also be obtained by starting from the good version +T[SU(2)] +SU(2)H +N F +(3.20) +of the theory (3.13), obtained from (3.13) by adding N fundamental hypers for SU(2)H. The +relationship between theories (3.19) and (3.20) is that the theory (3.19) flows very close to +the theory (3.20) at intermediate energy scales if the gauge coupling for SU(2)H is extremely +small compared to the gauge coupling for U(1). +We can thus use the symmetry properties of the theory (3.13) to deduce symmetry +properties for the IR SCFT associated to (3.19) as follows. The additional flavors in the theory +(3.20) screen the fundamental Wilson line of SU(2)H and thus the Z2 1-form symmetry of +– 24 – + +the theory (3.13) is lost in the theory (3.20). On the other hand, the only new genuine local +operators we obtain are gauge invariant combinations of these hypers. These carry trivial +charge under SO(3)C, and hence the Coulomb 0-form symmetry group of the theory (3.20) +is SO(3)C. For generic N, there is no enhancement of SO(3)C and the IR SCFT originating +from (3.19) (which is the same as the IR SCFT originating from (3.20)) has Coulomb 0-form +symmetry group +FIR +C = SO(3)C . +(3.21) +3.1.3 +SO(3)H Gauging +Consider now the 3d N = 4 theory +T[SU(2)] +SO(3)H +(3.22) +obtained by gauging the su(2)H flavor symmetry of T[SU(2)] by an SO(3)H gauge group. +One can reach a very closely related theory by flowing from a 3d N = 4 Lagrangian theory +u(1) +su(2)H +(3.23) +(where we have only displayed gauge algebras) with the gauge group +G = U(1) × SU(2)H +Z2 +, +(3.24) +where the Z2 being quotiented out is the diagonal combination of the Z2 subgroup of U(1) +and the ZH +2 center of SU(2)H. Note that (3.22) is again a bad theory just like (3.13). Later, +we also consider its good versions obtained by adding adjoint flavors for the SO(3)H gauge +group. +0-Form Symmetry Group. +In fact, this theory (3.22) can be obtained by gauging the Z2 +1-form symmetry of the theory (3.13). As a consequence, we expect the above theory (3.22) +to contain a dual Z2 0-form symmetry, alongside the residual SO(3)C 0-form symmetry +descending from (3.13). However, the combined group structure of the 0-form symmetry is +not Z2 × SO(3)C. Instead, the mixed anomaly (3.17) between Z2 1-form and SO(3)C 0- +form symmetries of the theory (3.13) dualizes to a non-trivial extension between the Z2 and +SO(3)C 0-form symmetries of the theory (3.22). Thus, in total the 0-form symmetry of the +theory (3.22) is +FC = SU(2)C , +(3.25) +which is a non-trivial extension of the form +1 → Z2 → SU(2)C → SO(3)C → 1 . +(3.26) +To see it more concretely, following [71] let us explicitly perform the gauging over B2 with +the term B2 ∪ B1 added to the 3d action, where B1 is background field for dual Z2 0-form +symmetry. This modifies the anomaly as +A4 → B2 ∪ wC +2 + δB2 ∪ B1 + B2 ∪ δB1 . +(3.27) +– 25 – + +The second term δB2 ∪ B1 = 0 as B2 is closed. The rest of the anomaly B2 ∪ (δB1 + wC +2 ) +is a gauge anomaly (because B2 is a gauge field now), so it must vanish. This gives us the +constraint +δB1 = wC +2 , +(3.28) +which implies that Z2 and SO(3)C 0-form symmetries indeed combine to form SU(2)C 0-form +symmetry. +The same conclusion can also be reached by studying the monopole operators discussed +above. Recall that (3.13) contains a fractional gauge monopole operator O living at the end +of topological line operator generating the Z2 1-form symmetry, such that O transforms in a +representation of SU(2)C which is not a representation of SO(3)C. As we gauge the 1-form +symmetry, the topological line generating the Z2 1-form symmetry disappears, and O becomes +a genuine local operator of the theory (3.22). Thus, the 0-form symmetry group associated +to su(2)C 0-form symmetry algebra in the theory (3.22) is SU(2)C. +Adding Flavors. +We are interested in understanding the global form of the Coulomb +symmetry group of the IR SCFT obtained (for large enough N) from the UV 3d N = 4 +Lagrangian theory +u(1) +su(2)H +N A +(3.29) +where we have N hypers transforming in adjoint representation of su(2)H along with the +bifundamental hyper between u(1) and su(2)H, and the gauge group is chosen to be (3.24). +Note that the corresponding IR SCFT can also be obtained by starting from the good version +T[SU(2)] +SO(3)H +N A +(3.30) +of the theory (3.22), obtained from (3.22) by adding N adjoint hypers for SO(3)H. The +relationship between theories (3.29) and (3.30) is that the theory (3.29) flows very close to +the theory (3.20) at intermediate energy scales if the gauge coupling for su(2)H is extremely +small compared to the gauge coupling for u(1). +We can thus use the symmetry properties of the theory (3.22) to deduce symmetry +properties for the IR SCFT associated to (3.29) as follows. Recall that (3.22) has a gauge +monopole operator transforming in SU(2)C representation that is not an SO(3)C representa- +tion. This operator is not impacted by the addition of adjoint hypers. Thus, we deduce that +the Coulomb 0-form symmetry group of the theory (3.30) is SU(2)C. For generic N, there is +no enhancement of SU(2)C and the IR SCFT originating from (3.29) (which is the same as +the IR SCFT originating from (3.30)) has Coulomb 0-form symmetry group +FIR +C = SU(2)C . +(3.31) +– 26 – + +3.1.4 +U(2) Gauging +We are interested in understanding the global form of Coulomb and Higgs symmetry groups +of the IR SCFT obtained (for large enough N) from the UV 3d N = 4 Lagrangian theory +U(1) +U(2) +[su(N)H] , +(3.32) +where we have N hypers transforming in fundamental representation of U(2) along with the +bifundamental hyper between U(1) and U(2). +0-Form Symmetry Algebras. +As shown in (3.32), there is a Higgs branch flavor symmetry +fH = su(N)H +(3.33) +rotating the N fundamental hypers. +We will focus on the case N > 3 for which the IR Coulomb symmetry algebra is +fIR +C = u(2)C = su(2)C ⊕ u(1)C . +(3.34) +The N = 3 case has a further enhancement of IR Coulomb symmetry algebra to su(3)C that +will be discussed in the following subsection on T[SU(n)] theories. +The su(2)C subalgebra of fIR +C is usually associated to the balanced node provided by the +U(1) gauge group, and the u(1)C subalgebra of fIR +C is usually associated to the unbalanced +node provided by the U(2) gauge group. However, we will need a more precise identification +of these subalgebras, which we discuss in what follows. +0-Form Symmetry Groups. +The Higgs 0-form symmetry group of the IR SCFT is the +same as that of the UV Lagrangian theory +FH = PSU(N)H = SU(N)H/ZN +(3.35) +and can be easily deduced by noticing that the gauge invariant combinations of hypermulti- +plets all form representations of PSU(N)H. +On the other hand, the Coulomb 0-form symmetry group of the IR SCFT is more subtle +to deduce. Let us begin with the Coulomb 0-form symmetry group of the UV theory. We have +a U(1)1 0-form symmetry associated to the U(1) gauge node and a U(1)2 0-form symmetry +associated to the U(2) gauge node. First, we need to understand the precise identification of +the U(1) subgroup of UV 0-form symmetry group +FUV +C += U(1)1 × U(1)2 , +(3.36) +which becomes the maximal torus of the Lie group SU(2)C associated to the su(2)C subalgebra +of fIR +C . This is done by studying FUV +C +charges of BPS monopole operators of low R-charge. +We look for a combination6 q = n1q1 + n2q2, where qi is the U(1)i charge of monopole and +6We thank Antoine Bourget for a discussion regarding this point. +– 27 – + +ni ∈ Z, such that the set of monopole operators at a fixed value of R-charge have q-charges +coinciding with the Dynkin coefficients of the weights of a representation of su(2)C. This fixes +q = 2q1 − q2 +(3.37) +Now the u(1)C factor is determined such that the BPS monopole operators furnishing the +weights of adjoint of su(2)C are uncharged under u(1)C. +This determines that the u(1)C +charge is proportional to q2, or more precisely there is a Lie group U(1)C with Lie algebra +u(1)C, such that the charge qC under U(1)C is +qC = q2 . +(3.38) +In order to determine the global form of the IR Coulomb flavor group FIR +C we need to deter- +mine the charges +� +q (mod 2), qC +� +(3.39) +of non-fractional gauge monopole operators, where the charge q (mod 2) is the charge of the +monopole operator under the Z2 center of the IR SU(2)C. The fundamental monopole oper- +ator from U(1) gauge node has (q1, q2) = (1, 0) implying +� +q (mod 2), qC +� += +� +0 (mod 2), 0 +� +, +while the fundamental monopole operators from U(2) gauge node have (q1, q2) = (0, 1) im- +plying +� +q (mod 2), qC +� += +� +1 (mod 2), 1 +� +. The only non-trivial charge is the latter one, which +implies that the Coulomb 0-form symmetry group of the IR SCFT is +FIR +C = U(2)C = SU(2)C × U(1)C +Z2 +. +(3.40) +3.2 +T[SU(n)] and its Gaugings +In this subsection we generalize to T[SU(n)] theories the results obtained in the previous +subsection regarding T[SU(2)] theories. +3.2.1 +T[SU(n)] +B-Type 0-Form Symmetry. +The T[SU(n)] theory arises as an IR fixed point of the +following 3d N = 4 Lagrangian theory +U(n − 1) +[su(n)H] +· · · +U(2) +U(1) +, +(3.41) +where we have a bifundamental hyper between adjacent unitary gauge nodes, and n funda- +mental hypers for U(n − 1) that are rotated by an +fH = su(n)H +(3.42) +Higgs flavor symmetry algebra. +The genuine local operators charged under su(n)H arise from gauge invariant combina- +tions of hypermultiplets. Such gauge invariant combinations all form representations of the +– 28 – + +Lie group PSU(n)H with Lie algebra su(n)H. +Thus, the Higgs branch 0-form symmetry +group is +FH = PSU(n)H = SU(n)H/Zn +(3.43) +where SU(n)H is the simply connected group with Lie algebra su(n)H. +Equivalently, we can deduce the above 0-form symmetry group by putting the theory +(3.41) on a non-trivial compact 3-manifold M3. The charges of the vector and hypermultiplets +allow us to turn on bundles for the group +S = U(1) × U(2) × · · · × U(n − 1) × SU(n)H +Zn +. +(3.44) +The Zn in the denominator is the diagonal combination of the Zn subgroups of U(1) centers of +U(i) gauge groups and the Zn center of SU(n)H. This diagonal combination leaves all vector +and hyper multiplets invariant. Thus the Higgs flavor symmetry group is PSU(n)H because, +according to (3.44), we can couple the theory (3.41) to non-trivial (in the sense that they +cannot be lifted to SU(n)H bundles) background bundles for PSU(n)H, provided we turn on +non-trivial (in the sense that they cannot be lifted to U(i) bundles) bundles for U(i)/Zn. +A-Type 0-Form Symmetry. +In addition to the PSU(n)H 0-form symmetry, the La- +grangian theory (3.41) admits a magnetic +FUV +C += U(1)n−1 +C += +n−1 +� +i=1 +U(1)C,i +(3.45) +0-form symmetry, where U(1)C,i is the magnetic symmetry arising from the U(i) gauge node. +The above Coulomb 0-form symmetry enhances in the IR to an su(n)C Coulomb 0-form +symmetry. The fundamental monopole operators associated to each node i describe roots of +su(n)C, which carry charge 0 (mod n) under the center Zn of SU(n)C. As a consequence, all +gauge monopole operators form representations of SU(n)C having charge 0 (mod n) under +the center Zn. Thus the Coulomb 0-form symmetry group of the IR SCFT is +FIR +C = PSU(n)C . +(3.46) +Mixed 0-Form Anomaly. +There is a mixed ’t Hooft anomaly between the PSU(n)H and +PSU(n)C 0-form symmetries of T[SU(n)]. The relevant flavor-gauge monopole operator is +associated to a co-character of the structure group S appearing in (3.44) with winding number +1/n around the U(1) center of each U(i) gauge group and winding number 1/n around a U(1) +subgroup of the maximal torus of SU(n)H. This is an allowed co-character because of the Zn +quotient appearing in the denominator of S. +The mixed flavor-gauge monopole operator O descends to a flavor monopole operator in +the IR SCFT T[SU(n)], which is a non-genuine local operator lying at the end of a flavor +vortex line defect. O carries a charge qi = i/n under U(1)C,i. This implies that the Dynkin +– 29 – + +coefficients di of the weight of su(n)C carried by O are +(d1, d2, · · · , dn−2, dn−1) = (2q1 − q2, −q1 + 2q2 − q3, · · · , −qn−3 + 2qn−2 − qn−1, −qn−2 + 2qn−1) += (0, 0, · · · , 0, 1) . +(3.47) +The charge of O under the Zn center of SU(n)C is computed in terms of di as +n−1 +� +i=1 +i × di (mod n) = −1 (mod n) . +(3.48) +Using the analysis of [12], this fact is equivalent to a mixed ’t Hooft anomaly between the +Coulomb and Higgs 0-form symmetry groups of the IR T[SU(n)] SCFT +AIR +4 = exp +� +−2πi +n +� +wH +2 ∪ wC +2 +� +, +(3.49) +where wC +2 , wH +2 +are Zn valued obstruction classes capturing the obstruction of lifting back- +ground PSU(n)C, PSU(n)H bundles to SU(n)C, SU(n)H bundles. +3.2.2 +SU(n)H Gauging +Consider now the 3d N = 4 theory +T[SU(n)] +SU(n)H +(3.50) +obtained by gauging the su(n)H flavor symmetry of T[SU(n)] by an SU(n)H gauge group. +We can reach very close to this theory by flowing from the 3d N = 4 quiver +U(n − 1) +SU(n)H +· · · +U(2) +U(1) +(3.51) +if the gauge coupling for SU(n)H is extremely small compared to the U(i) gauge couplings. +Note that (3.50) is a bad theory. We will later also discuss its good versions obtained by +adding flavors for the SU(n)H gauge group. +1-Form Symmetry. +This theory carries a +Γ(1) = Zn +(3.52) +1-form symmetry, as can be seen by the following argument. After gauging we obtain Wilson +line defects valued in representations of SU(n)H. A genuine local operator of T[SU(n)] trans- +forming in representation R of SU(n)H becomes a non-genuine local operator of the gauged +theory (3.50) that lives at the end of Wilson line defect in representation R. In other words, +Wilson line in representation R is screened in the gauged theory (3.50). From our previ- +ous discussion, we know that the genuine local operators transform only in representations +of PSU(n)H. Thus, Wilson lines transforming in representations of SU(n)H with non-zero +charge (modulo n) under its Zn center are left unscreened, implying that Zn center of SU(n)H +descends to a Zn 1-form symmetry of the gauged theory (3.50). +– 30 – + +0-Form Symmetry. +There are two types of genuine local operators of the gauged theory +(3.50) to consider: +1. The genuine local operators of T[SU(n)] that are uncharged under SU(n)H descend to +genuine local operators of the gauged theory (3.50). Since such operators form PSU(n)C +representations before gauging, they also form PSU(n)C representations after gauging. +2. The flavor monopole operators of T[SU(n)] associated to co-characters of SU(n)H be- +come genuine local operators (non-fractional gauge monopole operators) after gauging, +despite being non-genuine before gauging. Such monopole operators form PSU(n)C +representations before gauging, so only lead to PSU(n)C representations after gauging. +Thus, the PSU(n)C 0-form symmetry of T[SU(n)] is not impacted by the gauging procedure +and the gauged theory (3.50) also has +FC = PSU(n)C +(3.53) +0-form symmetry. +Is There a 2-Group Symmetry? +The question we would now like to address is whether +the above Zn 1-form and PSU(n)C 0-form symmetries combine to form a 2-group symmetry +with a non-trivial Postnikov class. We claim that the answer is negative, due to the following +reason. +The existence of such a 2-group symmetry requires the presence of a local operator O +in the gauged theory (3.50) which sits at the end of a Wilson line operator transforming in +a representation of PSU(n)H and transforms in a representation of SU(n)C that is not an +allowed representation of PSU(n)C. This means that in the ungauged theory T[SU(n)], O +must be a local operator (transforming in same representations of PSU(n)H and SU(n)C) of +one of the following two types: +1. O is a genuine local operator in T[SU(n)]. But then O must transform in a PSU(n)C +representation because the Coulomb 0-form symmetry group of T[SU(n)] is PSU(n)C. +2. O is a flavor monopole operator in T[SU(n)] associated to a co-character of the group +SU(n)H. But then O must transform in a PSU(n)C representation as already discussed +above. +Thus, we conclude that there is no non-trivial 2-group symmetry in the gauged theory (3.50). +Mixed 1-Form 0-Form ’t Hooft Anomaly. +Flavor monopole operators of T[SU(n)] +become (possibly fractional) gauge monopole operators in the gauged theory (3.50). This +includes the operator O discussed around equation (3.49), which becomes a fractional gauge +monopole operator after gauging and can be converted into a local operator living at the end +of the topological line defect generating the Zn 1-form symmetry. The fact that O transforms +in a representation of SU(n)C having charge −1 (mod n) under its Zn center is equivalent to +– 31 – + +the mixed ’t Hooft anomaly between the Zn 1-form symmetry and PSU(n)C 0-form symmetry +of the gauged theory (3.13) +A4 = exp +� +−2πi +n +� +B2 ∪ wC +2 +� +, +(3.54) +where B2 is the background field for the Zn 1-form symmetry. This anomaly can also be +derived as a consequence of the anomaly (3.49) using the identification B2 = wH +2 . +Adding Flavors. +We are interested in understanding the global form of the Coulomb +symmetry group of the IR SCFT obtained (for large enough N) from the UV 3d N = 4 +Lagrangian theory +U(n − 1) +SU(n)H +· · · +U(2) +U(1) +N F . +(3.55) +Note that the corresponding IR SCFT can also be obtained by starting from the good version +T[SU(n)] +SU(n)H +N F +(3.56) +of the theory (3.50), obtained from (3.50) by adding N fundamental hypers for SU(n)H. The +relationship between theories (3.55) and (3.56) is that the theory (3.55) flows very close to +the theory (3.56) at intermediate energy scales if the gauge coupling for SU(n)H is extremely +small compared to the U(i) gauge couplings. +We can thus use the symmetry properties of the theory (3.50) to deduce symmetry +properties for the IR SCFT associated to (3.55) as follows. The additional flavors in the +theory (3.56) screen the fundamental Wilson line of SU(n)H and thus the Zn 1-form symmetry +of the theory (3.50) is lost in the theory (3.56). On the other hand, the only new genuine +local operators we obtain are gauge invariant combinations of these hypers. +These carry +trivial charge under PSU(n)C, and hence the Coulomb 0-form symmetry group of the theory +(3.56) is PSU(n)C. For generic N, there is no enhancement of PSU(n)C and the IR SCFT +originating from (3.55) (which is the same as the IR SCFT originating from (3.56)) has +Coulomb 0-form symmetry group +FIR +C = PSU(n)C . +(3.57) +3.2.3 +Other su(n)H Gaugings +We can also consider gauging SU(n)H/Zm (where Zm < Zn is a subgroup, i.e. m|n) to obtain +the 3d N = 4 theory +T[SU(n)] +SU(n)H/Zm . +(3.58) +One can reach very close to this theory by flowing from a 3d N = 4 Lagrangian theory +u(n − 1) +su(n)H +· · · +u(2) +u(1) +, +(3.59) +– 32 – + +(where we have only displayed gauge algebras) with the gauge group +G = U(1) × U(2) × · · · × U(n − 1) × SU(n)H +Zm +, +(3.60) +where the Zm being quotiented out is the Zm subgroup of the Zn group appearing in the +denominator of (3.44). Note that (3.58) is again a bad theory just like (3.50). Later, we +also consider its good versions obtained by adding adjoint flavors for the SU(n)H/Zm gauge +group. +1-Form Symmetry Group. +In fact, this theory (3.58) can be obtained by gauging the +Zm 1-form symmetry of the theory (3.50). Consequently, there is a residual +Γ(1) = Zp +(3.61) +1-form symmetry in the theory (3.58) where p = n/m. +0-Form Symmetry Group. +There is also thus a dual Zm 0-form symmetry in (3.58) along- +side the residual PSU(n)C 0-form symmetry descending from (3.50). By similar arguments +as in the previous subsection on T[SU(2)], the two 0-form symmetries combine non-trivially +and the full 0-form symmetry group of (3.58) is +FC = SU(n)C/Zp . +(3.62) +Mixed 1-Form 0-Form ’t Hooft Anomaly. +There is a mixed ’t Hooft anomaly between +Zp 1-form and SU(n)C/Zp 0-form symmetries arising as a residue of the anomaly (3.54) +A4 = exp +� +−2πi +p +� +B2 ∪ wC +2 +� +, +(3.63) +where wC +2 is the Zp valued obstruction class for lifting SU(n)C/Zp bundles to SU(n)C bundles, +and B2 is the Zp valued background field for 1-form symmetry. +Particular Case: m = n. +In this particular case, we are studying a PSU(n)H gauging of +T[SU(n)]. There is no 1-form symmetry and the 0-form symmetry group is SU(n)C. +Adding Adjoint Flavors. +We can add N adjoint flavors for SU(n)C/Zm. For large enough +N, this flows to a 3d N = 4 SCFT in the IR. The global form of the Coulomb symmetry +group and mixed anomaly between Coulomb 0-form symmetry and 1-form symmetry in the +IR SCFT for generic N are the same as those described above. +3.2.4 +U(n) Gauging +We are interested in understanding the global form of Coulomb and Higgs symmetry groups +of the IR SCFT obtained (for large enough N) from the UV 3d N = 4 Lagrangian theory +U(n) +[su(n)H] +· · · +U(2) +U(1) +, +(3.64) +where we have N hypers transforming in fundamental representation of U(n) along with +bifundamental hypers between adjacent U(i) gauge nodes. +– 33 – + +0-Form Symmetry Algebras. +As shown in (3.64), there is a +fH = su(N)H +(3.65) +Higgs branch symmetry rotating the N fundamental hypers. +We will focus on the case N > n + 1 for which the IR Coulomb symmetry algebra is +fIR +C = u(n)C = su(n)C ⊕ u(1)C . +(3.66) +The N = n+1 case has a further enhancement of IR Coulomb symmetry algebra to su(n+1)C. +The su(n)C subalgebra of fIR +C is usually associated to the balanced nodes provided by the +U(i) gauge groups for 1 ≤ i ≤ n − 1, and the u(1)C subalgebra of fIR +C is usually associated to +the unbalanced node provided by the U(n) gauge group. However, as before we will need a +more precise identification of these subalgebras. +0-Form Symmetry Groups. +The Higgs 0-form symmetry group of the IR SCFT is is the +same as that of the UV Lagrangian theory +FH = PSU(N)H = SU(N)H/ZN . +(3.67) +To deduce the Coulomb 0-form symmetry group of the IR SCFT, we need a precise +identification of the Dynkin coefficients di for su(n)C weights in terms U(1)C,i charges. This +is the same as discussed around equation (3.47), except now dn−1 is modified to +dn−1 = −qn−2 + 2qn−1 − qn . +(3.68) +Moreover, the u(1)C factor has a global form U(1)C such that the charge qC under U(1)C is +qC = qn +(3.69) +Now we see that the fundamental monopole operators coming from the U(n) gauge node +transform in anti-fundamental representation of su(n)C and simultaneously have charge +1 +under u(1)C. Thus the Coulomb 0-form symmetry group of the IR SCFT is +FIR +C = U(n)C = SU(n)C × U(1)C +Zn +. +(3.70) +4 +General Symmetry and Anomaly Analysis for 3d N = 4 SCFTs +Now we are ready to describe the most general result of the considerations of this paper. +Consider a 3d N = 4 good quiver gauge theory composed of unitary and special unitary +gauge algebras and no Higgs flavor algebras. Let us write the gauge algebra as +g = +� +i +gi , +(4.1) +– 34 – + +where each gi is either su(ni) or u(ni). +We assume there is at least one special unitary +node and all special unitary nodes are unbalanced. Also let Gi be SU(ni) or U(ni) for the +two cases. The matter content is comprised entirely of bifundamental hypers. Say we have +mij ≥ 0 bifundamental hypers between gi and gj. We assume that the quiver is connected, +which means that we can go from any node i to any other node j by choosing a sequence of +nodes ia for 0 ≤ a ≤ b such that i0 = i, ib = j and miaia+1 > 0. Moreover, the balanced +unitary nodes are taken to form the Dynkin diagram of a finite semi-simple Lie algebra +f = +� +a +fa , +(4.2) +where each fa is a finite simple Lie algebra. Let Fa be the simply connected group associated +to fa with center Za and define F = � +a Fa with center Z = � +a Za. Let U be the set of +unbalanced unitary nodes. Let u(1)i for a node i ∈ U be the associated Coulomb 0-form +symmetry algebra and U(1)i be a group with Lie algebra u(1)i such that the fundamental +BPS monopoles associated to node i have charge gδij under U(1)j, where g is defined around +equation (4.5) below and δij is the Kronecker delta. We have scaled the U(1)i charges of +fundamental monopoles by g for later convenience in computing ’t Hooft anomalies. The +Coulomb 0-form symmetry algebra of the corresponding 3d N = 4 IR SCFT is +fIR +C = +� +i∈U +u(1)i ⊕ f +(4.3) +Let us define F IR +C = � +i∈U U(1)i × F with center ZIR +C = � +i∈U U(1)i × Z. +4.1 +Symmetries +1-Form Symmetry Group. +Consider choosing first the gauge group +G = +� +i +Gi . +(4.4) +Then the theory with the above-described matter content has a 1-form symmetry group given +by +Γ(1) = Zg , +(4.5) +where g is the GCD (greatest common divisor) of all ni for which gi = su(ni). +(Coulomb) 0-Form Symmetry Group. +Let us determine the Coulomb 0-form symmetry +group FIR +C of the IR SCFT. Each node i ∈ U provides a genuine local operator Oi in the IR +SCFT whose charge under ZIR +C +is what we want to determine. This local operator can be +chosen to be IR image of any fundamental BPS monopole associated to the node i. +First of all, Oi has charge qi,j = gδij under U(1)j factor of ZIR +C , where j ∈ U. The charge +qi,a of Oi under Za is given by +qi,a = − +� +j∈Na +mi,jna,j ∈ �Za , +(4.6) +– 35 – + +where Na is the set of nodes forming the Dynkin diagram of fa and na,j ∈ �Za is the charge +of the representation of fa with highest weight having Dynkin coefficients di = δij for i ∈ Na. +Combining the charges qi,j for all j ∈ U and charges qi,a for all a, we obtain a charge +qi ∈ �ZIR +C +(4.7) +of Oi under ZIR +C . The charges qi for all i ∈ U span a subgroup Y IR +C +of the abelian group �ZIR +C . +We thus obtain the information of a surjective map +�ZIR +C → � +ZIR +C := +�ZIR +C +Y IR +C +, +(4.8) +which can be Pontryagin dualized to an injective map +ZIR +C → ZIR +C +(4.9) +providing a subgroup ZIR +C of ZIR +C . The Coulomb 0-form symmetry group of the IR SCFT is +FIR +C = F IR +C +ZIR +C +. +(4.10) +Visual Representation. +The charges qi,a of operators Oi under the centers Za of Coulomb +flavor algebras fa can be deduced visually from the UV quiver. First of all, note that qi,a is +the same as the charge under Za of the representation +� +j∈Na +Fmi,j +j +, +(4.11) +of fa, where Fj is the oft-called ‘fundamental representation associated to node j’ of the Dynkin +diagram of fa, which is the representation with highest weight having Dynkin coefficients +di = δij for i ∈ Na. This representation is deduced visually from the UV quiver: we just see +how many times the node i ∈ U hits a node j in Na and include that many copies of the +representation Fj of fa. +4.2 +Anomaly +There is a mixed ’t Hooft anomaly between Γ(1) and FIR +C which is computed using the charge +of a fractional gauge monopole operator O associated to co-character of the group +G = G +Zg +, +(4.12) +where Zg being quotiented out is the diagonal combination of the Zg subgroup of the U(1) +center of each unitary gauge group and the Zg subgroup of the Zni center of the gauge group +SU(ni) for each special unitary gauge group. The co-character associated to O has winding +number 1/g around the U(1) center of each unitary gauge group and winding number 1/g +around a U(1) subgroup of the maximal torus of SU(ni) for each special unitary gauge group. +– 36 – + +The charge qi of O under U(1)i for i ∈ U is ni. The charge qa of O under Za is the same +as the charge of the representation of fa having highest weight with Dynkin coefficients +di = +� +j∈Na +Ma,ij +nj +g − +� +j∈U +mi,j +nj +g +(4.13) +for i ∈ Na, where Ma,ij is the Cartan matrix of fa and nj is the rank of the gauge algebra +u(nj) associated to the node j. Combining the qi and qa charges, we obtain a charge qO ∈ �ZIR +C +of O under ZIR +C . Projecting it using the map (4.8), we obtain an element q ∈ � +ZIR +C , letting us +define a homomorphism +γ : Γ(1) → � +ZIR +C +(4.14) +via +Γ(1) = Zg ∋ 1 �→ q ∈ � +ZIR +C +(4.15) +The ’t Hooft anomaly between the 1-form and 0-form symmetries of the IR SCFT is then +AIR +4 = exp +� +2πi +� +γ(B2) ∪ wC +2 +� +, +(4.16) +where B2 is the Γ(1) valued background field for the 1-form symmetry, wC +2 is the ZIR +C valued +class capturing the obstruction of lifting FIR +C +bundles to F IR +C +bundles, and the cup product +uses the natural pairing � +ZIR +C × ZIR +C → R/Z. +Visual Representation. +The charges qa of the operator O under the centers Za of Coulomb +flavor algebras fa can be deduced visually from the UV quiver. First of all, note that qa is +the same as the charge under Za of the representation +� +i∈S +� +j∈Na +F +nimi,j +g +j +(4.17) +of fa, where S is the set of special unitary gauge nodes. To see this, one needs to use the +balancing condition. +This representation is deduced visually from the UV quiver: for each special unitary node +i ∈ S, we just see how many times i hits a node j in Na and include that many copies of the +representation Fj of fa weighted by a factor of ni/g. +4.3 +Other Gauge Groups +We can change the gauge group by gauging a subgroup Zh ⊆ Zg of the 1-form symmetry. +The resulting gauge group is +Gh = G/Zh . +(4.18) +The 1-form symmetry group of the resulting IR SCFT is now +Γ(1) +h += Zg/Zh = Zk , +(4.19) +– 37 – + +where k = g/h. +Because of the extra Zh quotient in the new gauge group Gh, some of the gauge monopole +operators which were fractional (i.e. were non-genuine local operators) now become non- +fractional gauge monopole operators (i.e. +become genuine local operators). +We need to +account for center charges of these new genuine local operators to compute the Coulomb +0-form symmetry group FIR +C,h of the new IR SCFT. To account for these new charges, it +is sufficient to consider the contribution of a single monopole operator Oh for Gh with co- +characters having winding number k times the winding numbers associated to the fractional +gauge monopole operator O considered above. The charge qi,h of Oh under U(1)i for i ∈ U +is k, and the charge qa,h of Oh under Za is the same as the charge of the representation of +fa having highest weight with Dynkin coefficients di,h = kdi, where di are defined in (4.13). +Combining the qi,h and qa,h charges for various values of i and a, we obtain a charge qh ∈ �ZIR +C . +Appending qh to Y IR +C +and spanning, we obtain a larger subgroup Y IR +C,h of �ZIR +C . This provides +a surjective map +�ZIR +C → � +ZIR +C,h := +�ZIR +C +Y IR +C,h +, +(4.20) +whose Pontryagin dual is an injective map +ZIR +C,h → ZIR +C +(4.21) +providing a subgroup ZIR +C,h of ZIR +C . The Coulomb 0-form symmetry group of the new IR SCFT +is +FIR +C,h = F IR +C +ZIR +C,h +. +(4.22) +There is a residual mixed ’t Hooft anomaly between Γ(1) +h +and FIR +C,h descending from (4.16). +This is still a consequence of the operator O which remains a fractional gauge monopole +operator for h < g. Its charge is still qO ∈ �ZIR +C +which projects to an element qh ∈ � +ZIR +C,h, +letting us define a homomorphism +γh : Γ(1) +h +→ � +ZIR +C,h +(4.23) +via +Γ(1) = Zk ∋ 1 �→ qh ∈ � +ZIR +C,h +(4.24) +The ’t Hooft anomaly between the 1-form and 0-form symmetries of the new IR SCFT is +then +AIR +4,h = exp +� +2πi +� +γh(B2,h) ∪ wC +2,h +� +, +(4.25) +where B2,h is the Γ(1) +h +valued background field for the new 1-form symmetry, wC +2,h is the ZIR +C,h +valued class capturing the obstruction of lifting FIR +C,h bundles to F IR +C +bundles, and the cup +product uses the natural pairing � +ZIR +C,h × ZIR +C,h → R/Z. +– 38 – + +4.4 +Including Flavors +Let us now ungauge a few special unitary gauge algebras in the above UV theory, converting +those gauge nodes into flavor nodes. The resulting theory is still a good theory and flows to +a 3d N = 4 SCFT in the IR. Let the set R parametrize the nodes which remain as gauge +nodes. We choose the gauge group to be +GR = +� +i∈R +Gi . +(4.26) +Because of the presence of flavors, the theory has no 1-form symmetry +Γ(1) = 0 . +(4.27) +The Higgs flavor symmetry algebra is taken to be (there might be some extra abelian factors +that we ignore) +fH = +� +i̸∈R +su(ni) . +(4.28) +The structure group of the UV theory including Higgs flavor symmetries is +SR = G/Zg . +(4.29) +In particular, the Higgs 0-form symmetry group of the UV theory and the corresponding IR +SCFT is +FH = +� +i̸∈R SU(ni) +Zg +. +(4.30) +The Coulomb 0-form symmetry group is the same as before FIR +C . +The fractional gauge monopole O discussed above is now instead a mixed flavor-gauge +monopole operator providing a mixed ’t Hooft anomaly between the Higgs and Coulomb +0-form symmetries +AIR +4,R = exp +� +2πi +� +γ(wH +2 ) ∪ wC +2 +� +, +(4.31) +where wH +2 is the Zg valued class capturing the obstruction of lifting FH bundles to � +i̸∈R SU(ni) +bundles and γ is the homomorphism appearing in (4.14). +4.5 +Special Case 1: Single Special Unitary Node +In this and the following subsections, we discuss two special cases for which the symmetry +groups and anomalies of the IR SCFT take a simple form. These two special cases will be +used frequently in the rest of this paper. +In this subsection, we consider the first special case, which occurs when we have a single +(unbalanced) special unitary gauge node carrying su(n)H gauge algebra. +First choose the gauge group +G = +� +i +U(ni) × SU(n)H . +(4.32) +– 39 – + +The 1-form symmetry is +Γ(1) = Zn . +(4.33) +As explained around (4.11), the 0-form symmetry group FIR +C is read simply from the positions +where the unbalanced unitary nodes hit the balanced unitary nodes. +The IR SCFT has a mixed ’t Hooft anomaly between the 1-form and 0-form symmetry +groups. The charge qa under Za of the fractional gauge monopole operator O is read simply +visually from the UV quiver, and coincides with the charge of the representation +� +i∈Na +Fmi +i +(4.34) +of fa, where mi is the number of bifundamental hypers between the SU(n)H gauge node +and the balanced unitary gauge node i ∈ Na. That is, we simply observe the number of +times the SU(n)H gauge node hits the balanced node i and include that many copies of the +representation Fi. Combining the charges qa ∈ �Za with the charge ni under each Coulomb +symmetry U(1)i associated to unbalanced unitary node i ∈ U, we obtain the charge q ∈ �Z of +O, which provides the homomorphism (4.14) appearing in the mixed anomaly (4.16). +Let us gauge the 1-form symmetry group Γ(1) = Zn. The new gauge group is +Gn = +� +i U(ni) × SU(n)H +Zn +. +(4.35) +There is no residual 1-form symmetry and the Coulomb 0-form group is modified by the +presence of the operator O which is now a genuine local operator. Its charges qa described +above need to be accounted to compute the new Coulomb 0-form symmetry group. +We can instead ungauge SU(n)H. The gauge group is now +GR = +� +i +U(ni) . +(4.36) +There is a Higgs 0-form symmetry algebra +fH = su(n)H . +(4.37) +Both the UV gauge theory and the IR SCFT have Higgs 0-form symmetry group as +FH = PSU(n)H . +(4.38) +The Coulomb 0-form symmetry group is still as before the ungauging. There is a mixed ’t +Hooft anomaly between the Higgs and Coulomb 0-form symmetry groups. The anomaly is +described completely by the charges qa of the operator O discussed above, which is now a +mixed flavor-gauge monopole operator. +– 40 – + +Example. +Many of the results of the previous section 3 can be arrived at by a simple +application of the above general analysis. The anomaly (3.49) of T[SU(n)] is a consequence +of the fact that the flavor node su(n)H intersects the su(n)C balanced quiver at the location +of the anti-fundamental node. +Similarly, the anomaly (3.54) is a consequence of the fact that the unbalanced SU(n)H +gauge node intersects the su(n)C balanced quiver at the location of the anti-fundamental +node. Upon gauging Zn 1-form symmetry, it modifies the PSU(n)C 0-form symmetry to +SU(n)C 0-form symmetry as discussed in the paragraph on special case m = n of section +3.2.3. +4.6 +Special Case 2: Single Unbalanced Unitary Node +In this subsection, we consider the second special case, which occurs when we have a single +unbalanced unitary gauge node U(n), and no special unitary gauge nodes. We require the +presence of at least one flavor node. The gauge group is taken to be +G = +� +i +U(ni) × U(n) . +(4.39) +There is no 1-form symmetry, and the Coulomb 0-form symmetry algebra of the IR SCFT is +fIR +C = f ⊕ u(1)C . +(4.40) +The only non-trivial charge under its center Z × U(1)C is provided by fundamental gauge +monopole operators associated to the U(1)C gauge node. The charge qC of such an operator +O under U(1)C is +1 and the charge qa under Za is the same as that of the representation +� +i∈Na +Fmi +i +(4.41) +of fa, where mi is the number of bifundamental hypers between the U(n) gauge node and the +balanced unitary gauge node i ∈ Na. That is, to compute qa we simply observe the number +of times the U(n) node hits the node i and include that many copies of the representation Fi. +This allows us to compute the Coulomb 0-form symmetry group FIR +C of the IR SCFT. +Example. +Some of the results of the previous section 3 can be arrived at by a simple +application of the above general analysis. The U(n)C 0-form symmetry group of U(n) gauging +of T[SU(n)] appearing in equation (3.70) is a straightforward consequence of the fact that +the unbalanced U(n) gauge node intersects the balanced su(n)C quiver at the location of +anti-fundamental node. +5 +Consistency Checks +In this section we will provide various consistency checks of our results detailed in the pre- +vious section. One central application is to magnetic quivers of 4d and 5d SCFTs with 8 +supercharges. +– 41 – + +5.1 +Class S +In this subsection, we apply the methods of previous section to deduce the Coulomb 0-form +symmetry groups of magnetic quivers (MQs) associated to Class S theories of An−1 type +[72]. These MQs, derived in [73], are 3d quiver gauge theories that are mirror to the circle +compactification of 4d N = 2 Class S theories. Thus the Coulomb 0-form symmetry groups +of these MQs should capture the usual Higgs flavor symmetry groups of 4d N = 2 Class +S theories. The computation of flavor symmetry groups of Class S theories was described +recently in [64]. We demonstrate a match of results obtained using our methods against the +results obtained using their methods, and illustrate it with three examples. +Alternatively, one can view this subsection as providing the correct global form of the +MQs of Class S theories of An−1 type. That is, we provide the global form of the gauge group +that should be associated to the gauge algebra of the MQ described in [73]. This global form +of MQ is deduced by matching symmetries of MQ with the symmetries of the Class S theory. +5.1.1 +General Matching +Symmetries of Class S Theory. +Consider a Class S theory arising from the sphere com- +pactification of a 6d N = (2, 0) SCFT of An−1 type with k regular untwisted punctures. Each +puncture Pi is characterized by a partition ρi of n. Let ρi,j be the elements of the partition +where j takes values in 1 ≤ j ≤ |ρi|, and |ρi| is the number of elements in the partition. We +order ρi,j such that ρi,j ≥ ρi,j+1. +The flavor symmetry algebra fi associated to the puncture Pi is encoded in the partition +ρi according to the following standard rule. Define j1 such that ρi,j = ρi,1 for all j ≤ j1. +Define j2 such that ρi,j = ρi,j1+1 for all j1 < j ≤ j2. We continue defining ja in this fashion +until we reach j = |ρi|. Let b be the total number of ja. Then, +fi = +b +� +a=1 +su(ja − ja−1) ⊕ u(1)b−1 +(5.1) +with j0 := 0, and su(1) being the trivial Lie algebra. +Let us define Fi to be the following Lie group associated to the algebra fi +Fi = +b +� +a=1 +SU(ja − ja−1) × U(1)b−1 +(5.2) +where SU(1) denotes the trivial group. The flavor symmetry group F of the Class S theory +is obtained as a quotient +F = +� +i Fi +Z +(5.3) +Let ZF be the center of F := � +i Fi. The group Z ⊆ ZF is obtained by taking Pontryagin +dual of a surjective map +�ZF → � +Z = �ZF /YF +(5.4) +– 42 – + +where YF is a subgroup of �ZF whose computation was described in [64]. First of all, there is +a contribution YF,Pi ⊆ YF coming from each puncture Pi, and then there is a contribution +�YF ⊆ YF coming from all punctures put together. The group YF is recovered as the combined +span of all YF,Pi and �YF inside �ZF . Finally, as discussed in [74] the 1-form symmetry group +of the Class S theory is trivial. +Review of the Magnetic Quiver. +Let us now review the magnetic quiver of the Class S +theory discussed in [73]. We first associate a sub-quiver to each puncture Pi which can be +described as the following 3d N = 4 Lagrangian theory +· · · +u(ni,|ρi|−1) +u(ni,2) +u(ni,1) +[su(n)H] +(5.5) +where +ni,J = n − +J +� +j=1 +ρi,j +(5.6) +We see that the gauge nodes for ja−1 < J < ja are balanced for 1 ≤ a ≤ b, and give rise to +an emergent �b +a=1 su(ja − ja−1)C Coulomb symmetry in the IR. Moreover, the gauge nodes +J = ja for 1 ≤ a ≤ jb−1 are unbalanced and give rise to u(1)⊕(b−1) +C +Coulomb symmetry in the +IR. Thus, the contribution of this sub-quiver to the IR Coulomb 0-form symmetry algebra +matches the flavor symmetry algebra fi associated to the puncture Pi shown in (5.1). +The full magnetic quiver of the Class S theory is then obtained by gauging the diagonal +su(n)H symmetry of all the sub-quivers, resulting in the theory +u(n1,1) +su(n)H +u(n2,1) +u(nk,1) +· · · +u(n1,|ρ1|−1) +· · · +u(n2,|ρ2|−1) +· · · +u(nk,|ρk|−1) +(5.7) +where the su(n)H node is unbalanced, and thus the IR Coulomb 0-form symmetry algebra is +f = � +i fi, matching with the flavor symmetry algebra of the Class S theory. +Global Form of the Magnetic Quiver. +What is the gauge group that we should choose? +Apriori there are many choices +�k +i=1 +�|ρi|−1 +J=1 +U(ni,J) × SU(n)H +Zm +(5.8) +parametrized by divisors m of n. The 1-form symmetry of the theory for such a choice is +Γ(1) = Zn/m . +(5.9) +– 43 – + +To match it with the trivial 1-form symmetry of the Class S theory, we are forced to pick the +gauge group +G = +�k +i=1 +�|ρi|−1 +J=1 +U(ni,J) × SU(n)H +Zn +(5.10) +for the choice m = n. +This global form is actually manifest in the following usual presentation of the MQ +U(n1,1) +U(n)H +U(n2,1) +U(nk,1) +· · · +U(n1,|ρ1|−1) +· · · +U(n2,|ρ2|−1) +· · · +U(nk,|ρk|−1) +(5.11) +with an additional instruction of ungauging a U(1). If we perform this U(1) ungauging on the +central U(n)H node, we are forced to remove also the Zn center of the SU(n)H component of +U(n)H as this Zn sits inside the U(1) center of U(n)H being removed. Due to the presence of +bifundamental matter, this Zn also sits inside the U(1) center of all the other unitary gauge +groups. Thus, performing the U(1) ungauging at U(n)H node indeed leaves behind the G +gauge group appearing in (5.10). Similar discussions also appeared in [75], where a discussion +regarding other U(1) ungaugings can also be found (see also [76]). +Computing Flavor Symmetry Group Using the Magnetic Quiver. +Since, by this +simple argument based on 1-form symmetry, we have no other possible choice for the gauge +group, it better be true that the IR Coulomb 0-form symmetry group for this choice of gauge +group matches the flavor symmetry group of the Class S theory. +This is a straightforward application of the special case of our general prescription de- +scribed in section 4.5. Recall there we also encountered two types of contributions. The first +type of contributions came from unbalanced unitary gauge nodes. Collecting all such contri- +butions from the unbalanced unitary gauge nodes situated along the sub-quiver leg associated +to the puncture Pi provide the contribution YF,Pi of [64]. The second type of contribution +comes from the sole special unitary unbalanced gauge node, which provides the contribution +�YF of [64]. In this way, we find that the IR Coulomb 0-form symmetry group of the magnetic +quiver (with the correct global form) described above matches the flavor symmetry group F +of the Class S theory. +5.1.2 +Examples +Let us see this matching explicitly for the following three examples. +– 44 – + +Example 1: Trinion Tn. +The first example we consider is the 4d trinion Tn theory, ob- +tained as a Class S theory by compactifying 6d N = (2, 0) SCFT of An−1 type on a sphere +with 3 maximal regular punctures. Each puncture provides an su(n) flavor symmetry algebra. +Thus, the total flavor symmetry algebra is +f = su(n)1 ⊕ su(n)2 ⊕ su(n)3 +(5.12) +with associated F = SU(n)1 × SU(n)2 × SU(n)3 with center +ZF = (Zn)1 × (Zn)2 × (Zn)3 . +(5.13) +We assume n > 3, because as is well known there is an enhancement of the above flavor +symmetry algebra to f = e6 for n = 3, in which case the T3 trinion theory coincides with the +E6 Minahan-Nemeschansky theory. This was discussed as the first example in section 4.3 of +[64]. +The associated magnetic quiver is +u(n − 1) +su(n)H +u(n − 1) +u(n − 1) +u(n − 2) +· · · +u(n − 2) +· · · +u(n − 2) +· · · +u(1) +u(1) +u(1) +(5.14) +with gauge group +G = +�n−1 +i=1 U(i)3 × SU(n)H +Zn +. +(5.15) +Let us now compute the IR Coulomb 0-form symmetry group using the arguments of +section 4.5. Since every unitary gauge node is balanced, we do not have any puncture depen- +dent contribution i.e. YF,Pi = 0. On the other hand, the contribution �YF is obtained from +the monopole operator O, whose charge under ZF is the same as that of the representation +F1 ⊗ F2 ⊗ F3 +(5.16) +of F, where Fi is the fundamental representation of SU(n)i ⊂ F. This is because the su(n)H +node hits each balanced sub-quiver i at the node of Dynkin diagram of su(n)i corresponding +to the fundamental representation of su(n)i. +These contributions match those appearing in [64], and as described there the flavor +symmetry group can written as +F = SU(n)1 × SU(n)2 × SU(n)3 +Zn × Zn +. +(5.17) +– 45 – + +We refer the reader to [64] for more details regarding the identity of the two Zn subgroups +appearing in the denominator. +We can also discuss the case of n = 3, for which it was argued in [64] that the full flavor +symmetry group must be E6/Z3 as that is the only possible enhancement of (5.17) for the +n = 3 case. We can see this flavor group directly from the MQ, which is obtained by a U(1) +ungauging of +U(2) +U(3)H +U(2) +U(2) +U(1) +U(1) +U(1) +. +(5.18) +Instead of performing the U(1) ungauging on the U(3)H gauge node, let us perform it on one +of the U(1) gauge nodes. The magnetic quiver can then be expressed as +U(2) +U(3)H +U(2) +U(2) +U(1) +U(1) +F +, +(5.19) +where we have a fundamental hyper charged under the top U(2) gauge node. Every unitary +gauge node is balanced and hence the IR Coulomb 0-form symmetry algebra is +f = e6 . +(5.20) +Since there are no unbalanced unitary or special unitary gauge nodes, the IR Coulomb 0-form +symmetry group is the centerless global form +F = E6/Z3 +(5.21) +of f = e6. +Example 2: +Free Bifundamental Hyper. +As a second example, consider the Class +S theory obtained by compactifying 6d N = (2, 0) SCFT of An−1 type on a sphere with +2 maximal regular punctures P1, P2 and 1 minimal regular puncture P3. +Each maximal +puncture provides an su(n) flavor symmetry algebra, while the minimal puncture provides +u(1) flavor symmetry algebra. Thus, the total flavor symmetry algebra is +f = su(n)1 ⊕ su(n)2 ⊕ u(1) +(5.22) +with associated F = SU(n)1 × SU(n)2 × U(1) with center +ZF = (Zn)1 × (Zn)2 × U(1) . +(5.23) +The resulting 4d N = 2 theory can be recognized as a free hypermultiplet transforming in +bifundamental representation of two su(n) factors of f, with the u(1) factor of f rotating the +bifundamental hyper. This was discussed as the second example in section 4.1 of [64]. +– 46 – + +The associated magnetic quiver is +u(n − 1) +su(n)H +u(n − 1) +u(1) +u(n − 2) +· · · +u(n − 2) +· · · +u(1) +u(1) +(5.24) +with gauge group +G = U(1)3 × �n−1 +i=2 U(i)2 × SU(n)H +Zn +. +(5.25) +Let us now compute the IR Coulomb 0-form symmetry group using the arguments of +section 4.5. +Since every unitary gauge node is balanced in the sub-quivers associated to +punctures P1 and P2, we have YF,P1 = YF,P2 = 0. On the other hand, the U(1) gauge node +comprising the sub-quiver associated to P3 contributes a genuine local operator of charge +n under U(1) factor of ZF (and charge 0 under each (Zn)i factor). +This is precisely the +contribution YF,P3 of [64]. +The contribution �YF is obtained from the monopole operator O, whose charge under +(Zn)1 × (Zn)2 factor of ZF is the same as that of the representation +F1 ⊗ F2 +(5.26) +of SU(n)1 × SU(n)2 factor of F, where Fi is the fundamental representation of SU(n)i ⊂ F. +This is because the su(n)H node hits each balanced sub-quiver i ∈ {1, 2} at the node of Dynkin +diagram of su(n)i corresponding to the fundamental representation of su(n)i. Moreover, O +also has charge +1 under U(1) factor of ZF . +These contributions match those appearing in [64], and as described there the flavor +symmetry group can written as +F = SU(n)1 × SU(n)2 × U(1) +Zn × Zn +. +(5.27) +We refer the reader to [64] for more details regarding the identity of the two Zn subgroups +appearing in the denominator. +Example 3. +As a final example, let us consider a Class S theory involving more general +regular punctures that are neither maximal nor minimal. We are compactifying A3 N = (2, 0) +theory on a sphere with three punctures. The puncture P1 has partition ρ1 = {2, 1, 1}, the +puncture P2 has partition ρ2 = {2, 2}, and the puncture P3 is a maximal puncture with +partition ρ3 = {1, 1, 1, 1}. This was discussed as the third example in section 4.1 of [64]. +The flavor symmetry algebras associated to the punctures are f1 = su(2)1 ⊕ u(1), f2 = +su(2)2 and f3 = su(4), with the total flavor symmetry algebra being +f = su(2)1 ⊕ u(1) ⊕ su(2)2 ⊕ su(4) +(5.28) +– 47 – + +with associated F = SU(2)1 × U(1) × SU(2)2 × SU(4) with center +ZF = (Z2)1 × U(1) × (Z2)2 × Z4 . +(5.29) +The associated magnetic quiver is +u(2) +su(4)H +u(3) +u(2) +u(1) +u(2) +u(1) +(5.30) +with gauge group +G = U(1)2 × U(2)3 × U(3) × SU(4)H +Z4 +. +(5.31) +Let us now compute the IR Coulomb 0-form symmetry group using the arguments of +section 4.5. +Since every unitary gauge node is balanced in the sub-quivers associated to +punctures P2 and P3, we have YF,P2 = YF,P3 = 0. On the other hand, the unbalanced u(2) +gauge node in the sub-quiver associated to P1 contributes a genuine local operator whose +charge under (Z2)1 factor of ZF is same as that of the fundamental representation F1 of +SU(2)1. This is because this unbalanced u(2) gauge node hits once the Dynkin diagram of +su(2)1 formed by the u(1) gauge node in the sub-quiver associated to P1. Moreover, this +genuine local operator also has charge 4 under the U(1) factor of ZF which arises from this +unbalanced u(2) gauge node. +The contribution �YF is obtained from the monopole operator O, whose charge under +(Z2)1 × (Z2)2 × Z4 factor of ZF is the same as that of the representation +F2 ⊗ F +(5.32) +of SU(2)1×SU(2)2×SU(4) factor of F, where F2 is the fundamental representation of SU(2)2 +and F is the fundamental representation of SU(4). This is because the su(4)H node does not +hit the Dynkin diagram of su(2)1, while hitting the su(2)2 and su(4) Dynkin diagrams at the +nodes corresponding to the fundamental representations of su(2)2 and su(4). Moreover, O +also has charge 2 under U(1) factor of ZF . +Since all charges under U(1) factor of ZF are even, we can scale them half. The scaled +contributions match those appearing in [64], and as described there the flavor symmetry group +can written as +F = SU(2)1 × U(1) × SU(2)2 × SU(4) +Z4 × Z2 +. +(5.33) +We refer the reader to [64] for more details regarding the identity of the Z4 and Z2 subgroups +appearing in the denominator. +– 48 – + +5.2 +5d SCFTs +5d superconformal field theories (SCFTs) with 8 supercharges are closely related to 3d N = 4 +theories. The proposed correspondence is that the Higgs branch of the 5d SCFT is given +by the Coulomb branch of the 3d N = 4 IR SCFT arising from the associated magnetic +quiver (MQ) [21, 22], which is again a 3d N = 4 quiver gauge theory. This conjecture has +passed numerous non-trivial tests. It can be motivated from the 5d brane-web realization +of 5d SCFTs [19, 22, 25, 26, 30, 31], but also via a geometric construction of the 5d theory +in M-theory on a canonical singularity: in the case of isolated hypersurface singularities, the +MQs for the 5d theories can be derived from the geometry [27, 34, 41]. +One salient feature of 5d SCFTs is the flavor symmetry, which often is enhanced compared +to the flavor symmetry of an IR gauge theory description obtained in the IR after performing +a mass deformation (i.e. moving onto the extended Coulomb branch) of the UV 5d SCFT. +The simplest class of such models are the Seiberg En+1 theories having UV flavor symmetry +en+1, which after a mass deformation give rise to SU(2) + nF gauge theories in the IR with +IR flavor symmetry so(2n) ⊕ u(1) [77]. +The flavor symmetry is encoded in terms of the magnetic quiver as well: for simplicity +let us consider MQs which are built from � +i U(ni) gauge nodes (along with the additional +instruction of a ungauging a U(1) for each connected component of the MQ), connected by +bifundamentals such that there is a single bifundamental between any two nodes and the +resulting quiver has no loops. Then the balanced unitary nodes give rise to the non-abelian +part of the flavor symmetry algebra and the unbalanced nodes give rise to the abelian part +of the flavor symmetry algebra. The global form can be obtained using the methods in this +paper, specifically section 4. Note that the analysis of that section is applied after making a +suitable choices of U(1) ungaugings. +In this section we will compare the global form of the flavor symmetries of the 5d SCFTs +and their associated 3d MQ theories. The examples we will focus on are rank 2 theories. A +complete list of all MQs for rank 2 theories can be found in [31], using the method of [30]. +The flavor symmetry can be determined alternatively from geometry as in [66, 78–83]. The +models we consider are shown in table 1 and their magnetic quivers are in table 2. The 5d +theories have gauge theory descriptions with SU(3) gauge groups and fundamental flavor and +thus no 1-form symmetry (which in principle upon dimensional reduction can contribute to +the 0-form symmetry of the MQ theory). +5.2.1 +Flavor Symmetry Groups from MQs +To derive the flavor symmetry from the MQs we simply apply the special cases of our general +analysis discussed in sections 4.5 and 4.6. +The MQs are all listed in table 2. +All nodes +are unitary: balanced nodes are white, unbalanced ones are black. The magnetic quiver is +obtained by ungauging a U(1) in each connected component of the listed quiver. There is a +choice in this, and we will pick the ungaugings that allow us to apply the analysis of sections +4.5 and 4.6. +– 49 – + +Model Gauge Theory Flavor algebra +2 +SU(3)1/2 + 9F +so(20) +5 +SU(3)1 + 8F +so(16) ⊕ su(2) +12 +SU(3)0 + 6F +su(6) ⊕ su(2)2 +9 +SU(3)3/2 + 7F so(14) ⊕ u(1) +26 +SU(3)2 + 4F +su(5) ⊕ u(1) +Table 1: Rank 2 5d SCFTs. We list the IR gauge theory description as well as the flavor +symmetry algebra. We determine the group structure from both 5d and the corresponding +3d magnetic quivers. +To determine the flavor symmetry group from the magnetic quiver for the cases listed in +the table 2, we have to simply follow the following rules: +1. Pick a connected component α of the listed magnetic quiver. +Determine the set of +balanced nodes, which form a non-abelian Lie algebra fα. The number of unbalanced +nodes Uα determines an abelian Lie algebra u(1)|Uα−1|. The Coulomb flavor symmetry +algebra f of the 3d IR SCFT associated to the full magnetic quiver is obtained by picking +the component β with the maximal associated algebra, i.e. f = fβ ⊕ u(1)|Uβ−1|. +2. Ungauge a u(1) in the each connected component α. This is done at the location of +an unbalanced node. If the unbalanced node is u(n) for n > 1, then we are left with +a special unitary gauge node su(n), and the gauge group associated to the connected +component α is +Gα = +� +i U(ni) × SU(n) +Zn +. +(5.34) +That is the center Zn of the numerator is automatically removed in the U(1) ungauging +process, as discussed in [75] and after equation (5.11). +On the other hand, if the unbalanced node is u(1), then we are left with a fundamental +flavor there and the gauge group associated to the connected component α is +Gα = +� +i +U(ni) . +(5.35) +After the ungauging, for the cases appearing in table 2, we are left with either no +unbalanced nodes or another unbalanced unitary gauge node. +3. The position of the unbalanced unitary or special unitary node determines the repre- +sentation under f of the gauge monopole operators as described in sections 4.5 and 4.6. +We account for monopole operators coming from all connected components. From this +the global form of the flavor symmetry group is determined, by quotienting out the part +of the flavor center (of the simply connected group associated to f) that acts trivially +on the monopole operators. +– 50 – + +Model +Magnetic Quiver +Flavor Group +2 +1 +2 +3 +4 +5 +6 +7 +8 +5 +2 +4 +Ss(20) +5 +1 +2 +3 +4 +5 +6 +4 +2 +1 +3 +Ss(16)×SU(2) +Z2 +12 +1 +2 +3 +2 +1 +1 +2 +1 +SU(6)/Z3×SO(4) +Z2 +9 +1 +2 +3 +4 +5 +3 +1 +3 +1 +Spin(14)×U(1) +Z4 +26 +1 +2 +2 +1 +1 +1 +1 +1 +1 +1 +1 +U(5) +Table 2: The Magnetic Quivers for the 5d SCFTs listed in table 1. Each node labeled by n +corresponds to a U(n) gauge node, and connection lines to bi-fundamentals. The subgraph +given by the white nodes is the Dynkin diagram of the non-abelian part of the flavor symmetry +algebra (i.e. the balanced nodes). The black are unbalanced nodes. The MQ is obtained by +the ungauging of one of the nodes. +We will now determine the global form of the flavor symmetry groups in the examples of +table 2. +Model 2. +The balanced (white) nodes in the magnetic quiver form the Dynkin diagram of +f = so(20) . +(5.36) +There is no abelian factor in the flavor algebra because there is a single unbalanced node +(shown in black). We ungauge the U(1) at the location of this unbalanced node and land in +– 51 – + +the special case of our general analysis discussed in section 4.5. The gauge group is +G = +�8 +i=1 U(i) × U(4) × U(5) × SU(2) +Z2 +. +(5.37) +The unbalanced special unitary node is attached to the spinor node of so(20), and thus we +have a monopole operator transforming in the spinor representation7 of so(20). Since we do +not have a cospinor representation, we can remove the Z2 subgroup of the Z2 × Z2 center of +Spin(20), which acts on cospinor representation, but leaves the spinor representation invariant. +Thus the flavor symmetry group is +F = Spin(20)/Z2 = Ss(20) , +(5.38) +where the group Ss(20) is a global form of so(20) which admits spinor representation but +does not admit cospinor or vector representations of so(20). +Model 5. +Here the balanced nodes form f = so(16) ⊕ su(2). There is no abelian factor in +the flavor algebra. After ungauging U(1) at the location of the unbalanced U(2) gauge node, +we obtain the MQ whose gauge group is +G = +�6 +i=1 U(i) × U(3) × U(4) × SU(2) × U(1) +Z2 +. +(5.39) +The unbalanced special unitary node is attached to the spinor node of so(16) and the funda- +mental node of su(2), implying that we have a monopole operator transforming in represen- +tation (S, F) of so(16)⊕su(2), where S is spinor representation of so(16) and F is fundamental +representation of su(2). There is again no monopole operator transforming in the cospinor, +so we can reduce Spin(16) to Ss(16). The center of Ss(16) is Z2 which acts non-trivially on +the spinor representation, and the Z2 center of SU(2) acts non-trivially on the fundamental +representation. Thus the (S, F) monopole operator is left invariant by the diagonal Z2 and +we can express the flavor symmetry group as +F = Ss(16) × SU(2) +Z2 +. +(5.40) +Model 12. +Similarly for model 12, we have +f = su(6) ⊕ su(2) ⊕ su(2) . +(5.41) +The unbalanced special unitary node (obtained after U(1) ungauging) intersects the Dynkin +diagrams of simple components of f such that we have a monopole operator in representation +(Λ3, F, F) of f, where Λ3 is the 3-index antisymmetric irreducible representation of dimension +7More precisely, we are only claiming that we have a monopole operator transforming in a representation +of so(20) having same charges under the center of the simply connected group Spin(20) as the spinor repre- +sentation of so(20). However, for brevity here and in what follows, we will blur this distinction, but the reader +should keep this in mind. +– 52 – + +20 of su(6). +Since only Λ3 of su(6) appears, we can begin with the smallest global form +SU(6)/Z3 of su(6) allowing this representation. The center of SU(6)/Z3 is Z2 which acts non- +trivially on Λ3. Similarly, since we only have the bifundamental (F, F) of su(2)⊕su(2) = so(4), +we can begin with its smallest global form SO(4) allowing this representation. The center +of SO(4) is Z2 which acts non-trivially on (F, F). Thus, the diagonal Z2 of the Z2 centers of +SU(6)/Z3 and SO(4) acts trivially on (Λ3, F, F) the monopole, leading to the flavor symmetry +group +F = SU(6)/Z3 × SO(4) +Z2 +. +(5.42) +Model 9. +The balanced nodes provide an so(14) flavor algebra and the unbalanced nodes +provide a u(1) flavor algebra, since we have 2 unbalanced nodes. The total flavor algebra is +thus +f = so(14) ⊕ u(1)C . +(5.43) +We choose to ungauge one of the unbalanced U(1) nodes. The gauge group is thus +G = +5 +� +i=1 +U(i) × U(3)2 × U(1) . +(5.44) +We have to now apply the analysis of section 4.6. Since the unbalanced U(1) gauge node is +attached to the Dynkin diagram of so(14) at the location of co-spinor node, the monopole +operator associated to the unbalanced U(1) node transforms in the cospinor irreducible repre- +sentation C of so(14). Simultaneously, the monopole operator also carries a charge +1 under +a global form U(1)C of the u(1)C factor of f arising from this unbalanced node. Since, the +representation C has charge −1 under the Z4 center of the simply connected group Spin(14) +associated to so(14), the monopole operator discussed above is uncharged under the diagonal +combination of the Z4 center of Spin(14) and the Z4 subgroup of the group U(1)C. The flavor +symmetry group is thus +F = Spin(14) × U(1) +Z4 +. +(5.45) +Model 26. +There are two connected components in the MQ corresponding to two branches +of the moduli space. The first component gives rise to algebra f1 = su(5), while the second +component gives rise to a larger algebra f2 = su(5) ⊕ u(1)C. The flavor symmetry algebra is +thus +f = su(5) ⊕ u(1)C +(5.46) +provided by the second component. Ungauging the unbalanced U(1) gauge node in the first +component results in an MQ with balanced unitary gauge nodes only, and thus we do not +obtain any monopole operator relevant for the analysis of flavor group. Ungauging an unbal- +anced U(1) gauge node in the second component leaves behind another unbalanced U(1) gauge +node which provides a monopole operator transforming in anti-fundamental representation of +– 53 – + +su(5) along with charge +1 under U(1)C, leading to the flavor group +F = SU(5) × U(1)C +Z5 += U(5) . +(5.47) +5.2.2 +Flavor Symmetry Groups from String Theory Constructions +Reference [66] described a computation of flavor symmetry groups of 5d SCFTs using their +string theory constructions. +The key physical idea involved is as follows. +Study the 5d +conformal theory on a flat spacetime with a non-conformal vacuum at infinity. More precisely, +one chooses a supersymmetric non-conformal vacuum lying in the Coulomb branch of vacua8 +of the 5d SCFT. The theory now flows and in the IR we obtain a 5d supersymmetric gauge +theory with an abelian gauge group U(1)r, where r is referred to as the rank of the 5d SCFT. +In addition, we have massive BPS particles9 charged under U(1)r and the flavor symmetry. +The flavor group is then obtained easily by computing flavor charges of gauge invariant +combinations of the above charges, namely those linear combinations which have zero U(1)r +gauge charge. +The charges under U(1)r can be read off from any string theory construction, but the +charges under flavor symmetry require us to use “good” string theory constructions, namely +those that manifest the full enhanced flavor symmetry algebra of the 5d SCFT. As is well- +known, there are two main kinds of string theory constructions of 5d SCFTs: the first kind +involves compactifications of M-theory on Calabi-Yau threefolds while the second kind in- +volves intersecting brane configurations in Type IIB superstring theory. There is always a +good M-theory construction, which we will now focus on to compute explicitly the flavor +symmetry group of the 5d SCFTs appearing in table 1. +In an M-theory construction, the charges of all BPS particles can be captured by charges +of a special set of BPS particles arising from M2 branes wrapping irreducible holomorphic +curves in the Calabi-Yau threefold. The gauge/flavor charges are described by intersection +numbers of these curves with compact/non-compact divisors. Thus the set of data required +for computation of flavor groups is a set C of holomorphic curves (whose charges span charges +of all other curves) along with their intersection numbers with compact and non-compact +divisors. Recently, a lot of work has been performed on the computation of such a set C +of curves along with their intersection numbers, and crucially the intersection numbers with +non-compact divisors capturing enhanced flavor symmetries. These works have used various +geometric techniques involving blow-downs of flat [84, 85] and non-flat [78–81, 86] resolutions +of non-minimal elliptic fibrations, along with more general local surface geometry structures +8It is important not to confuse this with the extended Coulomb branch, which is simply referred to as the +Coulomb branch in many studies on 5d SCFTs. The extended Coulomb branch is a space obtained by fibering +Coulomb branch of vacua on the base space comprising of a family of theories obtained from the 5d SCFT by +performing supersymmetric mass deformations. The Coulomb branch of vacua being referred here is the fiber +at the origin (namely the point with zero mass deformations) of the base space of extended Coulomb branch. +9One might worry about non-BPS excitations and whether they provide any additional charges that can +modify the computation. From the string theory constructions, it is possible to argue that they do not provide +any new charges. +– 54 – + +[82, 83, 87–93] and description of the associated Calabi-Yau K¨ahler cone structure [78–81, 94] +that interpolates between different resolutions via flop transitions. +We now describe the spanning set C of curves and the relevant charges of BPS particles +associated to these curves, using this information to compute the flavor group for all the +models appearing in table 1. Detailed computation of the set C and its intersection numbers +can be found in appendix A. +Model 2. +One can use any of the above described methods to compute a sufficient spanning +set C of curves and its intersection numbers. One finds that C contains four curves whose +charges are +q(C1) = +� +0, 1|0 (mod 2), 1 (mod 2) +� +q(C2) = +� +− 2, 2|0 (mod 2), 0 (mod 2) +� +q(C3) = +� +2, −1|1 (mod 2), 1 (mod 2) +� +q(C4) = +� +1, −1|1 (mod 2), 1 (mod 2) +� +, +(5.48) +where the first two charges are under the U(1)2 gauge group, and the last two charges are +under the ZS +2×ZC +2 center of Spin(20) simply connected group associated to the flavor symmetry +algebra f = so(20), where ZS +2 is the subgroup of the center under which spinor and vector are +charged and ZC +2 is the subgroup of the center under which co-spinor and vector are charged. +From this the reader easily sees that all gauge-invariant linear combinations of these curves +are either trivially charged under ZS +2 × ZC +2 or have charge +� +1 (mod 2), 0 (mod 2) +� +. Thus we +are again led to the flavor group F = Ss(20). +The above curves Ci have a nice physical interpretation as follows. +Perform a mass +deformation of the 5d SCFT such that it flows in the IR to 5d N = 1 gauge theory with +gauge group Sp(2) and 9 fundamental hypers. +Then, the curve C1 gives rise to a BPS +instanton of the gauge theory. The curves C2 and C3 give rise to W-bosons of Sp(2) upon +moving onto the Coulomb branch of this gauge theory. Finally, the curve C4 gives rise to one +of the 9 hypers. +Model 5. +The analysis is similar to the above case. We again have four curves Ci which +have same physical interpretation after mass deforming to 5d N = 1 gauge theory with gauge +group Sp(2) and 8 fundamental hypers. One can use any of the above described methods to +compute that these curves have charges +q(C1) = +� +0, 1|1 (mod 2), 0 (mod 2), 0 (mod 2) +� +q(C2) = +� +− 2, 2|0 (mod 2), 0 (mod 2), 0 (mod 2) +� +q(C3) = +� +2, −1|0 (mod 2), 0 (mod 2), 1 (mod 2) +� +q(C4) = +� +1, −1|1 (mod 2), 1 (mod 2), 0 (mod 2) +� +, +(5.49) +where the first two charges are under the U(1)2 gauge group, the next two charges are under +the ZS +2 × ZC +2 center of Spin(16) simply connected group associated to so(16) ⊂ f, and the last +charge is under the Z2 center of SU(2) simply connected group associated to su(2) ⊂ f. From +– 55 – + +this the reader easily sees that all gauge-invariant linear combinations of these curves are +either trivially charged under ZS +2 ×ZC +2 ×Z2 or have charge +� +1 (mod 2), 0 (mod 2), 1 (mod 2) +� +. +Thus we are again led to the flavor group described in (5.40). +Model 12. +It is again sufficient to consider four curves Ci which have similar physical +interpretation as above after mass deforming to 5d N = 1 gauge theory with gauge group +SU(3), Chern-Simons level 0, and 6 fundamental hypers. +One can use any of the above +described methods to compute that these curves have charges +q(C1) = +� +0, 0|0 (mod 6), 0 (mod 2), 0 (mod 2) +� +q(C2) = +� +− 1, 2|0 (mod 6), 0 (mod 2), 1 (mod 2) +� +q(C3) = +� +2, −1|0 (mod 6), 1 (mod 2), 0 (mod 2) +� +q(C4) = +� +1, −1|1 (mod 6), 0 (mod 2), 0 (mod 2) +� +, +(5.50) +where the first two charges are under the U(1)2 gauge group, the next charge is under the +Z6 center of SU(6) simply connected group associated to su(6) ⊂ f, and the last two charges +are under the ZS +2 × ZC +2 center of Spin(4) = SU(2) × SU(2) simply connected group associated +to so(4) = su(2) ⊕ su(2) ⊂ f. From this the reader can see that all gauge-invariant linear +combinations of these curves are either trivially charged under Z6 × ZS +2 × ZC +2 or have charge +� +3 (mod 6), 1 (mod 2), 1 (mod 2) +� +. Thus we are again led to the flavor group described in +(5.42). +Model 9. +It is again sufficient to consider four curves Ci which have similar physical inter- +pretation as above after mass deforming to 5d N = 1 gauge theory with gauge group Sp(2) +and 7 fundamental hypers. One can use any of the above described methods to compute that +these curves have charges +q(C1) = +� +0, 1|3 (mod 4), 0 +� +q(C2) = +� +− 2, 2|0 (mod 4), 0 +� +q(C3) = +� +2, −1|0 (mod 4), −1 +� +q(C4) = +� +1, −1|2 (mod 4), 0 +� +, +(5.51) +where the first two charges are under the U(1)2 gauge group, the next charge is under the +Z4 center of Spin(14) simply connected group associated to so(14) ⊂ f, and the last charge +is under the U(1) group associated to u(1) ⊂ f. From this the reader can see that all gauge- +invariant linear combinations of these curves are either trivially charged under Z4 × U(1) or +have charge a multiple of +� +1 (mod 4), −1 +� +. Thus we are again led to the flavor group described +in (5.45). +Model 26. +It is again sufficient to consider four curves Ci which have similar physical +interpretation as above after mass deforming to 5d N = 1 gauge theory with gauge group +SU(3), Chern-Simons level 2 and 7 fundamental hypers. +One can use any of the above +– 56 – + +described methods to compute that these curves have charges +q(C1) = +� +− 1, 1|4 (mod 5), 0 +� +q(C2) = +� +− 1, 2|1 (mod 5), 0 +� +q(C3) = +� +2, −1|0 (mod 5), −1 +� +q(C4) = +� +1, −1|1 (mod 5), 0 +� +, +(5.52) +where the first two charges are under the U(1)2 gauge group, the next charge is under the Z5 +center of SU(5) simply connected group associated to su(5) ⊂ f, and the last charge is under +the U(1) group associated to u(1) ⊂ f. From this the reader can see that all gauge-invariant +linear combinations of these curves are either trivially charged under Z5×U(1) or have charge +a multiple of +� +1 (mod 5), −1 +� +. Thus we are again led to the flavor group described in (5.47). +5.3 +3d Mirror Symmetry +As a final consistency check, we can apply our methods to compute and match symmetries +and anomalies of two 3d N = 4 gauge theories that are related by 3d mirror symmetry. Let +us illustrate this with the general example of 3d N = 4 gauge theory +U(m) +[su(2m)H] +(5.53) +namely U(m) gauge theory with 2m fundamental hypers. The Higgs flavor symmetry algebra +is +fH = su(2m)H +(5.54) +rotating the fundamental hypers, and the IR Coulomb flavor symmetry algebra is +fIR +C = su(2)C +(5.55) +because the U(m) gauge node is balanced. The Higgs 0-form symmetry group is +FH = PSU(2m)H +(5.56) +because the full center Z2m of SU(2m)H is a subgroup of the U(1) center of the U(m) gauge +group. From the analysis of section 4.5, the IR Coulomb 0-form symmetry group is +FIR +C = SO(3)H +(5.57) +as there are no unbalanced unitary or special unitary gauge nodes. From the analysis of section +4.5 we also see that there is mixed flavor-gauge monopole operator having winding number +1 around a U(1) subgroup of the maximal torus of PSU(2m)H and a charge under the Z2 +center of SU(2)C same as that of the fundamental representation of SU(2)C. This is because +the flavor node [su(2m)H] hits the Dynkin diagram of su(2)C at the node corresponding to +– 57 – + +fundamental representation of su(2)C. This translates to a mixed ’t Hooft anomaly of the IR +SCFT between the Higgs and Coulomb 0-form symmetry groups of the form +AIR +4 = exp +� +πi +� +wH +2 ∪ wC +2 +� +, +(5.58) +where wC +2 is the Z2 valued second Stiefel-Whitney class of the background SO(3)C bundle +and wH +2 is the Z2m valued obstruction class for lifting background PSU(2m)H bundles to +SU(2m)H bundles. +Now consider its 3d N = 4 mirror gauge theory [4] +U(m − 1) +U(m) +· · · +U(2) +U(1) +U(m − 1) +· · · +U(2) +U(1) +[su(2)H] +. +(5.59) +The Higgs flavor symmetry algebra is +fH = su(2)H +(5.60) +rotating the two fundamental hypers of U(m) gauge group, and the IR Coulomb flavor sym- +metry algebra is +fIR +C = su(2m)C +(5.61) +because all the unitary gauge nodes are balanced. The Higgs 0-form symmetry group is +FH = SO(3)H +(5.62) +because the center Z2 of SU(2)H is a subgroup of the U(1) center of each unitary gauge +group. From the analysis of section 4.5, the IR Coulomb 0-form symmetry group is +FIR +C = PSU(2m)H +(5.63) +as there are no unbalanced unitary or special unitary gauge nodes. From the analysis of section +4.5 we also see that there is mixed flavor-gauge monopole operator having winding number 1 +around the maximal torus of SO(3)H and a charge under the Z2m center of SU(2m)C same +as that of the irreducible representation of su(2m)C whose highest weight has a single non- +zero Dynkin coefficient, namely the m-th one with dm = 1. This is because the flavor node +[su(2)H] hits the Dynkin diagram of su(2m)C at the node corresponding to this representation +of su(2m)C. This translates to a mixed ’t Hooft anomaly of the IR SCFT between the Higgs +and Coulomb 0-form symmetry groups of the form +AIR +4 = exp +� +πi +� +wC +2 ∪ wH +2 +� +(5.64) +– 58 – + +where wH +2 is the Z2 valued second Stiefel-Whitney class of the background SO(3)H bundle +and wC +2 is the Z2m valued obstruction class for lifting background PSU(2m)C bundles to +SU(2m)C bundles. +Thus, we have seen explicitly that the symmetry and anomaly properties of the two +mirror theory are same upto the exchange of labels C ↔ H. +6 +Some Generalizations +In this section, we discuss a few generalizations of the general considerations of this paper. +We will discuss two different types of generalizations: +1. In the first generalization, we will allow ourselves to perform an N = 2 gauging of flavor +symmetries of 3d N = 4 theories along with the addition of a Chern-Simons level. We +will see that the Chern-Simons level induces a ’t Hooft anomaly purely for the 1-form +symmetry, which is novel feature that we have not encountered in the N = 4 theories +that we studied in this paper. Related models have appeared in [70], motived from the +study of T[M3] compactifications of 6d theories. +2. In the second generalization, we will allow ourselves to perform gaugings of discrete +subgroups of flavor symmetries of 3d N = 4 theories. We will see that this opens up +the possibility of having non-trivial 2-group symmetries within the context of the study +of 3d N = 4 theories involving unitary and special unitary gauge groups. We will also +encounter the presence of mixed ’t Hooft anomalies between these 2-group symmetries +and 0-form symmetries of the 3d N = 4 theory. +6.1 +N = 2 Gauging of T[SU(n)] +In this subsection we study N = 2 gaugings (possibly with Chern-Simons levels) of su(n)H +Higgs flavor symmetry of T[SU(n)]. We begin with n = 2 and later generalize to arbitrary n. +We find that none of these have a non-trivial 2-group symmetry. +Symmetries. +Consider the 3d N = 2 theory obtained by an N = 2 gauging of the su(2)H +flavor symmetry of the 3d N = 4 theory T[SU(2)] by an SU(2)H gauge group. In addition, +we turn on a Chern-Simons level k for the SU(2)H gauge group while preserving the N = 2 +supersymmetry. Due to this Chern-Simons level, non-fractional gauge monopole operators +(which are genuine local operators) start transforming in representations of SO(3)H. For such +monopole operators to be gauge invariant, they must arise at the ends of Wilson line defects +associated to representations of SO(3)H. However, this clearly does not impact the 1-form +symmetry, and we have +Γ(1) = Z2 +(6.1) +just as for the case of N = 4 SU(2)H gauging. The 0-form symmetry is also the same as for +the N = 4 gauging +F = SO(3)C . +(6.2) +– 59 – + +One can argue just as for the case of N = 4 SU(2)H gauging that Z2 and SO(3)C do not +combine to form a 2-group symmetry with a non-trivial Postnikov class. +Purely 1-Form Anomaly. +Consider instead a fractional gauge monopole operator O la- +beled by a co-character of SU(2)H having winding number half around its maximal torus. +Due to Chern-Simons level k, the operator O transforms in a representation R of SU(2)H +with charge +k (mod 2) +(6.3) +under the Z2 center of SU(2)H. For O to be gauge invariant, it must arise at the end of of +a Wilson line defect associated to representation R. Using the analysis of [12], this fact is +equivalent to a ’t Hooft anomaly for the 1-form symmetry of the form +A(1) +4 += exp +� +πik +� P(B2) +2 +� +, +(6.4) +where P(B2) is the Pontryagin square of B2 and is a Z4 valued class. This class is even on +spin manifolds, and one can hence define a Z2 valued class 1 +2P(B2). The anomaly is this Z2 +valued. It vanishes for k even, but is non-trivial for k odd. +Mixed 1-Form 0-Form Anomaly. +The fractional gauge monopole operator O also trans- +forms in a representation of SU(2)C that is not a representation of SO(3)C, just as for the +case of N = 4 gauging of T[SU(2)]. This is equivalent to a mixed ’t Hooft anomaly between +the Z2 1-form and SO(3)C 0-form symmetries, and the full ’t Hooft anomaly can be expressed +as +A4 = exp +� +πi +� +k P(B2) +2 ++ B2 ∪ wC +2 +� +, +(6.5) +Generalization to T[SU(n)]. +It is straightforward to generalize to arbitrary n. We are +studying 3d N = 2 theory obtained by an N = 2 gauging of the su(n)H Higgs flavor symmetry +of the 3d N = 4 theory T[SU(n)] by an SU(n)H gauge group. In addition, we turn on a +Chern-Simons level k for the SU(n)H gauge group while preserving the N = 2 supersymmetry. +The non-fractional monopole operators transform in representations of PSU(n)H and so the +1-form symmetry is +Γ(1) = Zn +(6.6) +just as for the case of N = 4 SU(n)H gauging. The 0-form symmetry is also the same as for +the N = 4 gauging +F = PSU(n)C . +(6.7) +One can argue just as for the case of N = 4 SU(n)H gauging that Zn and PSU(n)C do not +combine to form a 2-group symmetry with a non-trivial Postnikov class. +A fractional gauge monopole operator O labeled by a co-character of SU(n)H having +winding number 1/n around its maximal torus transforms, due to Chern-Simons level k, in a +representation R of SU(n)H with charge +k (mod n) +(6.8) +– 60 – + +under the Zn center of SU(n)H, implying a ’t Hooft anomaly for the 1-form symmetry of the +form +A(1) +4 += exp +�2πik +n +� Pσ(n)(B2) +2 +� +, +(6.9) +where σ(n) = 0, 1 depending on whether n is even, odd respectively, and +P0(B2) +2 +:= P(B2) +2 +P1(B2) +2 +:= B2 ∪ B2 . +(6.10) +The Pontryagin square P(B2) is Z2n valued and is even for spin manifolds. As a consequence, +its half P(B2)/2 is Zn valued. On the other hand, B2 ∪ B2 is naturally Zn valued. +The fractional gauge monopole operator O also transforms in a representation of SU(n)C +with charge −1 (mod n) under its Zn center, just as for the case of N = 4 gauging of T[SU(n)]. +This is equivalent to a mixed ’t Hooft anomaly between the Zn 1-form and PSU(n)C 0-form +symmetries, and the full ’t Hooft anomaly can be expressed as +A4 = exp +�2πi +n +� +k Pσ(n)(B2) +2 +− B2 ∪ wC +2 +� +. +(6.11) +6.2 +T[SU(2)]/ZC +2 and Its Gaugings +In this subsection, we study the gauging of a Z2 subgroup of the SO(3)C 0-form symmetry of +T[SU(2)], which leads to a theory with a non-trivial 2-group symmetry. We also find a mixed +anomaly between this 2-group symmetry and the residual Coulomb 0-form symmetry. Finally, +we study the N = 4 gauging of su(2)H Higgs flavor symmetry of the theory T[SU(2)]/ZC +2 +obtained after the Z2 gauging of T[SU(2)]. +Let us consider the 3d N = 4 Lagrangian theory +U(1) +[su(2)H] +2 +(6.12) +which denotes a theory having a U(1) gauge group along with 2 hypermultiplets of charge 2 +that are rotated by an su(2)H Higgs flavor symmetry algebra. This theory can be obtained by +gauging the Z2 subgroup, denoted ZC +2 , of U(1)C Coulomb 0-form symmetry of the Lagrangian +theory (3.1) discussed earlier. +We are also interested in the 3d N = 4 SCFT that the above Lagrangian theory flows +to. We call this SCFT T[SU(2)]/ZC +2 because it can be obtained by gauging a Z2 subgroup, +denoted ZC +2 , of the SO(3)C 0-form symmetry of the 3d N = 4 SCFT T[SU(2)]. +This is +a consequence of the fact that the UV theory (6.12) is obtained by gauging Z2 subgroup of +U(1)C 0-form symmetry of the theory (3.1), and this U(1)C symmetry embeds as the maximal +torus of SO(3)C 0-form symmetry of the corresponding IR SCFT T[SU(2)]. +– 61 – + +1-Form Symmetry. +The symmetries and anomalies of the UV theory were discussed in +detail in section 7.4 of [12], which we review and use to deduce symmetry and anomalies of +the IR SCFT T[SU(2)]/ZC +2 . First of all, there is a +Γ(1) = Z2 +(6.13) +1-form symmetry coming from the fact that Wilson lines of odd U(1) charges cannot be +screened in the UV theory. This becomes the 1-form symmetry of the IR SCFT. This 1-form +symmetry can also be understood as the dual symmetry arising from the perspective of ZC +2 +0-form gauging. +Higgs 0-Form Symmetry. +The Higgs 0-form symmetry algebra of (6.12) is +fH = su(2)H +(6.14) +and the Higgs 0-form symmetry group is +FH = SO(3)H , +(6.15) +because the genuine local operators charged under su(2)H are gauge-invariant combinations +of hypermultiplets which all have trivial charge under Z2 center of SU(2)H. The IR SCFT +admits the same Higgs 0-form symmetry group. +Coulomb 0-Form Symmetry. +We label the Coulomb 0-form symmetry group of (6.12) +arising from the U(1) gauge node as +FC = U(1)′ +C = U(1)C/Z2 +(6.16) +to distinguish it from the U(1)C 0-form symmetry of the theory (3.1). +The IR SCFT admits the same Coulomb 0-form symmetry group. It can be checked +easily using the analysis of [5] that there is no enhancement due to monopole operators. +Alternatively, one can understand it from the point of view of gauging ZC +2 subgroup of SO(3)C +0-form symmetry group of T[SU(2)]. To compute the residual 0-form symmetry after this +gauging, we first compute the commutant of ZC +2 in SO(3)C, which is the maximal torus +U(1)C of SO(3)C, and then we mod out the commutant by ZC +2 to find that the residual +0-form symmetry is U(1)′ +C. +2-Group Symmetry. +There is a non-trivial 2-group symmetry formed by Z2 1-form sym- +metry and SO(3)H 0-form symmetry, with Postnikov class +δB2 = wH +3 = Bock(wH +2 ) , +(6.17) +where wH +3 is the third Stiefel-Whitney class of background SO(3)H bundles, which can be +obtained by applying Bockstein homomorphism associated to the non-split short exact se- +quence +0 → Z2 → Z4 → Z2 → 0 +(6.18) +– 62 – + +on the second Stiefel-Whitney class wH +2 . +This 2-group symmetry is a consequence of the fact that even though Wilson line opera- +tors of even charge can be screened, the local operators responsible for screening them have +different su(2)H representations depending on whether the charge of Wilson line is a multiple +of 4 or not. The non-genuine local operators living at the ends of Wilson line operators of +charge 4m form SO(3)H representations, while the non-genuine local operators living at the +ends of Wilson line operators of charge 4m + 2 form su(2)H representations that are not +allowed representations of SO(3)H. +The IR SCFT T[SU(2)]/ZC +2 also carries this 2-group symmetry. +Mixed 2-Group 0-Form ’t Hooft Anomaly. +The structure group of the gauge theory +(6.12) involving the Higgs 0-form symmetry is +S = U(1) × SU(2)H +Z4 +, +(6.19) +where the Z4 in the denominator is obtained by combining the Z4 subgroup of U(1) with +the Z2 center of SU(2)H. The Z4 in the denominator can actually be identified with the +Z4 group appearing in the short exact sequence (6.18) appearing in the description of the +2-group symmetry of the theory. +We can thus consider a mixed flavor-gauge monopole operator O associated to a co- +character of S with winding number 1/4 around U(1) and winding number 1/2 around the +maximal torus of SU(2)H. This monopole operator O is a solitonic defect associated to the +2-group symmetry described above because its obstruction to being lifted to a combination +of purely (and non-fractional) gauge and purely flavor monopole operator is captured by the +Z4 group in the denominator of the structure group S which as remarked above is associated +to the 2-group symmetry. See [12] for a more details regarding such solitonic defects. +The monopole operator O has charge q = 1/4 under U(1)′ +C. From the analysis of [12], +this fact is equivalent to a mixed ’t Hooft anomaly between the 2-group and the Coulomb +0-form symmetry of the form +A4 = exp +�πi +2 +� +BH +w ∪ +� +c1 +� +U(1)′ +C +� +(mod 4) +�� +, +(6.20) +where Bw is a Z4 valued background field associated to the 2-group symmetry comprised out +of the Z2 valued background field B2 for 1-form symmetry and the second Stiefel-Whitney +class wH +2 for background SO(3)H 0-form symmetry bundles, and c1 +� +U(1)′ +C +� +is the first Chern +class for background U(1)′ +C 0-form symmetry bundles. +The above anomaly for (6.12) descends to an anomaly in the IR SCFT T[SU(2)]/ZC +2 . +Gauging SU(2)H. +Consider performing N = 4 gauging of su(2)H symmetry of T[SU(2)]/ZC +2 +by an SU(2)H gauge group +T[SU(2)]/ZC +2 +SU(2)H +(6.21) +– 63 – + +In the T[SU(2)]/ZC +2 theory we have a line operator L that cannot be screened, but its +square 2L can be screened such that a non-genuine local operator living at the end of 2L +transforms in a representation of SU(2)H that is not a representation of SO(3)H. +After +gauging SU(2)H, this non-genuine local operator needs to attached to a SU(2)H Wilson line +in the same representation. Thus, after gauging SU(2)H, even 2L is not screened. As a +consequence, the 1-form symmetry group is +Γ(1) = Z4 . +(6.22) +The 2-group background Bw in T[SU(2)]/ZC +2 can be identified with the 1-form symmetry +background B′ +2 in (6.21). +The 0-form symmetry group remains U(1)′ +C, and the mixed ’t Hooft anomaly between +2-group and Coulomb 0-form symmetries of T[SU(2)]/ZC +2 becomes a mixed ’t Hooft anomaly +between 1-form and 0-form symmetries of (6.21) of the form +A4 = exp +�πi +2 +� +B′ +2 ∪ +� +c1 +� +U(1)′ +C +� +(mod 4) +�� +. +(6.23) +Acknowledgments +We thank Antoine Bourget, Marcus Sperling, Jingxiang Wu and Zhenghao Zhong for dis- +cussions on related topics. The work of MB is supported by the EPSRC Early Career Fel- +lowship EP/T004746/1 “Supersymmetric Gauge Theory and Enumerative Geometry”, STFC +Research Grant ST/T000708/1 “Particles, Fields and Spacetime”, and the Simons Collabo- +ration on Global Categorical Symmetries. This work is supported by the European Union’s +Horizon 2020 Framework through the ERC grants 682608 (LB and SSN) and 787185 (LB). +SSN is supported in part by the “Simons Collaboration on Special Holonomy in Geometry, +Analysis and Physics” and the EPSRC Open Fellowship EP/X01276X/1. +Note. +A paper with related results will appear at the same time in [95]. We thank the +authors for coordinating submission. +A +Geometric Computations for 5d SCFTs +In this appendix, we provide more details on the geometric computations leading to the +charges q(Ci) of BPS particles arising from curves Ci in Calabi-Yau threefolds involved in the +construction of 5d SCFTs appearing in table 1. +All the models we consider can be obtained by decoupling from the following 6d geometry, +which is a collision between a D10 Kodaira singularity and an I1 smooth fiber, tuned so that +the collision results in a non-minimal singularity. The fibration is best described in terms of +a so-called Tate model +y2 + b1Uxy + b3U 5δ2 +1 = x3 + b2UV x2 + b4U 5δ1x + b6U 10δ4 +1 . +(A.1) +– 64 – + +S1 : +U +u10 u14 u11 u15 u12 u16 u13 +u9 +u6 +u5 +δ1(0) +V (0) +e11 +S2 : +U(0) +e1 +δ2(4) +V (0) +Figure 1: The surface geometry for the marginal theory of type D10 −I1, which gives rise to +the rank 2 5d SCFTs discussed in this section (the labels for the curves is chosen in accord +with the derivation of the geometry in [79]). The two compact surfaces are denoted by Si +and the collection of rational curves and their intersections are shown in the figure. The +self-intersection of the curves in each surface is either shown next to the sections ui, U, V δi, +or is −2 for green and −1 for blue curves. +Here U = 0 is the D10 Kodaira fiber, and V = 0 the locus above which the I1 singular fiber +is located. We furthermore blowup the locus U = V = 0 by inserting a rational curve, and +denote the exceptional section of that by δ1. Non-flat resolution of this model was performed +in [79] and the geometry is given in figure 1, which we reproduced from said paper. +The compact surfaces are denoted by Si and are glued along U = 0, which is a degree +(−2, 0) curve. The numbers in the bracket indicate the self-intersection of the curve in the +divisor. The geometry shown is that of the marginal theory, i.e. the 6d parent SCFT (on the +tensor branch, which is modeled by the curve δ1) on a circle. Only once we start decoupling +matter (which geometrically corresponds to performing blowdowns), do we get a theory that +is a genuinely 5d SCFT fixed point. +The flavor symmetry is obtained from the geometry as follows: The non-abelian flavor +symmetry algebra is obtained from the intersection of compact Sa and non-compact Ni +divisors. The complete intersection curves Sa·Ni will have a normal bundle degree, which if it +is (−2, 0) will indicate that the non-compact divisors are ruled by these curves and correspond +to the Cartans of the flavor symmetry algebra (for a more refined discussion see [79]). On +the other hand, curves that have normal bundle (−1, −1) are inside the compact divisors, +and correspond to hypermultiplets in any associated 5d N = 1 gauge theory description, e.g. +with gauge group Sp(2) and 10 fundamental hypers. +The intersection pattern of the (−2, 0) curves give the Dynkin diagram of the flavor +– 65 – + +symmetry algebra, however in order to determine the flavor symmetry group, we need to +also determine the charges of the hypers under the flavor and gauge symmetry. Additionally, +we need to determine the charges of W-bosons and instantons under the flavor and gauge +symmetry. A convenient way of doing so is to convert the resolution information present +in figure 1 into a local surface geometry describing explicitly the various compact and non- +compact divisors present in the Calabi-Yau along with the intersections of these divisors. +The surface geometry associated to figure 1, i.e. the KK-theory, is +19 +1 +2 +2l +h+f-� xi +N0 +N1 +1 +N2 +· · · +N8 +N3 +10 +N9 +f-x-y +f +e-x2 +f +x2-x3 +f +x8-x9 +f +2 +x9, +x10 +f-x, y +l +x-y-e1 +f-x2 +f-e11 +x9-x10 +f +e +e +e +e +e +e +e +e +e +e +, +(A.2) +where we used the notation e1 and e11 for the (−1) curves that can be flopped to decouple +hyper-multiplets – in accordance with the notation in figure 1. The figure shows the compact +surfaces Si having been represented as Hirzebruch surfaces ib +d where the subscript d is the +degree of the Hirzebruch surface and the superscript b is the number of additional blowups +performed on it. +An edge between two surfaces denotes an intersection between the two +surfaces and the labels on the edges describe the curves in the two surfaces that are glued at +the locus of the intersection. Multiple edges are denoted by a number between the edge. A +curve e inside a surface denotes the base of the corresponding P1 fibration. This is a compact +curve for the Hirzebruch surfaces and a non-compact curve for the non-compact surfaces +Ni. A curve f inside a surface denotes the fiber of the corresponding P1 fibration. This +is a compact curve for both compact and non-compact surfaces. The curves xi, x, y denote +various blowups, and we finally we have defined the curve h := e + df. Moving forward we +denote e curve of surface Si as ei and f curve of surface Si as fi. +– 66 – + +After a single flop (of the curve e11) the surface geometry can be expressed as +110 +0 +21 +2h +e+2f-� xi +N0 +N1 +N2 +· · · +N8 +N2 +10 +N9 +f-x-y +f +e-x1-x2 +f +x2-x3 +f +x8-x9 +f +2 +x9, +x10 +f-x, y +f +x-y +x1-x2 +f +x9-x10 +f +e +e +e +e +e +e +e +e +e +e +. +(A.3) +The non-compact surfaces Ni form the Dynkin diagram of the affine Lie algebra so(20)(1) +which is associated to the fact that this is actually a 6d SCFT with so(20) flavor symmetry +compactified on a circle. +Model 2. +This model is obtained by blowing down the curve f1 − x1. This means that we +first flop f1 − x1 and then taking the volume of the flopped curve infinity. This effectively +decouples a hypermultiplet. The affect of this blowdown is that a P1 fibered non-compact +surface intersecting f1 − x1 does not remain P1 fibered anymore. In (A.3), N1 is the only +such non-compact surface since +(f1 − x1) · N1 = (f1 − x1) ·S1 (x1 − x2) = 1 , +(A.4) +where the intersection number (f1−x1)·N1 is in the Calabi-Yau threefold and the intersection +number (f1 − x1) ·S1 (x1 − x2) is in the surface S1. +The above intersection number is a +consequence of the facts that f1 ·S1 xi = xj ·S1 xi = 0 and xi ·S1 xi = −1. Later we will also +use ei ·Si ei = −di, where di is the degree of the Hirzebruch surface Si. +– 67 – + +After the blowdown we obtain the surface geometry +19 +1 +21 +2h +h+f-� xi +N0 +N2 +· · · +N8 +N2 +10 +N9 +f-x-y +f +e-x2 +f +x2-x3 +f +x8-x9 +f +2 +x9, +x10 +f-x, y +f +x-y +x9-x10 +f +e +e +e +e +e +e +e +e +, +(A.5) +where we have only kept the P1 fibered non-compact surfaces. Taking the gluings into account, +we see that all compact curves are (possibly non-positive) linear combinations of the curves +ei, fi and the blowups xi. Additionally due to the gluing between S1 and S2, we can express +e1 as a linear combination of10 the e2, fi and xi. +We have the identifications +C1 = e2, +C2 = f2, +C3 = f1, +C4 = xi , +(A.6) +where we can choose any xi because the charges under gauge and flavor centers are same for +all xi. +Let us now compute the charges. The charge qi(C) of a compact curve C under U(1) +gauge group associated to Si is computed as +qi(C) = −C · Si , +(A.7) +which for a genus zero curve is 2 + C ·Si C if C lives in Si. This implies +q2(C1) = 1, +q2(C2) = 2, +q1(C3) = 2, +q1(C4) = 1 . +(A.8) +On the other hand, if C is in surface Sj then C · Si is computed as C ·Sj Cj,i where Cj,i is the +gluing curve in Sj to Si. This implies +q1(C1) = 0, +q1(C2) = −2, +q2(C3) = −1, +q2(C4) = −1 . +(A.9) +10Note that the labels 1 and 2 are not interchangeable here. We cannot write e2 in terms of e1, fi and xi. +– 68 – + +Additionally, we have +− C1 · Ni = δi,10 . +(A.10) +That is, C1 intersects the non-compact surfaces along the cospinor node N10 and hence it has +the same charges under the center of Spin(20) as the cospinor representation of Spin(20). +Similarly, since C2 has no non-trivial intersections with any Ni, it does not contribute any +non-trivial charge under the center of Spin(20). On the other hand, we have +− C3 · Ni = δi,0 +(A.11) +implying that it has same charges under the center of Spin(20) as the vector representation +of Spin(20). Finally, for any xi the reader can compute that it has same charges under the +center of Spin(20) as the vector representation of Spin(20). For example choosing C4 = x9, +we have +− C4 · Ni = δi,9 + δi,10 , +(A.12) +which means it transforms both under ZS +2 and ZC +2 subgroups of the center of Spin(20). This +reproduces the charges claimed in (5.48). +Model 5. +Its surface geometry can be obtained from the surface geometry for model 2 by +blowing down the curve f − x2, resulting in the surface geometry +18 +2 +21 +2h +h-� xi +N0 +N3 +· · · +N8 +N2 +10 +N9 +f-x-y +f +e +f +x3-x4 +f +x8-x9 +f +2 +x9, +x10 +f-x, y +f +x-y +x9-x10 +f +e +e +e +e +e +e +(A.13) +Now N0 generates the su(2) ⊂ f and the other Ni generate the so(16) ⊂ f. +We have the same identifications as in (A.6). The gauge charges are the same as for +model 2. The flavor center charge of C1 is the same as spinor of Spin(16) as it intersects +the spinor node N10. The flavor center charge of C2 is trivial as it does not intersect any +Ni. The flavor center charge of C3 is the same as fundamental of SU(2) as it intersects N0. +– 69 – + +Finally, the flavor center charge of C4 is the same for all xi and can be easily seen to be that +for vector of Spin(16) by choosing C4 = x3. +Model 9. +Its surface geometry can be obtained from the surface geometry for model 5 by +blowing down the curve f − x3, resulting in the surface geometry +17 +1 +21 +2h +h-� xi +N1 +0 +N4 +· · · +N8 +N2 +10 +N9 +f-x-y +f +e +f-x +x4-x5 +f +x8-x9 +f +2 +x9, +x10 +f-x, y +f +x-y +x9-x10 +f +e +e +e +e +e +e +(A.14) +The blowdown process causes N3 and N0 to become non-P1 fibered non-compact surfaces. N3 +intersects the P1 fibers of remaining P1 fibered non-compact surfaces, and hence cannot give +rise to an independent u(1) flavor. However, N0 does not intersect the P1 fibers of remaining +P1 fibered non-compact surfaces and generates a u(1) factor in the flavor symmetry algebra +f. It should be noted that the curves f and x of N0 both have infinite volume, but their +difference f − x has finite volume. The other Ni shown above generate the so(14) ⊂ f. +We have the same identifications as in (A.6). The gauge charges are the same as for +the previous models. The flavor center charge of C1 is the same as cospinor of Spin(14) +as it intersects the cospinor node N10. The flavor center charge of C2 is trivial as it does +not intersect any Ni. The curve C3 carries charge −1 under U(1) flavor as it intersects N0. +Finally, the flavor center charge of C4 is the same for all xi and can be easily seen to be that +for vector of Spin(14) by choosing C4 = x4. +– 70 – + +Model 12. +Its surface geometry can be obtained from the surface geometry for model 5 by +blowing down curves x9, x10, resulting in the surface geometry +16 +2 +22 +h +h-� xi +N0 +N3 +· · · +N7 +N9 +e +f +x3-x4 +f +x7-x8 +f +e +f +e +e +(A.15) +We have the same identifications as in (A.6). The gauge charges are modified for curves living +in S2. The flavor center charge of C1 is trivial as its intersection number N9 is even. The +flavor center charge of C2 is the same as fundamental of SU(2) corresponding to N9 as the +intersection number of C2 with N9 is odd. Similarly, The flavor center charge of C3 is the +same as fundamental of SU(2) corresponding to N0. Finally, the flavor center charge of C4 +is the same for all xi and can be easily seen to be that of fundamental of SU(6) by choosing +C4 = x3. +Model 26. +Its surface geometry can be obtained from the surface geometry for model 9 by +performing some blowdowns of both types of curves f − xi and xi, resulting in the surface +geometry +14 +1 +21 +h+f +e-� xi +N3 +0 +N6 +· · · +N8 +N1 +9 +h +f − � xi +x6-x7 +f +x8-x9 +f +e +x +e +e +x9 +f-x +e +e +(A.16) +N0 generates a u(1) factor in the flavor symmetry algebra f. Its curves f and xi are non- +compact, but f − � xi is compact. 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Zhong, 3d N = 4 mirror symmetry with 1-form +symmetry, . +– 77 – + diff --git a/RtE0T4oBgHgl3EQfUQDk/content/tmp_files/load_file.txt b/RtE0T4oBgHgl3EQfUQDk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9471abaf1c1d67439e0342cc52c28c5b5e8f8218 --- /dev/null +++ b/RtE0T4oBgHgl3EQfUQDk/content/tmp_files/load_file.txt @@ -0,0 +1,2284 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf,len=2283 +page_content='Generalized Symmetries and Anomalies of 3d N = 4 SCFTs Lakshya Bhardwaj1, Mathew Bullimore2, Andrea E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Ferrari2, Sakura Sch¨afer-Nameki1 1 Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, United Kingdom 2 Department of Mathematical Sciences, Durham University, Upper Mountjoy, Stockton Road, Durham, DH1 3LE, United Kingdom Abstract: We study generalized global symmetries and their ’t Hooft anomalies in 3d N = 4 superconformal field theories (SCFTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Following some general considerations, we focus on good quiver gauge theories, comprised of balanced unitary nodes and unbalanced unitary and special unitary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' While the global form of the Higgs branch symmetry group may be determined from the UV Lagrangian, the global form of Coulomb branch symmetry groups and associated mixed ’t Hooft anomalies are more subtle due to potential symmetry enhancement in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We describe how Coulomb branch symmetry groups and their mixed ’t Hooft anomalies can be deduced from the UV Lagrangian by studying center charges of various types of monopole operators, providing a concrete and unambiguous way to implement ’t Hooft anomaly matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The final expression for the symmetry group and ’t Hooft anomalies has a concise form that can be easily read off from the quiver data, specifically from the positions of the unbalanced and flavor nodes with respect to the positions of the balanced nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We provide consistency checks by applying our method to compute symmetry groups of 3d N = 4 theories corresponding to magnetic quivers of 4d Class S theories and 5d SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We are able to match these results against the flavor symmetry groups of the 4d and 5d theories computed using independent methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Another strong consistency check is provided by comparing symmetry groups and anomalies of two theories related by 3d mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='02249v1 [hep-th] 5 Jan 2023 Contents 1 Introduction 1 2 Generalized Symmetries of 3d N = 4 Theories 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 BPS Defects 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 A- and B-type Symmetries 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 0-form Symmetry 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 1-form Symmetry 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 2-group Symmetry 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 Solitonic defects 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4 ’t Hooft Anomalies 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5 Discrete Gauging 18 3 Warmup: T[SU(n)] and its Gaugings 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 T[SU(2)] and its Gaugings 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 T[SU(2)] 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 SU(2)H Gauging 22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 SO(3)H Gauging 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4 U(2) Gauging 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 T[SU(n)] and its Gaugings 28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 T[SU(n)] 28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 SU(n)H Gauging 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 Other su(n)H Gaugings 32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4 U(n) Gauging 33 4 General Symmetry and Anomaly Analysis for 3d N = 4 SCFTs 34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 Symmetries 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 Anomaly 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 Other Gauge Groups 37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4 Including Flavors 39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5 Special Case 1: Single Special Unitary Node 39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6 Special Case 2: Single Unbalanced Unitary Node 41 5 Consistency Checks 41 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 Class S 42 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 General Matching 42 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 Examples 44 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 5d SCFTs 49 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 Flavor Symmetry Groups from MQs 49 – i – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 Flavor Symmetry Groups from String Theory Constructions 54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 3d Mirror Symmetry 57 6 Some Generalizations 59 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 N = 2 Gauging of T[SU(n)] 59 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 T[SU(2)]/ZC 2 and Its Gaugings 61 A Geometric Computations for 5d SCFTs 64 1 Introduction Gauge theories in three spacetime dimensions are extremely interesting to study from a the- oretical viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since the gauge coupling has positive mass dimension, any gauge theory can be given an ultraviolet (UV) complete definition, but in the infrared (IR) the effective gauge coupling becomes strong, opening up the possibility of interesting strong coupling be- haviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The case of 3d gauge theories with eight supercharges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' N = 4 supersymmetry, has been very well studied in this context, where it is known that with enough matter the gauge theory flows in the IR to a 3d N = 4 superconformal field theory (SCFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There are many interesting non-perturbative phenomena that arise in the context of N = 4 supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Of these, the most well-known phenomenon is that of 3d mirror symmetry [1–4] which relates two different 3d N = 4 gauge theories such that the corresponding IR 3d N = 4 SCFTs are same, up to the exchange of Coulomb and Higgs branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Arguably the most interesting aspect of 3d N = 4 supersymmetric gauge theories and mirror symmetry is symmetry enhancement in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor symmetries of 3d N = 4 gauge theories arise from hyper-K¨ahler isometries of the Higgs and Coulomb branch moduli spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' While the Higgs branch and associated symmetries may be understood classically, the Coulomb branch receives 1-loop and non-perturbative corrections and the hyper-K¨ahler metric depends on the gauge coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This allows for the emergence of additional hyper- K¨ahler isometries and associated symmetry enhancement on the Coulomb branch in the IR SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This phenomenon plays a fundamental role in mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This symmetry enhancement was systematically explained in [5], using techniques developed in [6–8] based on the study of monopole operators [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In this paper, we extend the discussion of symmetries in 3d N = 4 supersymmetric gauge theories to include generalized symmetries [11], including global aspects of traditional 0-form symmetries, 1-form symmetries, 2-groups symmetries and discrete ’t Hooft anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A key result is to identify the monopole operators in the UV gauge theory that allow a determination of the global form of the IR Coulomb branch 0-form symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The result matches the global form of the Higgs branch 0-form symmetry group of the mirror UV – 1 – gauge theory, which can be computed classically from the matter content pf the mirror gauge theory without considering its monopole operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Once the global form of the IR Coulomb 0-form symmetry group is known, an impor- tant question is to determine the ’t Hooft anomalies of the Coulomb 0-form symmetry with the Higgs 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In three space-time dimensions, such anomalies are necessar- ily discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The general structure of such anomalies for 3d gauge theories was explored in our previous work [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This captured the information about the anomaly in terms of flavor charges carried by mixed flavor-gauge monopole operators, which are in general non-genuine local operators that arise at the end points of flavor-gauge vortex line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In N = 4 supersymmetric gauge theories, there exist BPS configurations of flavor-gauge monopole op- erators sitting at the ends of flavor-gauge vortex lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These configurations thus descend to configurations of local operators sitting at the ends of line operators in the corresponding IR N = 4 SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' During the flow the flavor charges of these non-genuine local operators get mixed, and the mixing can be deduced using the methods of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Finally, reversing the logic of [12], one can use the information about the charges of these non-genuine local operators to deduce the ’t Hooft anomaly of the IR SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This provides a concrete and unambiguous way of implementing ’t Hooft anomaly matching from UV 3d N = 4 gauge theories to IR 3d N = 4 SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In a similar fashion, we also determine the ’t Hooft anomalies of the IR Coulomb 0- form symmetry with 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The requisite operators are now what were dubbed fractional gauge monopole operators in [12], which are non-genuine local operators living at the ends of topological line operators generating the 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For 3d N = 4 gauge theories, there exist BPS configurations of fractional gauge monopole operators living at the ends of topological line operators1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These configurations survive in the IR SCFT and the charges of such non-genuine local operators under IR Coulomb symmetry determine the precise form of the mixed ’t Hooft anomaly between the Coulomb 0-form symmetry and the 1-form symmetry of the IR 3d N = 4 SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' While this work was being written, we received [13] which also used the analysis of [12] to obtain some of the results appearing in sections 3 and 6 of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The analysis of this paper also opens the door for the determination of symmetries and anomalies of higher-dimensional (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' in d > 3) SCFTs with at least eight supercharges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This can be done by applying the analysis of this paper to 3d N = 4 magnetic quivers (MQs) associated to these higher-dimensional SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' MQs are 3d N = 4 gauge theories whose IR behaviour captures information about the Higgs branch of vacua of the corresponding higher- dimensional SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' MQs have been a subject of much interest and exploration recently [14–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In particular they have been instrumental in studying Higgs branches for 4d N = 2 and 5d N = 1 SCFTs, which are otherwise more difficult to access due to quantum corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In some instances, we expect the symmetries of the 4d or 5d SCFT and that of its 3d MQ theories to agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is in particular the case, when the higher-dimensional theory only 1Note that a topological operator automatically preserves all supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 2 – exhibits 0-form symmetries, which then should then agree with the 0-form symmetries of the MQ theory2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We test our proposal by computing the global 0-form symmetries of MQs, and compare them to the global form of flavor symmetries of 4d class S theories and 5d SCFTs, and find agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This paper opens up many interesting avenues to explore in the connection be- tween generalized symmetries and their ’t Hooft anomalies and SCFTs (in particular with 8 supercharges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As mentioned above, 0-form global symmetries act by hyper-K¨ahler isometries of Higgs and Coulomb branch moduli spaces of vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A natural question is therefore whether there exists such a geometric realization for generalised symmetries such as 1-form symmetries and 2-groups and their ’t Hooft anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In forthcoming work [44], we will show that such a realization may be found in the algebraic setting by promoting the Higgs and Coulomb branch moduli spaces to moduli stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These stacky enhancements of moduli spaces keep track of unbroken discrete gauge symmetry when flowing to the IR at points on the underlying moduli space and carry actions of generalized symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Moreover, the ’t Hooft anomalies for generalised symmetries considered in this paper may be understood geometrically in terms of equivariance properties of distinguished line bundles on these moduli stacks associated to half-BPS line operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Another natural generalization in light of the fact that we consider invertible symmetries in 3d, is the extension to non-invertible symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is a multitude of realizations now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Most relevant for the field theoretic approach that was the focus in this paper are the following constructions, which have direct 3d realizations [45–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Non-invertible symmetries in 3d are of course very well explored in the context of modular tensor categories, however here the interesting question is related to the interplay between non-invertible symmetries and superconformal symmetry in 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One construction, which relies on the presence of mixed anomalies has been explored in 3d in [45, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' To explore these in full it will be useful to characterize systematically the symmetry topological field theories for SCFTs in 3d, as started in [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Finally one can extend the considerations of this paper involving mostly continuous 0- form symmetries to also include discrete 0-form symmetries and their associated ’t Hooft anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The two types of symmetries in general combine to form a disconnected 0-form symmetry (Lie) group, which when combined with 1-form symmetries generally gives rise to disconnected 2-group symmetries introduced recently in [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In section 2, we provide a discussion of supersymmetric defect operators and generalized symmetries of A/B-type (Coulomb/Higgs in the gauge theory setting) and their ’t Hooft anomalies in 3d N = 4 theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is an application of our general results in [12] to this supersymmetric setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2If the higher dimensional theory has higher-form symmetries, then these can in principle contribute to the 0-form symmetry of the 3d MQ theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 3 – In section 3, we study the simplest example, in great detail, where one can apply the considerations of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This concerns T[SU(n)] theories and related theories that can be obtained by gauging Higgs 0-form symmetry of T[SU(n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In section 4, we generalize the methods employed in the previous section 3 to study a large class of 3d N = 4 SCFTs that can be obtained in the IR of good 3d N = 4 quiver gauge theories composed of balanced unitary and unbalanced unitary and special unitary gauge groups along with matter hypermultiplets transforming in fundamental and bifundamental representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We describe how the Coulomb 0-form symmetry groups and its mixed ’t Hooft anomalies with 1-form and Higgs 0-forms symmetries can be obtained easily by a visual analysis of the UV quiver and noting the placement of unbalanced and flavor nodes with respect to the positions of balanced nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In section 5, we present a variety of consistency checks of our general results of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We check that the IR Coulomb 0-form symmetry groups of magnetic quivers of Class S theories and 5d SCFTs match the flavor symmetry groups of these higher dimensional SCFTs computed via other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We also check that the IR Coulomb 0-form symmetry group matches the Higgs 0-form symmetry group of the 3d mirror gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In the final section 6, we study a few interesting theories that lie outside the general class of theories studied in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Our methods presented in section 4 can be easily generalized to include such theories and lead to many interesting phenomena not observed in the class of theories studied in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These include pure ’t Hooft anomalies for 1-form symmetry, the existence of 2-group symmetries in 3d N = 4 SCFTs, and mixed ’t Hooft anomalies between 2-group and 0-form symmetries of 3d N = 4 SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The latter two phenomena are exhibited by the 3d N = 4 SCFT called T[SU(2)]/ZC 2 that can be obtained from T[SU(2)] by gauging a Z2 subgroup of SO(3)C Coulomb 0-form symmetry of T[SU(2)], and can be obtained as the IR SCFT corresponding to 3d N = 4 SQED with U(1) gauge group and 2 hypermultiplets of charge 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Appendix A provides details on the computation of global forms of flavor symmetry groups of 5d SCFTs from Calabi-Yau threefold singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2 Generalized Symmetries of 3d N = 4 Theories In this section, we consider general aspects of invertible generalized symmetries in 3d N = 4 supersymmetric theories, including 0-form symmetries, 1-form symmetries and 2-group symmetries as well as their ’t Hooft anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For ordinary 0-form symmetries, we distinguish between R-symmetries and flavor symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We focus on flavor symmetries, which commute with all of the supercharges, and consider possible 2-group symmetries that combine flavor symmetries and 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We will introduce two types of such symmetries called “A-type” and “B-type” depending on which class of BPS operators are charged under them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For continuous 0-form symmetries, this corresponds to the known the classification of supermultiplets that conserved currents or background gauge fields for continuous symmetries may transform in, or equivalently central – 4 – extensions of the supersymmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' However, we explain how this classification can be applied more broadly to both finite and continuous symmetry groups, in addition to 1-form and 2-group symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 These symmetries may have various ’t Hooft anomalies, which we study using the tech- niques introduced in our previous paper [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In particular, we consider “A-type” and “B- type” BPS solitonic local operators and line defects that source background fields for the above symmetries and explain how their properties capture different types of ’t Hooft anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We also explain how gauging discrete symmetries interchanges A-type and B-type symmetries in a manner compatible with such ’t Hooft anomalies and how this leads to examples of mirror symmetry involving generalised symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Our primary example throughout this section will be standard 3d N = 4 supersymmetric gauge theories built from vectormultiplets and hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In such case, the A-type and B-type symmetries are associated to Coulomb and Higgs branch geometry respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It is also possible to consider gauge theories with less supersymmetry that flow to N = 4 supersymmetry in the IR [57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In these cases, the identification of the A-type and B-type symmetries from a UV perspective is more intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 BPS Defects We begin with a discussion of BPS local operators, line defects and junctions that will play a role in the classification of flavor symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' First recall that a theory with 3d N = 4 supersymmetry has R-symmetry algebra so(4) ∼= su(2) ⊕ su(2) and supercharges QA ˙A α where α is a euclidean space-time spinor index and the indices A, ˙A denote the spinor representation of the two factors of the R-symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Local Operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The half-BPS genuine local operators come in two types: A-type: annihilated by the four supercharges QA ˙+ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' B-type: annihilated by the four supercharges Q+ ˙A α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The A-type operators are constructed from the bottom scalar components of vectormultiplets and twisted hypermultiplets, while the B-type operators are constructed from the bottom scalar components of hypermultiplets and twisted vectormultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This classification into A-type and B-type applies equally well to non-genuine twisted sector local operators attached to a topological line defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The two sets of half-BPS genuine local operators generate two chiral rings CA,CB whose spectra define complex affine moduli spaces XA := Spec(CA) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1) XB := Spec(CB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2) 3This does not preclude the existence of additional discrete global symmetries that are not of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Examples of such symmetries include outer automorphisms of gauge groups such as charge conjugation or automorphisms of quiver diagrams, and anomalous 1-form symmetries that arise when coupling to a 3d TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 5 – In a standard supersymmetric gauge theory constructed from vectormultiplets and hyper- multiplets, they coincide with the Coulomb and Higgs branch respectively, in the absence of resolution or deformation parameters, viewed as complex algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We might consider the possibility that there are local operators annihilated by all of the supercharges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such operators are necessarily topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We will assume that there is a unique (upto multiplication by a complex number) such topological local operator, namely the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is tantamount to the statement that the theory is irreducible or equivalently that there are no 2-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In the opposite direction, we may have occasion to consider more general quarter-BPS operators annihilated by two supercharges Q+ ˙+ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' They are half-BPS for the 3d N = 2 supersymmetry algebra generated by Q+ ˙+ α , Q− ˙− α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Line Operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We consider half-BPS line defects along the x3-axis preserving a 1d N = 4 supersymmetric quantum mechanics sub-algebra of the 3d N = 4 supersymmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such line operators were first introduced in supersymmetric gauge theories in [60] and have been further studied in [61–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There are two classes of half-BPS lines: A-type: annihilated by four supercharges QA ˙+ + , QA ˙− − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' B-type: annihilated by four supercharges Q+ ˙A + , Q− ˙A − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The line defects can be described uniformly by consistent couplings to 1d N = 4 supersym- metric quantum mechanics with super-multiplets obtained by dimensional reduction from 2d N = (2, 2) and N = (0, 4) supersymmetry respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Examples include B-type Wilson lines for dynamical vectormultiplets and A-type Wilson lines for dynamical twisted vector- multiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We note that this classification applies equally well to half-BPS twisted sector line defects that are attached to a topological surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A special case is line defects annihilated by all of the supercharges, which are simultane- ously A-type and B-type and therefore necessarily topological line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such line defects are normally considered as generators of 1-form symmetries, but may also be charged under them in the presence of ’t Hooft anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This situation may arise when coupling to a general 3d TQFT in a way that preserves N = 4 supersymmetry but does not arise in the- ories constructed from standard supermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Incorporating such topological line defects as charged objects will require a refinement of the classification of symmetries presented here and some examples are presented in subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In the opposite direction, we may have occasion to consider more general quarter-BPS line defects preserving the common pair of supercharges Q+ ˙+ + , Q− ˙− − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' They can be regarded as half-BPS line defects for the 3d N = 2 supersymmetry algebra generated by the supercharges Q+ ˙+ α , Q− ˙− α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Finally we consider various local junction operators between pairs of line de- fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We consider two classes of quarter-BPS junctions between pairs of A-type and B-type – 6 – lines and preserve two supercharges lying in the intersections of the two sets of four super- charges preserved by genuine local operators and line defects: A-type: annihilated by two supercharges QA ˙+ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' B-type: annihilated by two supercharges Q+ ˙A + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It is also possible to consider local junction operators between a half-BPS A-type and a B-type line defect, or alternatively between a pair of quarter-BPS line defects, which both preserve the single supercharge Q++ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Comment on relation to topological twist The above classification of BPS operators is related but distinct to the classification of operators in topological twists of 3d N = 4 supersymmetry, where A-type and B-type operators are defined as those in the cohomology of the nilpotent supercharges QA := Q+ ˙+ + + Q− ˙+ − (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3) QB := Q+ ˙+ + + Q+ ˙− − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4) Correspondingly, we are interested only in genuine symmetries generated by extended opera- tors that are topological in the full 3d N = 4 theory, not merely after performing a topological twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 A- and B-type Symmetries We now consider the classification of invertible flavor symmetries in 3d N = 4 theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As mentioned above, we assume that the theory is irreducible and therefore restrict ourselves to at most 2-group symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In particular, there is a unique genuine local operator that is simultaneously A-type and B-type, which is the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The proposal is then that the most general flavor symmetry is a product of A-type and B-type 2-group symmetries associated to the above classification of BPS defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 0-form Symmetry For continuous 0-form symmetries, it is well known that the flavor symmetry takes the form of a product FA × FB for compact Lie groups FA, FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The two factors are known as A-type and B-type symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' At the level of the associated Lie algebra fA ⊕ fB, this decomposition may be understood from the allowed central extensions of the 3d N = 4 supersymmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This admits a pair of central charges ZAB, Z ˙A ˙B transforming in the adjoint representations of the two su(2) R-symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The central charges are proportional to the generators of the A-type and B-type symmetries respectively with coefficients given by scalar fields σAB, σ ˙A ˙B in vector- multiplets and twisted vectormultiplets respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In summary, A-type symmetries couple to vectormultiplets and B-type symmetries to twisted vectormultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 7 – However, in order to provide a definition of the flavor symmetry group FA × FB, which also applies to discrete symmetries, and in addition to formulate obstruction classes that appear in ’t Hooft anomalies for these symmetries, it is convenient to define symmetries starting from the BPS operators on which they act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor symmetry groups FA, FB are defined as the maximal compact Lie groups with Lie algebras fA, fB that act faithfully on A-type and B-type genuine half- BPS local operators respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This definition also applies when fA, fB are trivial, in which case the flavor symmetry groups are discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These symmetry groups (or rather their complexification in the continuous case) will act by complex isometries on the moduli spaces XA, XB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In the construction of 2-group symmetries involving these flavor symmetries and their ’t Hooft anomalies, we will also need to consider non-genuine local operators that sit at the junctions between half-BPS line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us consider A-type or B-type half-BPS line defects that preserve the whole symmetry group FA or FB respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such line defects may then end on A-type or B-type quarter- BPS local operators that transform in representations of central extensions of FA, FB by discrete abelian groups, that are not representations of FA, FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It is convenient to write down the short exact sequences 0 −→ ZA −→ FA −→ FA −→ 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5) 0 −→ ZB −→ FB −→ FB −→ 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6) where ZA, ZB are finite abelian groups and FA, FB denotes the extended symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Equivalently, we have the quotients FA = FA/ZA, FB = FB/ZB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In summary, local operators at the end of line defects may be charged under ZA, ZB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There are associated obstruction classes wA 2 ∈ H2(BFA, ZA) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='7) wB 2 ∈ H2(BFB, ZB) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='8) for lifting FA, FB bundles to FA, FB bundles, which play an important role in the description of 2-groups and ’t Hooft anomalies involving these symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In particular, introducing background fields BA 1 , BB 1 : M → BFA, BFB, there are associated obstruction classes on spacetime via pull-back (BA 1 )∗wA 2 , (BB 1 )∗wB 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In what follows, we will often abuse notation and denote these spacetime obstruction classes also by wA 2 , wB 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us consider standard supersymmetric gauge theories constructed from vectormultiplets and hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In such cases, it is appropriate to replace the monikers A/B by C/H, which refer to Coulomb and Higgs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The B-type symmetry FH acts faithfully on gauge-invariant combinations of hypermul- tiplet fields, while the central extension FH is constructed by examining the charges of non- gauge invariant combinations of hypermultiplet fields attached to B-type half-BPS Wilson lines for the dynamical vectormultiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 8 – This is conveniently captured by introducing the structure group S, which captures the combination of gauge and B-type flavor symmetries acting faithfully on all supermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In other words, the bundles for S correspond to the most general combination of gauge and B-type flavor symmetry bundles (transforming in dynamical and background vectormultiplets respectively) to which the theory may be consistently coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It takes the form S = G × FH E , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='9) where G denotes the gauge group, which we assume is connected, and E is a subgroup of the center Z(G × FH) of G × FH such that pH(E) = ZH where pH : Z(G) × Z(FH) → Z(FH) is the natural projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' On the other hand, the A-type symmetry group FC acts faithfully on genuine half-BPS monopole operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is the topological symmetry FC = � π1(G) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='10) which measures the topological class of the G-bundles on a sphere surrounding the monopole operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It may be continuous or discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Unlike the B-type symmetry group, this may undergo enhancement at an IR superconformal fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Determining the precise global form of the enhanced symmetry group is a major goal of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The gauge theory may couple to bundles for the structure group S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Correspondingly, there exist A-type half-BPS line defects corresponding to gauge-flavour vortex lines for the structure group labelled by a co-character φ : U(1) → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This may involve a fractional gauge vortex lines, which by definition are vortices associated to co-characters for the quotient group G = G/Zg , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='11) where Zg = pg(E) where pg : Z(G) × Z(FH) → Z(G) is the natural projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Note that the co-character associated to a fractional gauge vortex must be a co-character simultaneously for G and for S, and a general co-character for G does not satisfy this criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Gauge- flavour vortex line defects may end on monopole operators of fractional magnetic charge, which results in a short exact sequence 1 −→ ZC −→ FC −→ FC −→ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) where FC = � π1(G) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) and we identify ZC = � Zg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us note that in general 3d gauge theories it is necessary to include such topological symmetries as part of the structure group, thus extending the above discussed structure group S into an extended structure group �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus is due to the fact that monopole operators receive charges under gauge and flavor symmetries due to effective Chern-Simons levels, as discussed in our previous paper [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' However, with N = 4 supersymmetry and only standard supermultiplets, this extended structure group factorises as �S = S × FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 9 – Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A basic example to illustrate these points is supersymmetric QED with G = U(1) and N hypermultiplets of charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We assume without loss of generality that q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The hypermultiplets contain complex scalar fields Xj, Yj of charge q, −q transforming in the fundamental and anti-fundamental representations of the flavor symmetry algebra fH = su(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The B-type genuine local operators are the gauge-invariant combinations XiYj transforming in the adjoint representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The B-type flavor symmetry group is therefore FH = PSU(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There are B-type Wilson lines Wn labelled by an integer charge n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Consider the case where n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' If n is a multiple of the minimal charge q, the Wilson line may end on local operators consisting of homogeneous polynomials in Xj of degree m = n/q, which transform in the m-th symmetric power of the fundamental representation of su(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This includes representations of the central extension FH = SU(N) that are not representations of FH = PSU(N) forming a short exact sequence 1 −→ ZN −→ SU(N) −→ PSU(N) −→ 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='14) with ZH = ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The associated obstruction class may be denoted by wH 2 ∈ H2(X, ZN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is reflected in the structure group S = U(1) × SU(N) ZqN , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='15) where the denominator is generated by the central element (e2πi/qN, e2πi/N1N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The genuine A-type local operators correspond to half-BPS monopole operators labelled by a co-character m : U(1) → G, which is an integer magnetic charge m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Correspondingly, the A-type symmetry group is the topological symmetry FC = � π1(G) = U(1) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='16) whose charge measures the topological type of a G-bundle on a two-sphere surrounding a monopole operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' However, the theory may in general be coupled to bundles for the structure group S and therefore there exist A-type gauge-flavor vortex lines labelled by co-characters φ : U(1) → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' They are conveniently labelled by a pair of co-characters φ = (φg, φH) ∈ Z × Z≥0 with obstructions αg = φg mod qN (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='17) αH = φH mod N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='18) For this to define a consistent co-character of S, the obstructions must arise as projections αg = pg(α) and αH = pH(α) of a common obstruction α ∈ E, where pg, ph are projections from Z(G) × Z(FH) to Z(G), Z(FH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This requires αg mod N = αH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us summarise some examples that will play a role in what follows: – 10 – Consider pure fractional gauge vortex lines φ = (φg, 0), which requires φg is a multiple of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' If φg is a multiple of qN, this lifts to dynamical vortex for the gauge group and corresponds to a trivial line defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' They end on genuine A-type gauge monopole operators of magnetic charge m = φ/qN ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The remaining fractional gauge vortex lines, modulo dynamical vortices, are indexed by φg = nN with n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' , q − 1 and end on A-type monopoles of fractional magnetic charge n/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' More general gauge-flavour vortex lines φ = (φg, φH) may end on A-type gauge-flavour monopoles of fractional magnetic charge in multiples of 1/qN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In particular, the final bullet point means that we must introduce the qN-fold cover of the topological symmetry FC = � π1(G) ∼= U(1), which is an extension of the topological symmetry by ZC = ZqN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The associated obstruction class for background fields is wC 2 = cC 1 mod qN, where cC 1 denotes the first Chern class of a background FC = U(1) bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 1-form Symmetry Following the same philosophy, we define A/B-type 1-form symmetries by applying the recipe studied in [12], but restricted to half-BPS A/B-type line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Definition The construction begins by considering equivalence classes of A-type or B-type line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We say that two line defects L1, L2 are equivalent L1 ∼ L2 if there exists a non-trivial quarter-BPS junction of the appropriate type connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In other words, equivalence classes capture the half-BPS line defects that cannot be screened by quarter-BPS junctions of A-type or B-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The equivalence classes of A-type and B-type lines inherit the structure of abelian groups �ΓA, �ΓB from the OPE of parallel line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The A-type and B-type 1-form symmetries are defined as the Pontryagin dual groups ΓA := Hom(�ΓA, U(1)) ΓB := Hom(�ΓB, U(1)) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='19) such that these 1-form symmetries act on half-BPS lines via the natural pairings Γ×�Γ → U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Correspondingly, we can introduce ΓA, ΓB-valued 2-cochain backgrounds BA 2 , BB 2 for these 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' If the 1-form symmetries do not participate in 2-groups, the back- ground field are closed and define ΓA, ΓB-valued 2-cocycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In standard gauge theories built from vectormultiplets and hypermulti- plets, the A-type and B-type 1-form symmetries may be determined from the properties of vortex lines and dynamical Wilson lines respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The B-type symmetry arises from half-BPS Wilson lines in representations of the gauge group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In the absence of hypermultiplets, quarter-BPS junctions may only arise from – 11 – vectormultiplet fields in the adjoint representation of the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In this case, Wilson lines in representations R1, R2 are equivalent if and only if the central characters of the representations coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Therefore �ΓH = Hom(Z(G), U(1)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20) is the abelian group of central characters and the 1-form symmetry coincides with the centre of the gauge group ΓH = Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' More generally, incorporating hypermultiplet fields, the 1-form symmetry ΓH is the sub- group of the center of the gauge group that acts trivially on hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This can be formulated in terms of the structure group S = G × FH E , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='21) where the B-type 1-form symmetry may be identified with the intersection ΓH = Z(G) ∩ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This naturally forms a short exact sequence 1 −→ ΓH −→ E −→ ZH −→ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) which will play a role in the construction of 2-group symmetries below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' An A-type 1-form symmetry may arise in gauge theories with discrete or continuous but disconnected gauge groups, which we will not discuss here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We will instead explain below how A-type 1-form symmetries arise generally when gauging discrete B-type 0-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us again consider supersymmetric QED with G = U(1) and N hypermulti- plets of charge q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It is a standard result that this has a B-type 1-form symmetry ΓB = Zq as B-type Wilson lines Wn cannot be screened unless n is a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 2-group Symmetry The 0-form and 1-form symmetries defined above may combine to form A-type and B-type 2-group symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In order to define the 2-group symmetry structure, we will need to consider a more refined equivalence relation for line defects that takes into account the fact that junctions may transform in representations of central extensions of symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For further background on this perspective see [12, 64–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We first define another equivalence relation such that L1 ∼′ L2 if the two line operators admit quarter-BPS junctions of the appropriate type transforming in honest representations of FA, FB that are not charged under ZA, ZB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These equivalence classes form larger abelian groups � EA, � EB sitting in short exact se- quences 0 −→ � ZA −→ � EA −→ � ΓA −→ 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='23) 0 −→ � ZB −→ � EB −→ � ΓB −→ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='24) – 12 – The first terms in the sequence can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The quarter-BPS local operators screening line operators in equivalence classes corresponding to elements �zA ∈ � ZA ⊂ � EA, �zB ∈ � ZB ⊂ � EB transform in representations of FA, FB with charges �zA, �zB under ZA, ZB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Pontryagin dual exact sequences are 1 −→ ΓA −→ EA −→ ZA −→ 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='25) 1 −→ ΓB −→ EB −→ ZB −→ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='26) The 0-form symmetries FA, FB and 1-form symmetries ΓA, ΓB now combine into 2-groups whose Postnikov classes are given by ΘA = Bock(wA 2 ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='27) ΘB = Bock(wB 2 ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='28) using the appropriate Bockstein homomorphisms Bock : H2(X, ZA) → H3(X, ΓA) or Bock : H2(X, ZB) → H3(X, ΓB) associated to the above short exact sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' If the Postnikov classes are trivial the 2-group symmetry is a product of a 0-form and a 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We may then introduce backgrounds for the 2-group symmetry given by the EA, EB-valued combinations BA w = i(BA 2 ) + �wA 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='29) BB w = i(BB 2 ) + �wB 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='30) where i : ΓA, ΓB → EA, EB denotes the relevant inclusion maps and �wA 2 , �wB 2 are co-chain lifts of wA 2 , wB 2 under the projections p : EA, EB → ZA, ZB in the above short exact sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These combinations are closed by construction and define EA, EB-valued co-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' If the Postnikov class is trivial, we may work independently with closed backgrounds BA 2 , BB 2 and wA 2 , wB 2 for the 1-form and 0-form symmetries respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Consider a standard supersymmetric 3d N = 4 gauge theory built from ordinary vectormultiplets and hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The data determining the B-type 2-group is encoded in the structure group S = G × FH E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='31) In particular, we have already identified ZH = pH(E) and the 1-form symmetry ΓB = Z(G)∩E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The remaining ingredient is simply the identification EH = E, which forms the appropriate short exact sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us consider again supersymmetric QED with G = U(1) and N hypermulti- plets of charge q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Recall that there are B-type symmetry groups FH = PSU(N) and ΓH = Zq sitting in short exact sequences 1 −→ ZN −→ SU(N) −→ PSU(N) −→ 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='32) – 13 – and 1 −→ Zq −→ ZqN −→ ZN −→ 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='33) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is therefore a potential Postnikov class Θ = Bock(wH 2 ) where wH 2 is the obstruction class for the first sequence and Bock : H2(PSU(N), ZN) → H3(PSU(N), Zq) is the Bockstein homomorphism for the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Bockstein homomorphism may or may not be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In the former case, there is no 2-group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' An example of vanishing Bockstein is provided if the first sequence splits, which requires gcd(q, N) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A non- supersymmetric version of this example was considered already in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is also an A-type 0-form symmetry FA = U(1), which does not participate in a 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' However, we will show later that it has a mixed ’t Hooft anomaly with the above B-type 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 Solitonic defects The A-type and B-type local, line and junctions operators may induce background fields for flavor symmetries and correspond to solitonic defects in the terminology of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' More specifically they induce vortex and monopole configurations for background fields associated to flavor symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such defects play a crucial role in determining ’t Hooft anomalies from the spectrum of BPS charged objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The proposal is that A-type defects may source background fields for B-type flavor sym- metries and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We substantiate this claim for vortex and monopole backgrounds in the remainder of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For concreteness, let us first consider A-type line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The most general situation is that they induce a background field configuration for the B-type 2-group symmetry such that � D2 BB w = αB , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='34) where D2 denotes a small disk intersecting the line defect transversely and αB ∈ EB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We refer to this as a background vortex configuration for the 2-group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such line defects may end on A-type quarter-BPS local operators with the property that � S2 BB w = αB , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='35) where S2 is now a small 2-sphere surrounding the local operator and intersecting the line defect transversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We refer to this as a background monopole configuration for the 2- group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Entirely analogous statements hold with A-type and B-type symmetries and defects interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This reduces to simpler statements in special cases of individual 0-form and 1-form sym- metry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For example, an A-type line defect may induce a vortex background for a B-type 0-form symmetry such that � D2 wB 2 = αB , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='36) – 14 – where now αB ∈ ZB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Similarly, if there is a B-type 1-form symmetry that does not participate in a 2-group symmetry then an A-type line defect may induce a vortex background for the 1-form symmetry such that � D2 BB 2 = αB , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='37) where now αB ∈ ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Similar comments apply to local operators and monopole backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Again, entirely analogous statements fold with A-type and B-type symmetries and defects interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In a standard supersymmetric gauge theory, the B-type symmetry back- ground field configurations sourced by A-type line defects can be understood systematically in terms of the structure group S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since the gauge theory may be consistently coupled to S-bundles, A-type line defects include gauge-flavor vortex defects Vφ labelled by co-characters φ : U(1) → S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='38) Such A-type gauge-flavor vortex defects source background field configurations for the B-type 2-group symmetry such that � D2 BH 2 = αH (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='39) where αH ∈ E is the obstruction for lifting φ to a co-character for G × F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This reduces to corresponding simpler statements for individual 0-form and 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There are many special cases of interest and further examples are considered in the example below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In the opposite direction, B-type Wilson lines for a dynamical vectormultiplet source a background vortex configuration for the dual A-type topological symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is discussed for G = U(1) in the example below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Consider again supersymmetric QED with N hypermultiplets of charge q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For simplicity, we assume here that gcd(q, N) = 1 so that the 2-group structure is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We consider general A-type gauge-flavor vortex lines labelled by a co-character of the structure group φ : U(1) → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This can be conveniently labelled by a pair of co-characters φ = (φg, φH) ∈ Z × Z≥0 as discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We can then summarise the backgrounds that they induce as follows: Consider pure fractional gauge vortex lines φ = (φg, 0), which requires φg is a multiple of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' If φg is a multiple of qN, this lifts to dynamical vortex for the gauge group and corresponds to a trivial line defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The remaining fractional gauge vortex lines, modulo dynamical vortices, are indexed by φ = nN with n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' , q − 1 and source backgrounds for the B-type 1-form symmetry ΓH = Zq such that � D2 BH 2 = n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='40) – 15 – General vortex lines φ = (φg, φH) induce combinations of 0-form and 1-form symmetry backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Note that a pure flavor vortex φ = (0, φH) cannot induce an obstruction background wH 2 for the B-type 0-form symmetry since this would require that αH = φH mod N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In the opposite direction, the B-type Wilson lines Wn source a background for the A-type topological symmetry FC = U(1) such that � D2 cC 1 = n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='41) These statements will be utilised to derive mixed ’t Hooft anomalies below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4 ’t Hooft Anomalies We now consider the ’t Hooft anomalies captured by BPS operators considered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These are primarily mixed ’t Hooft anomalies between A-type and B-type symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The most general situation assuming potential A-type and B-type 2-group symmetries is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us consider A-type line defects that source background field configurations for the B-type 2-group symmetry labelled by elements αB ∈ EB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such line defects define equivalence classes in �EA and this provides a homomorphism �γ : � EB → EA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='42) In this situation there is a mixed ’t Hooft anomaly represented by the four-dimensional SPT phase A4 = � BA w ∪ γ(BB w ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='43) This construction may be performed exchanging A-type and B-type defects and symmetries and these constructions must be compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There are various simpler special cases that are worth considering: Let us assume there are 1-form symmetries ΓA, ΓB that do not participate in 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The A-type line defects may source backgrounds for a B-type 1-form symmetry labelled by elements αB ∈ ΓB and simultaneously charged under the A-type 1-form symmetry ΓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This determined a homomorphism γ : ΓB → �ΓA (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='44) and mixed ’t Hooft anomaly A4 = � BA 2 ∪ γ(BB 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='45) Consider an A-type 0-form symmetry FA and a B-type 1-form symmetry ΓB not par- ticipating in a 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The A-type line defects may source backgrounds for a B-type – 16 – 1-form symmetry labelled by elements αB ∈ ΓB and simultaneously end on A-type local operators charged under ZA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This determines a homomorphism γ : ΓB → � ZA (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='46) and mixed ’t Hooft anomaly A4 = � wA 2 ∪ γ(BB 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='47) Consider an A-type 0-form symmetry FA and a B-type 0-form symmetry FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The A-type line defects may source backgrounds wB 2 for a B-type 0-form symmetry labelled by elements αH ∈ ZB and simultaneously end on A-type local operators charged under ZA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This determines a homomorphism γ : ZB → � ZA (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='48) and mixed ’t Hooft anomaly A4 = � wA 2 ∪ γ(wB 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='49) Gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For a standard supersymmetric gauge theory with connected gauge group, there may be a mixed ’t Hooft anomaly between the A-type topological symmetry and the B-type 2-group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This may be determined, for example, by examining gauge-flavor vortex line defects that induce backgrounds for the B-type flavor symmetry and the fractional A-type topological charges of the monopoles on which they end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' An example is presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us consider again supersymmetric QED with N hypermultiplets of charge q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The symmetries are summarises as follows: An A-type topological symmetry FA = U(1) whose background field has an ZqN-valued obstruction class wC 2 = c1 mod qN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A B-type 2-group symmetry with FH = PSU(N) and ΓH = Zq with ZqN-valued background field BH w = NB2 + �wH 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The mixed ’t Hooft anomaly between these symmetries is derived from the fractional topolog- ical charges of the A-type local operators on which gauge-flavour vortex lines end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A marginal generalisation of the examples presented in [12] shows that this is represented by the 4d SPT phase A4 = exp �2πi qN � (cC 1 mod qN) ∪ BH w � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50) When gcd(q, N) = 1 and the 2-group structure is trivial, this simplifies to a sum of mixed anomalies for the individual B-type 0-form and 1-form symmetries A4 = exp �2πi q � (cC 1 mod q) ∪ BH 2 + 2πi N � (cC 1 mod N) ∪ wB 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='51) – 17 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5 Discrete Gauging In three dimensions, gauging a discrete abelian 0/1-form symmetry Γ group results in a Pontryagin dual 1/0-form symmetry group �Γ := Hom(A, U(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In the context of symmetries in theories with N = 4 supersymmetry these operations interchange A-type and B-type symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In summary: Gauging an A-type discrete abelian 0/1-form symmetry Γ results in a B-type Pontryagin dual 1/0-form symmetry �Γ := Hom(Γ, U(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Gauging an B-type discrete abelian 0/1-form symmetry Γ results in a A-type Pontryagin dual 1/0-form symmetry �Γ := Hom(Γ, U(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is compatible with our discussion of mixed ’t Hooft anomalies between A-type and B- type symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A slight generalisation of the above is that gauging a normal subgroup Γ ⊂ Γ′ of a 0-form symmetry results in a 1-form symmetry �Γ with a mixed anomaly with the remaining quotient group Γ′/Γ controlled by the extension class [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4 In theories with N = 4 supersymmetry and with the above identifications, this is always a mixed anomaly between A-type and B-type symmetries considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This provides a clean and general method to construct new examples of mirror symmetry that involved 1-form symmetries and their anomalies can be explicitly matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Examples are presented below and in the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' An example of this phenomenon arises in U(1) supersymmetric gauge theories, which have an A-type topological symmetry FC = U(1) under which genuine monopole oper- ators are charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Gauging a subgroup Zq ⊂ U(1) of the topological symmetry is equivalent to multiplying the charges of all hypermultiplet fields by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This results in a B-type 1-form symmetry ΓH = Zq due since a subgroup of the gauge group now acts trivially on all hyper- multiplet fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This B-type symmetry has a mixed anomaly with the remaining topological symmetry after gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As an example consider supersymmetric QED with N hypermultiplets of charge 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This has A-type topological symmetry FC = U(1) and B-type symmetry FH = PSU(N) with mixed ’t Hooft anomaly A4 = exp �2πi N � (cC 1 mod N) ∪ wH 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='52) We now gauge Zq ⊂ FC assuming gcd(q, N) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This results in a dual B-type 1-form sym- metry ΓH = Zq and an additional mixed anomaly with the remaining topological symmetry such that the total anomaly is A4 = exp �2πi q � (cC 1 mod q) ∪ BH 2 + 2πi N � (cC 1 mod N) ∪ wH 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='53) 4This conclusion holds even when Γ′ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 18 – This is indeed the anomaly of supersymmetric QED with N hypermultiplets of charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A slightly more intricate argument is required when gcd(q, N) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The mirror of supersymmetric QED with N hypermultiplets of charge q can therefore be obtained from the mirror of supersymmetric QED with N hypermultiplets of charge 1 by gauging a Zq symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This results in a circular quiver gauge theory with (N − 1) U(1) nodes and 1 Zq node, which has an A-type 1-form symmetry ΓA = Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In particular, when N = 1, the mirror theory is a Zq-quotient of a free hypermultiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 3 Warmup: T[SU(n)] and its Gaugings In this section, we study the simplest example, which is provided by 3d N = 4 SCFTs known as T[SU(n)] theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We study the global forms of flavor symmetry groups of T[SU(n)] along with their ’t Hooft anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We also study some other 3d N = 4 SCFTs closely related to T[SU(n)], in that they can be obtained by gauging (along with the addition of extra flavors for balancing purposes) the Higgs branch flavor symmetries of the UV unitary quiver theory whose IR fixed point is T[SU(n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 T[SU(2)] and its Gaugings Let us begin with T[SU(2)] and theories related to it by gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Some of the results on the symmetries and anomalies of the T[SU(2)] theory are known already in the literature [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We derive them here using the perspective of our earlier paper [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 T[SU(2)] The T[SU(2)] theory arises as an IR fixed point of the following 3d N = 4 Lagrangian theory U(1) [su(2)H] , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1) having a U(1) gauge group and 2 hypermultiplets of charge 1 that are rotated by an su(2)H flavor symmetry algebra5, where the subscript H indicates that the su(2)H symmetry acts non-trivially (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' is spontaneously broken) on the Higgs branch of vacua of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' B-Type 0-Form Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The genuine local operators charged under su(2)H arise from gauge invariant combinations of hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such gauge invariant combinations all form representations of the Lie group SO(3)H with Lie algebra su(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, the Higgs branch 0-form symmetry group is FH = SO(3)H = SU(2)H/Z2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2) where SU(2)H is the simply connected group with Lie algebra su(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 5Note that the flavor symmetry is not u(2)H = su(2)H ⊕ u(1)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One way to see it is to note that the u(1)H part acts in the same way as the u(1) gauge algebra and hence is absorbed into that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 19 – Another way of deducing the above 0-form symmetry group is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us put the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1) on a non-trivial compact 3-manifold M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charges of the vector and hypermultiplets allow us to turn on bundles for the structure group S = U(1) × SU(2)H Z2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3) where U(1) is the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Z2 in the denominator is the diagonal combination of the Z2 element in U(1) and the non-trivial element in the Z2 center of SU(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This diagonal combination leaves all vector and hyper multiplets invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus the flavor symmetry group associated to su(2)H is SO(3)H = SU(2)H/Z2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4) because, according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3), we can couple the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1) to non-trivial background bundles for SO(3)H, provided we turn on non-trivial bundles for U(1)/Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In more detail, let wH 2 be the obstruction class for lifting the SO(3)H bundle to an SU(2)H bundle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' the second Stiefel-Whitney class of the SO(3)H bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The obstruction class for lifting the U(1)/Z2 bundle to a U(1) bundle can then be written as c1 (mod 2), where c1 is the first Chern class for the U(1)/Z2 bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The group (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3) then requires that wH 2 = c1 (mod 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5) That is, SO(3)H bundle can be lifted to SU(2)H bundle if and only if U(1)/Z2 bundle can be lifted to U(1) bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Once an SO(3)H bundle is specified, the gauge theory sums over all U(1)/Z2 bundles satisfying the constraint (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A-Type 0-Form Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In addition of the SO(3)H 0-form symmetry, the Lagrangian theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1) admits a magnetic FUV C = U(1)C (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6) 0-form symmetry whose associated topological operators are exp � iα � F � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='7) parametrized by α ∈ [0, 2π), where F is the field strength of the U(1) gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The subscript C denotes that the U(1)C symmetry acts non-trivially on the Coulomb branch of vacua of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Indeed, the monopole operators, whose vacuum expectation values parametrize the Coulomb branch, are charged under this symmetry, with the charges valued in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It is well-known, from the analysis of [5], that at the level of Lie algebras, the u(1)C 0- form symmetry enhances in the IR to an su(2)C 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In particular, the Cartan of the simply connected Lie group SU(2)C with Lie algebra su(2)C is a double cover of U(1)C, or in other words we have an inclusion map U(1)C �→ SO(3)C = SU(2)C/Z2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='8) – 20 – which embeds U(1)C as the maximal torus of SO(3)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since the gauge monopole operators have integer charges under U(1)C in the UV, they descend to genuine local operators of the IR SCFT transforming in representations of SO(3)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus the Coulomb 0-form symmetry group of the IR SCFT is FIR C = SO(3)C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='9) In other words, at the level of Lie groups the U(1)C 0-form symmetry group enhances in the IR to SO(3)C 0-form symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Mixed 0-Form Symmetry Anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As is well-known, there is a mixed ’t Hooft anomaly between the SO(3)H and SO(3)C 0-form symmetries of T[SU(2)], see [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Here we describe how this ’t Hooft anomaly can be derived as a consequence of a mixed ’t Hooft anomaly between the SO(3)H and U(1)C 0-form symmetries in the UV gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For this purpose, we will use the analysis of [12] which described a general way of deducing ’t Hooft anomalies for any 3d gauge theory by computing center charges of flavor-gauge monopole operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The relevant monopole operator is associated to a co-character U(1) → S , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='10) where S is the structure group of the gauge theory appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3), with winding number half around the U(1) factor in the numerator of S and winding number half around the U(1) maximal torus of SU(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is an allowed co-character because of the Z2 quotient appearing in the denominator of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The mixed flavor-gauge monopole operator O associated to such a co-character is neces- sarily a non-genuine local operator of the gauge theory lying at the end of a vortex line defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From the methods of [12], we can compute that O carries a charge 1/2 under U(1)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A quick way to see this is to note that a purely gauge monopole operator associated to a co-character with winding number 1 around U(1) gauge group has charge 1 under U(1)C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' a purely flavor monopole operator associated to a co-character with winding number 1 around U(1) maximal torus of SU(2)H is uncharged under U(1)C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' and twice of the co-character associated to O is a product of such purely gauge and purely flavor co-characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As explained in [12], the half-integral charge under U(1)C of the monopole operator O is equivalent to a mixed ’t Hooft anomaly between the Coulomb and Higgs 0-form symmetry groups of the UV gauge theory AUV 4 = exp � πi � wH 2 ∪ � c1 � U(1)C � (mod 2) �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='11) where wH 2 denotes the Stiefel-Whitney class for the background SO(3)H bundle and c1 � U(1)C � denotes the first Chern class of the background U(1)C bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' After flowing to the IR, the monopole operator O descends to a non-genuine local operator of the T[SU(2)] theory that is now a purely flavor monopole operator (because there is no – 21 – gauge group in the IR SCFT) associated to a co-character having winding number 1 around the SO(3)H 0-form symmetry group of T[SU(2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Any flavor monopole operator is a non- genuine local operator living at the end of a flavor vortex line defect associated to the same co-character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor monopole operator O must transform in a representation of su(2)C which is not an allowed representation of SO(3)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is a straightforward consequence of the embedding (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Again using the analysis of [12], this fact is equivalent to a mixed ’t Hooft anomaly between the Coulomb and Higgs 0-form symmetry groups of the IR SCFT T[SU(2)] AIR 4 = exp � πi � wH 2 ∪ wC 2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) where wC 2 is the second Stiefel-Whitney class of the background SO(3)C bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One can think of the IR anomaly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) as being obtained from the UV anomaly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='11) by ’t Hooft anomaly matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The above analysis in terms of charges of flavor-monopole operators thus provides a precise and unambiguous way of performing such a ’t Hooft anomaly matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 SU(2)H Gauging Consider now the 3d N = 4 theory T[SU(2)] SU(2)H (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) obtained by gauging the su(2)H flavor symmetry of T[SU(2)] by an SU(2)H gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We can reach a very closely related cousin of this theory by flowing from the 3d N = 4 quiver U(1) SU(2)H (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='14) if the gauge coupling for SU(2)H is extremely small compared to the U(1) gauge coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' However, in this way we never land precisely on the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) is a “bad” theory in the sense of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' After a discussion of the symmetries and anomalies of this theory, we will add flavors for the SU(2)H gauge group converting the above theory into a “good” theory and study its symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 1-Form Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This theory carries a Γ(1) = Z2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='15) 1-form symmetry, as can be seen by the following argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' After gauging we obtain Wilson line defects valued in representations of SU(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A genuine local operator of T[SU(2)] trans- forming in representation R of SU(2)H becomes a non-genuine local operator of the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) that lives at the end of Wilson line defect in representation R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In other words, the Wilson line in representation R is screened in the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From our previous discussion, we know that the genuine local operators transform only in those representations – 22 – of SU(2)H that are also representations of SO(3)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, only Wilson lines in representa- tions of SO(3)H are screened, but the Wilson lines in representations of SU(2)H that are not representations of SO(3)H are not screened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The non-screened Wilson lines are non-trivially charged under the ZH 2 center of SU(2)H, which descends to a Z2 1-form symmetry of the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 0-Form Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We claim that the SO(3)C 0-form symmetry of T[SU(2)] is not im- pacted by the gauging procedure and the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) also has FC = SO(3)C (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='16) 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' To see this, we need to show that all genuine local operators of the gauged theory form representations of SO(3)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It is sufficient to show this fact for the following two types of genuine local operators: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The genuine local operators of T[SU(2)] that are uncharged under SU(2)H descend to genuine local operators of the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since such operators form SO(3)C representations before gauging, they also form SO(3)C representations after gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor monopole operators for su(2)H are non-genuine in T[SU(2)] as they are attached to vortex line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Some of these vortex line defects become invisible after gauging and the attached flavor monopole operators thus become genuine local operators of the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The co-characters associated to such monopoles are those which have even winding numbers around the maximal torus of SO(3)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such monopole operators form SO(3)C representations before gauging, so only lead to SO(3)C representations after gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Is There a 2-Group Symmetry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The question we would now like to address is whether the above Z2 1-form and SO(3)C 0-form symmetries combine to form a 2-group symmetry with a non-trivial Postnikov class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We claim that the answer is negative, due to the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The existence of such a 2-group symmetry requires the presence of a local operator O in the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) which sits at the end of a Wilson line operator transforming in an allowed representation of SO(3)H and transforms in a representation of SU(2)C that is not an allowed representation of SO(3)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This means that in the ungauged theory T[SU(2)], O must be a local operator (transforming in same representations of SO(3)H and SU(2)C) of one of the following two types: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' O is a genuine local operator in T[SU(2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' But then O must transform in an SO(3)C representation because the Coulomb 0-form symmetry group of T[SU(2)] is SO(3)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' O is a flavor monopole operator in T[SU(2)] associated to a co-character with even winding number around the maximal torus of SO(3)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' But then O must transform in an SO(3)C representation as already discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, we conclude that there is no non-trivial 2-group symmetry in the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 23 – Mixed 0-/1-Form Symmetry ’t Hooft Anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A flavor monopole operator in T[SU(2)] with odd winding number around maximal torus of SO(3)H becomes a gauge monopole op- erator in the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such a gauge monopole operator is not a standard gauge monopole operator usually discussed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The latter monopole operators are genuine local operators while the former monopole operator must be non-genuine attached to a non-trivial solitonic line defect which induces a non-trivial flux for the background field B2 for the Z2 1-form symmetry [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For this reason, such monopole operators were referred to as fractional gauge monopole operators in [12] to distinguish them from the standard (non- fractional) gauge monopole operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Some fractional monopole operators lie in the twisted sector of the Z2 1-form symmetry, that is, they lie at the end of a topological line operator generating the Z2 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since such a fractional gauge monopole operator forms a representation of SU(2)C that is not an allowed representation of SO(3)C, using the analysis of [12] we learn that there is a mixed ’t Hooft anomaly A4 = exp � πi � B2 ∪ wC 2 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='17) between the Z2 1-form symmetry and SO(3)C 0-form symmetry of the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A more straightforward way of deriving the above anomaly is to first note that the background field B2 for the Z2 1-form symmetry can be identified with the obstruction class wH 2 B2 = wH 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='18) and then the anomaly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='17) follows simply from the anomaly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) of T[SU(2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Adding Flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We are interested in understanding the global form of the Coulomb symmetry group of the IR SCFT obtained (for large enough N) from the UV 3d N = 4 Lagrangian theory U(1) SU(2)H N F , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='19) where we have N hypermultiplets transforming in fundamental representation of SU(2)H along with the bifundamental hypermultiplet between U(1) and SU(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Note that the corresponding IR SCFT can also be obtained by starting from the good version T[SU(2)] SU(2)H N F (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20) of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13), obtained from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) by adding N fundamental hypers for SU(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The relationship between theories (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20) is that the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='19) flows very close to the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20) at intermediate energy scales if the gauge coupling for SU(2)H is extremely small compared to the gauge coupling for U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We can thus use the symmetry properties of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) to deduce symmetry properties for the IR SCFT associated to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='19) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The additional flavors in the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20) screen the fundamental Wilson line of SU(2)H and thus the Z2 1-form symmetry of – 24 – the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) is lost in the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' On the other hand, the only new genuine local operators we obtain are gauge invariant combinations of these hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These carry trivial charge under SO(3)C, and hence the Coulomb 0-form symmetry group of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20) is SO(3)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For generic N, there is no enhancement of SO(3)C and the IR SCFT originating from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='19) (which is the same as the IR SCFT originating from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20)) has Coulomb 0-form symmetry group FIR C = SO(3)C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='21) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 SO(3)H Gauging Consider now the 3d N = 4 theory T[SU(2)] SO(3)H (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) obtained by gauging the su(2)H flavor symmetry of T[SU(2)] by an SO(3)H gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One can reach a very closely related theory by flowing from a 3d N = 4 Lagrangian theory u(1) su(2)H (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='23) (where we have only displayed gauge algebras) with the gauge group G = U(1) × SU(2)H Z2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='24) where the Z2 being quotiented out is the diagonal combination of the Z2 subgroup of U(1) and the ZH 2 center of SU(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) is again a bad theory just like (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Later, we also consider its good versions obtained by adding adjoint flavors for the SO(3)H gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 0-Form Symmetry Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In fact, this theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) can be obtained by gauging the Z2 1-form symmetry of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As a consequence, we expect the above theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) to contain a dual Z2 0-form symmetry, alongside the residual SO(3)C 0-form symmetry descending from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' However, the combined group structure of the 0-form symmetry is not Z2 × SO(3)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Instead, the mixed anomaly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='17) between Z2 1-form and SO(3)C 0- form symmetries of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) dualizes to a non-trivial extension between the Z2 and SO(3)C 0-form symmetries of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, in total the 0-form symmetry of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) is FC = SU(2)C , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='25) which is a non-trivial extension of the form 1 → Z2 → SU(2)C → SO(3)C → 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='26) To see it more concretely, following [71] let us explicitly perform the gauging over B2 with the term B2 ∪ B1 added to the 3d action, where B1 is background field for dual Z2 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This modifies the anomaly as A4 → B2 ∪ wC 2 + δB2 ∪ B1 + B2 ∪ δB1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='27) – 25 – The second term δB2 ∪ B1 = 0 as B2 is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The rest of the anomaly B2 ∪ (δB1 + wC 2 ) is a gauge anomaly (because B2 is a gauge field now), so it must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This gives us the constraint δB1 = wC 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='28) which implies that Z2 and SO(3)C 0-form symmetries indeed combine to form SU(2)C 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The same conclusion can also be reached by studying the monopole operators discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Recall that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) contains a fractional gauge monopole operator O living at the end of topological line operator generating the Z2 1-form symmetry, such that O transforms in a representation of SU(2)C which is not a representation of SO(3)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As we gauge the 1-form symmetry, the topological line generating the Z2 1-form symmetry disappears, and O becomes a genuine local operator of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, the 0-form symmetry group associated to su(2)C 0-form symmetry algebra in the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) is SU(2)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Adding Flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We are interested in understanding the global form of the Coulomb symmetry group of the IR SCFT obtained (for large enough N) from the UV 3d N = 4 Lagrangian theory u(1) su(2)H N A (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='29) where we have N hypers transforming in adjoint representation of su(2)H along with the bifundamental hyper between u(1) and su(2)H, and the gauge group is chosen to be (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Note that the corresponding IR SCFT can also be obtained by starting from the good version T[SU(2)] SO(3)H N A (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='30) of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22), obtained from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) by adding N adjoint hypers for SO(3)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The relationship between theories (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='29) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='30) is that the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='29) flows very close to the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20) at intermediate energy scales if the gauge coupling for su(2)H is extremely small compared to the gauge coupling for u(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We can thus use the symmetry properties of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) to deduce symmetry properties for the IR SCFT associated to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='29) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Recall that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) has a gauge monopole operator transforming in SU(2)C representation that is not an SO(3)C representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This operator is not impacted by the addition of adjoint hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, we deduce that the Coulomb 0-form symmetry group of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='30) is SU(2)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For generic N, there is no enhancement of SU(2)C and the IR SCFT originating from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='29) (which is the same as the IR SCFT originating from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='30)) has Coulomb 0-form symmetry group FIR C = SU(2)C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='31) – 26 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4 U(2) Gauging We are interested in understanding the global form of Coulomb and Higgs symmetry groups of the IR SCFT obtained (for large enough N) from the UV 3d N = 4 Lagrangian theory U(1) U(2) [su(N)H] , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='32) where we have N hypers transforming in fundamental representation of U(2) along with the bifundamental hyper between U(1) and U(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 0-Form Symmetry Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As shown in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='32), there is a Higgs branch flavor symmetry fH = su(N)H (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='33) rotating the N fundamental hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We will focus on the case N > 3 for which the IR Coulomb symmetry algebra is fIR C = u(2)C = su(2)C ⊕ u(1)C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='34) The N = 3 case has a further enhancement of IR Coulomb symmetry algebra to su(3)C that will be discussed in the following subsection on T[SU(n)] theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The su(2)C subalgebra of fIR C is usually associated to the balanced node provided by the U(1) gauge group, and the u(1)C subalgebra of fIR C is usually associated to the unbalanced node provided by the U(2) gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' However, we will need a more precise identification of these subalgebras, which we discuss in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 0-Form Symmetry Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Higgs 0-form symmetry group of the IR SCFT is the same as that of the UV Lagrangian theory FH = PSU(N)H = SU(N)H/ZN (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='35) and can be easily deduced by noticing that the gauge invariant combinations of hypermulti- plets all form representations of PSU(N)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' On the other hand, the Coulomb 0-form symmetry group of the IR SCFT is more subtle to deduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us begin with the Coulomb 0-form symmetry group of the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We have a U(1)1 0-form symmetry associated to the U(1) gauge node and a U(1)2 0-form symmetry associated to the U(2) gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' First, we need to understand the precise identification of the U(1) subgroup of UV 0-form symmetry group FUV C = U(1)1 × U(1)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='36) which becomes the maximal torus of the Lie group SU(2)C associated to the su(2)C subalgebra of fIR C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is done by studying FUV C charges of BPS monopole operators of low R-charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We look for a combination6 q = n1q1 + n2q2, where qi is the U(1)i charge of monopole and 6We thank Antoine Bourget for a discussion regarding this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 27 – ni ∈ Z, such that the set of monopole operators at a fixed value of R-charge have q-charges coinciding with the Dynkin coefficients of the weights of a representation of su(2)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This fixes q = 2q1 − q2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='37) Now the u(1)C factor is determined such that the BPS monopole operators furnishing the weights of adjoint of su(2)C are uncharged under u(1)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This determines that the u(1)C charge is proportional to q2, or more precisely there is a Lie group U(1)C with Lie algebra u(1)C, such that the charge qC under U(1)C is qC = q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='38) In order to determine the global form of the IR Coulomb flavor group FIR C we need to deter- mine the charges � q (mod 2), qC � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='39) of non-fractional gauge monopole operators, where the charge q (mod 2) is the charge of the monopole operator under the Z2 center of the IR SU(2)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The fundamental monopole oper- ator from U(1) gauge node has (q1, q2) = (1, 0) implying � q (mod 2), qC � = � 0 (mod 2), 0 � , while the fundamental monopole operators from U(2) gauge node have (q1, q2) = (0, 1) im- plying � q (mod 2), qC � = � 1 (mod 2), 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The only non-trivial charge is the latter one, which implies that the Coulomb 0-form symmetry group of the IR SCFT is FIR C = U(2)C = SU(2)C × U(1)C Z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='40) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 T[SU(n)] and its Gaugings In this subsection we generalize to T[SU(n)] theories the results obtained in the previous subsection regarding T[SU(2)] theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 T[SU(n)] B-Type 0-Form Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The T[SU(n)] theory arises as an IR fixed point of the following 3d N = 4 Lagrangian theory U(n − 1) [su(n)H] · · U(2) U(1) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='41) where we have a bifundamental hyper between adjacent unitary gauge nodes, and n funda- mental hypers for U(n − 1) that are rotated by an fH = su(n)H (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='42) Higgs flavor symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The genuine local operators charged under su(n)H arise from gauge invariant combina- tions of hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such gauge invariant combinations all form representations of the – 28 – Lie group PSU(n)H with Lie algebra su(n)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, the Higgs branch 0-form symmetry group is FH = PSU(n)H = SU(n)H/Zn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='43) where SU(n)H is the simply connected group with Lie algebra su(n)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Equivalently, we can deduce the above 0-form symmetry group by putting the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='41) on a non-trivial compact 3-manifold M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charges of the vector and hypermultiplets allow us to turn on bundles for the group S = U(1) × U(2) × · · · × U(n − 1) × SU(n)H Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='44) The Zn in the denominator is the diagonal combination of the Zn subgroups of U(1) centers of U(i) gauge groups and the Zn center of SU(n)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This diagonal combination leaves all vector and hyper multiplets invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus the Higgs flavor symmetry group is PSU(n)H because, according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='44), we can couple the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='41) to non-trivial (in the sense that they cannot be lifted to SU(n)H bundles) background bundles for PSU(n)H, provided we turn on non-trivial (in the sense that they cannot be lifted to U(i) bundles) bundles for U(i)/Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A-Type 0-Form Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In addition to the PSU(n)H 0-form symmetry, the La- grangian theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='41) admits a magnetic FUV C = U(1)n−1 C = n−1 � i=1 U(1)C,i (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='45) 0-form symmetry, where U(1)C,i is the magnetic symmetry arising from the U(i) gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The above Coulomb 0-form symmetry enhances in the IR to an su(n)C Coulomb 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The fundamental monopole operators associated to each node i describe roots of su(n)C, which carry charge 0 (mod n) under the center Zn of SU(n)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As a consequence, all gauge monopole operators form representations of SU(n)C having charge 0 (mod n) under the center Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus the Coulomb 0-form symmetry group of the IR SCFT is FIR C = PSU(n)C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='46) Mixed 0-Form Anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is a mixed ’t Hooft anomaly between the PSU(n)H and PSU(n)C 0-form symmetries of T[SU(n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The relevant flavor-gauge monopole operator is associated to a co-character of the structure group S appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='44) with winding number 1/n around the U(1) center of each U(i) gauge group and winding number 1/n around a U(1) subgroup of the maximal torus of SU(n)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is an allowed co-character because of the Zn quotient appearing in the denominator of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The mixed flavor-gauge monopole operator O descends to a flavor monopole operator in the IR SCFT T[SU(n)], which is a non-genuine local operator lying at the end of a flavor vortex line defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' O carries a charge qi = i/n under U(1)C,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This implies that the Dynkin – 29 – coefficients di of the weight of su(n)C carried by O are (d1, d2, · · · , dn−2, dn−1) = (2q1 − q2, −q1 + 2q2 − q3, · · · , −qn−3 + 2qn−2 − qn−1, −qn−2 + 2qn−1) = (0, 0, · · · , 0, 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='47) The charge of O under the Zn center of SU(n)C is computed in terms of di as n−1 � i=1 i × di (mod n) = −1 (mod n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='48) Using the analysis of [12], this fact is equivalent to a mixed ’t Hooft anomaly between the Coulomb and Higgs 0-form symmetry groups of the IR T[SU(n)] SCFT AIR 4 = exp � −2πi n � wH 2 ∪ wC 2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='49) where wC 2 , wH 2 are Zn valued obstruction classes capturing the obstruction of lifting back- ground PSU(n)C, PSU(n)H bundles to SU(n)C, SU(n)H bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 SU(n)H Gauging Consider now the 3d N = 4 theory T[SU(n)] SU(n)H (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50) obtained by gauging the su(n)H flavor symmetry of T[SU(n)] by an SU(n)H gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We can reach very close to this theory by flowing from the 3d N = 4 quiver U(n − 1) SU(n)H · · U(2) U(1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='51) if the gauge coupling for SU(n)H is extremely small compared to the U(i) gauge couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50) is a bad theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We will later also discuss its good versions obtained by adding flavors for the SU(n)H gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 1-Form Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This theory carries a Γ(1) = Zn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='52) 1-form symmetry, as can be seen by the following argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' After gauging we obtain Wilson line defects valued in representations of SU(n)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A genuine local operator of T[SU(n)] trans- forming in representation R of SU(n)H becomes a non-genuine local operator of the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50) that lives at the end of Wilson line defect in representation R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In other words, Wilson line in representation R is screened in the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From our previ- ous discussion, we know that the genuine local operators transform only in representations of PSU(n)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, Wilson lines transforming in representations of SU(n)H with non-zero charge (modulo n) under its Zn center are left unscreened, implying that Zn center of SU(n)H descends to a Zn 1-form symmetry of the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 30 – 0-Form Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There are two types of genuine local operators of the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50) to consider: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The genuine local operators of T[SU(n)] that are uncharged under SU(n)H descend to genuine local operators of the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since such operators form PSU(n)C representations before gauging, they also form PSU(n)C representations after gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor monopole operators of T[SU(n)] associated to co-characters of SU(n)H be- come genuine local operators (non-fractional gauge monopole operators) after gauging, despite being non-genuine before gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Such monopole operators form PSU(n)C representations before gauging, so only lead to PSU(n)C representations after gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, the PSU(n)C 0-form symmetry of T[SU(n)] is not impacted by the gauging procedure and the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50) also has FC = PSU(n)C (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='53) 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Is There a 2-Group Symmetry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The question we would now like to address is whether the above Zn 1-form and PSU(n)C 0-form symmetries combine to form a 2-group symmetry with a non-trivial Postnikov class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We claim that the answer is negative, due to the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The existence of such a 2-group symmetry requires the presence of a local operator O in the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50) which sits at the end of a Wilson line operator transforming in a representation of PSU(n)H and transforms in a representation of SU(n)C that is not an allowed representation of PSU(n)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This means that in the ungauged theory T[SU(n)], O must be a local operator (transforming in same representations of PSU(n)H and SU(n)C) of one of the following two types: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' O is a genuine local operator in T[SU(n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' But then O must transform in a PSU(n)C representation because the Coulomb 0-form symmetry group of T[SU(n)] is PSU(n)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' O is a flavor monopole operator in T[SU(n)] associated to a co-character of the group SU(n)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' But then O must transform in a PSU(n)C representation as already discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, we conclude that there is no non-trivial 2-group symmetry in the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Mixed 1-Form 0-Form ’t Hooft Anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Flavor monopole operators of T[SU(n)] become (possibly fractional) gauge monopole operators in the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This includes the operator O discussed around equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='49), which becomes a fractional gauge monopole operator after gauging and can be converted into a local operator living at the end of the topological line defect generating the Zn 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The fact that O transforms in a representation of SU(n)C having charge −1 (mod n) under its Zn center is equivalent to – 31 – the mixed ’t Hooft anomaly between the Zn 1-form symmetry and PSU(n)C 0-form symmetry of the gauged theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) A4 = exp � −2πi n � B2 ∪ wC 2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='54) where B2 is the background field for the Zn 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This anomaly can also be derived as a consequence of the anomaly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='49) using the identification B2 = wH 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Adding Flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We are interested in understanding the global form of the Coulomb symmetry group of the IR SCFT obtained (for large enough N) from the UV 3d N = 4 Lagrangian theory U(n − 1) SU(n)H · · U(2) U(1) N F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='55) Note that the corresponding IR SCFT can also be obtained by starting from the good version T[SU(n)] SU(n)H N F (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='56) of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50), obtained from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50) by adding N fundamental hypers for SU(n)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The relationship between theories (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='55) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='56) is that the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='55) flows very close to the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='56) at intermediate energy scales if the gauge coupling for SU(n)H is extremely small compared to the U(i) gauge couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We can thus use the symmetry properties of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50) to deduce symmetry properties for the IR SCFT associated to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='55) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The additional flavors in the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='56) screen the fundamental Wilson line of SU(n)H and thus the Zn 1-form symmetry of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50) is lost in the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' On the other hand, the only new genuine local operators we obtain are gauge invariant combinations of these hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These carry trivial charge under PSU(n)C, and hence the Coulomb 0-form symmetry group of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='56) is PSU(n)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For generic N, there is no enhancement of PSU(n)C and the IR SCFT originating from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='55) (which is the same as the IR SCFT originating from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='56)) has Coulomb 0-form symmetry group FIR C = PSU(n)C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='57) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 Other su(n)H Gaugings We can also consider gauging SU(n)H/Zm (where Zm < Zn is a subgroup, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' m|n) to obtain the 3d N = 4 theory T[SU(n)] SU(n)H/Zm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='58) One can reach very close to this theory by flowing from a 3d N = 4 Lagrangian theory u(n − 1) su(n)H · · u(2) u(1) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='59) – 32 – (where we have only displayed gauge algebras) with the gauge group G = U(1) × U(2) × · · · × U(n − 1) × SU(n)H Zm , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='60) where the Zm being quotiented out is the Zm subgroup of the Zn group appearing in the denominator of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='58) is again a bad theory just like (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Later, we also consider its good versions obtained by adding adjoint flavors for the SU(n)H/Zm gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 1-Form Symmetry Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In fact, this theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='58) can be obtained by gauging the Zm 1-form symmetry of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Consequently, there is a residual Γ(1) = Zp (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='61) 1-form symmetry in the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='58) where p = n/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 0-Form Symmetry Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is also thus a dual Zm 0-form symmetry in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='58) along- side the residual PSU(n)C 0-form symmetry descending from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' By similar arguments as in the previous subsection on T[SU(2)], the two 0-form symmetries combine non-trivially and the full 0-form symmetry group of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='58) is FC = SU(n)C/Zp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='62) Mixed 1-Form 0-Form ’t Hooft Anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is a mixed ’t Hooft anomaly between Zp 1-form and SU(n)C/Zp 0-form symmetries arising as a residue of the anomaly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='54) A4 = exp � −2πi p � B2 ∪ wC 2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='63) where wC 2 is the Zp valued obstruction class for lifting SU(n)C/Zp bundles to SU(n)C bundles, and B2 is the Zp valued background field for 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Particular Case: m = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In this particular case, we are studying a PSU(n)H gauging of T[SU(n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is no 1-form symmetry and the 0-form symmetry group is SU(n)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Adding Adjoint Flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We can add N adjoint flavors for SU(n)C/Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For large enough N, this flows to a 3d N = 4 SCFT in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The global form of the Coulomb symmetry group and mixed anomaly between Coulomb 0-form symmetry and 1-form symmetry in the IR SCFT for generic N are the same as those described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4 U(n) Gauging We are interested in understanding the global form of Coulomb and Higgs symmetry groups of the IR SCFT obtained (for large enough N) from the UV 3d N = 4 Lagrangian theory U(n) [su(n)H] · · U(2) U(1) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='64) where we have N hypers transforming in fundamental representation of U(n) along with bifundamental hypers between adjacent U(i) gauge nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 33 – 0-Form Symmetry Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As shown in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='64), there is a fH = su(N)H (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='65) Higgs branch symmetry rotating the N fundamental hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We will focus on the case N > n + 1 for which the IR Coulomb symmetry algebra is fIR C = u(n)C = su(n)C ⊕ u(1)C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='66) The N = n+1 case has a further enhancement of IR Coulomb symmetry algebra to su(n+1)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The su(n)C subalgebra of fIR C is usually associated to the balanced nodes provided by the U(i) gauge groups for 1 ≤ i ≤ n − 1, and the u(1)C subalgebra of fIR C is usually associated to the unbalanced node provided by the U(n) gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' However, as before we will need a more precise identification of these subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 0-Form Symmetry Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Higgs 0-form symmetry group of the IR SCFT is is the same as that of the UV Lagrangian theory FH = PSU(N)H = SU(N)H/ZN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='67) To deduce the Coulomb 0-form symmetry group of the IR SCFT, we need a precise identification of the Dynkin coefficients di for su(n)C weights in terms U(1)C,i charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is the same as discussed around equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='47), except now dn−1 is modified to dn−1 = −qn−2 + 2qn−1 − qn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='68) Moreover, the u(1)C factor has a global form U(1)C such that the charge qC under U(1)C is qC = qn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='69) Now we see that the fundamental monopole operators coming from the U(n) gauge node transform in anti-fundamental representation of su(n)C and simultaneously have charge +1 under u(1)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus the Coulomb 0-form symmetry group of the IR SCFT is FIR C = U(n)C = SU(n)C × U(1)C Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='70) 4 General Symmetry and Anomaly Analysis for 3d N = 4 SCFTs Now we are ready to describe the most general result of the considerations of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Consider a 3d N = 4 good quiver gauge theory composed of unitary and special unitary gauge algebras and no Higgs flavor algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us write the gauge algebra as g = � i gi , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1) – 34 – where each gi is either su(ni) or u(ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We assume there is at least one special unitary node and all special unitary nodes are unbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Also let Gi be SU(ni) or U(ni) for the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The matter content is comprised entirely of bifundamental hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Say we have mij ≥ 0 bifundamental hypers between gi and gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We assume that the quiver is connected, which means that we can go from any node i to any other node j by choosing a sequence of nodes ia for 0 ≤ a ≤ b such that i0 = i, ib = j and miaia+1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Moreover, the balanced unitary nodes are taken to form the Dynkin diagram of a finite semi-simple Lie algebra f = � a fa , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2) where each fa is a finite simple Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let Fa be the simply connected group associated to fa with center Za and define F = � a Fa with center Z = � a Za.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let U be the set of unbalanced unitary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let u(1)i for a node i ∈ U be the associated Coulomb 0-form symmetry algebra and U(1)i be a group with Lie algebra u(1)i such that the fundamental BPS monopoles associated to node i have charge gδij under U(1)j, where g is defined around equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5) below and δij is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We have scaled the U(1)i charges of fundamental monopoles by g for later convenience in computing ’t Hooft anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Coulomb 0-form symmetry algebra of the corresponding 3d N = 4 IR SCFT is fIR C = � i∈U u(1)i ⊕ f (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3) Let us define F IR C = � i∈U U(1)i × F with center ZIR C = � i∈U U(1)i × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 Symmetries 1-Form Symmetry Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Consider choosing first the gauge group G = � i Gi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4) Then the theory with the above-described matter content has a 1-form symmetry group given by Γ(1) = Zg , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5) where g is the GCD (greatest common divisor) of all ni for which gi = su(ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (Coulomb) 0-Form Symmetry Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us determine the Coulomb 0-form symmetry group FIR C of the IR SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Each node i ∈ U provides a genuine local operator Oi in the IR SCFT whose charge under ZIR C is what we want to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This local operator can be chosen to be IR image of any fundamental BPS monopole associated to the node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' First of all, Oi has charge qi,j = gδij under U(1)j factor of ZIR C , where j ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charge qi,a of Oi under Za is given by qi,a = − � j∈Na mi,jna,j ∈ �Za , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6) – 35 – where Na is the set of nodes forming the Dynkin diagram of fa and na,j ∈ �Za is the charge of the representation of fa with highest weight having Dynkin coefficients di = δij for i ∈ Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Combining the charges qi,j for all j ∈ U and charges qi,a for all a, we obtain a charge qi ∈ �ZIR C (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='7) of Oi under ZIR C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charges qi for all i ∈ U span a subgroup Y IR C of the abelian group �ZIR C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We thus obtain the information of a surjective map �ZIR C → � ZIR C := �ZIR C Y IR C , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='8) which can be Pontryagin dualized to an injective map ZIR C → ZIR C (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='9) providing a subgroup ZIR C of ZIR C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Coulomb 0-form symmetry group of the IR SCFT is FIR C = F IR C ZIR C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='10) Visual Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charges qi,a of operators Oi under the centers Za of Coulomb flavor algebras fa can be deduced visually from the UV quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' First of all, note that qi,a is the same as the charge under Za of the representation � j∈Na Fmi,j j , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='11) of fa, where Fj is the oft-called ‘fundamental representation associated to node j’ of the Dynkin diagram of fa, which is the representation with highest weight having Dynkin coefficients di = δij for i ∈ Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This representation is deduced visually from the UV quiver: we just see how many times the node i ∈ U hits a node j in Na and include that many copies of the representation Fj of fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 Anomaly There is a mixed ’t Hooft anomaly between Γ(1) and FIR C which is computed using the charge of a fractional gauge monopole operator O associated to co-character of the group G = G Zg , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) where Zg being quotiented out is the diagonal combination of the Zg subgroup of the U(1) center of each unitary gauge group and the Zg subgroup of the Zni center of the gauge group SU(ni) for each special unitary gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The co-character associated to O has winding number 1/g around the U(1) center of each unitary gauge group and winding number 1/g around a U(1) subgroup of the maximal torus of SU(ni) for each special unitary gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 36 – The charge qi of O under U(1)i for i ∈ U is ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charge qa of O under Za is the same as the charge of the representation of fa having highest weight with Dynkin coefficients di = � j∈Na Ma,ij nj g − � j∈U mi,j nj g (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) for i ∈ Na, where Ma,ij is the Cartan matrix of fa and nj is the rank of the gauge algebra u(nj) associated to the node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Combining the qi and qa charges, we obtain a charge qO ∈ �ZIR C of O under ZIR C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Projecting it using the map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='8), we obtain an element q ∈ � ZIR C , letting us define a homomorphism γ : Γ(1) → � ZIR C (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='14) via Γ(1) = Zg ∋ 1 �→ q ∈ � ZIR C (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='15) The ’t Hooft anomaly between the 1-form and 0-form symmetries of the IR SCFT is then AIR 4 = exp � 2πi � γ(B2) ∪ wC 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='16) where B2 is the Γ(1) valued background field for the 1-form symmetry, wC 2 is the ZIR C valued class capturing the obstruction of lifting FIR C bundles to F IR C bundles, and the cup product uses the natural pairing � ZIR C × ZIR C → R/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Visual Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charges qa of the operator O under the centers Za of Coulomb flavor algebras fa can be deduced visually from the UV quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' First of all, note that qa is the same as the charge under Za of the representation � i∈S � j∈Na F nimi,j g j (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='17) of fa, where S is the set of special unitary gauge nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' To see this, one needs to use the balancing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This representation is deduced visually from the UV quiver: for each special unitary node i ∈ S, we just see how many times i hits a node j in Na and include that many copies of the representation Fj of fa weighted by a factor of ni/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 Other Gauge Groups We can change the gauge group by gauging a subgroup Zh ⊆ Zg of the 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The resulting gauge group is Gh = G/Zh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='18) The 1-form symmetry group of the resulting IR SCFT is now Γ(1) h = Zg/Zh = Zk , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='19) – 37 – where k = g/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Because of the extra Zh quotient in the new gauge group Gh, some of the gauge monopole operators which were fractional (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' were non-genuine local operators) now become non- fractional gauge monopole operators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' become genuine local operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We need to account for center charges of these new genuine local operators to compute the Coulomb 0-form symmetry group FIR C,h of the new IR SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' To account for these new charges, it is sufficient to consider the contribution of a single monopole operator Oh for Gh with co- characters having winding number k times the winding numbers associated to the fractional gauge monopole operator O considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charge qi,h of Oh under U(1)i for i ∈ U is k, and the charge qa,h of Oh under Za is the same as the charge of the representation of fa having highest weight with Dynkin coefficients di,h = kdi, where di are defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Combining the qi,h and qa,h charges for various values of i and a, we obtain a charge qh ∈ �ZIR C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Appending qh to Y IR C and spanning, we obtain a larger subgroup Y IR C,h of �ZIR C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This provides a surjective map �ZIR C → � ZIR C,h := �ZIR C Y IR C,h , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20) whose Pontryagin dual is an injective map ZIR C,h → ZIR C (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='21) providing a subgroup ZIR C,h of ZIR C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Coulomb 0-form symmetry group of the new IR SCFT is FIR C,h = F IR C ZIR C,h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) There is a residual mixed ’t Hooft anomaly between Γ(1) h and FIR C,h descending from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is still a consequence of the operator O which remains a fractional gauge monopole operator for h < g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Its charge is still qO ∈ �ZIR C which projects to an element qh ∈ � ZIR C,h, letting us define a homomorphism γh : Γ(1) h → � ZIR C,h (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='23) via Γ(1) = Zk ∋ 1 �→ qh ∈ � ZIR C,h (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='24) The ’t Hooft anomaly between the 1-form and 0-form symmetries of the new IR SCFT is then AIR 4,h = exp � 2πi � γh(B2,h) ∪ wC 2,h � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='25) where B2,h is the Γ(1) h valued background field for the new 1-form symmetry, wC 2,h is the ZIR C,h valued class capturing the obstruction of lifting FIR C,h bundles to F IR C bundles, and the cup product uses the natural pairing � ZIR C,h × ZIR C,h → R/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 38 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4 Including Flavors Let us now ungauge a few special unitary gauge algebras in the above UV theory, converting those gauge nodes into flavor nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The resulting theory is still a good theory and flows to a 3d N = 4 SCFT in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let the set R parametrize the nodes which remain as gauge nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We choose the gauge group to be GR = � i∈R Gi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='26) Because of the presence of flavors, the theory has no 1-form symmetry Γ(1) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='27) The Higgs flavor symmetry algebra is taken to be (there might be some extra abelian factors that we ignore) fH = � i̸∈R su(ni) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='28) The structure group of the UV theory including Higgs flavor symmetries is SR = G/Zg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='29) In particular, the Higgs 0-form symmetry group of the UV theory and the corresponding IR SCFT is FH = � i̸∈R SU(ni) Zg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='30) The Coulomb 0-form symmetry group is the same as before FIR C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The fractional gauge monopole O discussed above is now instead a mixed flavor-gauge monopole operator providing a mixed ’t Hooft anomaly between the Higgs and Coulomb 0-form symmetries AIR 4,R = exp � 2πi � γ(wH 2 ) ∪ wC 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='31) where wH 2 is the Zg valued class capturing the obstruction of lifting FH bundles to � i̸∈R SU(ni) bundles and γ is the homomorphism appearing in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5 Special Case 1: Single Special Unitary Node In this and the following subsections, we discuss two special cases for which the symmetry groups and anomalies of the IR SCFT take a simple form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These two special cases will be used frequently in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In this subsection, we consider the first special case, which occurs when we have a single (unbalanced) special unitary gauge node carrying su(n)H gauge algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' First choose the gauge group G = � i U(ni) × SU(n)H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='32) – 39 – The 1-form symmetry is Γ(1) = Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='33) As explained around (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='11), the 0-form symmetry group FIR C is read simply from the positions where the unbalanced unitary nodes hit the balanced unitary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The IR SCFT has a mixed ’t Hooft anomaly between the 1-form and 0-form symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charge qa under Za of the fractional gauge monopole operator O is read simply visually from the UV quiver, and coincides with the charge of the representation � i∈Na Fmi i (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='34) of fa, where mi is the number of bifundamental hypers between the SU(n)H gauge node and the balanced unitary gauge node i ∈ Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' That is, we simply observe the number of times the SU(n)H gauge node hits the balanced node i and include that many copies of the representation Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Combining the charges qa ∈ �Za with the charge ni under each Coulomb symmetry U(1)i associated to unbalanced unitary node i ∈ U, we obtain the charge q ∈ �Z of O, which provides the homomorphism (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='14) appearing in the mixed anomaly (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us gauge the 1-form symmetry group Γ(1) = Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The new gauge group is Gn = � i U(ni) × SU(n)H Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='35) There is no residual 1-form symmetry and the Coulomb 0-form group is modified by the presence of the operator O which is now a genuine local operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Its charges qa described above need to be accounted to compute the new Coulomb 0-form symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We can instead ungauge SU(n)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The gauge group is now GR = � i U(ni) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='36) There is a Higgs 0-form symmetry algebra fH = su(n)H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='37) Both the UV gauge theory and the IR SCFT have Higgs 0-form symmetry group as FH = PSU(n)H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='38) The Coulomb 0-form symmetry group is still as before the ungauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is a mixed ’t Hooft anomaly between the Higgs and Coulomb 0-form symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The anomaly is described completely by the charges qa of the operator O discussed above, which is now a mixed flavor-gauge monopole operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 40 – Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Many of the results of the previous section 3 can be arrived at by a simple application of the above general analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The anomaly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='49) of T[SU(n)] is a consequence of the fact that the flavor node su(n)H intersects the su(n)C balanced quiver at the location of the anti-fundamental node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Similarly, the anomaly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='54) is a consequence of the fact that the unbalanced SU(n)H gauge node intersects the su(n)C balanced quiver at the location of the anti-fundamental node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Upon gauging Zn 1-form symmetry, it modifies the PSU(n)C 0-form symmetry to SU(n)C 0-form symmetry as discussed in the paragraph on special case m = n of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6 Special Case 2: Single Unbalanced Unitary Node In this subsection, we consider the second special case, which occurs when we have a single unbalanced unitary gauge node U(n), and no special unitary gauge nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We require the presence of at least one flavor node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The gauge group is taken to be G = � i U(ni) × U(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='39) There is no 1-form symmetry, and the Coulomb 0-form symmetry algebra of the IR SCFT is fIR C = f ⊕ u(1)C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='40) The only non-trivial charge under its center Z × U(1)C is provided by fundamental gauge monopole operators associated to the U(1)C gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charge qC of such an operator O under U(1)C is +1 and the charge qa under Za is the same as that of the representation � i∈Na Fmi i (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='41) of fa, where mi is the number of bifundamental hypers between the U(n) gauge node and the balanced unitary gauge node i ∈ Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' That is, to compute qa we simply observe the number of times the U(n) node hits the node i and include that many copies of the representation Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This allows us to compute the Coulomb 0-form symmetry group FIR C of the IR SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Some of the results of the previous section 3 can be arrived at by a simple application of the above general analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The U(n)C 0-form symmetry group of U(n) gauging of T[SU(n)] appearing in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='70) is a straightforward consequence of the fact that the unbalanced U(n) gauge node intersects the balanced su(n)C quiver at the location of anti-fundamental node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 5 Consistency Checks In this section we will provide various consistency checks of our results detailed in the pre- vious section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One central application is to magnetic quivers of 4d and 5d SCFTs with 8 supercharges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 41 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 Class S In this subsection, we apply the methods of previous section to deduce the Coulomb 0-form symmetry groups of magnetic quivers (MQs) associated to Class S theories of An−1 type [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These MQs, derived in [73], are 3d quiver gauge theories that are mirror to the circle compactification of 4d N = 2 Class S theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus the Coulomb 0-form symmetry groups of these MQs should capture the usual Higgs flavor symmetry groups of 4d N = 2 Class S theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The computation of flavor symmetry groups of Class S theories was described recently in [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We demonstrate a match of results obtained using our methods against the results obtained using their methods, and illustrate it with three examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Alternatively, one can view this subsection as providing the correct global form of the MQs of Class S theories of An−1 type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' That is, we provide the global form of the gauge group that should be associated to the gauge algebra of the MQ described in [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This global form of MQ is deduced by matching symmetries of MQ with the symmetries of the Class S theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 General Matching Symmetries of Class S Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Consider a Class S theory arising from the sphere com- pactification of a 6d N = (2, 0) SCFT of An−1 type with k regular untwisted punctures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Each puncture Pi is characterized by a partition ρi of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let ρi,j be the elements of the partition where j takes values in 1 ≤ j ≤ |ρi|, and |ρi| is the number of elements in the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We order ρi,j such that ρi,j ≥ ρi,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor symmetry algebra fi associated to the puncture Pi is encoded in the partition ρi according to the following standard rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Define j1 such that ρi,j = ρi,1 for all j ≤ j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Define j2 such that ρi,j = ρi,j1+1 for all j1 < j ≤ j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We continue defining ja in this fashion until we reach j = |ρi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let b be the total number of ja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Then, fi = b � a=1 su(ja − ja−1) ⊕ u(1)b−1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1) with j0 := 0, and su(1) being the trivial Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us define Fi to be the following Lie group associated to the algebra fi Fi = b � a=1 SU(ja − ja−1) × U(1)b−1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2) where SU(1) denotes the trivial group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor symmetry group F of the Class S theory is obtained as a quotient F = � i Fi Z (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3) Let ZF be the center of F := � i Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The group Z ⊆ ZF is obtained by taking Pontryagin dual of a surjective map �ZF → � Z = �ZF /YF (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4) – 42 – where YF is a subgroup of �ZF whose computation was described in [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' First of all, there is a contribution YF,Pi ⊆ YF coming from each puncture Pi, and then there is a contribution �YF ⊆ YF coming from all punctures put together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The group YF is recovered as the combined span of all YF,Pi and �YF inside �ZF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Finally, as discussed in [74] the 1-form symmetry group of the Class S theory is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Review of the Magnetic Quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us now review the magnetic quiver of the Class S theory discussed in [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We first associate a sub-quiver to each puncture Pi which can be described as the following 3d N = 4 Lagrangian theory · · u(ni,|ρi|−1) u(ni,2) u(ni,1) [su(n)H] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5) where ni,J = n − J � j=1 ρi,j (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6) We see that the gauge nodes for ja−1 < J < ja are balanced for 1 ≤ a ≤ b, and give rise to an emergent �b a=1 su(ja − ja−1)C Coulomb symmetry in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Moreover, the gauge nodes J = ja for 1 ≤ a ≤ jb−1 are unbalanced and give rise to u(1)⊕(b−1) C Coulomb symmetry in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, the contribution of this sub-quiver to the IR Coulomb 0-form symmetry algebra matches the flavor symmetry algebra fi associated to the puncture Pi shown in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The full magnetic quiver of the Class S theory is then obtained by gauging the diagonal su(n)H symmetry of all the sub-quivers, resulting in the theory u(n1,1) su(n)H u(n2,1) u(nk,1) · · u(n1,|ρ1|−1) · · u(n2,|ρ2|−1) · · u(nk,|ρk|−1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='7) where the su(n)H node is unbalanced, and thus the IR Coulomb 0-form symmetry algebra is f = � i fi, matching with the flavor symmetry algebra of the Class S theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Global Form of the Magnetic Quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' What is the gauge group that we should choose?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Apriori there are many choices �k i=1 �|ρi|−1 J=1 U(ni,J) × SU(n)H Zm (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='8) parametrized by divisors m of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The 1-form symmetry of the theory for such a choice is Γ(1) = Zn/m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='9) – 43 – To match it with the trivial 1-form symmetry of the Class S theory, we are forced to pick the gauge group G = �k i=1 �|ρi|−1 J=1 U(ni,J) × SU(n)H Zn (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='10) for the choice m = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This global form is actually manifest in the following usual presentation of the MQ U(n1,1) U(n)H U(n2,1) U(nk,1) · · U(n1,|ρ1|−1) · · U(n2,|ρ2|−1) · · U(nk,|ρk|−1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='11) with an additional instruction of ungauging a U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' If we perform this U(1) ungauging on the central U(n)H node, we are forced to remove also the Zn center of the SU(n)H component of U(n)H as this Zn sits inside the U(1) center of U(n)H being removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Due to the presence of bifundamental matter, this Zn also sits inside the U(1) center of all the other unitary gauge groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, performing the U(1) ungauging at U(n)H node indeed leaves behind the G gauge group appearing in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Similar discussions also appeared in [75], where a discussion regarding other U(1) ungaugings can also be found (see also [76]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Computing Flavor Symmetry Group Using the Magnetic Quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since, by this simple argument based on 1-form symmetry, we have no other possible choice for the gauge group, it better be true that the IR Coulomb 0-form symmetry group for this choice of gauge group matches the flavor symmetry group of the Class S theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is a straightforward application of the special case of our general prescription de- scribed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Recall there we also encountered two types of contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The first type of contributions came from unbalanced unitary gauge nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Collecting all such contri- butions from the unbalanced unitary gauge nodes situated along the sub-quiver leg associated to the puncture Pi provide the contribution YF,Pi of [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The second type of contribution comes from the sole special unitary unbalanced gauge node, which provides the contribution �YF of [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In this way, we find that the IR Coulomb 0-form symmetry group of the magnetic quiver (with the correct global form) described above matches the flavor symmetry group F of the Class S theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 Examples Let us see this matching explicitly for the following three examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 44 – Example 1: Trinion Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The first example we consider is the 4d trinion Tn theory, ob- tained as a Class S theory by compactifying 6d N = (2, 0) SCFT of An−1 type on a sphere with 3 maximal regular punctures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Each puncture provides an su(n) flavor symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, the total flavor symmetry algebra is f = su(n)1 ⊕ su(n)2 ⊕ su(n)3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) with associated F = SU(n)1 × SU(n)2 × SU(n)3 with center ZF = (Zn)1 × (Zn)2 × (Zn)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) We assume n > 3, because as is well known there is an enhancement of the above flavor symmetry algebra to f = e6 for n = 3, in which case the T3 trinion theory coincides with the E6 Minahan-Nemeschansky theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This was discussed as the first example in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 of [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The associated magnetic quiver is u(n − 1) su(n)H u(n − 1) u(n − 1) u(n − 2) · · u(n − 2) · · u(n − 2) · · u(1) u(1) u(1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='14) with gauge group G = �n−1 i=1 U(i)3 × SU(n)H Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='15) Let us now compute the IR Coulomb 0-form symmetry group using the arguments of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since every unitary gauge node is balanced, we do not have any puncture depen- dent contribution i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' YF,Pi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' On the other hand, the contribution �YF is obtained from the monopole operator O, whose charge under ZF is the same as that of the representation F1 ⊗ F2 ⊗ F3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='16) of F, where Fi is the fundamental representation of SU(n)i ⊂ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is because the su(n)H node hits each balanced sub-quiver i at the node of Dynkin diagram of su(n)i corresponding to the fundamental representation of su(n)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These contributions match those appearing in [64], and as described there the flavor symmetry group can written as F = SU(n)1 × SU(n)2 × SU(n)3 Zn × Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='17) – 45 – We refer the reader to [64] for more details regarding the identity of the two Zn subgroups appearing in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We can also discuss the case of n = 3, for which it was argued in [64] that the full flavor symmetry group must be E6/Z3 as that is the only possible enhancement of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='17) for the n = 3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We can see this flavor group directly from the MQ, which is obtained by a U(1) ungauging of U(2) U(3)H U(2) U(2) U(1) U(1) U(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='18) Instead of performing the U(1) ungauging on the U(3)H gauge node, let us perform it on one of the U(1) gauge nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The magnetic quiver can then be expressed as U(2) U(3)H U(2) U(2) U(1) U(1) F , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='19) where we have a fundamental hyper charged under the top U(2) gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Every unitary gauge node is balanced and hence the IR Coulomb 0-form symmetry algebra is f = e6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20) Since there are no unbalanced unitary or special unitary gauge nodes, the IR Coulomb 0-form symmetry group is the centerless global form F = E6/Z3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='21) of f = e6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Example 2: Free Bifundamental Hyper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As a second example, consider the Class S theory obtained by compactifying 6d N = (2, 0) SCFT of An−1 type on a sphere with 2 maximal regular punctures P1, P2 and 1 minimal regular puncture P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Each maximal puncture provides an su(n) flavor symmetry algebra, while the minimal puncture provides u(1) flavor symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, the total flavor symmetry algebra is f = su(n)1 ⊕ su(n)2 ⊕ u(1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) with associated F = SU(n)1 × SU(n)2 × U(1) with center ZF = (Zn)1 × (Zn)2 × U(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='23) The resulting 4d N = 2 theory can be recognized as a free hypermultiplet transforming in bifundamental representation of two su(n) factors of f, with the u(1) factor of f rotating the bifundamental hyper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This was discussed as the second example in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 of [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 46 – The associated magnetic quiver is u(n − 1) su(n)H u(n − 1) u(1) u(n − 2) · · u(n − 2) · · u(1) u(1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='24) with gauge group G = U(1)3 × �n−1 i=2 U(i)2 × SU(n)H Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='25) Let us now compute the IR Coulomb 0-form symmetry group using the arguments of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since every unitary gauge node is balanced in the sub-quivers associated to punctures P1 and P2, we have YF,P1 = YF,P2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' On the other hand, the U(1) gauge node comprising the sub-quiver associated to P3 contributes a genuine local operator of charge n under U(1) factor of ZF (and charge 0 under each (Zn)i factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is precisely the contribution YF,P3 of [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The contribution �YF is obtained from the monopole operator O, whose charge under (Zn)1 × (Zn)2 factor of ZF is the same as that of the representation F1 ⊗ F2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='26) of SU(n)1 × SU(n)2 factor of F, where Fi is the fundamental representation of SU(n)i ⊂ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is because the su(n)H node hits each balanced sub-quiver i ∈ {1, 2} at the node of Dynkin diagram of su(n)i corresponding to the fundamental representation of su(n)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Moreover, O also has charge +1 under U(1) factor of ZF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These contributions match those appearing in [64], and as described there the flavor symmetry group can written as F = SU(n)1 × SU(n)2 × U(1) Zn × Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='27) We refer the reader to [64] for more details regarding the identity of the two Zn subgroups appearing in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As a final example, let us consider a Class S theory involving more general regular punctures that are neither maximal nor minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We are compactifying A3 N = (2, 0) theory on a sphere with three punctures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The puncture P1 has partition ρ1 = {2, 1, 1}, the puncture P2 has partition ρ2 = {2, 2}, and the puncture P3 is a maximal puncture with partition ρ3 = {1, 1, 1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This was discussed as the third example in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 of [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor symmetry algebras associated to the punctures are f1 = su(2)1 ⊕ u(1), f2 = su(2)2 and f3 = su(4), with the total flavor symmetry algebra being f = su(2)1 ⊕ u(1) ⊕ su(2)2 ⊕ su(4) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='28) – 47 – with associated F = SU(2)1 × U(1) × SU(2)2 × SU(4) with center ZF = (Z2)1 × U(1) × (Z2)2 × Z4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='29) The associated magnetic quiver is u(2) su(4)H u(3) u(2) u(1) u(2) u(1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='30) with gauge group G = U(1)2 × U(2)3 × U(3) × SU(4)H Z4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='31) Let us now compute the IR Coulomb 0-form symmetry group using the arguments of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since every unitary gauge node is balanced in the sub-quivers associated to punctures P2 and P3, we have YF,P2 = YF,P3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' On the other hand, the unbalanced u(2) gauge node in the sub-quiver associated to P1 contributes a genuine local operator whose charge under (Z2)1 factor of ZF is same as that of the fundamental representation F1 of SU(2)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is because this unbalanced u(2) gauge node hits once the Dynkin diagram of su(2)1 formed by the u(1) gauge node in the sub-quiver associated to P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Moreover, this genuine local operator also has charge 4 under the U(1) factor of ZF which arises from this unbalanced u(2) gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The contribution �YF is obtained from the monopole operator O, whose charge under (Z2)1 × (Z2)2 × Z4 factor of ZF is the same as that of the representation F2 ⊗ F (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='32) of SU(2)1×SU(2)2×SU(4) factor of F, where F2 is the fundamental representation of SU(2)2 and F is the fundamental representation of SU(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is because the su(4)H node does not hit the Dynkin diagram of su(2)1, while hitting the su(2)2 and su(4) Dynkin diagrams at the nodes corresponding to the fundamental representations of su(2)2 and su(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Moreover, O also has charge 2 under U(1) factor of ZF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since all charges under U(1) factor of ZF are even, we can scale them half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The scaled contributions match those appearing in [64], and as described there the flavor symmetry group can written as F = SU(2)1 × U(1) × SU(2)2 × SU(4) Z4 × Z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='33) We refer the reader to [64] for more details regarding the identity of the Z4 and Z2 subgroups appearing in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 48 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 5d SCFTs 5d superconformal field theories (SCFTs) with 8 supercharges are closely related to 3d N = 4 theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The proposed correspondence is that the Higgs branch of the 5d SCFT is given by the Coulomb branch of the 3d N = 4 IR SCFT arising from the associated magnetic quiver (MQ) [21, 22], which is again a 3d N = 4 quiver gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This conjecture has passed numerous non-trivial tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It can be motivated from the 5d brane-web realization of 5d SCFTs [19, 22, 25, 26, 30, 31], but also via a geometric construction of the 5d theory in M-theory on a canonical singularity: in the case of isolated hypersurface singularities, the MQs for the 5d theories can be derived from the geometry [27, 34, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One salient feature of 5d SCFTs is the flavor symmetry, which often is enhanced compared to the flavor symmetry of an IR gauge theory description obtained in the IR after performing a mass deformation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' moving onto the extended Coulomb branch) of the UV 5d SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The simplest class of such models are the Seiberg En+1 theories having UV flavor symmetry en+1, which after a mass deformation give rise to SU(2) + nF gauge theories in the IR with IR flavor symmetry so(2n) ⊕ u(1) [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor symmetry is encoded in terms of the magnetic quiver as well: for simplicity let us consider MQs which are built from � i U(ni) gauge nodes (along with the additional instruction of a ungauging a U(1) for each connected component of the MQ), connected by bifundamentals such that there is a single bifundamental between any two nodes and the resulting quiver has no loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Then the balanced unitary nodes give rise to the non-abelian part of the flavor symmetry algebra and the unbalanced nodes give rise to the abelian part of the flavor symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The global form can be obtained using the methods in this paper, specifically section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Note that the analysis of that section is applied after making a suitable choices of U(1) ungaugings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In this section we will compare the global form of the flavor symmetries of the 5d SCFTs and their associated 3d MQ theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The examples we will focus on are rank 2 theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A complete list of all MQs for rank 2 theories can be found in [31], using the method of [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor symmetry can be determined alternatively from geometry as in [66, 78–83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The models we consider are shown in table 1 and their magnetic quivers are in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The 5d theories have gauge theory descriptions with SU(3) gauge groups and fundamental flavor and thus no 1-form symmetry (which in principle upon dimensional reduction can contribute to the 0-form symmetry of the MQ theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 Flavor Symmetry Groups from MQs To derive the flavor symmetry from the MQs we simply apply the special cases of our general analysis discussed in sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The MQs are all listed in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' All nodes are unitary: balanced nodes are white, unbalanced ones are black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The magnetic quiver is obtained by ungauging a U(1) in each connected component of the listed quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is a choice in this, and we will pick the ungaugings that allow us to apply the analysis of sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 49 – Model Gauge Theory Flavor algebra 2 SU(3)1/2 + 9F so(20) 5 SU(3)1 + 8F so(16) ⊕ su(2) 12 SU(3)0 + 6F su(6) ⊕ su(2)2 9 SU(3)3/2 + 7F so(14) ⊕ u(1) 26 SU(3)2 + 4F su(5) ⊕ u(1) Table 1: Rank 2 5d SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We list the IR gauge theory description as well as the flavor symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We determine the group structure from both 5d and the corresponding 3d magnetic quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' To determine the flavor symmetry group from the magnetic quiver for the cases listed in the table 2, we have to simply follow the following rules: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Pick a connected component α of the listed magnetic quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Determine the set of balanced nodes, which form a non-abelian Lie algebra fα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The number of unbalanced nodes Uα determines an abelian Lie algebra u(1)|Uα−1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Coulomb flavor symmetry algebra f of the 3d IR SCFT associated to the full magnetic quiver is obtained by picking the component β with the maximal associated algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' f = fβ ⊕ u(1)|Uβ−1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Ungauge a u(1) in the each connected component α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is done at the location of an unbalanced node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' If the unbalanced node is u(n) for n > 1, then we are left with a special unitary gauge node su(n), and the gauge group associated to the connected component α is Gα = � i U(ni) × SU(n) Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='34) That is the center Zn of the numerator is automatically removed in the U(1) ungauging process, as discussed in [75] and after equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' On the other hand, if the unbalanced node is u(1), then we are left with a fundamental flavor there and the gauge group associated to the connected component α is Gα = � i U(ni) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='35) After the ungauging, for the cases appearing in table 2, we are left with either no unbalanced nodes or another unbalanced unitary gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The position of the unbalanced unitary or special unitary node determines the repre- sentation under f of the gauge monopole operators as described in sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We account for monopole operators coming from all connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From this the global form of the flavor symmetry group is determined, by quotienting out the part of the flavor center (of the simply connected group associated to f) that acts trivially on the monopole operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 50 – Model Magnetic Quiver Flavor Group 2 1 2 3 4 5 6 7 8 5 2 4 Ss(20) 5 1 2 3 4 5 6 4 2 1 3 Ss(16)×SU(2) Z2 12 1 2 3 2 1 1 2 1 SU(6)/Z3×SO(4) Z2 9 1 2 3 4 5 3 1 3 1 Spin(14)×U(1) Z4 26 1 2 2 1 1 1 1 1 1 1 1 U(5) Table 2: The Magnetic Quivers for the 5d SCFTs listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Each node labeled by n corresponds to a U(n) gauge node, and connection lines to bi-fundamentals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The subgraph given by the white nodes is the Dynkin diagram of the non-abelian part of the flavor symmetry algebra (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' the balanced nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The black are unbalanced nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The MQ is obtained by the ungauging of one of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We will now determine the global form of the flavor symmetry groups in the examples of table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The balanced (white) nodes in the magnetic quiver form the Dynkin diagram of f = so(20) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='36) There is no abelian factor in the flavor algebra because there is a single unbalanced node (shown in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We ungauge the U(1) at the location of this unbalanced node and land in – 51 – the special case of our general analysis discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The gauge group is G = �8 i=1 U(i) × U(4) × U(5) × SU(2) Z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='37) The unbalanced special unitary node is attached to the spinor node of so(20), and thus we have a monopole operator transforming in the spinor representation7 of so(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since we do not have a cospinor representation, we can remove the Z2 subgroup of the Z2 × Z2 center of Spin(20), which acts on cospinor representation, but leaves the spinor representation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus the flavor symmetry group is F = Spin(20)/Z2 = Ss(20) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='38) where the group Ss(20) is a global form of so(20) which admits spinor representation but does not admit cospinor or vector representations of so(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Here the balanced nodes form f = so(16) ⊕ su(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is no abelian factor in the flavor algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' After ungauging U(1) at the location of the unbalanced U(2) gauge node, we obtain the MQ whose gauge group is G = �6 i=1 U(i) × U(3) × U(4) × SU(2) × U(1) Z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='39) The unbalanced special unitary node is attached to the spinor node of so(16) and the funda- mental node of su(2), implying that we have a monopole operator transforming in represen- tation (S, F) of so(16)⊕su(2), where S is spinor representation of so(16) and F is fundamental representation of su(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is again no monopole operator transforming in the cospinor, so we can reduce Spin(16) to Ss(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The center of Ss(16) is Z2 which acts non-trivially on the spinor representation, and the Z2 center of SU(2) acts non-trivially on the fundamental representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus the (S, F) monopole operator is left invariant by the diagonal Z2 and we can express the flavor symmetry group as F = Ss(16) × SU(2) Z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='40) Model 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Similarly for model 12, we have f = su(6) ⊕ su(2) ⊕ su(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='41) The unbalanced special unitary node (obtained after U(1) ungauging) intersects the Dynkin diagrams of simple components of f such that we have a monopole operator in representation (Λ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' F) of f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' where Λ3 is the 3-index antisymmetric irreducible representation of dimension 7More precisely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' we are only claiming that we have a monopole operator transforming in a representation of so(20) having same charges under the center of the simply connected group Spin(20) as the spinor repre- sentation of so(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' However, for brevity here and in what follows, we will blur this distinction, but the reader should keep this in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 52 – 20 of su(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since only Λ3 of su(6) appears, we can begin with the smallest global form SU(6)/Z3 of su(6) allowing this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The center of SU(6)/Z3 is Z2 which acts non- trivially on Λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Similarly, since we only have the bifundamental (F, F) of su(2)⊕su(2) = so(4), we can begin with its smallest global form SO(4) allowing this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The center of SO(4) is Z2 which acts non-trivially on (F, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, the diagonal Z2 of the Z2 centers of SU(6)/Z3 and SO(4) acts trivially on (Λ3, F, F) the monopole, leading to the flavor symmetry group F = SU(6)/Z3 × SO(4) Z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='42) Model 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The balanced nodes provide an so(14) flavor algebra and the unbalanced nodes provide a u(1) flavor algebra, since we have 2 unbalanced nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The total flavor algebra is thus f = so(14) ⊕ u(1)C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='43) We choose to ungauge one of the unbalanced U(1) nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The gauge group is thus G = 5 � i=1 U(i) × U(3)2 × U(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='44) We have to now apply the analysis of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since the unbalanced U(1) gauge node is attached to the Dynkin diagram of so(14) at the location of co-spinor node, the monopole operator associated to the unbalanced U(1) node transforms in the cospinor irreducible repre- sentation C of so(14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Simultaneously, the monopole operator also carries a charge +1 under a global form U(1)C of the u(1)C factor of f arising from this unbalanced node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Since, the representation C has charge −1 under the Z4 center of the simply connected group Spin(14) associated to so(14), the monopole operator discussed above is uncharged under the diagonal combination of the Z4 center of Spin(14) and the Z4 subgroup of the group U(1)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor symmetry group is thus F = Spin(14) × U(1) Z4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='45) Model 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There are two connected components in the MQ corresponding to two branches of the moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The first component gives rise to algebra f1 = su(5), while the second component gives rise to a larger algebra f2 = su(5) ⊕ u(1)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor symmetry algebra is thus f = su(5) ⊕ u(1)C (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='46) provided by the second component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Ungauging the unbalanced U(1) gauge node in the first component results in an MQ with balanced unitary gauge nodes only, and thus we do not obtain any monopole operator relevant for the analysis of flavor group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Ungauging an unbal- anced U(1) gauge node in the second component leaves behind another unbalanced U(1) gauge node which provides a monopole operator transforming in anti-fundamental representation of – 53 – su(5) along with charge +1 under U(1)C, leading to the flavor group F = SU(5) × U(1)C Z5 = U(5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='47) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 Flavor Symmetry Groups from String Theory Constructions Reference [66] described a computation of flavor symmetry groups of 5d SCFTs using their string theory constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The key physical idea involved is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Study the 5d conformal theory on a flat spacetime with a non-conformal vacuum at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' More precisely, one chooses a supersymmetric non-conformal vacuum lying in the Coulomb branch of vacua8 of the 5d SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The theory now flows and in the IR we obtain a 5d supersymmetric gauge theory with an abelian gauge group U(1)r, where r is referred to as the rank of the 5d SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In addition, we have massive BPS particles9 charged under U(1)r and the flavor symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor group is then obtained easily by computing flavor charges of gauge invariant combinations of the above charges, namely those linear combinations which have zero U(1)r gauge charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charges under U(1)r can be read off from any string theory construction, but the charges under flavor symmetry require us to use “good” string theory constructions, namely those that manifest the full enhanced flavor symmetry algebra of the 5d SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As is well- known, there are two main kinds of string theory constructions of 5d SCFTs: the first kind involves compactifications of M-theory on Calabi-Yau threefolds while the second kind in- volves intersecting brane configurations in Type IIB superstring theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is always a good M-theory construction, which we will now focus on to compute explicitly the flavor symmetry group of the 5d SCFTs appearing in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In an M-theory construction, the charges of all BPS particles can be captured by charges of a special set of BPS particles arising from M2 branes wrapping irreducible holomorphic curves in the Calabi-Yau threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The gauge/flavor charges are described by intersection numbers of these curves with compact/non-compact divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus the set of data required for computation of flavor groups is a set C of holomorphic curves (whose charges span charges of all other curves) along with their intersection numbers with compact and non-compact divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Recently, a lot of work has been performed on the computation of such a set C of curves along with their intersection numbers, and crucially the intersection numbers with non-compact divisors capturing enhanced flavor symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' These works have used various geometric techniques involving blow-downs of flat [84, 85] and non-flat [78–81, 86] resolutions of non-minimal elliptic fibrations, along with more general local surface geometry structures 8It is important not to confuse this with the extended Coulomb branch, which is simply referred to as the Coulomb branch in many studies on 5d SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The extended Coulomb branch is a space obtained by fibering Coulomb branch of vacua on the base space comprising of a family of theories obtained from the 5d SCFT by performing supersymmetric mass deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Coulomb branch of vacua being referred here is the fiber at the origin (namely the point with zero mass deformations) of the base space of extended Coulomb branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 9One might worry about non-BPS excitations and whether they provide any additional charges that can modify the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From the string theory constructions, it is possible to argue that they do not provide any new charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 54 – [82, 83, 87–93] and description of the associated Calabi-Yau K¨ahler cone structure [78–81, 94] that interpolates between different resolutions via flop transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We now describe the spanning set C of curves and the relevant charges of BPS particles associated to these curves, using this information to compute the flavor group for all the models appearing in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Detailed computation of the set C and its intersection numbers can be found in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One can use any of the above described methods to compute a sufficient spanning set C of curves and its intersection numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One finds that C contains four curves whose charges are q(C1) = � 0, 1|0 (mod 2), 1 (mod 2) � q(C2) = � − 2, 2|0 (mod 2), 0 (mod 2) � q(C3) = � 2, −1|1 (mod 2), 1 (mod 2) � q(C4) = � 1, −1|1 (mod 2), 1 (mod 2) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='48) where the first two charges are under the U(1)2 gauge group, and the last two charges are under the ZS 2×ZC 2 center of Spin(20) simply connected group associated to the flavor symmetry algebra f = so(20), where ZS 2 is the subgroup of the center under which spinor and vector are charged and ZC 2 is the subgroup of the center under which co-spinor and vector are charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From this the reader easily sees that all gauge-invariant linear combinations of these curves are either trivially charged under ZS 2 × ZC 2 or have charge � 1 (mod 2), 0 (mod 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus we are again led to the flavor group F = Ss(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The above curves Ci have a nice physical interpretation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Perform a mass deformation of the 5d SCFT such that it flows in the IR to 5d N = 1 gauge theory with gauge group Sp(2) and 9 fundamental hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Then, the curve C1 gives rise to a BPS instanton of the gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The curves C2 and C3 give rise to W-bosons of Sp(2) upon moving onto the Coulomb branch of this gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Finally, the curve C4 gives rise to one of the 9 hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The analysis is similar to the above case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We again have four curves Ci which have same physical interpretation after mass deforming to 5d N = 1 gauge theory with gauge group Sp(2) and 8 fundamental hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One can use any of the above described methods to compute that these curves have charges q(C1) = � 0, 1|1 (mod 2), 0 (mod 2), 0 (mod 2) � q(C2) = � − 2, 2|0 (mod 2), 0 (mod 2), 0 (mod 2) � q(C3) = � 2, −1|0 (mod 2), 0 (mod 2), 1 (mod 2) � q(C4) = � 1, −1|1 (mod 2), 1 (mod 2), 0 (mod 2) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='49) where the first two charges are under the U(1)2 gauge group, the next two charges are under the ZS 2 × ZC 2 center of Spin(16) simply connected group associated to so(16) ⊂ f, and the last charge is under the Z2 center of SU(2) simply connected group associated to su(2) ⊂ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From – 55 – this the reader easily sees that all gauge-invariant linear combinations of these curves are either trivially charged under ZS 2 ×ZC 2 ×Z2 or have charge � 1 (mod 2), 0 (mod 2), 1 (mod 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus we are again led to the flavor group described in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Model 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It is again sufficient to consider four curves Ci which have similar physical interpretation as above after mass deforming to 5d N = 1 gauge theory with gauge group SU(3), Chern-Simons level 0, and 6 fundamental hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One can use any of the above described methods to compute that these curves have charges q(C1) = � 0, 0|0 (mod 6), 0 (mod 2), 0 (mod 2) � q(C2) = � − 1, 2|0 (mod 6), 0 (mod 2), 1 (mod 2) � q(C3) = � 2, −1|0 (mod 6), 1 (mod 2), 0 (mod 2) � q(C4) = � 1, −1|1 (mod 6), 0 (mod 2), 0 (mod 2) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='50) where the first two charges are under the U(1)2 gauge group, the next charge is under the Z6 center of SU(6) simply connected group associated to su(6) ⊂ f, and the last two charges are under the ZS 2 × ZC 2 center of Spin(4) = SU(2) × SU(2) simply connected group associated to so(4) = su(2) ⊕ su(2) ⊂ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From this the reader can see that all gauge-invariant linear combinations of these curves are either trivially charged under Z6 × ZS 2 × ZC 2 or have charge � 3 (mod 6), 1 (mod 2), 1 (mod 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus we are again led to the flavor group described in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Model 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It is again sufficient to consider four curves Ci which have similar physical inter- pretation as above after mass deforming to 5d N = 1 gauge theory with gauge group Sp(2) and 7 fundamental hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One can use any of the above described methods to compute that these curves have charges q(C1) = � 0, 1|3 (mod 4), 0 � q(C2) = � − 2, 2|0 (mod 4), 0 � q(C3) = � 2, −1|0 (mod 4), −1 � q(C4) = � 1, −1|2 (mod 4), 0 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='51) where the first two charges are under the U(1)2 gauge group, the next charge is under the Z4 center of Spin(14) simply connected group associated to so(14) ⊂ f, and the last charge is under the U(1) group associated to u(1) ⊂ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From this the reader can see that all gauge- invariant linear combinations of these curves are either trivially charged under Z4 × U(1) or have charge a multiple of � 1 (mod 4), −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus we are again led to the flavor group described in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Model 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It is again sufficient to consider four curves Ci which have similar physical interpretation as above after mass deforming to 5d N = 1 gauge theory with gauge group SU(3), Chern-Simons level 2 and 7 fundamental hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' One can use any of the above – 56 – described methods to compute that these curves have charges q(C1) = � − 1, 1|4 (mod 5), 0 � q(C2) = � − 1, 2|1 (mod 5), 0 � q(C3) = � 2, −1|0 (mod 5), −1 � q(C4) = � 1, −1|1 (mod 5), 0 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='52) where the first two charges are under the U(1)2 gauge group, the next charge is under the Z5 center of SU(5) simply connected group associated to su(5) ⊂ f, and the last charge is under the U(1) group associated to u(1) ⊂ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From this the reader can see that all gauge-invariant linear combinations of these curves are either trivially charged under Z5×U(1) or have charge a multiple of � 1 (mod 5), −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus we are again led to the flavor group described in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3 3d Mirror Symmetry As a final consistency check, we can apply our methods to compute and match symmetries and anomalies of two 3d N = 4 gauge theories that are related by 3d mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us illustrate this with the general example of 3d N = 4 gauge theory U(m) [su(2m)H] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='53) namely U(m) gauge theory with 2m fundamental hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Higgs flavor symmetry algebra is fH = su(2m)H (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='54) rotating the fundamental hypers, and the IR Coulomb flavor symmetry algebra is fIR C = su(2)C (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='55) because the U(m) gauge node is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Higgs 0-form symmetry group is FH = PSU(2m)H (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='56) because the full center Z2m of SU(2m)H is a subgroup of the U(1) center of the U(m) gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From the analysis of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5, the IR Coulomb 0-form symmetry group is FIR C = SO(3)H (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='57) as there are no unbalanced unitary or special unitary gauge nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From the analysis of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5 we also see that there is mixed flavor-gauge monopole operator having winding number 1 around a U(1) subgroup of the maximal torus of PSU(2m)H and a charge under the Z2 center of SU(2)C same as that of the fundamental representation of SU(2)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is because the flavor node [su(2m)H] hits the Dynkin diagram of su(2)C at the node corresponding to – 57 – fundamental representation of su(2)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This translates to a mixed ’t Hooft anomaly of the IR SCFT between the Higgs and Coulomb 0-form symmetry groups of the form AIR 4 = exp � πi � wH 2 ∪ wC 2 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='58) where wC 2 is the Z2 valued second Stiefel-Whitney class of the background SO(3)C bundle and wH 2 is the Z2m valued obstruction class for lifting background PSU(2m)H bundles to SU(2m)H bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Now consider its 3d N = 4 mirror gauge theory [4] U(m − 1) U(m) · · U(2) U(1) U(m − 1) · · U(2) U(1) [su(2)H] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='59) The Higgs flavor symmetry algebra is fH = su(2)H (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='60) rotating the two fundamental hypers of U(m) gauge group, and the IR Coulomb flavor sym- metry algebra is fIR C = su(2m)C (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='61) because all the unitary gauge nodes are balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Higgs 0-form symmetry group is FH = SO(3)H (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='62) because the center Z2 of SU(2)H is a subgroup of the U(1) center of each unitary gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From the analysis of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5, the IR Coulomb 0-form symmetry group is FIR C = PSU(2m)H (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='63) as there are no unbalanced unitary or special unitary gauge nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From the analysis of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5 we also see that there is mixed flavor-gauge monopole operator having winding number 1 around the maximal torus of SO(3)H and a charge under the Z2m center of SU(2m)C same as that of the irreducible representation of su(2m)C whose highest weight has a single non- zero Dynkin coefficient, namely the m-th one with dm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is because the flavor node [su(2)H] hits the Dynkin diagram of su(2m)C at the node corresponding to this representation of su(2m)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This translates to a mixed ’t Hooft anomaly of the IR SCFT between the Higgs and Coulomb 0-form symmetry groups of the form AIR 4 = exp � πi � wC 2 ∪ wH 2 � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='64) – 58 – where wH 2 is the Z2 valued second Stiefel-Whitney class of the background SO(3)H bundle and wC 2 is the Z2m valued obstruction class for lifting background PSU(2m)C bundles to SU(2m)C bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, we have seen explicitly that the symmetry and anomaly properties of the two mirror theory are same upto the exchange of labels C ↔ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 6 Some Generalizations In this section, we discuss a few generalizations of the general considerations of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We will discuss two different types of generalizations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In the first generalization, we will allow ourselves to perform an N = 2 gauging of flavor symmetries of 3d N = 4 theories along with the addition of a Chern-Simons level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We will see that the Chern-Simons level induces a ’t Hooft anomaly purely for the 1-form symmetry, which is novel feature that we have not encountered in the N = 4 theories that we studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Related models have appeared in [70], motived from the study of T[M3] compactifications of 6d theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In the second generalization, we will allow ourselves to perform gaugings of discrete subgroups of flavor symmetries of 3d N = 4 theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We will see that this opens up the possibility of having non-trivial 2-group symmetries within the context of the study of 3d N = 4 theories involving unitary and special unitary gauge groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We will also encounter the presence of mixed ’t Hooft anomalies between these 2-group symmetries and 0-form symmetries of the 3d N = 4 theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1 N = 2 Gauging of T[SU(n)] In this subsection we study N = 2 gaugings (possibly with Chern-Simons levels) of su(n)H Higgs flavor symmetry of T[SU(n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We begin with n = 2 and later generalize to arbitrary n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We find that none of these have a non-trivial 2-group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Consider the 3d N = 2 theory obtained by an N = 2 gauging of the su(2)H flavor symmetry of the 3d N = 4 theory T[SU(2)] by an SU(2)H gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In addition, we turn on a Chern-Simons level k for the SU(2)H gauge group while preserving the N = 2 supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Due to this Chern-Simons level, non-fractional gauge monopole operators (which are genuine local operators) start transforming in representations of SO(3)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For such monopole operators to be gauge invariant, they must arise at the ends of Wilson line defects associated to representations of SO(3)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' However, this clearly does not impact the 1-form symmetry, and we have Γ(1) = Z2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1) just as for the case of N = 4 SU(2)H gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The 0-form symmetry is also the same as for the N = 4 gauging F = SO(3)C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2) – 59 – One can argue just as for the case of N = 4 SU(2)H gauging that Z2 and SO(3)C do not combine to form a 2-group symmetry with a non-trivial Postnikov class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Purely 1-Form Anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Consider instead a fractional gauge monopole operator O la- beled by a co-character of SU(2)H having winding number half around its maximal torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Due to Chern-Simons level k, the operator O transforms in a representation R of SU(2)H with charge k (mod 2) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3) under the Z2 center of SU(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For O to be gauge invariant, it must arise at the end of of a Wilson line defect associated to representation R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Using the analysis of [12], this fact is equivalent to a ’t Hooft anomaly for the 1-form symmetry of the form A(1) 4 = exp � πik � P(B2) 2 � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4) where P(B2) is the Pontryagin square of B2 and is a Z4 valued class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This class is even on spin manifolds, and one can hence define a Z2 valued class 1 2P(B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The anomaly is this Z2 valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It vanishes for k even, but is non-trivial for k odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Mixed 1-Form 0-Form Anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The fractional gauge monopole operator O also trans- forms in a representation of SU(2)C that is not a representation of SO(3)C, just as for the case of N = 4 gauging of T[SU(2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is equivalent to a mixed ’t Hooft anomaly between the Z2 1-form and SO(3)C 0-form symmetries, and the full ’t Hooft anomaly can be expressed as A4 = exp � πi � k P(B2) 2 + B2 ∪ wC 2 � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5) Generalization to T[SU(n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It is straightforward to generalize to arbitrary n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We are studying 3d N = 2 theory obtained by an N = 2 gauging of the su(n)H Higgs flavor symmetry of the 3d N = 4 theory T[SU(n)] by an SU(n)H gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In addition, we turn on a Chern-Simons level k for the SU(n)H gauge group while preserving the N = 2 supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The non-fractional monopole operators transform in representations of PSU(n)H and so the 1-form symmetry is Γ(1) = Zn (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6) just as for the case of N = 4 SU(n)H gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The 0-form symmetry is also the same as for the N = 4 gauging F = PSU(n)C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='7) One can argue just as for the case of N = 4 SU(n)H gauging that Zn and PSU(n)C do not combine to form a 2-group symmetry with a non-trivial Postnikov class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A fractional gauge monopole operator O labeled by a co-character of SU(n)H having winding number 1/n around its maximal torus transforms, due to Chern-Simons level k, in a representation R of SU(n)H with charge k (mod n) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='8) – 60 – under the Zn center of SU(n)H, implying a ’t Hooft anomaly for the 1-form symmetry of the form A(1) 4 = exp �2πik n � Pσ(n)(B2) 2 � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='9) where σ(n) = 0, 1 depending on whether n is even, odd respectively, and P0(B2) 2 := P(B2) 2 P1(B2) 2 := B2 ∪ B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='10) The Pontryagin square P(B2) is Z2n valued and is even for spin manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As a consequence, its half P(B2)/2 is Zn valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' On the other hand, B2 ∪ B2 is naturally Zn valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The fractional gauge monopole operator O also transforms in a representation of SU(n)C with charge −1 (mod n) under its Zn center, just as for the case of N = 4 gauging of T[SU(n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is equivalent to a mixed ’t Hooft anomaly between the Zn 1-form and PSU(n)C 0-form symmetries, and the full ’t Hooft anomaly can be expressed as A4 = exp �2πi n � k Pσ(n)(B2) 2 − B2 ∪ wC 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='11) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2 T[SU(2)]/ZC 2 and Its Gaugings In this subsection, we study the gauging of a Z2 subgroup of the SO(3)C 0-form symmetry of T[SU(2)], which leads to a theory with a non-trivial 2-group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We also find a mixed anomaly between this 2-group symmetry and the residual Coulomb 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Finally, we study the N = 4 gauging of su(2)H Higgs flavor symmetry of the theory T[SU(2)]/ZC 2 obtained after the Z2 gauging of T[SU(2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us consider the 3d N = 4 Lagrangian theory U(1) [su(2)H] 2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) which denotes a theory having a U(1) gauge group along with 2 hypermultiplets of charge 2 that are rotated by an su(2)H Higgs flavor symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This theory can be obtained by gauging the Z2 subgroup, denoted ZC 2 , of U(1)C Coulomb 0-form symmetry of the Lagrangian theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1) discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We are also interested in the 3d N = 4 SCFT that the above Lagrangian theory flows to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We call this SCFT T[SU(2)]/ZC 2 because it can be obtained by gauging a Z2 subgroup, denoted ZC 2 , of the SO(3)C 0-form symmetry of the 3d N = 4 SCFT T[SU(2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is a consequence of the fact that the UV theory (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) is obtained by gauging Z2 subgroup of U(1)C 0-form symmetry of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1), and this U(1)C symmetry embeds as the maximal torus of SO(3)C 0-form symmetry of the corresponding IR SCFT T[SU(2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 61 – 1-Form Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The symmetries and anomalies of the UV theory were discussed in detail in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4 of [12], which we review and use to deduce symmetry and anomalies of the IR SCFT T[SU(2)]/ZC 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' First of all, there is a Γ(1) = Z2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) 1-form symmetry coming from the fact that Wilson lines of odd U(1) charges cannot be screened in the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This becomes the 1-form symmetry of the IR SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This 1-form symmetry can also be understood as the dual symmetry arising from the perspective of ZC 2 0-form gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Higgs 0-Form Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Higgs 0-form symmetry algebra of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) is fH = su(2)H (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='14) and the Higgs 0-form symmetry group is FH = SO(3)H , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='15) because the genuine local operators charged under su(2)H are gauge-invariant combinations of hypermultiplets which all have trivial charge under Z2 center of SU(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The IR SCFT admits the same Higgs 0-form symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Coulomb 0-Form Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We label the Coulomb 0-form symmetry group of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) arising from the U(1) gauge node as FC = U(1)′ C = U(1)C/Z2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='16) to distinguish it from the U(1)C 0-form symmetry of the theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The IR SCFT admits the same Coulomb 0-form symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It can be checked easily using the analysis of [5] that there is no enhancement due to monopole operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Alternatively, one can understand it from the point of view of gauging ZC 2 subgroup of SO(3)C 0-form symmetry group of T[SU(2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' To compute the residual 0-form symmetry after this gauging, we first compute the commutant of ZC 2 in SO(3)C, which is the maximal torus U(1)C of SO(3)C, and then we mod out the commutant by ZC 2 to find that the residual 0-form symmetry is U(1)′ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' 2-Group Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' There is a non-trivial 2-group symmetry formed by Z2 1-form sym- metry and SO(3)H 0-form symmetry, with Postnikov class δB2 = wH 3 = Bock(wH 2 ) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='17) where wH 3 is the third Stiefel-Whitney class of background SO(3)H bundles, which can be obtained by applying Bockstein homomorphism associated to the non-split short exact se- quence 0 → Z2 → Z4 → Z2 → 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='18) – 62 – on the second Stiefel-Whitney class wH 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This 2-group symmetry is a consequence of the fact that even though Wilson line opera- tors of even charge can be screened, the local operators responsible for screening them have different su(2)H representations depending on whether the charge of Wilson line is a multiple of 4 or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The non-genuine local operators living at the ends of Wilson line operators of charge 4m form SO(3)H representations, while the non-genuine local operators living at the ends of Wilson line operators of charge 4m + 2 form su(2)H representations that are not allowed representations of SO(3)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The IR SCFT T[SU(2)]/ZC 2 also carries this 2-group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Mixed 2-Group 0-Form ’t Hooft Anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The structure group of the gauge theory (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) involving the Higgs 0-form symmetry is S = U(1) × SU(2)H Z4 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='19) where the Z4 in the denominator is obtained by combining the Z4 subgroup of U(1) with the Z2 center of SU(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The Z4 in the denominator can actually be identified with the Z4 group appearing in the short exact sequence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='18) appearing in the description of the 2-group symmetry of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We can thus consider a mixed flavor-gauge monopole operator O associated to a co- character of S with winding number 1/4 around U(1) and winding number 1/2 around the maximal torus of SU(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This monopole operator O is a solitonic defect associated to the 2-group symmetry described above because its obstruction to being lifted to a combination of purely (and non-fractional) gauge and purely flavor monopole operator is captured by the Z4 group in the denominator of the structure group S which as remarked above is associated to the 2-group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' See [12] for a more details regarding such solitonic defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The monopole operator O has charge q = 1/4 under U(1)′ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' From the analysis of [12], this fact is equivalent to a mixed ’t Hooft anomaly between the 2-group and the Coulomb 0-form symmetry of the form A4 = exp �πi 2 � BH w ∪ � c1 � U(1)′ C � (mod 4) �� , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='20) where Bw is a Z4 valued background field associated to the 2-group symmetry comprised out of the Z2 valued background field B2 for 1-form symmetry and the second Stiefel-Whitney class wH 2 for background SO(3)H 0-form symmetry bundles, and c1 � U(1)′ C � is the first Chern class for background U(1)′ C 0-form symmetry bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The above anomaly for (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) descends to an anomaly in the IR SCFT T[SU(2)]/ZC 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Gauging SU(2)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Consider performing N = 4 gauging of su(2)H symmetry of T[SU(2)]/ZC 2 by an SU(2)H gauge group T[SU(2)]/ZC 2 SU(2)H (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='21) – 63 – In the T[SU(2)]/ZC 2 theory we have a line operator L that cannot be screened, but its square 2L can be screened such that a non-genuine local operator living at the end of 2L transforms in a representation of SU(2)H that is not a representation of SO(3)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' After gauging SU(2)H, this non-genuine local operator needs to attached to a SU(2)H Wilson line in the same representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Thus, after gauging SU(2)H, even 2L is not screened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' As a consequence, the 1-form symmetry group is Γ(1) = Z4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='22) The 2-group background Bw in T[SU(2)]/ZC 2 can be identified with the 1-form symmetry background B′ 2 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The 0-form symmetry group remains U(1)′ C, and the mixed ’t Hooft anomaly between 2-group and Coulomb 0-form symmetries of T[SU(2)]/ZC 2 becomes a mixed ’t Hooft anomaly between 1-form and 0-form symmetries of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='21) of the form A4 = exp �πi 2 � B′ 2 ∪ � c1 � U(1)′ C � (mod 4) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='23) Acknowledgments We thank Antoine Bourget, Marcus Sperling, Jingxiang Wu and Zhenghao Zhong for dis- cussions on related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The work of MB is supported by the EPSRC Early Career Fel- lowship EP/T004746/1 “Supersymmetric Gauge Theory and Enumerative Geometry”, STFC Research Grant ST/T000708/1 “Particles, Fields and Spacetime”, and the Simons Collabo- ration on Global Categorical Symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This work is supported by the European Union’s Horizon 2020 Framework through the ERC grants 682608 (LB and SSN) and 787185 (LB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' SSN is supported in part by the “Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics” and the EPSRC Open Fellowship EP/X01276X/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A paper with related results will appear at the same time in [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We thank the authors for coordinating submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A Geometric Computations for 5d SCFTs In this appendix, we provide more details on the geometric computations leading to the charges q(Ci) of BPS particles arising from curves Ci in Calabi-Yau threefolds involved in the construction of 5d SCFTs appearing in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' All the models we consider can be obtained by decoupling from the following 6d geometry, which is a collision between a D10 Kodaira singularity and an I1 smooth fiber, tuned so that the collision results in a non-minimal singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The fibration is best described in terms of a so-called Tate model y2 + b1Uxy + b3U 5δ2 1 = x3 + b2UV x2 + b4U 5δ1x + b6U 10δ4 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='1) – 64 – S1 : U u10 u14 u11 u15 u12 u16 u13 u9 u6 u5 δ1(0) V (0) e11 S2 : U(0) e1 δ2(4) V (0) Figure 1: The surface geometry for the marginal theory of type D10 −I1, which gives rise to the rank 2 5d SCFTs discussed in this section (the labels for the curves is chosen in accord with the derivation of the geometry in [79]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The two compact surfaces are denoted by Si and the collection of rational curves and their intersections are shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The self-intersection of the curves in each surface is either shown next to the sections ui, U, V δi, or is −2 for green and −1 for blue curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Here U = 0 is the D10 Kodaira fiber, and V = 0 the locus above which the I1 singular fiber is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We furthermore blowup the locus U = V = 0 by inserting a rational curve, and denote the exceptional section of that by δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Non-flat resolution of this model was performed in [79] and the geometry is given in figure 1, which we reproduced from said paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The compact surfaces are denoted by Si and are glued along U = 0, which is a degree (−2, 0) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The numbers in the bracket indicate the self-intersection of the curve in the divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The geometry shown is that of the marginal theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' the 6d parent SCFT (on the tensor branch, which is modeled by the curve δ1) on a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Only once we start decoupling matter (which geometrically corresponds to performing blowdowns), do we get a theory that is a genuinely 5d SCFT fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor symmetry is obtained from the geometry as follows: The non-abelian flavor symmetry algebra is obtained from the intersection of compact Sa and non-compact Ni divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The complete intersection curves Sa·Ni will have a normal bundle degree, which if it is (−2, 0) will indicate that the non-compact divisors are ruled by these curves and correspond to the Cartans of the flavor symmetry algebra (for a more refined discussion see [79]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' On the other hand, curves that have normal bundle (−1, −1) are inside the compact divisors, and correspond to hypermultiplets in any associated 5d N = 1 gauge theory description, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' with gauge group Sp(2) and 10 fundamental hypers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The intersection pattern of the (−2, 0) curves give the Dynkin diagram of the flavor – 65 – symmetry algebra, however in order to determine the flavor symmetry group, we need to also determine the charges of the hypers under the flavor and gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Additionally, we need to determine the charges of W-bosons and instantons under the flavor and gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A convenient way of doing so is to convert the resolution information present in figure 1 into a local surface geometry describing explicitly the various compact and non- compact divisors present in the Calabi-Yau along with the intersections of these divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The surface geometry associated to figure 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' the KK-theory, is 19 1 2 2l h+f-� xi N0 N1 1 N2 · · N8 N3 10 N9 f-x-y f e-x2 f x2-x3 f x8-x9 f 2 x9, x10 f-x, y l x-y-e1 f-x2 f-e11 x9-x10 f e e e e e e e e e e , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2) where we used the notation e1 and e11 for the (−1) curves that can be flopped to decouple hyper-multiplets – in accordance with the notation in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The figure shows the compact surfaces Si having been represented as Hirzebruch surfaces ib d where the subscript d is the degree of the Hirzebruch surface and the superscript b is the number of additional blowups performed on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' An edge between two surfaces denotes an intersection between the two surfaces and the labels on the edges describe the curves in the two surfaces that are glued at the locus of the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Multiple edges are denoted by a number between the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A curve e inside a surface denotes the base of the corresponding P1 fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is a compact curve for the Hirzebruch surfaces and a non-compact curve for the non-compact surfaces Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' A curve f inside a surface denotes the fiber of the corresponding P1 fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This is a compact curve for both compact and non-compact surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The curves xi, x, y denote various blowups, and we finally we have defined the curve h := e + df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Moving forward we denote e curve of surface Si as ei and f curve of surface Si as fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 66 – After a single flop (of the curve e11) the surface geometry can be expressed as 110 0 21 2h e+2f-� xi N0 N1 N2 · · N8 N2 10 N9 f-x-y f e-x1-x2 f x2-x3 f x8-x9 f 2 x9, x10 f-x, y f x-y x1-x2 f x9-x10 f e e e e e e e e e e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3) The non-compact surfaces Ni form the Dynkin diagram of the affine Lie algebra so(20)(1) which is associated to the fact that this is actually a 6d SCFT with so(20) flavor symmetry compactified on a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This model is obtained by blowing down the curve f1 − x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This means that we first flop f1 − x1 and then taking the volume of the flopped curve infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This effectively decouples a hypermultiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The affect of this blowdown is that a P1 fibered non-compact surface intersecting f1 − x1 does not remain P1 fibered anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' In (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='3), N1 is the only such non-compact surface since (f1 − x1) · N1 = (f1 − x1) ·S1 (x1 − x2) = 1 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='4) where the intersection number (f1−x1)·N1 is in the Calabi-Yau threefold and the intersection number (f1 − x1) ·S1 (x1 − x2) is in the surface S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The above intersection number is a consequence of the facts that f1 ·S1 xi = xj ·S1 xi = 0 and xi ·S1 xi = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Later we will also use ei ·Si ei = −di, where di is the degree of the Hirzebruch surface Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 67 – After the blowdown we obtain the surface geometry 19 1 21 2h h+f-� xi N0 N2 · · N8 N2 10 N9 f-x-y f e-x2 f x2-x3 f x8-x9 f 2 x9, x10 f-x, y f x-y x9-x10 f e e e e e e e e , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='5) where we have only kept the P1 fibered non-compact surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Taking the gluings into account, we see that all compact curves are (possibly non-positive) linear combinations of the curves ei, fi and the blowups xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Additionally due to the gluing between S1 and S2, we can express e1 as a linear combination of10 the e2, fi and xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We have the identifications C1 = e2, C2 = f2, C3 = f1, C4 = xi , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6) where we can choose any xi because the charges under gauge and flavor centers are same for all xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Let us now compute the charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The charge qi(C) of a compact curve C under U(1) gauge group associated to Si is computed as qi(C) = −C · Si , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='7) which for a genus zero curve is 2 + C ·Si C if C lives in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This implies q2(C1) = 1, q2(C2) = 2, q1(C3) = 2, q1(C4) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='8) On the other hand, if C is in surface Sj then C · Si is computed as C ·Sj Cj,i where Cj,i is the gluing curve in Sj to Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This implies q1(C1) = 0, q1(C2) = −2, q2(C3) = −1, q2(C4) = −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='9) 10Note that the labels 1 and 2 are not interchangeable here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We cannot write e2 in terms of e1, fi and xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 68 – Additionally, we have − C1 · Ni = δi,10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='10) That is, C1 intersects the non-compact surfaces along the cospinor node N10 and hence it has the same charges under the center of Spin(20) as the cospinor representation of Spin(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Similarly, since C2 has no non-trivial intersections with any Ni, it does not contribute any non-trivial charge under the center of Spin(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' On the other hand, we have − C3 · Ni = δi,0 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='11) implying that it has same charges under the center of Spin(20) as the vector representation of Spin(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Finally, for any xi the reader can compute that it has same charges under the center of Spin(20) as the vector representation of Spin(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' For example choosing C4 = x9, we have − C4 · Ni = δi,9 + δi,10 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12) which means it transforms both under ZS 2 and ZC 2 subgroups of the center of Spin(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' This reproduces the charges claimed in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Its surface geometry can be obtained from the surface geometry for model 2 by blowing down the curve f − x2, resulting in the surface geometry 18 2 21 2h h-� xi N0 N3 · · N8 N2 10 N9 f-x-y f e f x3-x4 f x8-x9 f 2 x9, x10 f-x, y f x-y x9-x10 f e e e e e e (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13) Now N0 generates the su(2) ⊂ f and the other Ni generate the so(16) ⊂ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We have the same identifications as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The gauge charges are the same as for model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor center charge of C1 is the same as spinor of Spin(16) as it intersects the spinor node N10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor center charge of C2 is trivial as it does not intersect any Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor center charge of C3 is the same as fundamental of SU(2) as it intersects N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 69 – Finally, the flavor center charge of C4 is the same for all xi and can be easily seen to be that for vector of Spin(16) by choosing C4 = x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Model 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Its surface geometry can be obtained from the surface geometry for model 5 by blowing down the curve f − x3, resulting in the surface geometry 17 1 21 2h h-� xi N1 0 N4 · · N8 N2 10 N9 f-x-y f e f-x x4-x5 f x8-x9 f 2 x9, x10 f-x, y f x-y x9-x10 f e e e e e e (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='14) The blowdown process causes N3 and N0 to become non-P1 fibered non-compact surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' N3 intersects the P1 fibers of remaining P1 fibered non-compact surfaces, and hence cannot give rise to an independent u(1) flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' However, N0 does not intersect the P1 fibers of remaining P1 fibered non-compact surfaces and generates a u(1) factor in the flavor symmetry algebra f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' It should be noted that the curves f and x of N0 both have infinite volume, but their difference f − x has finite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The other Ni shown above generate the so(14) ⊂ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' We have the same identifications as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The gauge charges are the same as for the previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor center charge of C1 is the same as cospinor of Spin(14) as it intersects the cospinor node N10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor center charge of C2 is trivial as it does not intersect any Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The curve C3 carries charge −1 under U(1) flavor as it intersects N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Finally, the flavor center charge of C4 is the same for all xi and can be easily seen to be that for vector of Spin(14) by choosing C4 = x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 70 – Model 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Its surface geometry can be obtained from the surface geometry for model 5 by blowing down curves x9, x10, resulting in the surface geometry 16 2 22 h h-� xi N0 N3 · · N7 N9 e f x3-x4 f x7-x8 f e f e e (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='15) We have the same identifications as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The gauge charges are modified for curves living in S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor center charge of C1 is trivial as its intersection number N9 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor center charge of C2 is the same as fundamental of SU(2) corresponding to N9 as the intersection number of C2 with N9 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Similarly, The flavor center charge of C3 is the same as fundamental of SU(2) corresponding to N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Finally, the flavor center charge of C4 is the same for all xi and can be easily seen to be that of fundamental of SU(6) by choosing C4 = x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Model 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Its surface geometry can be obtained from the surface geometry for model 9 by performing some blowdowns of both types of curves f − xi and xi, resulting in the surface geometry 14 1 21 h+f e-� xi N3 0 N6 · · N8 N1 9 h f − � xi x6-x7 f x8-x9 f e x e e x9 f-x e e (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='16) N0 generates a u(1) factor in the flavor symmetry algebra f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Its curves f and xi are non- compact, but f − � xi is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The other Ni shown above generate the su(5) ⊂ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 71 – We have the same identifications as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The gauge charges are computed as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor center charge of C1 is the same as anti-fundamental of SU(5) as it has intersection −1 with the anti-fundamental node N9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The flavor center charge of C2 is the same as fundamental of SU(5) as it has intersection +1 with the anti-fundamental node N9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' The curve C3 carries charge −1 under U(1) flavor as it intersects N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Finally, the flavor center charge of C4 is the same for all xi and can be easily seen to be that for fundamental of SU(5) by choosing C4 = x6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' More generally for all descendants of the marginal geometry figure 1, we can determine the flavor symmetry group following similar reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' See tables in appendix A of [79] for the flavor algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Sch¨afer-Nameki and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Wang, Fibers add Flavor, Part I: Classification of 5d SCFTs, Flavor Symmetries and BPS States, JHEP 11 (2019) 068, [1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='05404].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' [80] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Apruzzi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Lawrie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Sch¨afer-Nameki and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Wang, Fibers add Flavor, Part II: 5d SCFTs, Gauge Theories, and Dualities, JHEP 03 (2020) 052, [1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='09128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' [81] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Apruzzi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Schafer-Nameki and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Wang, 5d SCFTs from Decoupling and Gluing, JHEP 08 (2020) 153, [1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='04264].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' [82] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Bhardwaj, Flavor symmetry of 5d SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Part I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' General setup, JHEP 09 (2021) 186, [2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13230].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' [83] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Bhardwaj, Flavor symmetry of 5d SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Part II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Applications, JHEP 04 (2021) 221, [2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='13235].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' [84] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Bhardwaj and P.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' [86] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Apruzzi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Lin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Mayrhofer, Phases of 5d SCFTs from M-/F-theory on Non-Flat Fibrations, JHEP 05 (2019) 187, [1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='12400].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' [87] P.' metadata={'source': 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and Singular Fibers, – 76 – JHEP 05 (2014) 048, [1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content='2653].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' [95] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Nawata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Sperling, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Wang and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' Zhong, 3d N = 4 mirror symmetry with 1-form symmetry, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} +page_content=' – 77 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfUQDk/content/2301.02249v1.pdf'} diff --git a/UNAzT4oBgHgl3EQf0_5k/content/tmp_files/2301.01792v1.pdf.txt b/UNAzT4oBgHgl3EQf0_5k/content/tmp_files/2301.01792v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf09499621b6933ae2f39156d905305f2ce8dfc9 --- /dev/null +++ b/UNAzT4oBgHgl3EQf0_5k/content/tmp_files/2301.01792v1.pdf.txt @@ -0,0 +1,1048 @@ +Saturation of fishbone modes by self-generated zonal flows in +tokamak plasmas +G. Brochard, C. Liu, X. Wei, W. Heidbrink, Z. Lin, N. Gorelenkov, +S.D. Pinches, P. Liu, J. H. Nicolau, H. L¨utjens +Abstract +Gyrokinetic and kinetic-MHD simulations of n=1 fishbone modes in DIII-D plasmas find that self-generated zonal flows +can dominate the fishbone saturation. The saturation mechanism is identified in phase space, where the zonal flows +prevent holes and clumps from persisting or drifting in phase space with mode down-chirping, reducing the wave-particle +resonant drive. This saturation is confirmed by quantitative agreement with experimental measurements for both mode +saturation amplitude and neutron emissivity. Zonal flows shearing rate exceeds the drift-wave growth rate, consistent +with the ITB observed in DIII-D plasmas. The deliberate destabilization of fishbones for the development of high +performance scenarios in ITER is then proposed. +Introduction. - Energetic Particles (EPs) in tokamak plas- +mas can destabilize a large spatial range of instabilities that +may lead to their outward transport. This is a critical issue +for burning plasmas as in ITER [1] since such a transport +can degrade the fusion performances, the plasma confine- +ment as well as threaten the reactor’s integrity. This trans- +port therefore needs to be predicted for mitigation strategies +to be incorporated in plasma scenarios. +Fortunately, it was discovered theoretically [2][3][4][5] and +shown numerically [6][7][8][9][10] that instabilities arising at +the microscopic and mesoscopic scales such as drift-waves +and Alfv´en eigenmodes (AEs) are able to excite zonal flows +(ZFs), that can mitigate the saturation amplitudes of these +modes, and therefore the associated EP transport. Besides +this mitigation, the destabilisation of zonal flows can gener- +ate strongly sheared poloidal flows that suppress turbulent +transport by damping drift-waves turbulence [11], resulting +in the formation of an internal transport barrier (ITB) that +greatly enhances plasma confinement [12][13]. Macroscopic +MHD modes triggered by energetic particles such as the fish- +bone instability [14][15] however were not self-consistently +observed to trigger n = m = 0 zonal flows so far. +The +mechanism dominating the fishbone saturation was identi- +fied in nonlinear simulations [16][17][18][19] to be the res- +onant wave-particle trapping due to kinetic nonlinearities, +mode-mode nonlinearities playing a secondary role. +In this Letter, we report the first self-consistent gyrokinetic +simulations finding fishbone saturation by the self-generated +zonal flows, in a DIII-D discharge. This discharge is chosen +for validation purposes to predict the EP transport in a +ITER baseline prefusion scenario [20]. The zonal flows are +found to be force-driven by the fishbone and are the main +mechanism for the fishbone saturation. This mechanism is +observed for the first time in phase space, where zonal flows +prevent hole and clump structures from persisting or drift- +ing in the nonlinear phase, reducing the EP resonant drive. +This saturation by zonal flows is confirmed by experimental +measurements, as simulations including zonal flows are able +to recover quantitatively, for the first time, the mode satura- +tion amplitude and the neutron emissivity drop. Moreover, +the shearing rate generated by the fishbone-induced zonal +flows exceeds the linear growth rate of unstable drift-wave +modes, similar to recent numerical work based on EAST +discharges [21]. This strong E × B suppression is consistent +with the ITB arising experimentally after fishbone bursts in +the DIII-D discharge. It confirms the long suspected role of +fishbones in ITB formation [22], fishbone bursts having been +observed to precede ITBs in ASDEX [23], MAST [24][25], +HL-2A [4] and EAST [26][27] plasmas. Finally, gyrokinetic +simulations find that the fishbone-induced EP transport in +the ITER scenario is marginal, 2% of the core EPs being re- +distributed, similar to previous studies on the alpha fishbone +in ITER DT scenarios [19]. The intentional destabilization +of fishbone modes in ITER scenarios is therefore possibly a +way to enhance fusion performances. +Experimental setup. +- The selected DIII-D discharge +#178631 [28] has a nearly circular oval shape (elongation +κ = 1.17, triangularity δ = 0.07) that is limited on the car- +bon inner wall. The major radius is R0 = 1.74 m, the minor +radius is a = 0.64 m, the toroidal field is 2.0 T, the plasma +current is 0.88 MA, and the line-average electron density is +∼ 2.0 × 1019 m−3. This discharge was chosen primarily be- +cause it has an accurately known, weakly reversed, q profile +with q0 = 1.2, qmin = 1.09, and q95 = 3.8 values that re- +semble the profile predicted for the ITER baseline scenario. +The deuterium, L-mode plasma is heated by 3.8 MW of 81 +1 +arXiv:2301.01792v1 [physics.plasm-ph] 4 Jan 2023 + +keV deuterium beams that are injected in the midplane in +the direction of the plasma current and by 1.0 MW of 2nd +harmonic, central electron cyclotron heating. +Numerical setups. +- The DIII-D discharge #178631 is +studied numerically mostly through gyrokinetic simulations +with the GTC code [6][29][30][31], and with kinetic-MHD +simulations using the M3D-C1 [32][33][34] and XTOR-K +[35][36][37] codes. +GTC capability at simulating MHD +modes was recently verified and validated on DIII-D ex- +periments [38]. The magnetic configuration is reproduced +from the EFIT code at t=1580ms. Plasma profiles are ob- +tained from TRANSP simulations. +To simulate properly +MHD modes, the sum of partial pressures need to add up +to the total pressure in EFIT, which is not always the case +using TRANSP profiles. To ensure it, the EP pressure is +constrained as pf = ptot − pi − pe, given that the uncer- +tainty on EP profiles in TRANSP is the highest. The exper- +imental NBI distribution is reproduced from the NUBEAM +code. Such a distribution is described in our first-principle +simulations with an anisotropic slowing-down model, us- +ing a zero-th order Legendre expansion [39]. +A superpo- +sition of three slowing-downs is used to reproduce the in- +jection energies at nominal, half and third energies. +The +critical velocity is artificially set to recover similar gradi- +ents in the (E, v||/v) phase space. All nonlinear simulations +cover the whole simulation domain, with an edge buffer after +ρT = +� +ψT /ψT,edge = 0.8 in GTC suppressing equilibrium +gradients. GTC retains only the n=1 mode in its simula- +tions, with or without the n=m=0 zonal component, using +kinetic thermal/fast ions and fluid electrons. M3D-C1 cov- +ers low n modes n ∈ [0, 2] with both thermal and fast ions +kinetic effects. Due to the anisotropic nature of the cho- +sen configuration that has βf/βtot = 54% on axis, XTOR-K +only evolves the n=1 mode, as the n=0 mode contains both +equilibrium and perturbed fields in the code, contrarily to +GTC and M3D-C1. +XTOR-K treats kinetically only the +fast ion specie. Convergence studies over spatial grid size, +time step and number of particles per cell were successfully +conducted. +Fishbone mitigation by self-induced zonal flows - The im- +pact of MHD nonlinearities on the n=1 fishbone were pre- +viously examined numerically by keeping side-band n=0-4 +modes, highlighting reduction of initial saturation ampli- +tude [18][21], and generation of n=m=0 sheared poloidal +flows [19][21]. +The role played specifically by zonal flows +in fishbone mitigation was however not identified. The ef- +fects of zonal flows on the fishbone instability are studied +here self-consistently for the first time with the gyrokinetic +GTC code. +A gyrokinetic treatment of zonal flows is es- +sential as it takes into account their collisionless damping +[40], which is absent in the kinetic-MHD formalism without +kinetic thermal ions effects. For the considered DIII-D con- +figuration, a n=1 fishbone mode is linearly unstable, close to +marginal stability at pf,thres = 0.8pf, with a growth rate of +γn=1 = 8.5×104 s−1 and a mode frequency of ω/2π = 17kHz +in GTC simulations. +(a) +(b) +(c) +(d) +Figure 1: Time evolution of (a) the volume-averaged per- +turbed electrostatic potential eφ/Te (n=0,1), and (b) the +the n=1 mode frequency ω, with and without zonal flows +in GTC simulations. The linearly resonant precessional fre- +quency plus the zonal E × B frequency is also displayed. +(c) eφ/Te mode structure in the poloidal plane after mode +saturation. (d) Zonal electric field eEr,00/Te after mode sat- +uration. +When the realistic beam is replaced by its equivalent +Maxwellian distribution, this mode is fully stabilized, high- +lighting the sensitivity of fishbone instabilities over EP dis- +tributions. +Nonlinear simulations are performed with and without the +n=m=0 component, as illustrated in Fig.1. The time evo- +lution of the volume-averaged electrostatic potential eφ/Te, +displayed on Fig.1a, shows that the n=1 fishbone mode is +able to force-drive the n=m=0 zonal flow, with a growth +rate twice that of the n=1. +As shown analytically in [5] +for TAEs, the mechanism for this zonal flow generation is +the charge separation induced by nonlinear EP redistribu- +tion, as opposed to the usual one relying on Reynolds and +Maxwell stresses [2][3][4][9]. As the n=0 amplitude exceeds +the n=1 at t=0.13ms, the zonal mode forces the fishbone +to +2 + +Mode amplitude +n=0, ZFs +-n=1. without ZFs +10-1 +-n=1, with ZFs +e +10° +e +10~3 +n=1 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +Time (ms)Mode frequency +24 +22 +一w/2π without ZFs +-w/2π with ZFs +20 +d.res +18 +(ZH) / +16 +14 +12 +10 +8 +6 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +Time (ms)ed +-/T-, t=0.19ms +n=1 +ed +n=1 +e +0.1 +0.6 +-q=2 +....q=3 +0.4 +0.05 +0.2 +0 +0 +N +-0.2 +-0.05 +-0.4 +-0.6 +-0.1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +R (m)eE +T +at t=0.19ms and q +r,00° +e +-0.2 +e +3 +-0.4 +eE. +2 +-0.6 +-0.8 +0 +0.2 +0.4 +0.6 +0.8 +ld(a) +(b) +(c) +(d) +Figure 2: Radial envelope of δTe after saturation without +(a) and with (b) zonal flows in GTC, M3D-C1 and XTOR- +K simulations, compared to the ECE measurement for the +DIII-D #178631 discharge. (c) Time evolution of the sim- +ulated neutron drop, with and without zonal flows. (d) EP +density profiles in GTC simulations before and after fishbone +burst. +saturate at δB/B0 ∼ 2 × 10−3, with a saturation amplitude +lower by a factor of 4 compared to the case without zonal +flows. The zonal flows saturates at an even larger amplitude, +about six times larger than the n=1 mode when including +zonal flows, with a spontaneous growth after t=0.15ms when +the n=1 is fully saturated. Such mitigation by zonal flows +have been theoretically predicted [2][3][5] and numerically +observed [7][8][9][10] for Alfv´en eigenmodes, but never so +far for the fishbone instability. +The zonal flows inclusion +also lowers significantly the EP diffusivity at saturation, +from 30 to 4 m2.s−1. +As shown in Figure 1b, the mode +frequency down-chirps after the n=1 mode saturation with +and without zonal flows, which is a typical fishbone signa- +ture, with similar chirping rates. Just before saturation, the +case without zonal flows experiences a notable up-chirping +of the mode frequency, that stops when the mode starts +saturating. This increase may be attributed to the larger +mode amplitude near saturation. The n=1 electrostatic po- +tential and the n=0 radial electric field after saturation at +t=0.19ms are displayed on Fig.1c-d. +The n=1 mode fea- +tures a dominant m=1 harmonic centered around qmin, as +well as a significant m=2 side-band that vanishes after q = 2. +The zonal electric field exhibits a macroscopic structure cen- +tered near qmin as well, which differs from the usual mi- +croscopic/mesoscopic scale observed with drift-waves/AEs- +induced zonal flows. This large structure can be attributed +to the charge separation provoked by the outward drift of +resonant EPs within the n=1 mode. It leads to a strongly +sheared poloidal rotation in the electron direction, which is +opposite to the n=1 fishbone rotation, and a weak toroidal +rotation. +This fishbone mitigation by self-generated zonal flows is ex- +perimentally confirmed by DIII-D measurements as can be +seen in Fig.2. The δTe envelope obtained from GTC, M3D- +C1 and XTOR-K nonlinear simulations at saturation are +compared with the ECE measurements on Fig.2 (a-b), with +and without zonal flows inclusion. The δTe envelope is de- +fined here as the n=1 sum of all poloidal harmonics. With- +out zonal flows, XTOR-K and GTC results have compa- +rable saturation amplitudes with δTe,max ∼ 500 − 600 eV, +which are three time larger than the experimental satura- +tion. The simulated envelopes differ however, GTC results +having a dominant m=2 harmonic after ρ = 0.34. When +including zonal flows however, M3D-C1 and GTC satura- +tion amplitudes at δTe,max ∼ 200 eV match very well with +the experimental one. +The significant m=2 harmonic in +GTC simulations leads to a quantitative agreement with +the ECE measurement, which provides a nonlinear valida- +tion for GTC regarding fishbone instabilities, completing the +linear one obtained in [38] for kink instabilities. +Nonlin- +ear scans for the fishbone saturation amplitude performed +over the radial position and amplitude of qmin recover the +same significant mitigation by zonal flows. This nonlinear +validation is further demonstrated by comparing the sim- +ulated and experimental volume-averaged neutron emissiv- +ity. In GTC the volume-averaged neutron flux is defined as +ΓN = ni +�N +k δ(x − xf,k)δ(v − vf,k)σ(vf,k)vf,k with ni the +thermal ion density profile, xk and vk the position and ve- +locity of EPs and σ the D-D nuclear fusion cross section, +assuming reasonably that vi ≪ vf. +As shown on Fig.2c, +without zonal flows GTC recovers a neutron drop at satura- +tion of about 6%, much higher than the experimental one at +δΓN = 0.9% ± 0.3%. When including zonal flows however, +the neutron drop yields δΓN ∼ 1.1%, which falls within the +experimental interval. As expected from these neutron drop +values, the fishbone-induced EP transport with zonal flows +is rather weak as shown on Fig. 2d, with about 3% of EPs +inside of the qmin volume redistributed outward. The redis- +tribution is more significant without zonal flows, as it affects +15% of EPs in the core plasma. +Mechanism for fishbone mitigation by zonal flows - Beyond +the additional dissipation brought by the inclusion of the +n=0 toroidal mode [7], phase-space analysis reveals that +zonal flows influence the time evolution of coherent phase +space structures, impacting the n=1 fishbone mode satu- +ration. On Fig.3, the instantaneous EP transport ∂tδf is +displayed in the invariants phase space diagram (Pζ, λ = +µB0/E) at fixed magnetic momentum µB0 = 45keV before +3 + + T.(eV), without ZFs +600 +=1.09 +q=2 +min +500 +-XTOR-K n=1 +-GTC n=1 ++ECE +400 +(eV) +300 +OS +200 +100 +0 +0 +0.2 +0.4 +0.6 +0.8 +PT T.(eV), with ZFs +600 +. +=1.09 +q=2 +min +M3D-C1, n=0,1,2 +500 +GTC n=0,1 +ECE +400 +(eV) +e +300 +OS +200 +100 +0 +0 +0.2 +0.4 +0.6 +0.8 +PTNeutron drop +0 +Experimental +-1 +neutron drop +-2 +Neutron drop (%) +-Without ZFs +3 +_With ZFs +-6 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +Time (ms)X1018 +EP density profiles +10 +-t = Oms +...t = 0.19ms with ZFs +-t = 0.19ms without ZFs +8 +6 +EP +=1.09 +9 +min +q=2 +n +4 +: +2 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +ldand after the fishbone saturation, with and without zonal +flows. The instantaneous transport is chosen rather than the +usual perturbed EP distribution δf as the fishbone mode +frequency is chirping in the nonlinear phase, which leads +phase space structure to drift in time. In the linear phase, +the mode is driven by two resonances, the precessional one +ω = ωd linked to trapped particles, and a drift-transit one +ω = ωζ −ωb due to passing particles, with ωζ = qωb +ωd the +drift frequency and ωb the bounce/transit frequency. The +passing and trapped phase space zones are separated by a +black line on the diagrams. +(a) +(b) +(c) +(d) +Figure 3: Time evolution of the instantaneous EP transport +∂tδf without (left) and with (right) zonal flows, in the invari- +ants (Pζ, λ) phase space diagram at fixed µ (µB0 = 45keV ). +As can be observed on Fig.3 (a-b), a hole and clump struc- +ture develops around each resonances in the linear phase, +indicating a resonant outward EP redistribution. In the non- +linear phase, the dynamical evolution of these phase space +structures differ significantly with and without zonals flows. +In their absence, the hole and clump in the trapped region +drifts to higher ψ positions, under the influence of the mode +down-chirping as ωd ∝ 1/ψ, while the one in the passing part +does not move. However with zonal flows, the phase space +structure in the trapped region remains static, even thought +the mode is chirping down, and the hole and clump in the +passing part vanishes. Such behaviours prevent the fishbone +mode from affecting resonantly new EPs, which leads to its +weaker saturation due to the absence of drive. +These differences in dynamical evolution can be explained +by the influence of the zonal flows on the EPs wave- +particle resonance. The perturbed radial electric field as- +sociated with zonal flows generates an additional drift ve- +locity δvE,00 = δE00 ×B/B2. This additional velocity leads +to an E × B drift frequency defined in general geometry +as ωE,00 = ⟨vE,00 · (∇ζ − q∇θ)⟩ with ⟨· · ·⟩ the bounce- +average operator, which yields δωE,00 = δEψ = −∇φ00 us- +ing a thin-orbit width approximation for simplicity. This +is similar to the so-called ”orbit-squeezing” effects in neo- +classical theory [41], EPs have an overall decrease of their +precessional frequency due to their large orbit width over a +strongly sheared radial electric field. As can be observed on +Fig.1b, the time evolution of the precessional frequency of +linearly resonant EPs plus the perturbed E×B frequency at +ρ = ρqmin matches almost exactly the time evolution of the +fishbone frequency with zonal flows, which explains why the +phase space structure in the trapped region remains static. +The strongly sheared E × B poloidal flow can also perturb +the EPs transit frequencies due to their large orbit width, +leading to a resonance detuning and the disappearance of +the ω = ωζ − ωb hole and clump. Zonal flows are therefore +able to dominate the fishbone saturation by strongly reduc- +ing the resonant wave-particle trapping. +Fishbone-induced ion ITB formation - On top of affecting +the fishbone mode mitigation, the zonal flows also generate a +strong shearing rate within ρT ∈ [0.1, 0.5] with γE ∼ 3×105 +s−1. +High-n electrostatic GTC simulations with kinetic +trapped electrons were performed for this DIII-D configu- +ration, finding that the most unstable drift-wave is a TEM +mode at ρ = 0.4, shown on Fig.4a, with a linear growth rate +of γT EM = 1.38 × 105 s−1. The shearing rate being larger +than the TEM growth rate, as displayed on Fig.4b, the sim- +ulated fishbone mode could then lead to turbulence modu- +lation by suppression the TEM growth through zonal flows +[11], confirming the speculated role of fishbones in the emer- +gence of ITBs [22]. This modulation is supported experi- +mentally in DIII-D by the charge exchange recombination +spectroscopy diagnostic. The formation of an ion ITB after +fishbone bursts occurring at t=1581,1594,1607 and 1615ms +can indeed be observed on Fig.4 c. The core-increase of Ti +cannot be explained by additional heating from the beam, as +it was at constant power since t=300ms, multiple slowing- +down times before the onset of fishbones. Fishbone bursts +were also observed to precede ion-ITB in four others DIII-D +discharges with similar heating power, density, current and +qmin parameters. Electrons are not affected by the ITB, as +zonal flows are only able to mitigate ion-scale turbulence +[42]. +EP transport in ITER prefusion baseline - The GTC code +having been nonlinearly validated for fishbone simulations, +it can now be applied to the selected ITER scenario to pre- +dict the fishbone-induced EP transport. Similar to the DIII- +D simulations, the NBI beam is reproduced from an analyt- +ical anisotropic slowing-down distribution. +4 + +0,of, with ZFs, t=0.13ms +3 +1.1 +3 +2 +1 +1 +4 +0.9 +,=1.09 +入=μ/B。 +min +0 +T +0.8 +P +-1 +0.7 +-2 +0.6 +-3 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +P +wall0,of, no ZFs, t=0.2ms +60 +.. +1.1 +3 +3 +40 +1 +20 +0.9 +in=1.09 +入=μ/B。 +min +0 +T +0.8 +p +-20 +0.7 +Va +-40 +0.6 +-60 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +P +wall0,of, with ZFs, t=0.2ms +... +20 +1.1 +d +3=3: +3 +15 +1 +10 +5 +4 +0.9 +入=μ/B。 +0 +T +0.8 +P +-5 +-10 +0.7 +V +-15 +0.6 +-20 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +P +wall0,of, no ZFs, t=0.13ms +...8 +3 +1.1 +d +3 +3 +3 +2 +1 +1 +0.9 +n =1.09 +入=μ/B。 +min +0 +T +0.8 +P +-1 +0.7 +Vab +-2 +0.6 +-3 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +P +b +wall(a) +(b) +(c) +Figure 4: a) Electrostatic potential φ of unstable TEM mode +in the poloidal plane b) Fishbone-induced shearing rate pro- +file after saturation c) Ti profiles in eV before and after +fishbone bursts from charge exchange recombination spec- +troscopy, exhibiting an ion-ITB. +Linear GTC simulations show that the configuration is un- +stable to the n=1 fishbone with the realistic beam, with +a mode growth rate and frequency of γ = 4.4 × 104 s−1 +and ω/2π = 48 kHz, while simulations with equivalent +Maxwellian distributions find a configuration stable to n=1 +modes. +Similarly to DIII-D based simulations, the zonal flows inclu- +sion lowers the n=1 mode saturation. The zonal electric field +also peaks with negative values close to the qmin surface, +with a subdominant positive layer further in the plasma. +Electrostatic GTC simulations were also performed for this +ITER scenario, finding an unstable TEM mode at ρ = 0.71 +with γT EM = 3 × 104 s−1. At that location, the fishbone- +induced shearing rate is three times larger than the TEM +linear growth rate, suggesting that an ion-ITB can also be +triggered for this ITER scenario. +However after saturation with zonal flows, the n=1 mode +abruptly explodes. This numerical instability is due to how +zonal flows are computed in GTC. The flux-surface averaged +potential φ00 is computed over the equilibrium flux surfaces, +which can be a strong assumption in the nonlinear fishbone +phase as δB/B grows. This computation will soon be modi- +fied to include the perturbed flux surface to enable long time +cross-scale simulation between microturbulence and MHD +modes with GTC. The study of the fishbone-induced EP +transport for that scenario is then conducted without the +inclusion of zonal flows to achieve a long nonlinear phase. +The transport level will then represent the upper-bound as +zonal flows decrease it significantly. +After the end of the fishbone burst, the overall redistribu- +tion within qmin is of order 2% of the initial distribution, +with both inward and outward EP fluxes due to positive +and negative EP equilibrium pressure gradient. Such a re- +distribution tends to marginally flatten the initial pressure +gradient, the NBI pressure drive being too low to cause large +redistribution. Overall, the NBI fishbone should not impact +significantly the plasma heating of this ITER baseline pre- +fusion, similar to what was shown for the alpha-fishbone in +the ITER 15MA baseline DT scenario [19]. +Conclusion - Since fishbone oscillations may not cause signif- +icant EP redistribution in ITER plasmas, it can be of great +interest to design ITER scenarios to trigger them on purpose +rather than avoiding them. As was shown in this Letter, +fishbones can generate zonal flows which present two ad- +vantages : 1) mitigating the fishbone saturation and its im- +pact on EP transport and 2) creating strong shearing rates +that can damp drift-wave instabilities and hence reducing +the turbulent transport. While it was observed here and in +several tokamak discharges [23][24][43][26][27][25] that fish- +bone oscillations led to ITBs formation, that was not the +case in some others such as JET [44][45][42], despite efforts +to reproduce the fishbone-induced ITB formation observed +in ASDEX plasmas [23]. +It may therefore exist a para- +metric dependency for the fishbone instability that controls +the emergence of strongly sheared fishbone-induced zonal +flows. The numerical identification and experimental obser- +vation of such a dependency could enable the creation of +high-performance scenarios, of crucial importance for ITER +burning plasmas. +References +[1] ITER Physics Expert Group on Energe Drive and ITER +Physics Basis Editors. Chapter 5: Physics of energetic +ions. Nuclear Fusion, 39(12):2471–2495, dec 1999. +[2] Liu Chen, Zhihong Lin, and Roscoe White. Excitation +of zonal flow by drift waves in toroidal plasmas. 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Plasma Physics and Controlled Fusion, +44(8):1739–1752, aug 2002. +7 + diff --git a/UNAzT4oBgHgl3EQf0_5k/content/tmp_files/load_file.txt b/UNAzT4oBgHgl3EQf0_5k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..938680b4a10a04ebd984e25e130b479a3d76ef60 --- /dev/null +++ b/UNAzT4oBgHgl3EQf0_5k/content/tmp_files/load_file.txt @@ -0,0 +1,695 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf,len=694 +page_content='Saturation of fishbone modes by self-generated zonal flows in tokamak plasmas G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Brochard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Wei, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Heidbrink, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Lin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Gorelenkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Pinches, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Nicolau, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' L¨utjens Abstract Gyrokinetic and kinetic-MHD simulations of n=1 fishbone modes in DIII-D plasmas find that self-generated zonal flows can dominate the fishbone saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The saturation mechanism is identified in phase space, where the zonal flows prevent holes and clumps from persisting or drifting in phase space with mode down-chirping, reducing the wave-particle resonant drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This saturation is confirmed by quantitative agreement with experimental measurements for both mode saturation amplitude and neutron emissivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Zonal flows shearing rate exceeds the drift-wave growth rate, consistent with the ITB observed in DIII-D plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The deliberate destabilization of fishbones for the development of high performance scenarios in ITER is then proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' - Energetic Particles (EPs) in tokamak plas- mas can destabilize a large spatial range of instabilities that may lead to their outward transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This is a critical issue for burning plasmas as in ITER [1] since such a transport can degrade the fusion performances, the plasma confine- ment as well as threaten the reactor’s integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This trans- port therefore needs to be predicted for mitigation strategies to be incorporated in plasma scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Fortunately, it was discovered theoretically [2][3][4][5] and shown numerically [6][7][8][9][10] that instabilities arising at the microscopic and mesoscopic scales such as drift-waves and Alfv´en eigenmodes (AEs) are able to excite zonal flows (ZFs), that can mitigate the saturation amplitudes of these modes, and therefore the associated EP transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Besides this mitigation, the destabilisation of zonal flows can gener- ate strongly sheared poloidal flows that suppress turbulent transport by damping drift-waves turbulence [11], resulting in the formation of an internal transport barrier (ITB) that greatly enhances plasma confinement [12][13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Macroscopic MHD modes triggered by energetic particles such as the fish- bone instability [14][15] however were not self-consistently observed to trigger n = m = 0 zonal flows so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The mechanism dominating the fishbone saturation was identi- fied in nonlinear simulations [16][17][18][19] to be the res- onant wave-particle trapping due to kinetic nonlinearities, mode-mode nonlinearities playing a secondary role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' In this Letter, we report the first self-consistent gyrokinetic simulations finding fishbone saturation by the self-generated zonal flows, in a DIII-D discharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This discharge is chosen for validation purposes to predict the EP transport in a ITER baseline prefusion scenario [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The zonal flows are found to be force-driven by the fishbone and are the main mechanism for the fishbone saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This mechanism is observed for the first time in phase space, where zonal flows prevent hole and clump structures from persisting or drift- ing in the nonlinear phase, reducing the EP resonant drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This saturation by zonal flows is confirmed by experimental measurements, as simulations including zonal flows are able to recover quantitatively, for the first time, the mode satura- tion amplitude and the neutron emissivity drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Moreover, the shearing rate generated by the fishbone-induced zonal flows exceeds the linear growth rate of unstable drift-wave modes, similar to recent numerical work based on EAST discharges [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This strong E × B suppression is consistent with the ITB arising experimentally after fishbone bursts in the DIII-D discharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' It confirms the long suspected role of fishbones in ITB formation [22], fishbone bursts having been observed to precede ITBs in ASDEX [23], MAST [24][25], HL-2A [4] and EAST [26][27] plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Finally, gyrokinetic simulations find that the fishbone-induced EP transport in the ITER scenario is marginal, 2% of the core EPs being re- distributed, similar to previous studies on the alpha fishbone in ITER DT scenarios [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The intentional destabilization of fishbone modes in ITER scenarios is therefore possibly a way to enhance fusion performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The selected DIII-D discharge #178631 [28] has a nearly circular oval shape (elongation κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='17, triangularity δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='07) that is limited on the car- bon inner wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The major radius is R0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='74 m, the minor radius is a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='64 m, the toroidal field is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='0 T, the plasma current is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='88 MA, and the line-average electron density is ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='0 × 1019 m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This discharge was chosen primarily be- cause it has an accurately known, weakly reversed, q profile with q0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2, qmin = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='09, and q95 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 values that re- semble the profile predicted for the ITER baseline scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The deuterium, L-mode plasma is heated by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 MW of 81 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='01792v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='plasm-ph] 4 Jan 2023 keV deuterium beams that are injected in the midplane in the direction of the plasma current and by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='0 MW of 2nd harmonic, central electron cyclotron heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Numerical setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The DIII-D discharge #178631 is studied numerically mostly through gyrokinetic simulations with the GTC code [6][29][30][31], and with kinetic-MHD simulations using the M3D-C1 [32][33][34] and XTOR-K [35][36][37] codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' GTC capability at simulating MHD modes was recently verified and validated on DIII-D ex- periments [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The magnetic configuration is reproduced from the EFIT code at t=1580ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Plasma profiles are ob- tained from TRANSP simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' To simulate properly MHD modes, the sum of partial pressures need to add up to the total pressure in EFIT, which is not always the case using TRANSP profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' To ensure it, the EP pressure is constrained as pf = ptot − pi − pe, given that the uncer- tainty on EP profiles in TRANSP is the highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The exper- imental NBI distribution is reproduced from the NUBEAM code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Such a distribution is described in our first-principle simulations with an anisotropic slowing-down model, us- ing a zero-th order Legendre expansion [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' A superpo- sition of three slowing-downs is used to reproduce the in- jection energies at nominal, half and third energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The critical velocity is artificially set to recover similar gradi- ents in the (E, v||/v) phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' All nonlinear simulations cover the whole simulation domain, with an edge buffer after ρT = � ψT /ψT,edge = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 in GTC suppressing equilibrium gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' GTC retains only the n=1 mode in its simula- tions, with or without the n=m=0 zonal component, using kinetic thermal/fast ions and fluid electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' M3D-C1 cov- ers low n modes n ∈ [0, 2] with both thermal and fast ions kinetic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Due to the anisotropic nature of the cho- sen configuration that has βf/βtot = 54% on axis, XTOR-K only evolves the n=1 mode, as the n=0 mode contains both equilibrium and perturbed fields in the code, contrarily to GTC and M3D-C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' XTOR-K treats kinetically only the fast ion specie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Convergence studies over spatial grid size, time step and number of particles per cell were successfully conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Fishbone mitigation by self-induced zonal flows - The im- pact of MHD nonlinearities on the n=1 fishbone were pre- viously examined numerically by keeping side-band n=0-4 modes, highlighting reduction of initial saturation ampli- tude [18][21], and generation of n=m=0 sheared poloidal flows [19][21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The role played specifically by zonal flows in fishbone mitigation was however not identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The ef- fects of zonal flows on the fishbone instability are studied here self-consistently for the first time with the gyrokinetic GTC code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' A gyrokinetic treatment of zonal flows is es- sential as it takes into account their collisionless damping [40], which is absent in the kinetic-MHD formalism without kinetic thermal ions effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' For the considered DIII-D con- figuration, a n=1 fishbone mode is linearly unstable, close to marginal stability at pf,thres = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8pf, with a growth rate of γn=1 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='5×104 s−1 and a mode frequency of ω/2π = 17kHz in GTC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 1: Time evolution of (a) the volume-averaged per- turbed electrostatic potential eφ/Te (n=0,1), and (b) the the n=1 mode frequency ω, with and without zonal flows in GTC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The linearly resonant precessional fre- quency plus the zonal E × B frequency is also displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' (c) eφ/Te mode structure in the poloidal plane after mode saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' (d) Zonal electric field eEr,00/Te after mode sat- uration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' When the realistic beam is replaced by its equivalent Maxwellian distribution, this mode is fully stabilized, high- lighting the sensitivity of fishbone instabilities over EP dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Nonlinear simulations are performed with and without the n=m=0 component, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The time evo- lution of the volume-averaged electrostatic potential eφ/Te, displayed on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1a, shows that the n=1 fishbone mode is able to force-drive the n=m=0 zonal flow, with a growth rate twice that of the n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' As shown analytically in [5] for TAEs, the mechanism for this zonal flow generation is the charge separation induced by nonlinear EP redistribu- tion, as opposed to the usual one relying on Reynolds and Maxwell stresses [2][3][4][9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' As the n=0 amplitude exceeds the n=1 at t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='13ms, the zonal mode forces the fishbone to 2 Mode amplitude n=0, ZFs n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' without ZFs 10-1 n=1, with ZFs e 10° e 10~3 n=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 Time (ms)Mode frequency 24 22 一w/2π without ZFs w/2π with ZFs 20 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='res 18 (ZH) / 16 14 12 10 8 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 Time (ms)ed /T-, t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='19ms n=1 ed n=1 e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 q=2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='.q=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0 0 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 R (m)eE T at t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='19ms and q r,00° e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 e 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 eE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 ld(a) (b) (c) (d) Figure 2: Radial envelope of δTe after saturation without (a) and with (b) zonal flows in GTC, M3D-C1 and XTOR- K simulations, compared to the ECE measurement for the DIII-D #178631 discharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' (c) Time evolution of the sim- ulated neutron drop, with and without zonal flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' (d) EP density profiles in GTC simulations before and after fishbone burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' saturate at δB/B0 ∼ 2 × 10−3, with a saturation amplitude lower by a factor of 4 compared to the case without zonal flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The zonal flows saturates at an even larger amplitude, about six times larger than the n=1 mode when including zonal flows, with a spontaneous growth after t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='15ms when the n=1 is fully saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Such mitigation by zonal flows have been theoretically predicted [2][3][5] and numerically observed [7][8][9][10] for Alfv´en eigenmodes, but never so far for the fishbone instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The zonal flows inclusion also lowers significantly the EP diffusivity at saturation, from 30 to 4 m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' As shown in Figure 1b, the mode frequency down-chirps after the n=1 mode saturation with and without zonal flows, which is a typical fishbone signa- ture, with similar chirping rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Just before saturation, the case without zonal flows experiences a notable up-chirping of the mode frequency, that stops when the mode starts saturating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This increase may be attributed to the larger mode amplitude near saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The n=1 electrostatic po- tential and the n=0 radial electric field after saturation at t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='19ms are displayed on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1c-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The n=1 mode fea- tures a dominant m=1 harmonic centered around qmin, as well as a significant m=2 side-band that vanishes after q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The zonal electric field exhibits a macroscopic structure cen- tered near qmin as well, which differs from the usual mi- croscopic/mesoscopic scale observed with drift-waves/AEs- induced zonal flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This large structure can be attributed to the charge separation provoked by the outward drift of resonant EPs within the n=1 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' It leads to a strongly sheared poloidal rotation in the electron direction, which is opposite to the n=1 fishbone rotation, and a weak toroidal rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This fishbone mitigation by self-generated zonal flows is ex- perimentally confirmed by DIII-D measurements as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The δTe envelope obtained from GTC, M3D- C1 and XTOR-K nonlinear simulations at saturation are compared with the ECE measurements on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 (a-b), with and without zonal flows inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The δTe envelope is de- fined here as the n=1 sum of all poloidal harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' With- out zonal flows, XTOR-K and GTC results have compa- rable saturation amplitudes with δTe,max ∼ 500 − 600 eV, which are three time larger than the experimental satura- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The simulated envelopes differ however, GTC results having a dominant m=2 harmonic after ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' When including zonal flows however, M3D-C1 and GTC satura- tion amplitudes at δTe,max ∼ 200 eV match very well with the experimental one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The significant m=2 harmonic in GTC simulations leads to a quantitative agreement with the ECE measurement, which provides a nonlinear valida- tion for GTC regarding fishbone instabilities, completing the linear one obtained in [38] for kink instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Nonlin- ear scans for the fishbone saturation amplitude performed over the radial position and amplitude of qmin recover the same significant mitigation by zonal flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This nonlinear validation is further demonstrated by comparing the sim- ulated and experimental volume-averaged neutron emissiv- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' In GTC the volume-averaged neutron flux is defined as ΓN = ni �N k δ(x − xf,k)δ(v − vf,k)σ(vf,k)vf,k with ni the thermal ion density profile, xk and vk the position and ve- locity of EPs and σ the D-D nuclear fusion cross section, assuming reasonably that vi ≪ vf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' As shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2c, without zonal flows GTC recovers a neutron drop at satura- tion of about 6%, much higher than the experimental one at δΓN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='9% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' When including zonal flows however, the neutron drop yields δΓN ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1%, which falls within the experimental interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' As expected from these neutron drop values, the fishbone-induced EP transport with zonal flows is rather weak as shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' 2d, with about 3% of EPs inside of the qmin volume redistributed outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The redis- tribution is more significant without zonal flows, as it affects 15% of EPs in the core plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Mechanism for fishbone mitigation by zonal flows - Beyond the additional dissipation brought by the inclusion of the n=0 toroidal mode [7], phase-space analysis reveals that zonal flows influence the time evolution of coherent phase space structures, impacting the n=1 fishbone mode satu- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' On Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3, the instantaneous EP transport ∂tδf is displayed in the invariants phase space diagram (Pζ, λ = µB0/E) at fixed magnetic momentum µB0 = 45keV before 3 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='(eV), without ZFs 600 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='09 q=2 min 500 XTOR-K n=1 GTC n=1 +ECE 400 (eV) 300 OS 200 100 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 PT T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='(eV), with ZFs 600 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='09 q=2 min M3D-C1, n=0,1,2 500 GTC n=0,1 ECE 400 (eV) e 300 OS 200 100 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 PTNeutron drop 0 Experimental 1 neutron drop 2 Neutron drop (%) Without ZFs 3 _With ZFs 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 Time (ms)X1018 EP density profiles 10 t = Oms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='19ms with ZFs t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='19ms without ZFs 8 6 EP =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='09 9 min q=2 n 4 : 2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 1 ldand after the fishbone saturation, with and without zonal flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The instantaneous transport is chosen rather than the usual perturbed EP distribution δf as the fishbone mode frequency is chirping in the nonlinear phase, which leads phase space structure to drift in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' In the linear phase, the mode is driven by two resonances, the precessional one ω = ωd linked to trapped particles, and a drift-transit one ω = ωζ −ωb due to passing particles, with ωζ = qωb +ωd the drift frequency and ωb the bounce/transit frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The passing and trapped phase space zones are separated by a black line on the diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 3: Time evolution of the instantaneous EP transport ∂tδf without (left) and with (right) zonal flows, in the invari- ants (Pζ, λ) phase space diagram at fixed µ (µB0 = 45keV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' As can be observed on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3 (a-b), a hole and clump struc- ture develops around each resonances in the linear phase, indicating a resonant outward EP redistribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' In the non- linear phase, the dynamical evolution of these phase space structures differ significantly with and without zonals flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' In their absence, the hole and clump in the trapped region drifts to higher ψ positions, under the influence of the mode down-chirping as ωd ∝ 1/ψ, while the one in the passing part does not move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' However with zonal flows, the phase space structure in the trapped region remains static, even thought the mode is chirping down, and the hole and clump in the passing part vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Such behaviours prevent the fishbone mode from affecting resonantly new EPs, which leads to its weaker saturation due to the absence of drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' These differences in dynamical evolution can be explained by the influence of the zonal flows on the EPs wave- particle resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The perturbed radial electric field as- sociated with zonal flows generates an additional drift ve- locity δvE,00 = δE00 ×B/B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This additional velocity leads to an E × B drift frequency defined in general geometry as ωE,00 = ⟨vE,00 · (∇ζ − q∇θ)⟩ with ⟨· · ·⟩ the bounce- average operator, which yields δωE,00 = δEψ = −∇φ00 us- ing a thin-orbit width approximation for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This is similar to the so-called ”orbit-squeezing” effects in neo- classical theory [41], EPs have an overall decrease of their precessional frequency due to their large orbit width over a strongly sheared radial electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' As can be observed on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1b, the time evolution of the precessional frequency of linearly resonant EPs plus the perturbed E×B frequency at ρ = ρqmin matches almost exactly the time evolution of the fishbone frequency with zonal flows, which explains why the phase space structure in the trapped region remains static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The strongly sheared E × B poloidal flow can also perturb the EPs transit frequencies due to their large orbit width, leading to a resonance detuning and the disappearance of the ω = ωζ − ωb hole and clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Zonal flows are therefore able to dominate the fishbone saturation by strongly reduc- ing the resonant wave-particle trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Fishbone-induced ion ITB formation - On top of affecting the fishbone mode mitigation, the zonal flows also generate a strong shearing rate within ρT ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='5] with γE ∼ 3×105 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' High-n electrostatic GTC simulations with kinetic trapped electrons were performed for this DIII-D configu- ration, finding that the most unstable drift-wave is a TEM mode at ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4, shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4a, with a linear growth rate of γT EM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='38 × 105 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The shearing rate being larger than the TEM growth rate, as displayed on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4b, the sim- ulated fishbone mode could then lead to turbulence modu- lation by suppression the TEM growth through zonal flows [11], confirming the speculated role of fishbones in the emer- gence of ITBs [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This modulation is supported experi- mentally in DIII-D by the charge exchange recombination spectroscopy diagnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The formation of an ion ITB after fishbone bursts occurring at t=1581,1594,1607 and 1615ms can indeed be observed on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The core-increase of Ti cannot be explained by additional heating from the beam, as it was at constant power since t=300ms, multiple slowing- down times before the onset of fishbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Fishbone bursts were also observed to precede ion-ITB in four others DIII-D discharges with similar heating power, density, current and qmin parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Electrons are not affected by the ITB, as zonal flows are only able to mitigate ion-scale turbulence [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' EP transport in ITER prefusion baseline - The GTC code having been nonlinearly validated for fishbone simulations, it can now be applied to the selected ITER scenario to pre- dict the fishbone-induced EP transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Similar to the DIII- D simulations, the NBI beam is reproduced from an analyt- ical anisotropic slowing-down distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' 4 0,of, with ZFs, t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='13ms 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 3 2 1 1 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='9 ,=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='09 入=μ/B。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' min 0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 P 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='7 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3 P wall0,of, no ZFs, t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2ms 60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='. 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 3 3 40 1 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='9 in=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='09 入=μ/B。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' min 0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 p 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='7 Va 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3 P wall0,of, with ZFs, t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2ms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 d 3=3: 3 15 1 10 5 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='9 入=μ/B。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' 0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 P 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='7 V 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3 P wall0,of, no ZFs, t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='13ms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 d 3 3 3 2 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='9 n =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='09 入=μ/B。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' min 0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 P 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='7 Vab 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3 P b wall(a) (b) (c) Figure 4: a) Electrostatic potential φ of unstable TEM mode in the poloidal plane b) Fishbone-induced shearing rate pro- file after saturation c) Ti profiles in eV before and after fishbone bursts from charge exchange recombination spec- troscopy, exhibiting an ion-ITB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Linear GTC simulations show that the configuration is un- stable to the n=1 fishbone with the realistic beam, with a mode growth rate and frequency of γ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 × 104 s−1 and ω/2π = 48 kHz, while simulations with equivalent Maxwellian distributions find a configuration stable to n=1 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Similarly to DIII-D based simulations, the zonal flows inclu- sion lowers the n=1 mode saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The zonal electric field also peaks with negative values close to the qmin surface, with a subdominant positive layer further in the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Electrostatic GTC simulations were also performed for this ITER scenario, finding an unstable TEM mode at ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='71 with γT EM = 3 × 104 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' At that location, the fishbone- induced shearing rate is three times larger than the TEM linear growth rate, suggesting that an ion-ITB can also be triggered for this ITER scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' However after saturation with zonal flows, the n=1 mode abruptly explodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This numerical instability is due to how zonal flows are computed in GTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The flux-surface averaged potential φ00 is computed over the equilibrium flux surfaces, which can be a strong assumption in the nonlinear fishbone phase as δB/B grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' This computation will soon be modi- fied to include the perturbed flux surface to enable long time cross-scale simulation between microturbulence and MHD modes with GTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The study of the fishbone-induced EP transport for that scenario is then conducted without the inclusion of zonal flows to achieve a long nonlinear phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The transport level will then represent the upper-bound as zonal flows decrease it significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' After the end of the fishbone burst, the overall redistribu- tion within qmin is of order 2% of the initial distribution, with both inward and outward EP fluxes due to positive and negative EP equilibrium pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Such a re- distribution tends to marginally flatten the initial pressure gradient, the NBI pressure drive being too low to cause large redistribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Overall, the NBI fishbone should not impact significantly the plasma heating of this ITER baseline pre- fusion, similar to what was shown for the alpha-fishbone in the ITER 15MA baseline DT scenario [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Conclusion - Since fishbone oscillations may not cause signif- icant EP redistribution in ITER plasmas, it can be of great interest to design ITER scenarios to trigger them on purpose rather than avoiding them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' As was shown in this Letter, fishbones can generate zonal flows which present two ad- vantages : 1) mitigating the fishbone saturation and its im- pact on EP transport and 2) creating strong shearing rates that can damp drift-wave instabilities and hence reducing the turbulent transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' While it was observed here and in several tokamak discharges [23][24][43][26][27][25] that fish- bone oscillations led to ITBs formation, that was not the case in some others such as JET [44][45][42], despite efforts to reproduce the fishbone-induced ITB formation observed in ASDEX plasmas [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' It may therefore exist a para- metric dependency for the fishbone instability that controls the emergence of strongly sheared fishbone-induced zonal flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' The numerical identification and experimental obser- vation of such a dependency could enable the creation of high-performance scenarios, of crucial importance for ITER burning plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' References [1] ITER Physics Expert Group on Energe Drive and ITER Physics Basis Editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Chapter 5: Physics of energetic ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Nuclear Fusion, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Nonlinear generation of zonal fields by the beta-induced alfv´en eigenmode in tokamak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Plasma Science and Technology, 15(10):969– 973, oct 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' 5 Φ, TEM mode ×10~3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 d =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='09 2 min p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='q=2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='5 0 0 N 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 R (m)Fishbone-induced ZFs shearing rate 6 TEM min location 5 4 YTEM 3 B x 3 2 YTEM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='8 don ITB in D-D #178631 5000 t=1580ms 4500 t=1620ms 4000 TEM =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='09 min location 3500 (eV) 3000 2500 2000 1500 1000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content='6 PT[9] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQf0_5k/content/2301.01792v1.pdf'} +page_content=' Chen, G.' metadata={'source': 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+1,1370 @@ +Thermal curvature perturbations +in thermal inflation +Mar Bastero-Gil,a Joaquim M. Gomes,b and Jo˜ao G. Rosac +aDepartamento de F´ısica Te´orica y del Cosmos, Universidad de Granada, +Granada-18071, Spain +bDepartment of Mathematical Sciences, University of Liverpool, +Liverpool L69 7ZL, United Kingdom +cUniv Coimbra, Faculdade de Ciˆencias e Tecnologia da Universidade de Coimbra and CFisUC, +Rua Larga, 3004-516 Coimbra, Portugal +E-mail: mbg@ugr.es, j.m.gomes@liverpool.ac.uk, jgrosa@uc.pt +Abstract. We compute the power spectrum of super-horizon curvature perturbations gen- +erated during a late period of thermal inflation, taking into account fluctuation-dissipation +effects resulting from the scalar flaton field’s interactions with the ambient radiation bath. +We find that, at the onset of thermal inflation, the flaton field may reach an equilibrium +with the radiation bath even for relatively small coupling constants, maintaining a spectrum +of thermal fluctuations until the critical temperature Tc, below which thermal effects stop +holding the field at the false potential minimum. This enhances the field variance compared +to purely quantum fluctuations, therefore increasing the average energy density during ther- +mal inflation and damping the induced curvature perturbations. In particular, we find that +this inhibits the later formation of primordial black holes, at least on scales that leave the +horizon for T > Tc. The larger thermal field variance also reduces the duration of a period +of fast-roll inflation below Tc, as the field rolls to the true potential minimum, which should +also affect the generation of (large) curvature perturbations on even smaller scales. +arXiv:2301.11666v1 [hep-ph] 27 Jan 2023 + +Contents +1 +Introduction +1 +2 +Thermal inflation +2 +3 +Curvature Perturbations +5 +4 +Comparison between the thermal and quantum power spectra +10 +5 +Conclusion +12 +A Evolution of the temperature during thermal inflation +13 +B Field correlation functions +14 +1 +Introduction +It is widely believed that the universe went through a period of inflation in its early stages +[1–4], thus explaining its observed homogeneity and isotropy on large scales, as well as its +apparently small spatial curvature. +Most importantly, inflation in principle provided the +seeds for the small curvature perturbations that grew into the large-scale structure that we +observe in the Universe. +Although the simplest models postulate a single period of slow-roll inflation lasting for at +least 50-60 e-folds after the largest presently observable scales became super-horizon, there is +a priori no reason to exclude scenarios with multiple inflation periods with different dynamics. +In particular, it is well known that reheating after inflation may lead to the production of e.g. +topological defects if the associated reheating temperature exceeds the grand unification scale +(∼ 1016 GeV) [5] or other unwanted relics such as moduli or gravitinos in supersymmetric +(SUSY) models [6–8]. Such relics could have overclosed the Universe or spoiled the successful +predictions of primordial nucleosynthesis through their late decay [9]. This and the fact that +currently there is no evidence for such relics motivates considering scenarios with additional +inflationary stages that could have diluted their abundances [10–14]. +One of the most appealing possibilities is a late period of thermal inflation, where a +scalar flaton field is trapped in a false vacuum by thermal effects above a certain critical +temperature. Candidates to drive such a secondary inflation period are ubiquitous in SUSY +and supergravity theories, in particular given the many flat directions in the scalar potential +that characterize such models at the renormalizable level [15]. The spectrum of curvature +perturbations generated during such a period (or possibly multiple periods) need not be +nearly as scale-invariant as the one generated by the first period of slow-roll inflation, during +which the large-scale perturbations observable in the Cosmic Microwave Background (CMB) +anisotropies became super-horizon. In fact, this spectrum was recently computed in [16], +where it was shown that large curvature perturbations could have been generated (on small +scales) during a period of thermal inflation and a fast roll inflation period [17] that potentially +followed it once thermal effects stopped trapping the field in the false vacuum state. These +large curvature/density perturbations could have then collapsed into a significant population +of primordial black holes upon horizon-reentry later in the radiation-dominated epoch. Such a +– 1 – + +possibility has attracted a substantial interest in the recent literature given the latter’s appeal +as dark matter candidates and the possibility that these may explain the recent LIGO/Virgo +detections of heavy black hole binaries (see e.g. [18]). +The analysis in [16] considered, however, only the part of the curvature spectrum gen- +erated by quantum fluctuations of the flaton scalar field. Since thermal effects are a crucial +aspect in the dynamics of thermal inflation, one should investigate whether thermal fluctua- +tions also play an important role, which is our goal with this work. We note, in particular, +that the flaton field is trapped in a false vacuum at temperatures above a certain critical tem- +perature, as we review in the next section, due to the large thermal mass resulting from its +interactions with the ambient thermal bath. It is well-known that such interactions also lead +to fluctuation-dissipation effects, resulting in an effective Langevin-like equation describing +the dynamics of the scalar field. Such effects have been thoroughly analyzed in the context +of warm inflation scenarios [19–36], in setting initial conditions for slow-roll inflation in a +pre-inflationary radiation epoch [37], and in cosmological phase transitions both after and +during (warm) inflation [38,39]. Our objective is then to apply the techniques developed in +these contexts to the case of thermal inflation, and investigate their role in the generation of +curvature perturbations during this period. +Surprisingly, we find that for thermal flaton fluctuations the amplitude of the curvature +power spectrum is suppressed with respect to the purely quantum case analyzed in [16], at +least for scales exiting the horizon before the temperature decreases below the critical value. +This is essentially due to the fact that, as we will show, thermal effects, by enhancing flaton +density fluctuations, also increase the time-dependent part of the average energy density +during thermal inflation. This effect overcomes the enhancement of individual perturbation +modes, therefore suppressing the corresponding power spectrum. +This work is organized as follows. We will start by constructing a generic model for +thermal inflation in Section 2. The curvature perturbation spectrum induced by the thermal +flaton fluctuations is computed in Section 3. In Section 4 we compare our result with the +purely quantum computation performed in [16], discussing and summarizing our conclusions +in Section 5. We use natural units throughout this work, ℏ = c = kB = 1 and the reduced +Planck mass MP = 2.435 × 1018 GeV. +2 +Thermal inflation +Let us consider a scalar field φ interacting with a thermal radiation bath at temperature +T, with energy density ρR = π2 +30g∗T 4, where g∗ denotes the number of relativistic degrees +of freedom. For concreteness, we consider a radiation bath made up of NF Dirac fermion +species ψi, which interact with the scalar field through Yukawa interactions with universal +coupling constant g: +LY = −gφ +NF +� +i=1 +¯ψiψi . +(2.1) +We take the mass of the fermions mψi ≪ T, so that they can be treated as relativistic degrees +of freedom, but such that mψi > H so that flat quantum field theory calculations for the +decay width of scalars into fermions are valid [37]. +We assume that the scalar field φ corresponds to a renormalizable flat direction, or flaton +field, common in several SUSY/supergravity scenarios [11,12,14,40,41], such that its potential +is only lifted by soft terms such as a mass term from SUSY breaking, and non-renormalizable +– 2 – + +terms. We are interested in the case where the squared mass term is negative, such that the +field acquires a large expectation value M0 at zero temperature from the latter’s interplay +with the non-renormalizable operators. +The interaction with the radiation bath induces, +however, a thermal mass correction such that the field’s effective mass is of the form [42]: +m2 +eff = α2T 2 − m2 , +(2.2) +where m corresponds to the zero temperature (tachyonic) mass and α is the effective coupling +to the thermal bath. For the Yukawa interactions described above we have α2 = g2NF /6 at +one-loop order. This implies, in particular, that for temperatures above the critical value, +Tc ≡ m/α, the origin is a stable minimum of the scalar potential, whereas for lower tempera- +tures the minimum is non-trivial and asymptotes to M0 in the limit of vanishing temperature. +The origin thus constitutes a false vacuum state, near which we may write the scalar potential +as: +V (φ) = 1 +3M2 +0 m2 + 1 +2m2 +effφ2 + · · · , +(2.3) +where for concreteness we have chosen the constant term such that, if the leading non- +renormalizable term is ∼ φ6 the cosmological constant vanishes at the minimum, V (φ = +M0) = 0, although this is not crucial to our analysis. For typical flat directions, M0 ≫ m, +since the scale at which the non-renormalizable operators become relevant is generically large +(around the grand unification or even the Planck scale). +If, after the first period of slow-roll inflation, the Universe is reheated to attain a tem- +perature T > Tc, the flaton field will thus be driven to the false minimum at the origin by +Hubble friction, where it is trapped and gives a contribution V0 = M2 +0 m2/3 to the vacuum +energy. Since the temperature drops as the universe expands, i.e. ρR ∝ a−4, eventually this +vacuum energy may become dominant, thus triggering a new period of inflation, with expan- +sion rate H ≃ mM0/3MP ≲ m. Thermal inflation thus begins when the temperature drops +below: +Ti = +� 10 +g∗π2 +� 1 +4 � +M0m . +(2.4) +Assuming that there is no significant entropy production during thermal inflation, as we +confirm in Appendix A, the temperature of the radiation bath drops as T ∝ a−1 during +thermal inflation, eventually reaching the critical value Tc below which the minimum at the +origin is destabilized. The nature of the phase transition (or smooth crossover) that ensues is +model-dependent and irrelevant to our discussion (see e.g. [43]), since we are mostly interested +in what happens for temperatures Tc < T < Ti. +We note that thermal inflation is only possible if the flaton field has a non-negligible +interaction with the thermal bath, and in particular Ti > Tc imposes: +α > +�g∗π2 +10 +� 1 +4 � m +M0 +. +(2.5) +For instance, for m ∼ 10 TeV and M0 ∼ MP , this imposes the lower bound α ≳ 10−7 for +g∗ = 10−100. Although this may not seem too restrictive, we note that the number of e-folds +of thermal inflation is given by: +N(TI) +e += ln +�Ti +Tc +� += 1 +2 ln +�M0 +m +� ++ 1 +4 ln +� 10 +π2g∗ +� ++ ln(α) . +(2.6) +– 3 – + +For the reference values given above, we see that a period of thermal inflation lasting more +than 10 e-folds is only possible for α ≳ 0.01, with even larger effective couplings required for +scenarios with a smaller hierarchy between the mass scales m and M0. +We note that inflation does not necessarily end when the temperature falls below Tc, +since expansion only stops accelerating once the flaton’s kinetic energy surpasses its potential +energy. Below Tc the field develops a tachyonic instability, since m2 +eff ≃ −m2 < 0 once T ≪ Tc, +and its value moves away from the origin as ∼ emt ∼ e +m +H Ne for H ≲ m, and there may be +a period of fast-roll inflation [17] until the field gets close to the minimum at M0 and its +kinetic energy takes over. Note that, in the opposite regime m ≲ H, thermal inflation would +be followed by an additional period of slow-roll inflation, but we will not consider this regime +in our discussion. The duration of the fast-roll period is, of course, model dependent and, +moreover, dependent on the mean field value at the critical temperature. +In [16,17] it was shown that this period may last for as much as, or even longer than, the +thermal inflation period for H/m ≲ 1, depending on the flaton’s mass value. This assumed, +however, that the mean field value at the critical temperature is set by quantum fluctuations, +which as we will see is not necessarily the case. In particular, thermal fluctuations typically +enhance the field’s variance at Tc, therefore reducing the duration of the subsequent fast- +roll period. +For this reason, we will restrict our analysis to the thermal inflation period +(Tc < T < Ti), discussing the implications of our results to the subsequent cosmological +evolution at the end of our discussion. +Independently of whether or not there is a significant period of inflation below Tc, the +field will eventually begin oscillating about the minimum of its potential and decay away +through the Yukawa interactions in Eq. (2.1) [44]. Although we do not specify the exact +nature of the fermion fields in the thermal bath, since we are only modelling the interactions +between the flaton and the ambient radiation and our discussion is largely independent of the +particular interactions considered, it is implicit that such interactions will eventually lead to +the reheating of the Standard Model degrees of freedom at temperatures exceeding at least +a few MeV to ensure the correct conditions for primordial nucleosynthesis. +We note that having late thermal inflation and fast-roll inflation periods alters the +predictions of inflationary cosmology [45], since the largest CMB scales leave the horizon +50-60 e-folds before the end of the full inflationary epoch, including the primary slow-roll +inflation period, which therefore must necessarily be shorter. +Although the leading effect of the interactions between the flaton and the thermal bath +is the thermal mass correction responsible for its trapping at the origin, it also induces +fluctuation-dissipation effects in the flaton’s dynamics that, as we will see, can play an im- +portant role in the evolution of field perturbations during thermal inflation. These have been +considered in [46] to analyze the nature of the phase transition at Tc, but their effects on +the associated spectrum of curvature perturbations have so far been overlooked. To study +them, we consider the full Langevin-like equation for the flaton field modes φk of comoving +momentum k, which can be obtained through standard techniques in linear response theory +assuming the ambient radiation bath is close to an equilibrium state, and is given by (see +e.g. [25,47]): +¨φk + (3H + Γφ) ˙φk + ω2 +kφk = ξk , +(2.7) +where ω2 +k = k2/a2 +m2 +eff and Γφ is the dissipation coefficient, which for a field oscillating near +a local minimum of its potential (in this case the false minimum at the origin for T > Tc) +coincides with its finite-temperature decay width [48]. On the right hand side of (2.7), ξk +is a stochastic noise term which encodes the randomness of the field’s interactions with the +– 4 – + +thermal bath. For modes with physical momentum p = k/a ≲ πT it is well approximated by +a gaussian white noise term with a two-point correlator given by the fluctuation-dissipation +relation [46,49].: +⟨ξk(t)ξk′(t′)⟩ = 2ΓφT (2π)3 +a3 +δ3(k + k′)δ(t − t′) . +(2.8) +We note that physically this is reminiscent of the Brownian motion of a heavy particle in an +gas, for which random collisions with the gas molecules induce an effective friction that damps +its motion. However, the particle never actually comes to rest due to the very same random +collisions, eventually reaching an equilibrium with the gas. We expect something very similar +to occur to the flaton field modes, with the combined effects of dissipation (Γφ) and thermal +fluctuations (ξk) driving the field towards a thermal equilibrium with the radiation bath. +This behaviour has been observed for scalar fields interacting with a radiation bath both +in an inflationary and non-inflationary context [37, 39], so we anticipate that the same will +occur in the case of thermal inflation. +At finite temperature the flaton decay width into relativistic fermions is given by [27,37]: +Γφ(p) = 3m2 +effα2 +4πωp +� +1 + 2T +p ln +�1 + exp(− ω+ +T ) +1 + exp(− ω− +T ) +�� +, +(2.9) +where ω± = |ωp±p| +2 +and we have neglected the mass of the fermions, T ≫ mψi. Note that +fermions acquire a mass through their interaction with the flaton field but, as we will obtain +bellow, +� +⟨φ2⟩ ≲ T for perturbative couplings. +Since the thermal bath will excite field modes p ≲ T and meff ≲ T, the decay width can +be well approximated by: +Γφ ≃ 3m2 +effα2 +16πT +≃ 3α4 +16πT , +(2.10) +where in the last step we have used meff ≃ αT for T ≳ Tc. At the onset of thermal inflation, +we then have: +Γφ +H +���� +Ti +≃ +9 +16π +� 10 +g∗π2 , +�1/4 +α4 +MP +√M0m +≃ 2.3g−1/4 +∗ +� α +0.03 +�4 �MP +M0 +�1/2 � +m +10 TeV +�−1/2 +, +(2.11) +so that we expect dissipative effects to play an important role in the field’s dynamics roughly +for the same range of the effective coupling α leading to a period of thermal inflation lasting +for more than 10 e-folds, as we have seen above. In the next section we compute the thermal +field correlators and associated curvature perturbation power spectrum to better quantify +this statement. +3 +Curvature Perturbations +Let us consider the gauge-invariant curvature perturbation on uniform density hypersurfaces, +which in the flat gauge can be written as [50,51]: +ζ = − H +˙⟨ρ⟩ +δρ , +(3.1) +– 5 – + +where the perturbation of a generic function is given by δf(t, x) ≡ f(t, x) − ⟨f(t, x)⟩, and +brackets denote its thermal averaged value. The dimensionless power spectrum of ζ is defined +as [16], +∆2 +ζ(k) = k3 +2π2 +� +d3x exp(−ik · x) ⟨ζ(0)ζ(x)⟩ , += +2k3 +(2π)2 +� H +˙⟨ρ⟩ +�2 � +d3x exp(−ik · x) ⟨δρ(0)δρ(x)⟩ . +(3.2) +The total energy density ρ during thermal inflation includes the contributions from both the +flaton field and the radiation fluid [52]: +ρ = ρφ + ρR = 1 +2 +˙φ2 + V (φ) + 1 +2a−2(t)∂iφ∂iφ + π2 +30g∗T 4 , +(3.3) +and so we have +⟨ρ⟩ = π2 +30g∗T 4 + 1 +3m2M2 +0 + 1 +2m2 +eff ⟨φ2⟩ + 1 +2 ⟨ ˙φ2⟩ + 1 +2a−2 ⟨∂iφ∂iφ⟩ , +(3.4a) +δρ = 1 +2m2 +effδ(φ2) + 1 +2δ( ˙φ2) + 1 +2a−2δ(∂iφ∂iφ) . +(3.4b) +Since density perturbations involve perturbations of quadratic functions of the field and its +derivatives, the power spectrum, Eq. (3.2), involves contributions of the form: +⟨δ(Xi(0)2)δ(Xj(x)2)⟩ = ⟨Xi(0)2Xj(x)2⟩ − ⟨Xi(0)2⟩ ⟨Xj(x)2⟩ , +(3.5) +where Xi generically denotes the field perturbations and their derivatives. The first term +on the right-hand side corresponds to 4th moments involving the gaussian variables Xi. +According to Isserlis’ theorem [53] it is possible to write a kth moment of zero-average +gaussian variables in terms of their variances. Thus, the correlators can be simply written +as [54]: +⟨δ(Xi(0)2)δ(Xj(x)2)⟩ = 2 ⟨Xi(0)Xj(x)⟩2 . +(3.6) +The two-point correlation function for the energy density is then: +⟨δρ(0)δρ(x)⟩ = m4 +eff +2 +⟨φ(0)φ(x)⟩2 + m2 +eff ⟨φ(0) ˙φ(x)⟩ +2 + a−2m2 +eff ⟨φ(0)∂iφ(x)⟩2 , ++ 1 +2 ⟨ ˙φ(0) ˙φ(x)⟩ +2 + a−2 ⟨ ˙φ(0)∂iφ(x)⟩ +2 + a−4 +2 +⟨∂iφ(0)∂jφ(x)⟩2 , +(3.7) +that is, contributions from all possible correlation functions involving φ, ˙φ and ∂iφ. +We note that we are interested in computing the curvature perturbation power spectrum +on super-horizon scales, k ≪ aH. To do this we need to compute the field variance ⟨φ2⟩ and +the average kinetic and gradient energies appearing in Eq. (3.4a), which involve integrating +over all thermally excited field modes. Since the noise term correlator is exponentially sup- +pressed for physical momentum scales p ≳ πT [46], we use this value as a hard cutoff. This +can be translated into a comoving momentum cutoff kc = πTc if we set a(Tc) = 1, following +the conventions of [16] to allow for a better comparison with the purely quantum calculation. +To compute the power spectrum we need the three combinations of the correlations +between φk and ˙φk, i.e. ⟨φkφk⟩, ⟨φk ˙φk⟩ and ⟨ ˙φk ˙φk⟩. These are the building blocks of all the +remaining correlation functions involved in the power spectrum. We will explicitly compute +– 6 – + +the correlator of the field modes and list all others in Appendix B as their computation +follows similar steps. +The equal-time two-point correlation function of the field modes can be written in terms +of the Green’s function associated with (2.7) and the noise correlator: +⟨φk(z)φk′(z)⟩ = H−4 +� z +zi +ds1 +� z +zi +ds2 s−2 +1 s−2 +2 Gs(z, s1)Gs(z, s2) ⟨ξk(s1)ξk′(s2)⟩ , +(3.8) +where we have traded the time-dependence for a dependence on the variable z = T/H, with +zi = Ti/H. Note that z ∝ a−1 during thermal inflation, so that it is a decreasing function +of time. We have ignored the contributions from the homogeneous solutions of (2.7) since, +as we will see bellow, they quickly become subdominant. These are required, however, to +compute the Green’s function, which is given by the usual expression: +Gs(z, s) = φ(1) +k (s)φ(2) +k (z) − φ(1) +k (z)φ(2) +k (s) +W(φ(1) +k , φ(2) +k )(s) +, +(3.9) +where φ(1) +k +and φ(2) +k +are the homogeneous solutions of equation (2.7) and W denotes their +Wronskian. +During most of thermal inflation, except for temperatures close to the critical value, +the thermal mass dominates over the field’s zero temperature mass, αT ≫ m. This allows +us to compute analytically the field modes, and thus obtain the field’s two-point correlation +function with a decay width of the form (2.10). +The homogeneous equation of motion for the flaton field modes (2.7) can be written in +terms of the z variable as: +z2φ′′ +k − z (2 + γz) φ′ +k + z2¯ω2 +kφk = 0 , +(3.10) +where ¯ω2 +k ≡ ω2 +k/T 2 ≃ k2/T 2 +c + α2 and γ ≡ 3α4/16π, such that Γφ/H = γz. Let us define +φk = zeγz/2χk, such that: +χ′′ +k + +� +¯ω2 +k − γ2 +4 − γ +z − 2 +z2 +� +χk = 0 . +(3.11) +Even though we can express the exact solutions of the above equation in terms of Whittaker +functions [55], it is more instructive to note that, since γ ≪ ¯ω2 +k for α ≲ 1 and z > zc = +m/αH > α−1 > 1, we may neglect all the terms inside the brackets in Eq. (3.11) except for +the one involving ¯ω2 +k to a good approximation. This means that the homogeneous solutions +are approximately given by: +φ(1) +k (z) ≃ ze +γ +2 z sin(¯ωkz) , +φ(2) +k (z) ≃ ze +γ +2 z cos(¯ωkz) , +(3.12) +thus constituting oscillatory functions in the z variable with an amplitude decreasing due to +both Hubble expansion (z ∝ a−1) and the field’s decay into the light fermions. This yields +the Green’s function: +Gs(z, s) = 1 +¯ωk +z +s exp +�γ +2(z − s) +� +sin +� +¯ωk(z − s) +� +. +(3.13) +– 7 – + +The noise correlation function can be written in terms of the z variable as: +⟨ξk(z1)ξk′(z2)⟩ = 2Hz1ΓφT (2π)3 +a3 +δ3(k + k′)δ(z1 − z2) , +≃ 2γH3z6 +1 +(2π)3 +z3c +δ3(k + k′)δ(z1 − z2) , +(3.14) +where in the second line we used the dominance of the thermal mass for T > Tc. +We may now substitute Eqs. (3.13) and (3.14) into Eq. (3.8) to obtain the field’s two- +point correlation function: +⟨φk(z)φk′(z)⟩ = (2π)3δ3(k + k′) T +a3ω2 +k +(1 − δ) , +δ = exp +� +− 3α4 +16π +Ti +H +� +1 − T +Ti +�� +, +(3.15) +where again we used that ¯ωk ≫ γ. Note that for Γφ/H(Ti) ≳ 1, we have δ ≪ 1 for all +temperatures below Ti (but above Tc), thus yielding a thermal equilibrium distribution for +the field modes that is independent of the decay width. This means that if the field decays +efficiently at the onset of thermal inflation it will attain an equilibrium distribution that +simplify redshifts with expansion (with corresponding decrease in temperature). +This is +a generic result obtained in other cosmological contexts [37, 39] that we now recover also +within thermal inflation – it simply states that if the field interacts significantly with the +thermal bath at some point during its evolution it reaches a near-thermal configuration that +is subsequently maintained unless there is some significant change in the field’s properties +(in our case the tachyonic instability just below the critical temperature). +We note that the two-point correlation function vanishes at the onset of thermal inflation +by construction, since the integral Eq. (3.8) is zero at z = zi. +This assumes that field +modes were not excited when thermal inflation begins, which need not be the case since +interactions with the thermal bath are present in the prior radiation-dominated epoch. If field +modes thermalize before its vacuum energy becomes dominant, Eq. (3.15) will nevertheless +hold (with δ ≃ 0), since this result is also valid for a radiation-dominated cosmological +background [37]. However, we note that during the radiation era Γφ/H ∝ T/H ∝ a, while +Γφ/H ∝ a−1 during thermal inflation, so that this ratio attains its maximum value at the +onset of thermal inflation. Recalling Eq. (2.11), we conclude that α ≳ 0.01 is required for +field thermalization if the zero temperature mass m is not far from the TeV scale at which +new physics may be expected. As discussed in the previous section, this is exactly the regime +where a period of thermal inflation lasting more than 10 e-folds (and which can in particular +sufficiently dilute unwanted relics of the first reheating process) can occur. We will thus +henceforth focus our analysis on this parametric regime, in which the field thermalizes either +before or at the onset of the thermal inflation epoch. +We may now use Eq. (3.15) to compute the field variance and related correlation func- +tions, as we detail in Appendix B. We obtain for the total average energy density: +⟨ρ⟩ = π2 +30 +� +g∗ + 5 +π(1 − δ) +� +T 4 + 1 +3m2M2 +0 , +(3.16) +where we note that the field contributes essentially as an additional bosonic degree of freedom +to the radiation energy density if thermalization is efficient (δ ≪ 1). Its contribution is not +exactly one degree of freedom since we have considered a hard-cutoff on the momentum of +the modes that are excited by interactions with the thermal bath at kc = πTc. +This is +– 8 – + +only an approximation to the smooth cutoff associated with the noise correlator [46], which +nevertheless captures the essential physics of the problem. +Using the values of each component of the power spectrum (3.5) given in Appendix B, +the density perturbations are: +� +d3x exp(−ik · x) ⟨δρ(0)δρ(x)⟩ ≈ πT 5 +6a3 +� +1 + 3 +� 3α4 +32π2 +�2� +1 − α +π arctan +�π +α +� �� +(1 − δ)2 , +(3.17) +to leading order on super-horizon scales k < aH < αTc. We note that the first term within +the square brackets dominates over the second one. This then yields for the power spectrum +on super-horizon scales: +∆2 +ζ +(therm)(k) = +150 +(2π)5 +k3 +T 3c +(1 − δ)2 +� +g∗ + 5 +π(1 − δ) − 5 +π +3α4 +64π +T +H δ +�2 , +≃ +150 +(2π)5 +α3 +g2 +∗,f +�H +m +�3� k +kc +�3 +, +(3.18) +where in the second line we have taken the prompt thermalization limit, i.e. δ ≪ 1, in +which case the flaton field contributes to the total number of relativistic degrees of freedom, +given by g∗,f ≃ g∗ + 5/π. Note that this result is time-independent, reflecting the freeze-out +of curvature perturbations on super-horizon scales and thus the single-fluid nature of the +dynamics, i.e. the fact that the flaton field thermalized with the radiation bath. +The power spectrum is blue-tilted so its maximum value is attained for the last scale to +leave the horizon during thermal inflation, i.e. kc = H which leaves at T = Tc. Although our +calculation assumes the dominance of the thermal piece of the flaton’s mass, an approximation +that breaks down close to the critical temperature, we may extrapolate our results with a +reasonable accuracy to kc, thus yielding an upper bound on the power spectrum of scales +leaving the horizon before the phase transition, in the thermal equilibrium limit: +∆2 +ζ +(therm, max)(k) ≃ 150 +(2π)5 +α3 +g2 +∗,f +�H +m +�3 +. +(3.19) +The power spectrum would, thus, be maximized for g∗,f ∼ α ∼ H +m ∼ 1, yielding ∆2 +ζ +(therm, max) ∼ +10−2, but in realistic scenarios with perturbative couplings and at least one fermionic degree +of freedom in the ambient thermal bath the power spectrum should have a parametrically +smaller amplitude. +Hence, if the flaton field has significant interactions with the radiation bath, α ≳ 0.01 (as +expected in scenarios with a significant number of e-folds of thermal inflation), the thermal +nature of its fluctuations suppresses the amplitude of the induced curvature perturbations +on super-horizon scales, which is the main result of this work. While this may seem sur- +prising, given that thermal fluctuations generically have a larger amplitude than quantum +vacuum fluctuations (as considered in [16]), it has a simple physical explanation: fluctuation- +dissipation effects increase not only the density fluctuations on super-horizon scales but also +the field variance and the average gradient and kinetic energies, thus, the average energy den- +sity. The latter effect turns out to be more significant and, hence, decreases the amplitude +of the associated curvature power spectrum with respect to the quantum case. +– 9 – + +A relevant consequence of our analysis is that, in realistic scenarios, we do not expect the +amplitude of the curvature power spectrum to be sufficiently large to lead to the formation of +primordial black holes, which would require ∆2 +ζ ≳ 10−2 [56–58], at least on scales that become +super-horizon above the critical temperature. This motivates a better comparison with the +results obtained in [16] for quantum flaton fluctuations, where larger curvature perturbations +were obtained. We pursue this comparison in the next Section. +4 +Comparison between the thermal and quantum power spectra +The linear approximation to the quantum power spectrum is given in [16] by: +∆2 +ζ +(quan)(k) = +4 +√π +Γ(ν) +ν2Γ +� +ν − 3 +2 +� +�H +m +�3−2ν� k +kc +�3�� k +kc +�2 ++ m2 +H2 +�−ν +, +(4.1) +where ν = +� +m2/H2 + 9/4. To better compare our results with those obtained assuming +purely quantum flaton fluctuations in [16], we plot both power spectra as a function of +comoving momentum in Figure 1. We show the case of H/m = 0.3 (which according to the +analysis in [16] yields all dark matter in the form of primordial black holes) and taking α = 1, +NF = 1 and δ = 0 to maximize the thermal power spectrum. We note that the thermal power +spectrum is only shown up to k = kc, since our calculation is only valid for modes that exit +the horizon before the phase transition; whereas the quantum calculation can be extended to +larger momentum, assuming a subsequent period of fast-roll inflation as mentioned earlier. +quantum +thermal +0.5 +1 +5 +10 +10-5 +10-4 +10-3 +10-2 +k / kc +Δζ +2 +Figure 1. The quantum power spectrum (blue) and the thermal power spectrum (red) as a function +of k for H/m = 0.3, α = 1, mNF = 1 and δ = 0. +As one can clearly see in this figure, thermal fluctuations significantly suppress the cur- +vature perturbation spectrum with respect to the quantum case, for the reasons explained +in the above section. Furthermore, whereas quantum vacuum fluctuations may yield a suffi- +ciently large amplitude to lead to primordial black hole formation, a thermalized flaton field +induces much smaller perturbations, although they may nevertheless exceed the even smaller +fluctuations observed on large scales in the CMB anisotropies spectrum. +We should note that the quantum power spectrum peaks at scales that leave the horizon +for T < Tc, where our approximations break down. Extending our calculation to this regime +– 10 – + +would involve a different form of the dissipation coefficient, since as the field experiences +a tachyonic instability the latter no longer corresponds to the perturbative decay width +at finite temperature. Let us note, however, that fluctuation-dissipation effects are more +pronounced at the start of thermal inflation as discussed earlier, so that they no longer +play a significant role near Tc. If the field thermalizes at the onset of thermal inflation, it +will nevertheless maintain an equilibrium distribution with a decreasing temperature due to +inflationary expansion. Let us then compare the magnitude of field fluctuations at Tc in both +the quantum vacuum and thermal cases. The thermal variance is obtained by expanding the +field in terms of its modes +⟨φ(x)φ(y)⟩ = +� +d3k +(2π)3 +d3k′ +(2π)3 ⟨φkφk′⟩ exp(ik · x) exp(ik · y) , +(4.2) +and using the field modes correlator (3.15), we obtain for the field variance in the thermalized +limit: +⟨φ2⟩therm = +2 +(2π)2 +T +a +� kcutoff +0 +dk +k2 +k2 + α2T 2c += T 2 +2π +� +1 − α +π arctan +�π +α +�� +, +(4.3) +which we note is only mildly dependent on the effective coupling α, while the quantum one +is given by [16]: +⟨φ2⟩quan = +� H +2π +�2 Γ2(ν)22ν +6π +�aH +m +�2ν +F +� +ν, 3 +2; 5 +2; − +�aH +m +�2� +, +(4.4) +where F(a, b, c, z) denotes the Hypergeometric function. The field variance in both cases is +shown in Figure 2, where we extrapolate the thermal variance beyond the phase transition +purely for comparison purposes. +quantum +thermal +0.1 +0.5 +1 +5 +10 +10-8 +10-4 +1 +a / ac +ϕ2 / H2 +Figure 2. Quantum (blue) and thermal (red) field variance as a function of the scale factor for +H/m = 0.3, α = 1 and δ = 0. The critical temperature corresponds to the dashed vertical line, below +which the thermal variance is extrapolated, as indicated by the dashed red line. +As one can clearly observe in this figure, the quantum field variance is several orders of +magnitude smaller than the thermal variance before the phase transition, which validates our +calculation in neglecting vacuum fluctuations in the thermalized flaton scenario. While at +the critical temperature this is still true, if one extrapolates the thermal variance for T < Tc +– 11 – + +(a > ac = 1), we see that quantum fluctuations become dominant less than one e-fold after +the critical temperature is attained. +While this extrapolation is non-trivial, since the fluctuation-dissipation effects would +have to be re-computed, it may suggest that vacuum perturbations may become dominant +after the phase transition, in which case the computation in [16] would hold. In fact, the peak +in the quantum power spectrum is obtained for modes with k = H +2 +� +3(2ν + 3) > kc = H, +which leave the horizon for temperatures below the critical value and thus, in the example +shown above, already in the regime where the quantum variance is dominant. +This would, in fact, suggest that large enough curvature perturbations leading to pri- +mordial black hole formation may be generated after thermal inflation (from quantum fluc- +tuations), but it is not clear that quantum and thermal fluctuations may be examined in- +dependently nor that the thermal variance maintains its form below Tc. In addition, and +perhaps most importantly, the fact that the thermal variance is still typically a few orders of +magnitude larger than the quantum one at the critical temperature indicates that the flaton +field should reach the minimum of its potential much more quickly if it thermalizes, therefore +considerably shortening, or even possibly, precluding an ensuing period of fast-roll inflation. +A more complete analysis of the problem including both thermal and quantum fluctua- +tions in the analysis, potentially along the lines of [59], is required to compute the spectrum +of curvature perturbations on scales that leave the horizon at temperatures below Tc, and is +left for future work. +5 +Conclusion +We have computed the spectrum of curvature perturbations generated during thermal in- +flation taking into account the thermal fluctuations of the flaton field driving this period. +These are associated with fluctuation-dissipation effects driven by the flaton’s interactions +with the ambient radiation bath. Our analysis involved solving the Langevin-like equation +effectively describing the evolution of the flaton’s Fourier modes. We computed the associ- +ated correlation functions in the approximation of a gaussian white noise and a dominant +thermal contribution to the flaton’s mass, for temperatures above the critical value at which +the flaton is held at the false vacuum at the origin. +We have concluded that, if the flaton’s (finite-temperature) decay width exceeds the +Hubble parameter at the onset of thermal inflation, the field essentially thermalizes with +the ambient radiation bath, contributing approximately as an extra relativistic degree of +freedom. This occurs when the effective coupling between the flaton and the thermalized +degrees of freedom α ≳ 0.01, which roughly corresponds to the parametric regime where over +10 e-folds of thermal inflation (above Tc) occur. We found that the consequent increase in +the field variance and the average gradient and kinetic energies enhances the background +energy density (namely its time-dependent part that determines curvature perturbations) +with respect to a field with purely quantum vacuum fluctuations analyzed in [16]. Despite the +enhancement of super-horizon density fluctuations in the thermal case, the overall amplitude +of the curvature power spectrum is significantly reduced with respect to the quantum case, so +that thermal fluctuations behave very differently compared to their quantum counterparts, +regarding the generation of curvature perturbations during periods of thermal inflation. +While our analysis is not applicable for modes that leave the horizon once the tempera- +ture has fallen below the critical value and the field starts rolling towards the true minimum +of its potential, we expect thermal effects to become less relevant in this regime and quantum +– 12 – + +fluctuations to become dominant, potentially yielding large curvature perturbations at such +scales as computed in [16]. However, a full analysis including both quantum and thermal +fluctuations in the dynamics of the flaton field is required to accurately describe the puta- +tive fast-roll inflation phase below the critical temperature. It must be noted, in any case, +that such a phase is necessarily shortened by the fact that the field variance at the critical +temperature, which sets the typical field value at this stage, is much larger if the field ther- +malizes with the radiation bath. It is therefore unclear whether super-horizon fluctuations +with k > kc can be generated in this phase. +We have modelled the thermal bath through a set of fermion species coupled to the +flaton field, but we expect our main conclusions to hold with the inclusion of other bosonic +fields, like scalars or vector bosons: if Γφ > H at some stage during thermal inflation, the +field will be driven towards a thermal fluctuation spectrum. Only the details of how and when +this equilibrium is attained may depend on the types of light fields that interact with the flat +direction. Our analysis shows that thermalization does not require large coupling constants +describing the interaction between the flaton and the radiation bath. +In any case such +couplings cannot be too suppressed to sustain a sufficiently long period of thermal inflation +that may, in particular, dilute any unwanted relics generated after the primary slow-roll +inflation period. Hence, fluctuation-dissipation effects cannot in general be neglected in the +dynamics of the flaton field and on the curvature perturbations they induce during thermal +inflation. This is particularly relevant if one wishes to understand whether thermal inflation +periods may leave behind a sizeable population of primordial black holes, and we hope that +our work motivates further exploration of these and related issues, including other potential +implications for structure formation in our Universe [60]. +Acknowledgements +M.B.G. work has been partially supported by MICINN (PID2019-105943GB-I00/AEI/10.130 +39/501100011033) and “Junta de Andaluc´ıa” grant P18-FR-4314. JMG acknowledges the +support from the Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P. (FCT) through the Research +Fellowship No. +2021.05180.BD derived from Portuguese national funds. +This work was +supported by the CFisUC project No. UID/FIS/04564/2020 and by the FCT-CERN grant +No. CERN/FIS-PAR/0027/2021. +A +Evolution of the temperature during thermal inflation +In our calculation we assumed that no significant entropy is produced during thermal inflation +as a result of fluctuation-dissipation effects, i.e. that T ∝ a−1. In this appendix we aim to +verify this assumption. The flaton field satisfies the Langevin-like equation [25]: +¨φ + (3H + Γφ) ˙φ − a−2∇2φ + m2 +effφ = ξ , +(A.1) +and by multiplying both sides by ˙φ we obtain: +˙ρφ + 3H(ρφ + pφ) = ξ ˙φ − Γφ ˙φ2 + α2T ˙Tφ2 + a−2∂i( ˙φ∂iφ) , +(A.2) +where the field’s energy density and pressure are given by: +ρφ = 1 +2 +˙φ2 + 1 +2a−2∂iφ∂iφ + V (φ) , +pφ = 1 +2 +˙φ2 − 1 +6a−2∂iφ∂iφ − V (φ) . +(A.3) +– 13 – + +Conservation of the full energy-momentum tensor then yields the following continuity equa- +tion for the radiation energy density: +˙ρR + 4HρR = − ⟨ξ ˙φ⟩ + Γφ ⟨ ˙φ2⟩ − α2T ˙T ⟨φ2⟩ − a−2 ⟨∂i( ˙φ∂iφ⟩) . +(A.4) +We note that the radiation energy density is an ensemble average over the energy density +of the relativistic degrees of freedom, which justifies considering also the thermal average +of the terms on the right-hand side of the above equation. Here we have also neglected the +sub-leading corrections to the radiation energy and entropy densities from the fermions’ finite +mass, ∼ g ⟨ +� +φ2⟩ ∼ gT ≪ T. +Using the field solutions we obtained for the correlators1: +⟨ξ ˙φ⟩ = π +6 ΓφT 4 , +Γφ ⟨ ˙φ2⟩ = π +6 ΓφT 4(1 − δ) , +α2T ˙T ⟨φ2⟩ = α2 +2πT 3 ˙T +� +1 − α +π arctan +�π +α +�� +(1 − δ) ≈ 15α2 +4π3g∗ +(1 − δ) ˙ρR , +⟨∂i( ˙φ∂iφ⟩ = 0 . +(A.5) +Note that the third term is related to the time-dependence of the thermal flaton mass, and +yields a contribution to the variation of the radiation energy density comparable to the above- +mentioned sub-leading corrections from the fermions’ non-vanishing mass. For consistency, +we thus neglect this term, and obtain: +˙ρR + 4HρR = −5/π +g∗ +ΓφρRδ . +(A.6) +From this we immediately see that the right-hand side can only be significant if Γφ ≳ H, +but this implies a quick thermalization of the flaton field such that δ → 0 exponentially +fast, thus making this term negligible. This simply reflects the balance between the effects +of fluctuations and dissipation as the flaton field reaches an equilibrium with the radiation +bath. Note, furthermore, that the term on the right-hand side is suppressed by the relative +contribution of the flaton to the number of relativistic species in equilibrium, (g∗,f − g∗)/g∗, +as obtained in Section 3. We therefore conclude that one may consistently assume ρR ∝ a−4 +and hence that T ∝ a−1 during thermal inflation. +B +Field correlation functions +Here we list the field correlation functions used to compute the curvature perturbation power +spectrum. As we mentioned above, when integrating over momentum modes we consider +a sharp cut-off at k = πTc, which constitutes a good approximation to the behaviour of +the noise correlation function [46]. +To compute the curvature perturbation power spec- +trum on super-horizon scales, k ≪ aH, we consider the leading order results in k/αTc ∼ +(k/aH)(M0/MP )a ≪ 1 considering M0 < MP and noting that in our convention a < ac = 1 +above the critical temperature. +1⟨∇( ˙φ∇φ⟩) = − +� +d3k1 +(2π)3 +d3k2 +(2π)3 ⟨ ˙φk1φk2⟩ k2 · (k1 + k2) exp [ix · (k1 + k2)] = 0 , since the integral of this delta +function is non-zero if and only if k1 = −k2. +– 14 – + +Mode correlators +The building blocks of all field correlators are the equal time correlators between the field +modes and their time derivatives. Writing φk and ˙φk in terms of the Green’s function (3.13): +φk = H−2 +� z +zi +ds s−2Gs(z, s)ξk(s) , +˙φk = H−1z +� z +zi +ds s−2∂zGs(z, s)ξk(s) , +(B.1) +one finds: +⟨φkφk′⟩ = (2π)3δ3(k + k′) T +a3ω2 +k +(1 − δ) , +⟨ ˙φk ˙φk′⟩ = (2π)3δ3(k + k′) T +a3 (1 − δ) , +⟨φk ˙φk′⟩ = −Γφ +2 ⟨φkφk′⟩ = −(2π)3δ3(k + k′) TΓφ +2a3ω2 +k +(1 − δ) . +(B.2) +Field correlators +To compute the total energy density (3.4a) one needs to determine the field variance and the +average kinetic and gradient energies. Expanding the field in terms of comoving momentum +modes, these are given by: +⟨φ2⟩ = +� +d3k +(2π)3 +d3k′ +(2π)3 ⟨φkφk′⟩ exp(ix · (k + k′)) , +⟨ ˙φ2⟩ = +� +d3k +(2π)3 +d3k′ +(2π)3 ⟨ ˙φk ˙φk′⟩ exp(ix · (k + k′)) , +⟨∂iφ∂iφ⟩ = +� +d3k +(2π)3 +d3k′ +(2π)3 ⟨φkφk′⟩ k · k′ exp(ix · (k + k′)) . +(B.3) +Inserting the mode correlation functions (B.2) and integrating over comoving momenta up +to πTc one obtains: +⟨φ2⟩ = T 2 +2π (1 − δ) +� +1 − α +π arctan +�π +α +�� +, +⟨ ˙φ2⟩ = πT 4 +6 (1 − δ) , +⟨∂iφ∂iφ⟩ = π +2 a2T 4(1 − δ) +�1 +3 − +�α +π +�2 ++ +�α +π +�3 +arctan +�π +α +�� +. +(B.4) +Contributions to the power spectrum +Consider the power spectrum of a generic correlator ⟨Xi(0)Xj(x)⟩, for example X1 = φ, +X2 = ˙φ and X3 = ∂iφ, that appears in (3.7): +� +d3x exp(−ik · x) ⟨Xi(0)Xj(x)⟩2 . +(B.5) +Note that, upon expanding each quantity Xj(x) in terms of comoving momentum modes, this +yields four momentum integrals and a volume integral. Two of the momentum integrals can +– 15 – + +be performed using the two delta functions appearing in the mode correlators (B.2). Then, +the volume integral will generate a delta function with the two surviving momentum modes: +� +d3x exp[−ix · (k1 + k2 + k)] = (2π)3δ3(k1 + k2 + k) . +(B.6) +After integrating this delta function over another of the 3-momentum variables, we are left +with a single 3-dimensional integral over k that we need to compute in each case. In the +following table we give the different contributions to the power spectrum in terms of their +corresponding momentum integrals: +Table 1. Contributions to the power spectrum in Eq. (3.17). +field-field +m4 +eff +2 +� +d3x exp(−ik · x) ⟨φ(0)φ(x)⟩2 +(1 − δ)2 +α3 +2(2π)3 T 5 +a3 I1(k) +field-kinetic +m2 +eff +� +d3x exp(−ik · x) ⟨φ(0) ˙φ(x)⟩ +2 +(1 − δ)2 +α +(2π)3 +� 3α4 +32π +�2 T 5 +a3 I1(k) +field-gradient +a−2m2 +eff +� +d3x exp(−ik · x) ⟨φ(0)∂iφ(x)⟩2 +(1 − δ)2 +α3 +(2π)3 T 5 +a3 I2(k) +kinetic-kinetic +1 +2 +� +d3x exp(−ik · x) ⟨ ˙φ(0) ˙φ(x)⟩ +2 +(1 − δ)2 +α3 +2(2π)3 T 5 +a3 I3(k) +kinetic-gradient +a−2 � +d3x exp(−ik · x) ⟨ ˙φ(0)∂iφ(x)⟩ +2 +(1 − δ)2 +α +(2π)3 +� 3α4 +32π +�2 T 5 +a3 I2(k) +gradient-gradient +a−4 +2 +� +d3x exp(−ik · x) ⟨∂iφ(0)∂jφ(x)⟩2 +(1 − δ)2 +α3 +2(2π)3 T 5 +a3 I4(k) +The momentum integrals can be expressed in terms of the normalized comoving mo- +mentum y = k/αTc with norm 0 < y < π/α. These are given by: +I1(k) = +� +dy dΩ +y2 +(y2 + 1)[(y + k/(αTc))2 + 1] , +I2(k) = +� +dy dΩ +y2y · (y + k/(αTc)) +(y2 + 1)[(y + k/(αTc))2 + 1] , +I3(k) = +� +dy dΩ y2 = 4π4 +3α3 , +I4(k) = +� +dy dΩ +y2[y · (y + k/(αTc))]2 +(y2 + 1)[(y + k/(αTc))2 + 1] , +(B.7) +where dΩ denotes integration over the solid angle in momentum space. Except for I3(k), all +integrals above depend non-trivially on k. To leading order in k/αTc these integrals are given +by: +I1(k) ≃ 4π +� +− 1 +2 +πα +α2 + π2 + 1 +2 arctan(π/α) +� +, +I2(k) ≃ 4π +�π +α + 1 +2 +απ +α2 + π2 − 3 +2 arctan(π/α) +� +, +I4(k) ≃ 4π +� +− 2π +α + 1 +3 +π3 +α3 − 1 +2 +απ +α2 + π2 + 5 +2 arctan(π/α) +� +, +(B.8) +which are the expressions used to compute the curvature perturbation power spectrum (3.17) +given in the main body of this article. +– 16 – + +References +[1] A.A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. +Lett. B 91 (1980) 99. +[2] K. 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Pascoli, N-body simulations of structure formation in +thermal inflation cosmologies, JCAP 12 (2018) 010 [1807.04980]. +– 20 – + diff --git a/WdFJT4oBgHgl3EQf4i1g/content/tmp_files/load_file.txt b/WdFJT4oBgHgl3EQf4i1g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f995a5f1e61f34b1f3dc8ff1c32af03179f00aa --- /dev/null +++ b/WdFJT4oBgHgl3EQf4i1g/content/tmp_files/load_file.txt @@ -0,0 +1,832 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf,len=831 +page_content='Thermal curvature perturbations in thermal inflation Mar Bastero-Gil,a Joaquim M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Gomes,b and Jo˜ao G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Rosac aDepartamento de F´ısica Te´orica y del Cosmos, Universidad de Granada, Granada-18071, Spain bDepartment of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom cUniv Coimbra, Faculdade de Ciˆencias e Tecnologia da Universidade de Coimbra and CFisUC, Rua Larga, 3004-516 Coimbra, Portugal E-mail: mbg@ugr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='es, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='gomes@liverpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='uk, jgrosa@uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='pt Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We compute the power spectrum of super-horizon curvature perturbations gen- erated during a late period of thermal inflation, taking into account fluctuation-dissipation effects resulting from the scalar flaton field’s interactions with the ambient radiation bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We find that, at the onset of thermal inflation, the flaton field may reach an equilibrium with the radiation bath even for relatively small coupling constants, maintaining a spectrum of thermal fluctuations until the critical temperature Tc, below which thermal effects stop holding the field at the false potential minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This enhances the field variance compared to purely quantum fluctuations, therefore increasing the average energy density during ther- mal inflation and damping the induced curvature perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In particular, we find that this inhibits the later formation of primordial black holes, at least on scales that leave the horizon for T > Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The larger thermal field variance also reduces the duration of a period of fast-roll inflation below Tc, as the field rolls to the true potential minimum, which should also affect the generation of (large) curvature perturbations on even smaller scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='11666v1 [hep-ph] 27 Jan 2023 Contents 1 Introduction 1 2 Thermal inflation 2 3 Curvature Perturbations 5 4 Comparison between the thermal and quantum power spectra 10 5 Conclusion 12 A Evolution of the temperature during thermal inflation 13 B Field correlation functions 14 1 Introduction It is widely believed that the universe went through a period of inflation in its early stages [1–4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' thus explaining its observed homogeneity and isotropy on large scales,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' as well as its apparently small spatial curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Most importantly, inflation in principle provided the seeds for the small curvature perturbations that grew into the large-scale structure that we observe in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Although the simplest models postulate a single period of slow-roll inflation lasting for at least 50-60 e-folds after the largest presently observable scales became super-horizon, there is a priori no reason to exclude scenarios with multiple inflation periods with different dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In particular, it is well known that reheating after inflation may lead to the production of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' topological defects if the associated reheating temperature exceeds the grand unification scale (∼ 1016 GeV) [5] or other unwanted relics such as moduli or gravitinos in supersymmetric (SUSY) models [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Such relics could have overclosed the Universe or spoiled the successful predictions of primordial nucleosynthesis through their late decay [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This and the fact that currently there is no evidence for such relics motivates considering scenarios with additional inflationary stages that could have diluted their abundances [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' One of the most appealing possibilities is a late period of thermal inflation, where a scalar flaton field is trapped in a false vacuum by thermal effects above a certain critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Candidates to drive such a secondary inflation period are ubiquitous in SUSY and supergravity theories, in particular given the many flat directions in the scalar potential that characterize such models at the renormalizable level [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The spectrum of curvature perturbations generated during such a period (or possibly multiple periods) need not be nearly as scale-invariant as the one generated by the first period of slow-roll inflation, during which the large-scale perturbations observable in the Cosmic Microwave Background (CMB) anisotropies became super-horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In fact, this spectrum was recently computed in [16], where it was shown that large curvature perturbations could have been generated (on small scales) during a period of thermal inflation and a fast roll inflation period [17] that potentially followed it once thermal effects stopped trapping the field in the false vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' These large curvature/density perturbations could have then collapsed into a significant population of primordial black holes upon horizon-reentry later in the radiation-dominated epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Such a – 1 – possibility has attracted a substantial interest in the recent literature given the latter’s appeal as dark matter candidates and the possibility that these may explain the recent LIGO/Virgo detections of heavy black hole binaries (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The analysis in [16] considered, however, only the part of the curvature spectrum gen- erated by quantum fluctuations of the flaton scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Since thermal effects are a crucial aspect in the dynamics of thermal inflation, one should investigate whether thermal fluctua- tions also play an important role, which is our goal with this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We note, in particular, that the flaton field is trapped in a false vacuum at temperatures above a certain critical tem- perature, as we review in the next section, due to the large thermal mass resulting from its interactions with the ambient thermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' It is well-known that such interactions also lead to fluctuation-dissipation effects, resulting in an effective Langevin-like equation describing the dynamics of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Such effects have been thoroughly analyzed in the context of warm inflation scenarios [19–36], in setting initial conditions for slow-roll inflation in a pre-inflationary radiation epoch [37], and in cosmological phase transitions both after and during (warm) inflation [38,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Our objective is then to apply the techniques developed in these contexts to the case of thermal inflation, and investigate their role in the generation of curvature perturbations during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Surprisingly, we find that for thermal flaton fluctuations the amplitude of the curvature power spectrum is suppressed with respect to the purely quantum case analyzed in [16], at least for scales exiting the horizon before the temperature decreases below the critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This is essentially due to the fact that, as we will show, thermal effects, by enhancing flaton density fluctuations, also increase the time-dependent part of the average energy density during thermal inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This effect overcomes the enhancement of individual perturbation modes, therefore suppressing the corresponding power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We will start by constructing a generic model for thermal inflation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The curvature perturbation spectrum induced by the thermal flaton fluctuations is computed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In Section 4 we compare our result with the purely quantum computation performed in [16], discussing and summarizing our conclusions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We use natural units throughout this work, ℏ = c = kB = 1 and the reduced Planck mass MP = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='435 × 1018 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' 2 Thermal inflation Let us consider a scalar field φ interacting with a thermal radiation bath at temperature T, with energy density ρR = π2 30g∗T 4, where g∗ denotes the number of relativistic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' For concreteness, we consider a radiation bath made up of NF Dirac fermion species ψi, which interact with the scalar field through Yukawa interactions with universal coupling constant g: LY = −gφ NF � i=1 ¯ψiψi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='1) We take the mass of the fermions mψi ≪ T, so that they can be treated as relativistic degrees of freedom, but such that mψi > H so that flat quantum field theory calculations for the decay width of scalars into fermions are valid [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We assume that the scalar field φ corresponds to a renormalizable flat direction, or flaton field, common in several SUSY/supergravity scenarios [11,12,14,40,41], such that its potential is only lifted by soft terms such as a mass term from SUSY breaking, and non-renormalizable – 2 – terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We are interested in the case where the squared mass term is negative, such that the field acquires a large expectation value M0 at zero temperature from the latter’s interplay with the non-renormalizable operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The interaction with the radiation bath induces, however, a thermal mass correction such that the field’s effective mass is of the form [42]: m2 eff = α2T 2 − m2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2) where m corresponds to the zero temperature (tachyonic) mass and α is the effective coupling to the thermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' For the Yukawa interactions described above we have α2 = g2NF /6 at one-loop order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This implies, in particular, that for temperatures above the critical value, Tc ≡ m/α, the origin is a stable minimum of the scalar potential, whereas for lower tempera- tures the minimum is non-trivial and asymptotes to M0 in the limit of vanishing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The origin thus constitutes a false vacuum state, near which we may write the scalar potential as: V (φ) = 1 3M2 0 m2 + 1 2m2 effφ2 + · · · , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='3) where for concreteness we have chosen the constant term such that, if the leading non- renormalizable term is ∼ φ6 the cosmological constant vanishes at the minimum, V (φ = M0) = 0, although this is not crucial to our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' For typical flat directions, M0 ≫ m, since the scale at which the non-renormalizable operators become relevant is generically large (around the grand unification or even the Planck scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' If, after the first period of slow-roll inflation, the Universe is reheated to attain a tem- perature T > Tc, the flaton field will thus be driven to the false minimum at the origin by Hubble friction, where it is trapped and gives a contribution V0 = M2 0 m2/3 to the vacuum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Since the temperature drops as the universe expands, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' ρR ∝ a−4, eventually this vacuum energy may become dominant, thus triggering a new period of inflation, with expan- sion rate H ≃ mM0/3MP ≲ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Thermal inflation thus begins when the temperature drops below: Ti = � 10 g∗π2 � 1 4 � M0m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='4) Assuming that there is no significant entropy production during thermal inflation, as we confirm in Appendix A, the temperature of the radiation bath drops as T ∝ a−1 during thermal inflation, eventually reaching the critical value Tc below which the minimum at the origin is destabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The nature of the phase transition (or smooth crossover) that ensues is model-dependent and irrelevant to our discussion (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' [43]), since we are mostly interested in what happens for temperatures Tc < T < Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We note that thermal inflation is only possible if the flaton field has a non-negligible interaction with the thermal bath, and in particular Ti > Tc imposes: α > �g∗π2 10 � 1 4 � m M0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='5) For instance, for m ∼ 10 TeV and M0 ∼ MP , this imposes the lower bound α ≳ 10−7 for g∗ = 10−100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Although this may not seem too restrictive, we note that the number of e-folds of thermal inflation is given by: N(TI) e = ln �Ti Tc � = 1 2 ln �M0 m � + 1 4 ln � 10 π2g∗ � + ln(α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='6) – 3 – For the reference values given above, we see that a period of thermal inflation lasting more than 10 e-folds is only possible for α ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='01, with even larger effective couplings required for scenarios with a smaller hierarchy between the mass scales m and M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We note that inflation does not necessarily end when the temperature falls below Tc, since expansion only stops accelerating once the flaton’s kinetic energy surpasses its potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Below Tc the field develops a tachyonic instability, since m2 eff ≃ −m2 < 0 once T ≪ Tc, and its value moves away from the origin as ∼ emt ∼ e m H Ne for H ≲ m, and there may be a period of fast-roll inflation [17] until the field gets close to the minimum at M0 and its kinetic energy takes over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Note that, in the opposite regime m ≲ H, thermal inflation would be followed by an additional period of slow-roll inflation, but we will not consider this regime in our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The duration of the fast-roll period is, of course, model dependent and, moreover, dependent on the mean field value at the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In [16,17] it was shown that this period may last for as much as, or even longer than, the thermal inflation period for H/m ≲ 1, depending on the flaton’s mass value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This assumed, however, that the mean field value at the critical temperature is set by quantum fluctuations, which as we will see is not necessarily the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In particular, thermal fluctuations typically enhance the field’s variance at Tc, therefore reducing the duration of the subsequent fast- roll period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' For this reason, we will restrict our analysis to the thermal inflation period (Tc < T < Ti), discussing the implications of our results to the subsequent cosmological evolution at the end of our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Independently of whether or not there is a significant period of inflation below Tc, the field will eventually begin oscillating about the minimum of its potential and decay away through the Yukawa interactions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='1) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Although we do not specify the exact nature of the fermion fields in the thermal bath, since we are only modelling the interactions between the flaton and the ambient radiation and our discussion is largely independent of the particular interactions considered, it is implicit that such interactions will eventually lead to the reheating of the Standard Model degrees of freedom at temperatures exceeding at least a few MeV to ensure the correct conditions for primordial nucleosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We note that having late thermal inflation and fast-roll inflation periods alters the predictions of inflationary cosmology [45], since the largest CMB scales leave the horizon 50-60 e-folds before the end of the full inflationary epoch, including the primary slow-roll inflation period, which therefore must necessarily be shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Although the leading effect of the interactions between the flaton and the thermal bath is the thermal mass correction responsible for its trapping at the origin, it also induces fluctuation-dissipation effects in the flaton’s dynamics that, as we will see, can play an im- portant role in the evolution of field perturbations during thermal inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' These have been considered in [46] to analyze the nature of the phase transition at Tc, but their effects on the associated spectrum of curvature perturbations have so far been overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' To study them, we consider the full Langevin-like equation for the flaton field modes φk of comoving momentum k, which can be obtained through standard techniques in linear response theory assuming the ambient radiation bath is close to an equilibrium state, and is given by (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' [25,47]): ¨φk + (3H + Γφ) ˙φk + ω2 kφk = ξk , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='7) where ω2 k = k2/a2 +m2 eff and Γφ is the dissipation coefficient, which for a field oscillating near a local minimum of its potential (in this case the false minimum at the origin for T > Tc) coincides with its finite-temperature decay width [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' On the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='7), ξk is a stochastic noise term which encodes the randomness of the field’s interactions with the – 4 – thermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' For modes with physical momentum p = k/a ≲ πT it is well approximated by a gaussian white noise term with a two-point correlator given by the fluctuation-dissipation relation [46,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' : ⟨ξk(t)ξk′(t′)⟩ = 2ΓφT (2π)3 a3 δ3(k + k′)δ(t − t′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='8) We note that physically this is reminiscent of the Brownian motion of a heavy particle in an gas, for which random collisions with the gas molecules induce an effective friction that damps its motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' However, the particle never actually comes to rest due to the very same random collisions, eventually reaching an equilibrium with the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We expect something very similar to occur to the flaton field modes, with the combined effects of dissipation (Γφ) and thermal fluctuations (ξk) driving the field towards a thermal equilibrium with the radiation bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This behaviour has been observed for scalar fields interacting with a radiation bath both in an inflationary and non-inflationary context [37, 39], so we anticipate that the same will occur in the case of thermal inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' At finite temperature the flaton decay width into relativistic fermions is given by [27,37]: Γφ(p) = 3m2 effα2 4πωp � 1 + 2T p ln �1 + exp(− ω+ T ) 1 + exp(− ω− T ) �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='9) where ω± = |ωp±p| 2 and we have neglected the mass of the fermions, T ≫ mψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Note that fermions acquire a mass through their interaction with the flaton field but, as we will obtain bellow, � ⟨φ2⟩ ≲ T for perturbative couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Since the thermal bath will excite field modes p ≲ T and meff ≲ T, the decay width can be well approximated by: Γφ ≃ 3m2 effα2 16πT ≃ 3α4 16πT , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='10) where in the last step we have used meff ≃ αT for T ≳ Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' At the onset of thermal inflation, we then have: Γφ H ���� Ti ≃ 9 16π � 10 g∗π2 , �1/4 α4 MP √M0m ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='3g−1/4 ∗ � α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='03 �4 �MP M0 �1/2 � m 10 TeV �−1/2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='11) so that we expect dissipative effects to play an important role in the field’s dynamics roughly for the same range of the effective coupling α leading to a period of thermal inflation lasting for more than 10 e-folds, as we have seen above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In the next section we compute the thermal field correlators and associated curvature perturbation power spectrum to better quantify this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' 3 Curvature Perturbations Let us consider the gauge-invariant curvature perturbation on uniform density hypersurfaces, which in the flat gauge can be written as [50,51]: ζ = − H ˙⟨ρ⟩ δρ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='1) – 5 – where the perturbation of a generic function is given by δf(t, x) ≡ f(t, x) − ⟨f(t, x)⟩, and brackets denote its thermal averaged value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The dimensionless power spectrum of ζ is defined as [16], ∆2 ζ(k) = k3 2π2 � d3x exp(−ik · x) ⟨ζ(0)ζ(x)⟩ , = 2k3 (2π)2 � H ˙⟨ρ⟩ �2 � d3x exp(−ik · x) ⟨δρ(0)δρ(x)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2) The total energy density ρ during thermal inflation includes the contributions from both the flaton field and the radiation fluid [52]: ρ = ρφ + ρR = 1 2 ˙φ2 + V (φ) + 1 2a−2(t)∂iφ∂iφ + π2 30g∗T 4 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='3) and so we have ⟨ρ⟩ = π2 30g∗T 4 + 1 3m2M2 0 + 1 2m2 eff ⟨φ2⟩ + 1 2 ⟨ ˙φ2⟩ + 1 2a−2 ⟨∂iφ∂iφ⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='4a) δρ = 1 2m2 effδ(φ2) + 1 2δ( ˙φ2) + 1 2a−2δ(∂iφ∂iφ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='4b) Since density perturbations involve perturbations of quadratic functions of the field and its derivatives, the power spectrum, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2), involves contributions of the form: ⟨δ(Xi(0)2)δ(Xj(x)2)⟩ = ⟨Xi(0)2Xj(x)2⟩ − ⟨Xi(0)2⟩ ⟨Xj(x)2⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='5) where Xi generically denotes the field perturbations and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The first term on the right-hand side corresponds to 4th moments involving the gaussian variables Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' According to Isserlis’ theorem [53] it is possible to write a kth moment of zero-average gaussian variables in terms of their variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Thus, the correlators can be simply written as [54]: ⟨δ(Xi(0)2)δ(Xj(x)2)⟩ = 2 ⟨Xi(0)Xj(x)⟩2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='6) The two-point correlation function for the energy density is then: ⟨δρ(0)δρ(x)⟩ = m4 eff 2 ⟨φ(0)φ(x)⟩2 + m2 eff ⟨φ(0) ˙φ(x)⟩ 2 + a−2m2 eff ⟨φ(0)∂iφ(x)⟩2 , + 1 2 ⟨ ˙φ(0) ˙φ(x)⟩ 2 + a−2 ⟨ ˙φ(0)∂iφ(x)⟩ 2 + a−4 2 ⟨∂iφ(0)∂jφ(x)⟩2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='7) that is, contributions from all possible correlation functions involving φ, ˙φ and ∂iφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We note that we are interested in computing the curvature perturbation power spectrum on super-horizon scales, k ≪ aH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' To do this we need to compute the field variance ⟨φ2⟩ and the average kinetic and gradient energies appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='4a), which involve integrating over all thermally excited field modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Since the noise term correlator is exponentially sup- pressed for physical momentum scales p ≳ πT [46], we use this value as a hard cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This can be translated into a comoving momentum cutoff kc = πTc if we set a(Tc) = 1, following the conventions of [16] to allow for a better comparison with the purely quantum calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' To compute the power spectrum we need the three combinations of the correlations between φk and ˙φk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' ⟨φkφk⟩, ⟨φk ˙φk⟩ and ⟨ ˙φk ˙φk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' These are the building blocks of all the remaining correlation functions involved in the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We will explicitly compute – 6 – the correlator of the field modes and list all others in Appendix B as their computation follows similar steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The equal-time two-point correlation function of the field modes can be written in terms of the Green’s function associated with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='7) and the noise correlator: ⟨φk(z)φk′(z)⟩ = H−4 � z zi ds1 � z zi ds2 s−2 1 s−2 2 Gs(z, s1)Gs(z, s2) ⟨ξk(s1)ξk′(s2)⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='8) where we have traded the time-dependence for a dependence on the variable z = T/H, with zi = Ti/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Note that z ∝ a−1 during thermal inflation, so that it is a decreasing function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We have ignored the contributions from the homogeneous solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='7) since, as we will see bellow, they quickly become subdominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' These are required, however, to compute the Green’s function, which is given by the usual expression: Gs(z, s) = φ(1) k (s)φ(2) k (z) − φ(1) k (z)φ(2) k (s) W(φ(1) k , φ(2) k )(s) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='9) where φ(1) k and φ(2) k are the homogeneous solutions of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='7) and W denotes their Wronskian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' During most of thermal inflation, except for temperatures close to the critical value, the thermal mass dominates over the field’s zero temperature mass, αT ≫ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This allows us to compute analytically the field modes, and thus obtain the field’s two-point correlation function with a decay width of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The homogeneous equation of motion for the flaton field modes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='7) can be written in terms of the z variable as: z2φ′′ k − z (2 + γz) φ′ k + z2¯ω2 kφk = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='10) where ¯ω2 k ≡ ω2 k/T 2 ≃ k2/T 2 c + α2 and γ ≡ 3α4/16π, such that Γφ/H = γz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Let us define φk = zeγz/2χk, such that: χ′′ k + � ¯ω2 k − γ2 4 − γ z − 2 z2 � χk = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='11) Even though we can express the exact solutions of the above equation in terms of Whittaker functions [55], it is more instructive to note that, since γ ≪ ¯ω2 k for α ≲ 1 and z > zc = m/αH > α−1 > 1, we may neglect all the terms inside the brackets in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='11) except for the one involving ¯ω2 k to a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This means that the homogeneous solutions are approximately given by: φ(1) k (z) ≃ ze γ 2 z sin(¯ωkz) , φ(2) k (z) ≃ ze γ 2 z cos(¯ωkz) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='12) thus constituting oscillatory functions in the z variable with an amplitude decreasing due to both Hubble expansion (z ∝ a−1) and the field’s decay into the light fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This yields the Green’s function: Gs(z, s) = 1 ¯ωk z s exp �γ 2(z − s) � sin � ¯ωk(z − s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='13) – 7 – The noise correlation function can be written in terms of the z variable as: ⟨ξk(z1)ξk′(z2)⟩ = 2Hz1ΓφT (2π)3 a3 δ3(k + k′)δ(z1 − z2) , ≃ 2γH3z6 1 (2π)3 z3c δ3(k + k′)δ(z1 − z2) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='14) where in the second line we used the dominance of the thermal mass for T > Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We may now substitute Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='14) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='8) to obtain the field’s two- point correlation function: ⟨φk(z)φk′(z)⟩ = (2π)3δ3(k + k′) T a3ω2 k (1 − δ) , δ = exp � − 3α4 16π Ti H � 1 − T Ti �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='15) where again we used that ¯ωk ≫ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Note that for Γφ/H(Ti) ≳ 1, we have δ ≪ 1 for all temperatures below Ti (but above Tc), thus yielding a thermal equilibrium distribution for the field modes that is independent of the decay width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This means that if the field decays efficiently at the onset of thermal inflation it will attain an equilibrium distribution that simplify redshifts with expansion (with corresponding decrease in temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This is a generic result obtained in other cosmological contexts [37, 39] that we now recover also within thermal inflation – it simply states that if the field interacts significantly with the thermal bath at some point during its evolution it reaches a near-thermal configuration that is subsequently maintained unless there is some significant change in the field’s properties (in our case the tachyonic instability just below the critical temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We note that the two-point correlation function vanishes at the onset of thermal inflation by construction, since the integral Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='8) is zero at z = zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This assumes that field modes were not excited when thermal inflation begins, which need not be the case since interactions with the thermal bath are present in the prior radiation-dominated epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' If field modes thermalize before its vacuum energy becomes dominant, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='15) will nevertheless hold (with δ ≃ 0), since this result is also valid for a radiation-dominated cosmological background [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' However, we note that during the radiation era Γφ/H ∝ T/H ∝ a, while Γφ/H ∝ a−1 during thermal inflation, so that this ratio attains its maximum value at the onset of thermal inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Recalling Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='11), we conclude that α ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='01 is required for field thermalization if the zero temperature mass m is not far from the TeV scale at which new physics may be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' As discussed in the previous section, this is exactly the regime where a period of thermal inflation lasting more than 10 e-folds (and which can in particular sufficiently dilute unwanted relics of the first reheating process) can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We will thus henceforth focus our analysis on this parametric regime, in which the field thermalizes either before or at the onset of the thermal inflation epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We may now use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='15) to compute the field variance and related correlation func- tions, as we detail in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We obtain for the total average energy density: ⟨ρ⟩ = π2 30 � g∗ + 5 π(1 − δ) � T 4 + 1 3m2M2 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='16) where we note that the field contributes essentially as an additional bosonic degree of freedom to the radiation energy density if thermalization is efficient (δ ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Its contribution is not exactly one degree of freedom since we have considered a hard-cutoff on the momentum of the modes that are excited by interactions with the thermal bath at kc = πTc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This is – 8 – only an approximation to the smooth cutoff associated with the noise correlator [46], which nevertheless captures the essential physics of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Using the values of each component of the power spectrum (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='5) given in Appendix B, the density perturbations are: � d3x exp(−ik · x) ⟨δρ(0)δρ(x)⟩ ≈ πT 5 6a3 � 1 + 3 � 3α4 32π2 �2� 1 − α π arctan �π α � �� (1 − δ)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='17) to leading order on super-horizon scales k < aH < αTc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We note that the first term within the square brackets dominates over the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This then yields for the power spectrum on super-horizon scales: ∆2 ζ (therm)(k) = 150 (2π)5 k3 T 3c (1 − δ)2 � g∗ + 5 π(1 − δ) − 5 π 3α4 64π T H δ �2 , ≃ 150 (2π)5 α3 g2 ∗,f �H m �3� k kc �3 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='18) where in the second line we have taken the prompt thermalization limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' δ ≪ 1, in which case the flaton field contributes to the total number of relativistic degrees of freedom, given by g∗,f ≃ g∗ + 5/π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Note that this result is time-independent, reflecting the freeze-out of curvature perturbations on super-horizon scales and thus the single-fluid nature of the dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' the fact that the flaton field thermalized with the radiation bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The power spectrum is blue-tilted so its maximum value is attained for the last scale to leave the horizon during thermal inflation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' kc = H which leaves at T = Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Although our calculation assumes the dominance of the thermal piece of the flaton’s mass, an approximation that breaks down close to the critical temperature, we may extrapolate our results with a reasonable accuracy to kc, thus yielding an upper bound on the power spectrum of scales leaving the horizon before the phase transition, in the thermal equilibrium limit: ∆2 ζ (therm, max)(k) ≃ 150 (2π)5 α3 g2 ∗,f �H m �3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='19) The power spectrum would, thus, be maximized for g∗,f ∼ α ∼ H m ∼ 1, yielding ∆2 ζ (therm, max) ∼ 10−2, but in realistic scenarios with perturbative couplings and at least one fermionic degree of freedom in the ambient thermal bath the power spectrum should have a parametrically smaller amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Hence, if the flaton field has significant interactions with the radiation bath, α ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='01 (as expected in scenarios with a significant number of e-folds of thermal inflation), the thermal nature of its fluctuations suppresses the amplitude of the induced curvature perturbations on super-horizon scales, which is the main result of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' While this may seem sur- prising, given that thermal fluctuations generically have a larger amplitude than quantum vacuum fluctuations (as considered in [16]), it has a simple physical explanation: fluctuation- dissipation effects increase not only the density fluctuations on super-horizon scales but also the field variance and the average gradient and kinetic energies, thus, the average energy den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The latter effect turns out to be more significant and, hence, decreases the amplitude of the associated curvature power spectrum with respect to the quantum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' – 9 – A relevant consequence of our analysis is that, in realistic scenarios, we do not expect the amplitude of the curvature power spectrum to be sufficiently large to lead to the formation of primordial black holes, which would require ∆2 ζ ≳ 10−2 [56–58], at least on scales that become super-horizon above the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This motivates a better comparison with the results obtained in [16] for quantum flaton fluctuations, where larger curvature perturbations were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We pursue this comparison in the next Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' 4 Comparison between the thermal and quantum power spectra The linear approximation to the quantum power spectrum is given in [16] by: ∆2 ζ (quan)(k) = 4 √π Γ(ν) ν2Γ � ν − 3 2 � �H m �3−2ν� k kc �3�� k kc �2 + m2 H2 �−ν , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='1) where ν = � m2/H2 + 9/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' To better compare our results with those obtained assuming purely quantum flaton fluctuations in [16], we plot both power spectra as a function of comoving momentum in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We show the case of H/m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='3 (which according to the analysis in [16] yields all dark matter in the form of primordial black holes) and taking α = 1, NF = 1 and δ = 0 to maximize the thermal power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We note that the thermal power spectrum is only shown up to k = kc, since our calculation is only valid for modes that exit the horizon before the phase transition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' whereas the quantum calculation can be extended to larger momentum, assuming a subsequent period of fast-roll inflation as mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' quantum thermal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='5 1 5 10 10-5 10-4 10-3 10-2 k / kc Δζ 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The quantum power spectrum (blue) and the thermal power spectrum (red) as a function of k for H/m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='3, α = 1, mNF = 1 and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' As one can clearly see in this figure, thermal fluctuations significantly suppress the cur- vature perturbation spectrum with respect to the quantum case, for the reasons explained in the above section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Furthermore, whereas quantum vacuum fluctuations may yield a suffi- ciently large amplitude to lead to primordial black hole formation, a thermalized flaton field induces much smaller perturbations, although they may nevertheless exceed the even smaller fluctuations observed on large scales in the CMB anisotropies spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We should note that the quantum power spectrum peaks at scales that leave the horizon for T < Tc, where our approximations break down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Extending our calculation to this regime – 10 – would involve a different form of the dissipation coefficient, since as the field experiences a tachyonic instability the latter no longer corresponds to the perturbative decay width at finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Let us note, however, that fluctuation-dissipation effects are more pronounced at the start of thermal inflation as discussed earlier, so that they no longer play a significant role near Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' If the field thermalizes at the onset of thermal inflation, it will nevertheless maintain an equilibrium distribution with a decreasing temperature due to inflationary expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Let us then compare the magnitude of field fluctuations at Tc in both the quantum vacuum and thermal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The thermal variance is obtained by expanding the field in terms of its modes ⟨φ(x)φ(y)⟩ = � d3k (2π)3 d3k′ (2π)3 ⟨φkφk′⟩ exp(ik · x) exp(ik · y) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2) and using the field modes correlator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='15), we obtain for the field variance in the thermalized limit: ⟨φ2⟩therm = 2 (2π)2 T a � kcutoff 0 dk k2 k2 + α2T 2c = T 2 2π � 1 − α π arctan �π α �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='3) which we note is only mildly dependent on the effective coupling α, while the quantum one is given by [16]: ⟨φ2⟩quan = � H 2π �2 Γ2(ν)22ν 6π �aH m �2ν F � ν, 3 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' 5 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' − �aH m �2� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='4) where F(a, b, c, z) denotes the Hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The field variance in both cases is shown in Figure 2, where we extrapolate the thermal variance beyond the phase transition purely for comparison purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' quantum thermal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='5 1 5 10 10-8 10-4 1 a / ac \uf0b3ϕ2\uf0b6 / H2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Quantum (blue) and thermal (red) field variance as a function of the scale factor for H/m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='3, α = 1 and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The critical temperature corresponds to the dashed vertical line, below which the thermal variance is extrapolated, as indicated by the dashed red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' As one can clearly observe in this figure, the quantum field variance is several orders of magnitude smaller than the thermal variance before the phase transition, which validates our calculation in neglecting vacuum fluctuations in the thermalized flaton scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' While at the critical temperature this is still true, if one extrapolates the thermal variance for T < Tc – 11 – (a > ac = 1), we see that quantum fluctuations become dominant less than one e-fold after the critical temperature is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' While this extrapolation is non-trivial, since the fluctuation-dissipation effects would have to be re-computed, it may suggest that vacuum perturbations may become dominant after the phase transition, in which case the computation in [16] would hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In fact, the peak in the quantum power spectrum is obtained for modes with k = H 2 � 3(2ν + 3) > kc = H, which leave the horizon for temperatures below the critical value and thus, in the example shown above, already in the regime where the quantum variance is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This would, in fact, suggest that large enough curvature perturbations leading to pri- mordial black hole formation may be generated after thermal inflation (from quantum fluc- tuations), but it is not clear that quantum and thermal fluctuations may be examined in- dependently nor that the thermal variance maintains its form below Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In addition, and perhaps most importantly, the fact that the thermal variance is still typically a few orders of magnitude larger than the quantum one at the critical temperature indicates that the flaton field should reach the minimum of its potential much more quickly if it thermalizes, therefore considerably shortening, or even possibly, precluding an ensuing period of fast-roll inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' A more complete analysis of the problem including both thermal and quantum fluctua- tions in the analysis, potentially along the lines of [59], is required to compute the spectrum of curvature perturbations on scales that leave the horizon at temperatures below Tc, and is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' 5 Conclusion We have computed the spectrum of curvature perturbations generated during thermal in- flation taking into account the thermal fluctuations of the flaton field driving this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' These are associated with fluctuation-dissipation effects driven by the flaton’s interactions with the ambient radiation bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Our analysis involved solving the Langevin-like equation effectively describing the evolution of the flaton’s Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We computed the associ- ated correlation functions in the approximation of a gaussian white noise and a dominant thermal contribution to the flaton’s mass, for temperatures above the critical value at which the flaton is held at the false vacuum at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We have concluded that, if the flaton’s (finite-temperature) decay width exceeds the Hubble parameter at the onset of thermal inflation, the field essentially thermalizes with the ambient radiation bath, contributing approximately as an extra relativistic degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This occurs when the effective coupling between the flaton and the thermalized degrees of freedom α ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='01, which roughly corresponds to the parametric regime where over 10 e-folds of thermal inflation (above Tc) occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We found that the consequent increase in the field variance and the average gradient and kinetic energies enhances the background energy density (namely its time-dependent part that determines curvature perturbations) with respect to a field with purely quantum vacuum fluctuations analyzed in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Despite the enhancement of super-horizon density fluctuations in the thermal case, the overall amplitude of the curvature power spectrum is significantly reduced with respect to the quantum case, so that thermal fluctuations behave very differently compared to their quantum counterparts, regarding the generation of curvature perturbations during periods of thermal inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' While our analysis is not applicable for modes that leave the horizon once the tempera- ture has fallen below the critical value and the field starts rolling towards the true minimum of its potential, we expect thermal effects to become less relevant in this regime and quantum – 12 – fluctuations to become dominant, potentially yielding large curvature perturbations at such scales as computed in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' However, a full analysis including both quantum and thermal fluctuations in the dynamics of the flaton field is required to accurately describe the puta- tive fast-roll inflation phase below the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' It must be noted, in any case, that such a phase is necessarily shortened by the fact that the field variance at the critical temperature, which sets the typical field value at this stage, is much larger if the field ther- malizes with the radiation bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' It is therefore unclear whether super-horizon fluctuations with k > kc can be generated in this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We have modelled the thermal bath through a set of fermion species coupled to the flaton field, but we expect our main conclusions to hold with the inclusion of other bosonic fields, like scalars or vector bosons: if Γφ > H at some stage during thermal inflation, the field will be driven towards a thermal fluctuation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Only the details of how and when this equilibrium is attained may depend on the types of light fields that interact with the flat direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Our analysis shows that thermalization does not require large coupling constants describing the interaction between the flaton and the radiation bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In any case such couplings cannot be too suppressed to sustain a sufficiently long period of thermal inflation that may, in particular, dilute any unwanted relics generated after the primary slow-roll inflation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Hence, fluctuation-dissipation effects cannot in general be neglected in the dynamics of the flaton field and on the curvature perturbations they induce during thermal inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This is particularly relevant if one wishes to understand whether thermal inflation periods may leave behind a sizeable population of primordial black holes, and we hope that our work motivates further exploration of these and related issues, including other potential implications for structure formation in our Universe [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Acknowledgements M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' work has been partially supported by MICINN (PID2019-105943GB-I00/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='130 39/501100011033) and “Junta de Andaluc´ıa” grant P18-FR-4314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' JMG acknowledges the support from the Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (FCT) through the Research Fellowship No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='05180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='BD derived from Portuguese national funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This work was supported by the CFisUC project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' UID/FIS/04564/2020 and by the FCT-CERN grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' CERN/FIS-PAR/0027/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' A Evolution of the temperature during thermal inflation In our calculation we assumed that no significant entropy is produced during thermal inflation as a result of fluctuation-dissipation effects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' that T ∝ a−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In this appendix we aim to verify this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' The flaton field satisfies the Langevin-like equation [25]: ¨φ + (3H + Γφ) ˙φ − a−2∇2φ + m2 effφ = ξ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='1) and by multiplying both sides by ˙φ we obtain: ˙ρφ + 3H(ρφ + pφ) = ξ ˙φ − Γφ ˙φ2 + α2T ˙Tφ2 + a−2∂i( ˙φ∂iφ) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2) where the field’s energy density and pressure are given by: ρφ = 1 2 ˙φ2 + 1 2a−2∂iφ∂iφ + V (φ) , pφ = 1 2 ˙φ2 − 1 6a−2∂iφ∂iφ − V (φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='3) – 13 – Conservation of the full energy-momentum tensor then yields the following continuity equa- tion for the radiation energy density: ˙ρR + 4HρR = − ⟨ξ ˙φ⟩ + Γφ ⟨ ˙φ2⟩ − α2T ˙T ⟨φ2⟩ − a−2 ⟨∂i( ˙φ∂iφ⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='4) We note that the radiation energy density is an ensemble average over the energy density of the relativistic degrees of freedom, which justifies considering also the thermal average of the terms on the right-hand side of the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Here we have also neglected the sub-leading corrections to the radiation energy and entropy densities from the fermions’ finite mass, ∼ g ⟨ � φ2⟩ ∼ gT ≪ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Using the field solutions we obtained for the correlators1: ⟨ξ ˙φ⟩ = π 6 ΓφT 4 , Γφ ⟨ ˙φ2⟩ = π 6 ΓφT 4(1 − δ) , α2T ˙T ⟨φ2⟩ = α2 2πT 3 ˙T � 1 − α π arctan �π α �� (1 − δ) ≈ 15α2 4π3g∗ (1 − δ) ˙ρR , ⟨∂i( ˙φ∂iφ⟩ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='5) Note that the third term is related to the time-dependence of the thermal flaton mass, and yields a contribution to the variation of the radiation energy density comparable to the above- mentioned sub-leading corrections from the fermions’ non-vanishing mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' For consistency, we thus neglect this term, and obtain: ˙ρR + 4HρR = −5/π g∗ ΓφρRδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='6) From this we immediately see that the right-hand side can only be significant if Γφ ≳ H, but this implies a quick thermalization of the flaton field such that δ → 0 exponentially fast, thus making this term negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' This simply reflects the balance between the effects of fluctuations and dissipation as the flaton field reaches an equilibrium with the radiation bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Note, furthermore, that the term on the right-hand side is suppressed by the relative contribution of the flaton to the number of relativistic species in equilibrium, (g∗,f − g∗)/g∗, as obtained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' We therefore conclude that one may consistently assume ρR ∝ a−4 and hence that T ∝ a−1 during thermal inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' B Field correlation functions Here we list the field correlation functions used to compute the curvature perturbation power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' As we mentioned above, when integrating over momentum modes we consider a sharp cut-off at k = πTc, which constitutes a good approximation to the behaviour of the noise correlation function [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' To compute the curvature perturbation power spec- trum on super-horizon scales, k ≪ aH, we consider the leading order results in k/αTc ∼ (k/aH)(M0/MP )a ≪ 1 considering M0 < MP and noting that in our convention a < ac = 1 above the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' 1⟨∇( ˙φ∇φ⟩) = − � d3k1 (2π)3 d3k2 (2π)3 ⟨ ˙φk1φk2⟩ k2 · (k1 + k2) exp [ix · (k1 + k2)] = 0 , since the integral of this delta function is non-zero if and only if k1 = −k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' – 14 – Mode correlators The building blocks of all field correlators are the equal time correlators between the field modes and their time derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Writing φk and ˙φk in terms of the Green’s function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='13): φk = H−2 � z zi ds s−2Gs(z, s)ξk(s) , ˙φk = H−1z � z zi ds s−2∂zGs(z, s)ξk(s) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='1) one finds: ⟨φkφk′⟩ = (2π)3δ3(k + k′) T a3ω2 k (1 − δ) , ⟨ ˙φk ˙φk′⟩ = (2π)3δ3(k + k′) T a3 (1 − δ) , ⟨φk ˙φk′⟩ = −Γφ 2 ⟨φkφk′⟩ = −(2π)3δ3(k + k′) TΓφ 2a3ω2 k (1 − δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2) Field correlators To compute the total energy density (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='4a) one needs to determine the field variance and the average kinetic and gradient energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Expanding the field in terms of comoving momentum modes, these are given by: ⟨φ2⟩ = � d3k (2π)3 d3k′ (2π)3 ⟨φkφk′⟩ exp(ix · (k + k′)) , ⟨ ˙φ2⟩ = � d3k (2π)3 d3k′ (2π)3 ⟨ ˙φk ˙φk′⟩ exp(ix · (k + k′)) , ⟨∂iφ∂iφ⟩ = � d3k (2π)3 d3k′ (2π)3 ⟨φkφk′⟩ k · k′ exp(ix · (k + k′)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='3) Inserting the mode correlation functions (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2) and integrating over comoving momenta up to πTc one obtains: ⟨φ2⟩ = T 2 2π (1 − δ) � 1 − α π arctan �π α �� , ⟨ ˙φ2⟩ = πT 4 6 (1 − δ) , ⟨∂iφ∂iφ⟩ = π 2 a2T 4(1 − δ) �1 3 − �α π �2 + �α π �3 arctan �π α �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='4) Contributions to the power spectrum Consider the power spectrum of a generic correlator ⟨Xi(0)Xj(x)⟩, for example X1 = φ, X2 = ˙φ and X3 = ∂iφ, that appears in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='7): � d3x exp(−ik · x) ⟨Xi(0)Xj(x)⟩2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='5) Note that, upon expanding each quantity Xj(x) in terms of comoving momentum modes, this yields four momentum integrals and a volume integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Two of the momentum integrals can – 15 – be performed using the two delta functions appearing in the mode correlators (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Then, the volume integral will generate a delta function with the two surviving momentum modes: � d3x exp[−ix · (k1 + k2 + k)] = (2π)3δ3(k1 + k2 + k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='6) After integrating this delta function over another of the 3-momentum variables, we are left with a single 3-dimensional integral over k that we need to compute in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' In the following table we give the different contributions to the power spectrum in terms of their corresponding momentum integrals: Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Contributions to the power spectrum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='field-field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='eff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='d3x exp(−ik · x) ⟨φ(0)φ(x)⟩2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='(1 − δ)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2(2π)3 T 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='a3 I1(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='field-kinetic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='eff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='d3x exp(−ik · x) ⟨φ(0) ˙φ(x)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='(1 − δ)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='(2π)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='� 3α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='32π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='�2 T 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='a3 I1(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='field-gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='a−2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='eff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='d3x exp(−ik · x) ⟨φ(0)∂iφ(x)⟩2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='(1 − δ)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='(2π)3 T 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='a3 I2(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='kinetic-kinetic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='d3x exp(−ik · x) ⟨ ˙φ(0) ˙φ(x)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='(1 − δ)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2(2π)3 T 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='a3 I3(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='kinetic-gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='a−2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='d3x exp(−ik · x) ⟨ ˙φ(0)∂iφ(x)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='(1 − δ)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='(2π)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='� 3α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='32π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='�2 T 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='a3 I2(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='gradient-gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='a−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='d3x exp(−ik · x) ⟨∂iφ(0)∂jφ(x)⟩2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='(1 − δ)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='2(2π)3 T 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='a3 I4(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='The momentum integrals can be expressed in terms of the normalized comoving mo- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='mentum y = k/αTc with norm 0 < y < π/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' These are given by: I1(k) = � dy dΩ y2 (y2 + 1)[(y + k/(αTc))2 + 1] , I2(k) = � dy dΩ y2y · (y + k/(αTc)) (y2 + 1)[(y + k/(αTc))2 + 1] , I3(k) = � dy dΩ y2 = 4π4 3α3 , I4(k) = � dy dΩ y2[y · (y + k/(αTc))]2 (y2 + 1)[(y + k/(αTc))2 + 1] , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='7) where dΩ denotes integration over the solid angle in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' Except for I3(k), all integrals above depend non-trivially on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' To leading order in k/αTc these integrals are given by: I1(k) ≃ 4π � − 1 2 πα α2 + π2 + 1 2 arctan(π/α) � , I2(k) ≃ 4π �π α + 1 2 απ α2 + π2 − 3 2 arctan(π/α) � , I4(k) ≃ 4π � − 2π α + 1 3 π3 α3 − 1 2 απ α2 + π2 + 5 2 arctan(π/α) � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='8) which are the expressions used to compute the curvature perturbation power spectrum (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content='17) given in the main body of this article.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} +page_content=' – 20 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFJT4oBgHgl3EQf4i1g/content/2301.11666v1.pdf'} diff --git a/X9FRT4oBgHgl3EQf_DhA/content/tmp_files/2301.13693v1.pdf.txt b/X9FRT4oBgHgl3EQf_DhA/content/tmp_files/2301.13693v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..558ffb85efcde6fa050956bd7e772683bcfafc35 --- /dev/null +++ b/X9FRT4oBgHgl3EQf_DhA/content/tmp_files/2301.13693v1.pdf.txt @@ -0,0 +1,959 @@ +arXiv:2301.13693v1 [math.NA] 31 Jan 2023 +Application of dimension truncation error analysis to +high-dimensional function approximation +Philipp A. Guth† +Vesa Kaarnioja‡ +February 1, 2023 +Abstract +Parametric mathematical models such as partial differential equations with random +coefficients have received a lot of attention within the field of uncertainty quantifica- +tion. The model uncertainties are often represented via a series expansion in terms of +the parametric variables. In practice, this series expansion needs to be truncated to +a finite number of terms, introducing a dimension truncation error to the numerical +simulation of a parametric mathematical model. There have been several studies of +the dimension truncation error corresponding to different models of the input random +field in recent years, but many of these analyses have been carried out within the +context of numerical integration. In this paper, we study the L2 dimension truncation +error of the parametric model problem. Estimates of this kind arise in the assessment +of the dimension truncation error for function approximation in high dimensions. In +addition, we show that the dimension truncation error rate is invariant with respect to +certain transformations of the parametric variables. Numerical results are presented +which showcase the sharpness of the theoretical results. +1 +Introduction +In the field of uncertainty quantification it is common to study mathematical models with +uncertain influences parameterized by countably infinite sequences of random variables. +Consider, for instance, an abstract model M : X × U → Y such that +M(g(y), y) = 0, +(1) +where X and Y are separable Hilbert spaces and U is a nonempty subset of the infinite- +dimensional sequence space of parameters RN. The solution g(y) ∈ X to (1) for y ∈ U, if +it exists, may be computationally expensive to evaluate. To this end, it may be preferable +to instead approximate g using a surrogate which is cheap to evaluate and hence enables, +e.g., efficient sampling of the (approximated) solution. +Some possible surrogate models include, but are not limited to, Gaussian process +regression [3], reduced basis approaches [1, 21], generalized polynomial chaos expansions +[4, 23], neural network approximations [2, 7, 9, 22], and kernel interpolation based on +lattice point sets [16, 25, 26]. The results presented in this manuscript are particularly +well-suited to the analysis of kernel methods used in conjunction with the so-called periodic +model discussed in [13, 16, 17], and we will devote a section of this manuscript to explore +the application of our dimension truncation results within this framework. +†Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, +Altenbergerstraße 69, A-4040 Linz, Austria, philipp.guth@ricam.oeaw.ac.at +‡Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 6, 14195 +Berlin, Germany, vesa.kaarnioja@fu-berlin.de +1 + +Integration +Function approximation +Affine parametric +[6, 20] +[16] +operator equation setting +rate O(s− 2 +p +1) +rate O(s− 1 +p + 1 +2) +Non-affine parametric +[8, 12] +this paper +operator equation setting +rate O(s− 2 +p +1) +rate O(s− 1 +p + 1 +2) +Table 1: An overview of various dimension truncation results. +A natural first step for the numerical treatment of (1) is the approximation by a +dimensionally-truncated model Ms : X × Us → Y such that +Ms(gs(y≤s), y≤s) = 0, +where ∅ ̸= Us ⊆ Rs and gs(y≤s) ∈ X for all y≤s ∈ Us. Consider the problem of finding +a surrogate solution gs,n := An(gs) using an algorithm An which uses n point evaluations +of the s-dimensional function gs, where the surrogate belongs to X such that +∥gs − gs,n∥L2µ(U;X) +n→∞ +−−−→ 0 +with some known convergence rate and µ indicating a probability measure on U. The total +error of the approximation obtained in this fashion can be estimated using the triangle +inequality +∥g − gs,n∥L2µ(U;X) ≤ ∥g − gs∥L2µ(U;X) + ∥gs − gs,n∥L2µ(U;X). +In this manuscript we focus on the first term—the dimension truncation error—which is +independent of the chosen approximation scheme An. +Dimension truncation error rates are typically studied for problems involving partial +differential equations (PDEs) with random inputs. For integration problems a dimension +truncation rate is derived in [20] for the source problem with an affine parameterization +of the diffusion coefficient. This rate was then improved by [6] in the generalized context +of affine parametric operator equations. Dimension truncation has also been studied for +coupled PDE systems arising in optimal control problems under uncertainty [10], in the +context of the periodic model of uncertainty quantification for both numerical integra- +tion [17] and kernel interpolation [16], as well as for Bayesian inverse problems governed +by PDEs [5, 15]. The results in these papers have been proved using Neumann series, +which is known to work well in the affine parametric setting, but may lead to suboptimal +results if the problem depends nonlinearly on the parameters. +In the non-affine setting, using Taylor series makes it possible to derive dimension +truncation error rates by exploiting the parametric regularity of the problem, whereas the +Neumann series approach relies fundamentally on the parametric structure of the model. +The Taylor series approach was first applied in [8], and motivated the authors in [11] +and [12] to derive dimension truncation error rates for sufficiently smooth, Banach space +valued integrands, and with parameters following a generalized β-Gaussian distribution. +An overview of the various dimension truncation error bounds studied in the literature is +given in Table 1. +Our manuscript is structured as follows. Subsection 1.1 introduces the multi-index +notation used throughout the paper. +The problem setting is introduced in Section 2, +including the central assumptions for the ensuing dimension truncation analysis. Section 3 +contains the L2 dimension truncation theorem for Hilbert space valued functions, and +in Section 4 we discuss the invariance of the dimension truncation rate under certain +transformations of the variables. Numerical experiments assessing the sharpness of our +2 + +theoretical results are presented in Section 5. The paper ends with some conclusions in +Section 6. +1.1 +Notations and preliminaries +Throughout this manuscript, boldfaced symbols are used to denote multi-indices while the +subscript notation mj is used to refer to the j-th component of multi-index m. Let +F := {m ∈ NN +0 : |m| < ∞} +denote the set of finitely supported multi-indices, where the order of multi-index m is +defined as +|m| := +� +j≥1 +mj. +Moreover, we denote +|m|∞ := max +j≥1 mj, +and, for any sequence x := (xj)∞ +j=1 of real numbers and m ∈ F, we define +xm := +� +j≥1 +xmj +j , +where we use the convention 00 := 1. +2 +Problem setting +Let X be a real separable Hilbert space, U := [− 1 +2, 1 +2]N a set of parameters, and suppose +that g(y) ∈ X is a parameterized family of functions with smooth dependence on y ∈ +U. +We define gs(y) := g(y≤s, 0) := g(y1, . . . , ys, 0, 0, . . .) and assume that µ(dy) := +� +j≥1 µ(dyj) is a countable product probability measure, i.e., µ(U) = 1. We suppose that +1. For µ-a.e. y ∈ U, there holds +∥g(y) − gs(y)∥X +s→∞ +−−−→ 0. +2. Let (Θk)k≥0 and b := (bj)j≥1 be sequences of nonnegative numbers such that b ∈ +ℓp(N) for some p ∈ (0, 1) and b1 ≥ b2 ≥ · · · . +Suppose that g is continuously +differentiable up to order k + 1, with +∥∂νg(y)∥X ≤ Θ|ν|bν +for all y ∈ U and for all ν ∈ Fk := {ν ∈ NN +0 : |ν| ≤ k + 1}, where k := ⌈ +1 +1−p⌉. +3. There holds +� 1/2 +−1/2 yj µ(dyj) = 0 and there exists a constant Cµ ≥ 0 such that +� 1/2 +−1/2 |yj|k µ(dyj) ≤ Cµ for all k ≥ 2. +If Assumption 2 holds, then we infer that y �→ G(g(y)) for all G ∈ X′ is continuous as +a composition of continuous mappings. Hence y �→ G(g(y)) is measurable for all G ∈ X′, +i.e., y �→ g(y) is weakly measurable. +Since X is assumed to be a separable Hilbert +space, by Pettis’ theorem (cf., e.g., [24, Chapter 4]) we obtain that y �→ g(y) is strongly +measurable. The upper bound in Assumption 2 is µ-integrable. Thus we conclude from +Bochner’s theorem (cf., e.g., [24, Chapter 5]) and Assumption 2 that g is µ-integrable over +U. +3 + +Further, µ-a.e. equality defines an equivalence relation among strongly µ-measurable +functions. By L2 +µ(U; X) we denote the Hilbert space of equivalence classes of strongly +µ-measurable functions f : U → X with norm +∥f∥L2µ(U;X) := +� � +U +∥f(y)∥2 +X µ(dy) +� 1 +2 +< ∞. +Moreover, under the Assumptions 1 and 2 it can be shown that g, gs ∈ L2 +µ(U; X) and +lim +s→∞ ∥g(y) − g(y≤s, 0)∥L2µ(U;X) = lim +s→∞ +� � +U +∥g(y) − g(y≤s, 0)∥2 +X µ(dy) +� 1 +2 += 0, +by applying Lebesgue’s dominated convergence theorem (see, e.g., [18, Theorem 1] and +[14, Section 26]) to +F s(y) := ∥g(y) − g(y≤s, 0)∥2 +X, +which converges µ-a.e. to zero by Assumption 1, and can be bounded by (2Θ0)2 by As- +sumption 2. We use the superscript to avoid confusion with the notation used to denote +dimensionally-truncated functions elsewhere in the document. +3 +Dimension truncation error +We will require the following parametric regularity bound for the main dimension trunca- +tion result. +Lemma 1. Under Assumption 2, there holds +|∂ν∥g(y) − gs(y)∥2 +X| ≤ +� +max +0≤ℓ≤|ν| +2Θℓ +ℓ! +�2 +(|ν| + 1)!bν +for all ν ∈ Fk and y ∈ U. +Proof. Let ν ∈ Fk. We apply the Leibniz product rule with respect to the inner product +of the Hilbert space X to obtain +∂ν∥g(y) − gs(y)∥2 +X = ∂ν⟨g(y) − gs(y), g(y) − gs(y)⟩X += +� +m≤ν +� ν +m +� +⟨∂m(g(y) − gs(y)), ∂ν−m(g(y) − gs(y))⟩X. +Using the Cauchy–Schwarz inequality together with Assumption 2 yields +|∂ν∥g(y) − gs(y)∥2 +X| ≤ +� +m≤ν +� ν +m +� +∥∂m(g(y) − gs(y))∥X∥∂ν−m(g(y) − gs(y))∥X +≤ 4 +� +m≤ν +� ν +m +� +Θ|m|bmΘ|ν|−|m|bν−m += 4bν +|ν| +� +ℓ=0 +ΘℓΘ|ν|−ℓ +� +|m|=ℓ +m≤ν +� ν +m +� += 4bν +|ν| +� +ℓ=0 +ΘℓΘ|ν|−ℓ +|ν|! +ℓ!(|ν| − ℓ)! +≤ 4 +� +max +0≤ℓ≤|ν| +Θℓ +ℓ! +�2 +(|ν| + 1)!bν, +where we used the Vandermonde convolution � +|m|=ℓ +m≤ν +� ν +m +� += +�|ν| +ℓ +� += +|ν|! +ℓ!(|ν|−ℓ)!. +4 + +The main result of this document is stated below. +Theorem 1. Let g(y) ∈ X, y ∈ U, satisfy Assumptions 1–3. Then +∥g − gs∥L2µ(U;X) = O(s− 1 +p + 1 +2), +where the implied coefficient is independent of s. +Proof. Let s ≥ 1 and define +F s(y) := ∥g(y) − gs(y)∥2 +X +for y ∈ U. +In the special case of the uniform distribution µ(dy) = dy, we can apply [12, Theorem 4.2] +to obtain +∥g − gs∥2 +L2(U;X) = +���� +� +U +(F s(y) − F s(y≤s, 0)) dy +���� = O(s− 2 +p +1), +from which the claim follows. For completeness, we present the proof below for the prob- +ability measure µ and because parts of the argument will also be useful to establish the +invariance of the dimension truncation rate in Section 4. +Developing the Taylor expansion of F s about (y≤s, 0) and observing that F s(y≤s, 0) = +0, we obtain +F s(y) = +k +� +ℓ=1 +� +|ν|=ℓ +νj=0 ∀j≤s +yν +ν! ∂νF s(y≤s, 0) ++ +� +|ν|=k+1 +νj=0 ∀j≤s +k + 1 +ν! yν +� 1 +0 +(1 − t)k∂νF s(y≤s, ty>s) dt, +(2) +where y>s := (yj)j>s. Integrating both sides over y ∈ U yields +� +U +F s(y) µ(dy) = +k +� +ℓ=1 +� +|ν|=ℓ +νj=0 ∀j≤s +1 +ν! +� +U +yν∂νF s(y≤s, 0) µ(dy) ++ +� +|ν|=k+1 +νj=0 ∀j≤s +k + 1 +ν! +� +U +� 1 +0 +(1 − t)kyν∂νF s(y≤s, ty>s) dt µ(dy). +If ν ∈ Fk is such that νj = 1 for any j > s, then Fubini’s theorem together with Assump- +tion 3 imply for the summands appearing in the first term that +� +U +yν∂νF s(y≤s, 0) µ(dy) = +� � +j>s +� +1 +2 +− 1 +2 +yνj +j µ(dyj) +� +� +�� +� +=0 +� +[− 1 +2, 1 +2]s ∂νF s(y≤s, 0) µ(dy>s). +Therefore all multi-indices with any component equal to 1 can be removed from the first +sum (especially, we can omit all multi-indices with |ν| = 1). Further, applying the regu- +larity bound proved in Lemma 1 and writing open the definition of F s yields +� +U +∥g(y) − gs(y)∥2 +X µ(dy) ≤ Ck +µ +� +max +0≤ℓ≤k +2Θℓ +ℓ! +�2 +(k + 1)! +k +� +ℓ=2 +� +|ν|=ℓ +νj=0 ∀j≤s +νj̸=1 ∀j>s +bν ++ Ck+1 +µ +� +max +0≤ℓ≤k+1 +2Θℓ +ℓ! +�2 +(k + 2)! +� +|ν|=k+1 +νj=0 ∀j≤s +1 +ν!bν, +(3) +5 + +where we used +� 1 +0 (1 − t)k dt = +1 +k+1 and Assumption 3. +The second term in (3) can +be estimated from above using the multinomial theorem in conjunction with Stechkin’s +lemma: +� +|ν|=k+1 +νj=0 ∀j≤s +1 +ν!bν ≤ +� +|ν|=k+1 +νj=0 ∀j≤s +|ν|! +ν! bν = +� � +j>s +bj +�k+1 +≤ s(k+1)(− 1 +p +1) +� � +j≥1 +bp +j +� k+1 +p +. +On the other hand, the first term in (3) can be estimated similarly to [6]: +� +2≤|ν|≤k +νj=0 ∀j≤s +νj̸=1 ∀j>s +bν ≤ +� +0̸=|ν|∞≤k +νj=0 ∀j≤s +νj̸=1 ∀j>s +bν = −1 + +� +j>s +� +1 + +k +� +ℓ=2 +bℓ +j +� += −1 + +� +j>s +� +1 + b2 +j +k−2 +� +ℓ=0 +bℓ +j +� +≤ −1 + +� +j>s +� +1 + b2 +j +k−2 +� +ℓ=0 +bℓ +1 +� �� � +=:βk +� +≤ −1 + exp +� +βk +� +j>s +b2 +j +� += +� +ℓ≥1 +1 +ℓ! +� +βk +� +j>s +b2 +j +�ℓ +. +Using � +j>s b2 +j ≤ s− 2 +p +1(� +j≥1 bp +j) +2 +p , which follows from Stechkin’s lemma, we further +estimate +� +ℓ≥1 +1 +ℓ! +� +βk +� +j>s +b2 +j +�ℓ +≤ s− 2 +p +1 � +ℓ≥1 +1 +ℓ!(βk∥b∥2 +p)ℓ = s− 2 +p +1(−1 + exp(βk∥b∥2 +p) +since s− 2 +p +1 ≥ (s− 2 +p +1)ℓ for all ℓ ≥ 1. +Altogether, the above discussion yields the bound +∥g(y) − gs(y)∥2 +L2µ(U;X) = +� +U +∥g(y) − gs(y)∥2 +X µ(dy) = O(s− 2 +p +1 + s(k+1)(− 1 +p +1)), +where the implied coefficient is independent of s. Since we assumed that k = ⌈ +1 +1−p⌉, the +assertion follows by taking the square root on both sides. +4 +Invariance of the dimension truncation rate under trans- +formations of variables +An interesting consequence of the Taylor series argument used in Theorem 1 is that the di- +mension truncation rate remains invariant under certain transformations of the variables. +This has been previously observed in the context of dimension truncation for integration +problems under the periodic model [13]. To make this notion precise, let us consider a +mapping ξ: U → U, ξ(y) := (ξ(y1), ξ(y2), . . .), which satisfies the following conditions: +4. There hold ξ(0) = 0 and +� 1/2 +−1/2 ξ(y) dy = 0. +5. There exists Cξ ≥ 0 such that +� 1/2 +−1/2 |ξ(y)|k dy ≤ Cξ for all k ≥ 2. +Then we obtain the following as a consequence of Theorem 1. +Corollary 1. Let g(y) ∈ X, y ∈ U, satisfy Assumptions 1–3 and let ξ : U → U satisfy +Assumptions 4–5. Define the ξ-transformed function gξ by +gξ(y) := g(ξ(y)), +y ∈ U, +6 + +and its dimension truncation by gξ,s(y) := gξ(y≤s, 0) for y ∈ U. Then +∥gξ − gξ,s∥L2µ(U;X) = O(s− 1 +p + 1 +2 ), +where the implied coefficient is independent of s. +Proof. We introduce F s +ξ (y) := ∥gξ(y) − gξ,s(y)∥2 +X for y ∈ U. By carrying out the change +of variable y ← ξ(y) in (2), we obtain +F s +ξ (y) = +k +� +ℓ=1 +� +|ν|=ℓ +νj=0 ∀j≤s +ξ(y)ν +ν! +∂νF s(ξ(y≤s, 0)) ++ +� +|ν|=k+1 +νj=0 ∀j≤s +k + 1 +ν! ξ(y)ν +� 1 +0 +(1 − t)k∂νF s(ξ(y≤s, ty>s)) dt. +Integrating the above formula on both sides over y ∈ U and utilizing Lemma 1 as well as +Assumption 5, we obtain—in complete analogy with the proof of Theorem 1—that +� +U +∥gξ(y) − gξ,s(y)∥2 +X dy ≤ Ck +ξ +� +max +0≤ℓ≤k +2Θℓ +ℓ! +�2 +(k + 1)! +k +� +ℓ=2 +� +|ν|=ℓ +νj=0 ∀j≤s +νj̸=1 ∀j>s +bν ++ Ck+1 +ξ +� +max +0≤ℓ≤k+1 +2Θℓ +ℓ! +�2 +(k + 2)! +� +|ν|=k+1 +νj=0 ∀j≤s +1 +ν!bν. +The desired result follows by exactly the same argument as in the proof of Theorem 1. +As an application, with U := [− 1 +2, 1 +2]N, let ξ: U → U satisfy the Assumptions 4 and 5, +let D ⊂ Rd, d ∈ {1, 2, 3}, be a bounded Lipschitz domain, and let f : D → R be a fixed +source term. Consider the parametric PDE problem +� +−∇ · (aξ(x, y)∇uξ(x, y)) = f(x), +x ∈ D, y ∈ U, +uξ(x, y) = 0, +x ∈ ∂D, y ∈ U, +(4) +endowed with the ξ-transformed diffusion coefficient +aξ(x, y) := a0(x) + +∞ +� +i=1 +ξ(yi)ψi(x), +x ∈ D, y ∈ U, +which is assumed to satisfy the following: +6. There exist amin, amax > 0 such that 0 < amin ≤ aξ(x, y) ≤ amax < ∞ for all x ∈ D +and y ∈ U. +7. a0 ∈ L∞(D) and ψi ∈ L∞(D) for all i ∈ N. +8. �∞ +i=1 ∥ψi∥p +L∞(D) < ∞ for some p ∈ (0, 1). +In this case, the transformation ξ(y) := ( 1 +√ +6 sin(2πyj))j≥1 corresponds to the so-called +periodic model studied in [13, 16, 17]. Let X := H1 +0(D) be equipped with the norm ∥v∥X := +7 + +� +D ∥∇v(x)∥2 +Rd dx. In this special case, it is known that the weak solution u(·, y) ∈ X to (4) +for y ∈ U satisfies the parametric regularity bound +∥∂νuξ(·, y)∥X ≤ (2π)|ν|∥f∥X′ +amin +� +m≤ν +|m|!bm � +j≥1 +S(νj, mj) +for all ν ∈ F and y ∈ U, where the source term f ∈ X′, S(·, ·) denotes the Stirling number +of the second kind, and b := (bj)j≥1 is defined by setting bj := +∥ψj∥L∞(D) +√ +6amin +for all j ≥ 1. +Let µ(dµ) = dy. Then Corollary 1 can be used to deduce that +∥uξ − uξ,s∥L2µ(U;X) = O(s− 1 +p + 1 +2 ), +where the constant is independent of the dimension s. +In fact, if Xh is a conforming +finite element subspace of X, uξ,h(·, y) ∈ Xh denotes the finite element discretization of +uξ(·, y) ∈ X for all y ∈ U, and uξ,h,s(·, y) ∈ Xh denotes the dimension truncation of +uξ,h(·, y) for all y ∈ U, then we have +∥uξ,h − uξ,h,s∥L2µ(U;X) = O(s− 1 +p + 1 +2), +independently of s. +Finally, we present an example illustrating how our results can be applied to nonlinear +quantities of interest. +Example. Let X := H1 +0(D) as above. Consider the nonlinear quantity of interest +Gnl(v) := ∥v∥2 +X := +� +D +∥∇v(x)∥2 +Rd dx, +v ∈ X. +(5) +If u(·, y) ∈ X is the solution to (4) with U = [− 1 +2, 1 +2]N, µ(dy) := dy, and ξ(y) := y, then +it is known to satisfy Assumptions 1–3 with the regularity bound +∥∂νu(·, y)∥X ≤ C|ν|!bν, +where the constant C > 0 only depends on ∥f∥X′ and amin. By the Leibniz product rule, +there holds +∂νGnl(u(·, y)) = +� +D +� +m≤ν +� ν +m +� +∇∂mu(x, y) · ∇∂ν−mu(x, y) dx +≤ +� +m≤ν +� ν +m +� +∥∂mu(·, y)∥X∥∂ν−mu(·, y)∥X +≤ C2 � +m≤ν +� ν +m +� +|m|!bm|ν − m|!bν−m += C2bν +|ν| +� +ℓ=0 +ℓ!(|ν| − ℓ)! +� +m≤ν +|m|=ℓ +� ν +m +� += C2bν(|ν| + 1)!, +where we used the Vandermonde convolution � +|m|=ℓ +m≤ν +� ν +m +� += +�|ν| +ℓ +� += +|ν|! +ℓ!(|ν|−ℓ)!. +It follows from Theorem 1 that +∥Gnl(u) − Gnl(us)∥L2µ(U;R) = O(s− 1 +p + 1 +2), +independently of s. Moreover, if ξ(y) := ( 1 +√ +6 sin(2πyj))j≥1, then it follows from Corollary 1 +that +∥Gnl(uξ) − Gnl(uξ,s)∥L2µ(U;R) = O(s− 1 +p + 1 +2 ), +independently of s. +8 + +5 +Numerical experiments +Let D = (0, 1)2 be a spatial domain, U = [− 1 +2, 1 +2]N, and f(x) := x1 a fixed source term. +Let ξ: U → U, ξ(y) = ( 1 +√ +6 sin(2πyj))j≥1. We consider the PDE problem +� +−∇ · (aξ(x, y)∇uξ(x, y)) = f(x), +x ∈ D, y ∈ U, +uξ(x, y) = 0, +x ∈ ∂D, y ∈ U, +(6) +equipped with the diffusion coefficient +aξ(x, y) = 3 +2 + +� +j≥1 +ξ(yj)j−ϑ sin(jπx1) sin(jπx2), +x ∈ D, y ∈ U, ϑ > 1. +The PDE (6) is spatially discretized using a first-order conforming finite element method +with mesh size h = 2−5. +We consider the dimension truncation error for the full PDE solution using the formula +∥uξ − uξ,s∥L2(U;L2(D)) ≈ +� � +[− 1 +2 , 1 +2 ]s′ ∥uξ,s′(·, y) − uξ,s(·, y)∥2 +L2(D) dy +� 1 +2 +, +and we also consider the nonlinear quantity of interest (5), estimating the dimension +truncation error using the formula +∥Gnl(uξ) − Gnl(uξ,s)∥L2(U) ≈ +� � +[− 1 +2, 1 +2]s′ |Gnl(uξ,s′(·, y)) − Gnl(uξ,s(·, y))|2 dy +� 1 +2 +. +In both cases, we choose s′ ≫ s and the high-dimensional integrals are approximated using +a randomly shifted rank-1 lattice rule with 220 cubature nodes and a single random shift. +As the integration lattice, we use in both cases an off-the-shelf rank-1 lattice rule [19, +lattice-39101-1024-1048576.3600] and use the same random shift for each value of ϑ. As +the reference solution, we use the PDE solution corresponding to s′ = 211. +The numerical results for dimensions s ∈ {2k : k = 1, . . . , 9} and decay rates ϑ ∈ +{1.5, 2.0, 3.0} corresponding to the full PDE solution and the nonlinear quantity of interest +are displayed in Figures 1 and 2, respectively. The theoretical convergence rates in each +case are −1.0, −1.5, and −2.5, respectively, and they are displayed alongside the numerical +results. +The convergence graphs corresponding to the full PDE solution in Figure 1 display +an aliasing behavior between 10 ≤ s ≤ 100, which may be explained by the contributions +of the finite element discretization error as well as the use of an “off-the-shelf” lattice +rule (in contrast to a “tailored” lattice rule). This behavior appear to be exacerbated in +the convergence graphs corresponding to the nonlinear quantity of interest in Figure 2. +Nonetheless, in all cases the theoretically anticipated convergence rates are easily observed +in practice. +We remark that the convergence graphs corresponding to the affine and +uniform model with ξ(y) := (yj)j≥1 are extremely similar to the results corresponding to +the periodic model, and have thus been omitted. +9 + +Figure 1: The dimension truncation errors of the full PDE solution corresponding to a periodically parameterized +input random field with decay parameters ϑ ∈ {1.5, 2.0, 3.0}. The expected dimension truncation error rates are +−1.0, −1.5, and −2.5, respectively. +Figure 2: The dimension truncation errors of the nonlinear quantity of interest corresponding to a periodically +parameterized input random field with decay parameters ϑ ∈ {1.5, 2.0, 3.0}. The expected dimension truncation +error rates are −1.0, −1.5, and −2.5, respectively. +6 +Conclusions +Unlike many studies which have considered the dimension truncation error rate within +the context of high-dimensional numerical integration, we considered the L2 dimension +truncation error rate for parametric Hilbert space valued functions. Our theory covers +both affine parametric as well as non-affine parametric problems with sufficiently smooth +dependence on a sequence of bounded, parametric variables. The main dimension trun- +cation results presented in this work can be applied to nonlinear quantities of interest of +parametric model problems, provided that they satisfy the conditions of our framework. +10 + +In addition, the Hilbert space can be chosen to be a finite element subspace, indicating +that our dimension truncation results are also valid for conforming finite element approx- +imations of parametric PDEs. +The L2 dimension truncation error rates considered in this work arise, e.g., in the +study of high-dimensional function approximation of parametric PDEs. An example of +such an approximation scheme is the kernel method over lattice point sets considered +in [16]. The kernel method was analyzed in the context of the so-called periodic model, in +which a countable number of independent random variables enter the input random field +of the PDE as periodic functions. Our second main result shows that the L2 dimension +truncation error rate remains invariant under certain transformations of the parametric +variables: especially, the L2 dimension truncation rate considered in this work holds for +periodically parametrized model problems such as those studied in [13, 16, 17]. +References +[1] Bachmayr, M., Cohen, A., Dahmen, W.: Parametric PDEs: sparse or low-rank +approximations? IMA J. Numer. Anal., 38(4):1661–1708 (2017) +[2] Bhattacharya, K., Hosseini, B., Kovachki, N. B., Stuart, A. M.: Model reduction and +neural networks for parametric PDEs. SMAI J. Comput. Math., 7:121–157 (2021) +[3] Chen, Y., Hosseini, B., Owhadi, H., Stuart, A. M.: Solving and learning nonlinear +PDEs with Gaussian processes. J. Comput. Phys., 447:110668 (2021) +[4] Cohen, A., DeVore, R., Schwab, Ch.: Convergence rates of best N-term Galerkin +approximations for a class of elliptic sPDEs. Found. Comput. Math., 10:615–646 +(2010) +[5] Dick, J., Gantner, R. N., Le Gia, Q. 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T.: Reduced basis approximation and a pos- +teriori error estimation for affinely parametrized elliptic coercive partial differential +equations. Arch. Comput. Methods Eng., 15:229 (2008) +[22] Schwab, Ch., Zech, J.: Deep learning in high dimension: Neural network expression +rates for generalized polynomial chaos expansions in UQ. Anal. Appl. (Singap.), +17(1):19–55 (2019) +[23] Xiu, D., Karniadakis, G. E.: The Wiener-Askey polynomial chaos for stochastic +differential equations. SIAM J. Sci. Comput., 24:619–644 (2002) +[24] Yosida, K.: Functional Analysis. Springer, Heidelberg (1980) +[25] Zeng, X. Y., Leung, K. T., Hickernell, F. J.: Error analysis of splines for periodic +problems using lattice designs. In: Niederreiter, H., Talay, D. (eds.), Monte Carlo +and Quasi-Monte Carlo Methods 2004, pp. 501–514, Springer (2006) +[26] Zeng, X. Y., Kritzer, P., Hickernell, F. J.: Spline methods using integration lattices +and digital nets. Constr. Approx., 30:529–555 (2009) +12 + diff --git a/X9FRT4oBgHgl3EQf_DhA/content/tmp_files/load_file.txt b/X9FRT4oBgHgl3EQf_DhA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b23cf323a554c4f20df21c8126fcad256bb31ad --- /dev/null +++ b/X9FRT4oBgHgl3EQf_DhA/content/tmp_files/load_file.txt @@ -0,0 +1,487 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf,len=486 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='13693v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='NA] 31 Jan 2023 Application of dimension truncation error analysis to high-dimensional function approximation Philipp A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Guth† Vesa Kaarnioja‡ February 1, 2023 Abstract Parametric mathematical models such as partial differential equations with random coefficients have received a lot of attention within the field of uncertainty quantifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The model uncertainties are often represented via a series expansion in terms of the parametric variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' In practice, this series expansion needs to be truncated to a finite number of terms, introducing a dimension truncation error to the numerical simulation of a parametric mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' There have been several studies of the dimension truncation error corresponding to different models of the input random field in recent years, but many of these analyses have been carried out within the context of numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' In this paper, we study the L2 dimension truncation error of the parametric model problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Estimates of this kind arise in the assessment of the dimension truncation error for function approximation in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' In addition, we show that the dimension truncation error rate is invariant with respect to certain transformations of the parametric variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Numerical results are presented which showcase the sharpness of the theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 1 Introduction In the field of uncertainty quantification it is common to study mathematical models with uncertain influences parameterized by countably infinite sequences of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Consider, for instance, an abstract model M : X × U → Y such that M(g(y), y) = 0, (1) where X and Y are separable Hilbert spaces and U is a nonempty subset of the infinite- dimensional sequence space of parameters RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The solution g(y) ∈ X to (1) for y ∈ U, if it exists, may be computationally expensive to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' To this end, it may be preferable to instead approximate g using a surrogate which is cheap to evaluate and hence enables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', efficient sampling of the (approximated) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Some possible surrogate models include, but are not limited to, Gaussian process regression [3], reduced basis approaches [1, 21], generalized polynomial chaos expansions [4, 23], neural network approximations [2, 7, 9, 22], and kernel interpolation based on lattice point sets [16, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The results presented in this manuscript are particularly well-suited to the analysis of kernel methods used in conjunction with the so-called periodic model discussed in [13, 16, 17], and we will devote a section of this manuscript to explore the application of our dimension truncation results within this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' †Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria, philipp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='guth@ricam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='oeaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='at ‡Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 6, 14195 Berlin, Germany, vesa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='kaarnioja@fu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='de 1 Integration Function approximation Affine parametric [6, 20] [16] operator equation setting rate O(s− 2 p +1) rate O(s− 1 p + 1 2) Non-affine parametric [8, 12] this paper operator equation setting rate O(s− 2 p +1) rate O(s− 1 p + 1 2) Table 1: An overview of various dimension truncation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' A natural first step for the numerical treatment of (1) is the approximation by a dimensionally-truncated model Ms : X × Us → Y such that Ms(gs(y≤s), y≤s) = 0, where ∅ ̸= Us ⊆ Rs and gs(y≤s) ∈ X for all y≤s ∈ Us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Consider the problem of finding a surrogate solution gs,n := An(gs) using an algorithm An which uses n point evaluations of the s-dimensional function gs, where the surrogate belongs to X such that ∥gs − gs,n∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X) n→∞ −−−→ 0 with some known convergence rate and µ indicating a probability measure on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The total error of the approximation obtained in this fashion can be estimated using the triangle inequality ∥g − gs,n∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X) ≤ ∥g − gs∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X) + ∥gs − gs,n∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' In this manuscript we focus on the first term—the dimension truncation error—which is independent of the chosen approximation scheme An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Dimension truncation error rates are typically studied for problems involving partial differential equations (PDEs) with random inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' For integration problems a dimension truncation rate is derived in [20] for the source problem with an affine parameterization of the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' This rate was then improved by [6] in the generalized context of affine parametric operator equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Dimension truncation has also been studied for coupled PDE systems arising in optimal control problems under uncertainty [10], in the context of the periodic model of uncertainty quantification for both numerical integra- tion [17] and kernel interpolation [16], as well as for Bayesian inverse problems governed by PDEs [5, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The results in these papers have been proved using Neumann series, which is known to work well in the affine parametric setting, but may lead to suboptimal results if the problem depends nonlinearly on the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' In the non-affine setting, using Taylor series makes it possible to derive dimension truncation error rates by exploiting the parametric regularity of the problem, whereas the Neumann series approach relies fundamentally on the parametric structure of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The Taylor series approach was first applied in [8], and motivated the authors in [11] and [12] to derive dimension truncation error rates for sufficiently smooth, Banach space valued integrands, and with parameters following a generalized β-Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' An overview of the various dimension truncation error bounds studied in the literature is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Our manuscript is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='1 introduces the multi-index notation used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The problem setting is introduced in Section 2, including the central assumptions for the ensuing dimension truncation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Section 3 contains the L2 dimension truncation theorem for Hilbert space valued functions, and in Section 4 we discuss the invariance of the dimension truncation rate under certain transformations of the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Numerical experiments assessing the sharpness of our 2 theoretical results are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The paper ends with some conclusions in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='1 Notations and preliminaries Throughout this manuscript, boldfaced symbols are used to denote multi-indices while the subscript notation mj is used to refer to the j-th component of multi-index m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Let F := {m ∈ NN 0 : |m| < ∞} denote the set of finitely supported multi-indices, where the order of multi-index m is defined as |m| := � j≥1 mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Moreover, we denote |m|∞ := max j≥1 mj, and, for any sequence x := (xj)∞ j=1 of real numbers and m ∈ F, we define xm := � j≥1 xmj j , where we use the convention 00 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 2 Problem setting Let X be a real separable Hilbert space, U := [− 1 2, 1 2]N a set of parameters, and suppose that g(y) ∈ X is a parameterized family of functions with smooth dependence on y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' We define gs(y) := g(y≤s, 0) := g(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' , ys, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=') and assume that µ(dy) := � j≥1 µ(dyj) is a countable product probability measure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', µ(U) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' We suppose that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' For µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' y ∈ U, there holds ∥g(y) − gs(y)∥X s→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Let (Θk)k≥0 and b := (bj)j≥1 be sequences of nonnegative numbers such that b ∈ ℓp(N) for some p ∈ (0, 1) and b1 ≥ b2 ≥ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Suppose that g is continuously differentiable up to order k + 1, with ∥∂νg(y)∥X ≤ Θ|ν|bν for all y ∈ U and for all ν ∈ Fk := {ν ∈ NN 0 : |ν| ≤ k + 1}, where k := ⌈ 1 1−p⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' There holds � 1/2 −1/2 yj µ(dyj) = 0 and there exists a constant Cµ ≥ 0 such that � 1/2 −1/2 |yj|k µ(dyj) ≤ Cµ for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' If Assumption 2 holds, then we infer that y �→ G(g(y)) for all G ∈ X′ is continuous as a composition of continuous mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Hence y �→ G(g(y)) is measurable for all G ∈ X′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', y �→ g(y) is weakly measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Since X is assumed to be a separable Hilbert space, by Pettis’ theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', [24, Chapter 4]) we obtain that y �→ g(y) is strongly measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The upper bound in Assumption 2 is µ-integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Thus we conclude from Bochner’s theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', [24, Chapter 5]) and Assumption 2 that g is µ-integrable over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 3 Further, µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' equality defines an equivalence relation among strongly µ-measurable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' By L2 µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' X) we denote the Hilbert space of equivalence classes of strongly µ-measurable functions f : U → X with norm ∥f∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X) := � � U ∥f(y)∥2 X µ(dy) � 1 2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Moreover, under the Assumptions 1 and 2 it can be shown that g, gs ∈ L2 µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' X) and lim s→∞ ∥g(y) − g(y≤s, 0)∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X) = lim s→∞ � � U ∥g(y) − g(y≤s, 0)∥2 X µ(dy) � 1 2 = 0, by applying Lebesgue’s dominated convergence theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', [18, Theorem 1] and [14, Section 26]) to F s(y) := ∥g(y) − g(y≤s, 0)∥2 X, which converges µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' to zero by Assumption 1, and can be bounded by (2Θ0)2 by As- sumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' We use the superscript to avoid confusion with the notation used to denote dimensionally-truncated functions elsewhere in the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 3 Dimension truncation error We will require the following parametric regularity bound for the main dimension trunca- tion result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Under Assumption 2, there holds |∂ν∥g(y) − gs(y)∥2 X| ≤ � max 0≤ℓ≤|ν| 2Θℓ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' �2 (|ν| + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='bν for all ν ∈ Fk and y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Let ν ∈ Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' We apply the Leibniz product rule with respect to the inner product of the Hilbert space X to obtain ∂ν∥g(y) − gs(y)∥2 X = ∂ν⟨g(y) − gs(y), g(y) − gs(y)⟩X = � m≤ν � ν m � ⟨∂m(g(y) − gs(y)), ∂ν−m(g(y) − gs(y))⟩X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Using the Cauchy–Schwarz inequality together with Assumption 2 yields |∂ν∥g(y) − gs(y)∥2 X| ≤ � m≤ν � ν m � ∥∂m(g(y) − gs(y))∥X∥∂ν−m(g(y) − gs(y))∥X ≤ 4 � m≤ν � ν m � Θ|m|bmΘ|ν|−|m|bν−m = 4bν |ν| � ℓ=0 ΘℓΘ|ν|−ℓ � |m|=ℓ m≤ν � ν m � = 4bν |ν| � ℓ=0 ΘℓΘ|ν|−ℓ |ν|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' (|ν| − ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' ≤ 4 � max 0≤ℓ≤|ν| Θℓ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' �2 (|ν| + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='bν, where we used the Vandermonde convolution � |m|=ℓ m≤ν � ν m � = �|ν| ℓ � = |ν|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' (|ν|−ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='. 4 The main result of this document is stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Let g(y) ∈ X, y ∈ U, satisfy Assumptions 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Then ∥g − gs∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X) = O(s− 1 p + 1 2), where the implied coefficient is independent of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Let s ≥ 1 and define F s(y) := ∥g(y) − gs(y)∥2 X for y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' In the special case of the uniform distribution µ(dy) = dy, we can apply [12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='2] to obtain ∥g − gs∥2 L2(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X) = ���� � U (F s(y) − F s(y≤s, 0)) dy ���� = O(s− 2 p +1), from which the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' For completeness, we present the proof below for the prob- ability measure µ and because parts of the argument will also be useful to establish the invariance of the dimension truncation rate in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Developing the Taylor expansion of F s about (y≤s, 0) and observing that F s(y≤s, 0) = 0, we obtain F s(y) = k � ℓ=1 � |ν|=ℓ νj=0 ∀j≤s yν ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' ∂νF s(y≤s, 0) + � |ν|=k+1 νj=0 ∀j≤s k + 1 ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' yν � 1 0 (1 − t)k∂νF s(y≤s, ty>s) dt, (2) where y>s := (yj)j>s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Integrating both sides over y ∈ U yields � U F s(y) µ(dy) = k � ℓ=1 � |ν|=ℓ νj=0 ∀j≤s 1 ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' � U yν∂νF s(y≤s, 0) µ(dy) + � |ν|=k+1 νj=0 ∀j≤s k + 1 ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' � U � 1 0 (1 − t)kyν∂νF s(y≤s, ty>s) dt µ(dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' If ν ∈ Fk is such that νj = 1 for any j > s, then Fubini’s theorem together with Assump- tion 3 imply for the summands appearing in the first term that � U yν∂νF s(y≤s, 0) µ(dy) = � � j>s � 1 2 − 1 2 yνj j µ(dyj) � � �� � =0 � [− 1 2, 1 2]s ∂νF s(y≤s, 0) µ(dy>s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Therefore all multi-indices with any component equal to 1 can be removed from the first sum (especially, we can omit all multi-indices with |ν| = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Further, applying the regu- larity bound proved in Lemma 1 and writing open the definition of F s yields � U ∥g(y) − gs(y)∥2 X µ(dy) ≤ Ck µ � max 0≤ℓ≤k 2Θℓ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' �2 (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' k � ℓ=2 � |ν|=ℓ νj=0 ∀j≤s νj̸=1 ∀j>s bν + Ck+1 µ � max 0≤ℓ≤k+1 2Θℓ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' �2 (k + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' � |ν|=k+1 νj=0 ∀j≤s 1 ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='bν, (3) 5 where we used � 1 0 (1 − t)k dt = 1 k+1 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The second term in (3) can be estimated from above using the multinomial theorem in conjunction with Stechkin’s lemma: � |ν|=k+1 νj=0 ∀j≤s 1 ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='bν ≤ � |ν|=k+1 νj=0 ∀j≤s |ν|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' bν = � � j>s bj �k+1 ≤ s(k+1)(− 1 p +1) � � j≥1 bp j � k+1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' On the other hand, the first term in (3) can be estimated similarly to [6]: � 2≤|ν|≤k νj=0 ∀j≤s νj̸=1 ∀j>s bν ≤ � 0̸=|ν|∞≤k νj=0 ∀j≤s νj̸=1 ∀j>s bν = −1 + � j>s � 1 + k � ℓ=2 bℓ j � = −1 + � j>s � 1 + b2 j k−2 � ℓ=0 bℓ j � ≤ −1 + � j>s � 1 + b2 j k−2 � ℓ=0 bℓ 1 � �� � =:βk � ≤ −1 + exp � βk � j>s b2 j � = � ℓ≥1 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' � βk � j>s b2 j �ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Using � j>s b2 j ≤ s− 2 p +1(� j≥1 bp j) 2 p , which follows from Stechkin’s lemma, we further estimate � ℓ≥1 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' � βk � j>s b2 j �ℓ ≤ s− 2 p +1 � ℓ≥1 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' (βk∥b∥2 p)ℓ = s− 2 p +1(−1 + exp(βk∥b∥2 p) since s− 2 p +1 ≥ (s− 2 p +1)ℓ for all ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Altogether, the above discussion yields the bound ∥g(y) − gs(y)∥2 L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X) = � U ∥g(y) − gs(y)∥2 X µ(dy) = O(s− 2 p +1 + s(k+1)(− 1 p +1)), where the implied coefficient is independent of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Since we assumed that k = ⌈ 1 1−p⌉, the assertion follows by taking the square root on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 4 Invariance of the dimension truncation rate under trans- formations of variables An interesting consequence of the Taylor series argument used in Theorem 1 is that the di- mension truncation rate remains invariant under certain transformations of the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' This has been previously observed in the context of dimension truncation for integration problems under the periodic model [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' To make this notion precise, let us consider a mapping ξ: U → U, ξ(y) := (ξ(y1), ξ(y2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' ), which satisfies the following conditions: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' There hold ξ(0) = 0 and � 1/2 −1/2 ξ(y) dy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' There exists Cξ ≥ 0 such that � 1/2 −1/2 |ξ(y)|k dy ≤ Cξ for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Then we obtain the following as a consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Let g(y) ∈ X, y ∈ U, satisfy Assumptions 1–3 and let ξ : U → U satisfy Assumptions 4–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Define the ξ-transformed function gξ by gξ(y) := g(ξ(y)), y ∈ U, 6 and its dimension truncation by gξ,s(y) := gξ(y≤s, 0) for y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Then ∥gξ − gξ,s∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X) = O(s− 1 p + 1 2 ), where the implied coefficient is independent of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' We introduce F s ξ (y) := ∥gξ(y) − gξ,s(y)∥2 X for y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' By carrying out the change of variable y ← ξ(y) in (2), we obtain F s ξ (y) = k � ℓ=1 � |ν|=ℓ νj=0 ∀j≤s ξ(y)ν ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' ∂νF s(ξ(y≤s, 0)) + � |ν|=k+1 νj=0 ∀j≤s k + 1 ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' ξ(y)ν � 1 0 (1 − t)k∂νF s(ξ(y≤s, ty>s)) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Integrating the above formula on both sides over y ∈ U and utilizing Lemma 1 as well as Assumption 5, we obtain—in complete analogy with the proof of Theorem 1—that � U ∥gξ(y) − gξ,s(y)∥2 X dy ≤ Ck ξ � max 0≤ℓ≤k 2Θℓ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' �2 (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' k � ℓ=2 � |ν|=ℓ νj=0 ∀j≤s νj̸=1 ∀j>s bν + Ck+1 ξ � max 0≤ℓ≤k+1 2Θℓ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' �2 (k + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' � |ν|=k+1 νj=0 ∀j≤s 1 ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The desired result follows by exactly the same argument as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' As an application, with U := [− 1 2, 1 2]N, let ξ: U → U satisfy the Assumptions 4 and 5, let D ⊂ Rd, d ∈ {1, 2, 3}, be a bounded Lipschitz domain, and let f : D → R be a fixed source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Consider the parametric PDE problem � −∇ · (aξ(x, y)∇uξ(x, y)) = f(x), x ∈ D, y ∈ U, uξ(x, y) = 0, x ∈ ∂D, y ∈ U, (4) endowed with the ξ-transformed diffusion coefficient aξ(x, y) := a0(x) + ∞ � i=1 ξ(yi)ψi(x), x ∈ D, y ∈ U, which is assumed to satisfy the following: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' There exist amin, amax > 0 such that 0 < amin ≤ aξ(x, y) ≤ amax < ∞ for all x ∈ D and y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' a0 ∈ L∞(D) and ψi ∈ L∞(D) for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' �∞ i=1 ∥ψi∥p L∞(D) < ∞ for some p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' In this case, the transformation ξ(y) := ( 1 √ 6 sin(2πyj))j≥1 corresponds to the so-called periodic model studied in [13, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Let X := H1 0(D) be equipped with the norm ∥v∥X := 7 � D ∥∇v(x)∥2 Rd dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' In this special case, it is known that the weak solution u(·, y) ∈ X to (4) for y ∈ U satisfies the parametric regularity bound ∥∂νuξ(·, y)∥X ≤ (2π)|ν|∥f∥X′ amin � m≤ν |m|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='bm � j≥1 S(νj, mj) for all ν ∈ F and y ∈ U, where the source term f ∈ X′, S(·, ·) denotes the Stirling number of the second kind, and b := (bj)j≥1 is defined by setting bj := ∥ψj∥L∞(D) √ 6amin for all j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Let µ(dµ) = dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Then Corollary 1 can be used to deduce that ∥uξ − uξ,s∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X) = O(s− 1 p + 1 2 ), where the constant is independent of the dimension s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' In fact, if Xh is a conforming finite element subspace of X, uξ,h(·, y) ∈ Xh denotes the finite element discretization of uξ(·, y) ∈ X for all y ∈ U, and uξ,h,s(·, y) ∈ Xh denotes the dimension truncation of uξ,h(·, y) for all y ∈ U, then we have ∥uξ,h − uξ,h,s∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='X) = O(s− 1 p + 1 2), independently of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Finally, we present an example illustrating how our results can be applied to nonlinear quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Let X := H1 0(D) as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Consider the nonlinear quantity of interest Gnl(v) := ∥v∥2 X := � D ∥∇v(x)∥2 Rd dx, v ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' (5) If u(·, y) ∈ X is the solution to (4) with U = [− 1 2, 1 2]N, µ(dy) := dy, and ξ(y) := y, then it is known to satisfy Assumptions 1–3 with the regularity bound ∥∂νu(·, y)∥X ≤ C|ν|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='bν, where the constant C > 0 only depends on ∥f∥X′ and amin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' By the Leibniz product rule, there holds ∂νGnl(u(·, y)) = � D � m≤ν � ν m � ∇∂mu(x, y) · ∇∂ν−mu(x, y) dx ≤ � m≤ν � ν m � ∥∂mu(·, y)∥X∥∂ν−mu(·, y)∥X ≤ C2 � m≤ν � ν m � |m|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='bm|ν − m|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='bν−m = C2bν |ν| � ℓ=0 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' (|ν| − ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' � m≤ν |m|=ℓ � ν m � = C2bν(|ν| + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', where we used the Vandermonde convolution � |m|=ℓ m≤ν � ν m � = �|ν| ℓ � = |ν|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' (|ν|−ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='. It follows from Theorem 1 that ∥Gnl(u) − Gnl(us)∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='R) = O(s− 1 p + 1 2), independently of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Moreover, if ξ(y) := ( 1 √ 6 sin(2πyj))j≥1, then it follows from Corollary 1 that ∥Gnl(uξ) − Gnl(uξ,s)∥L2µ(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='R) = O(s− 1 p + 1 2 ), independently of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 8 5 Numerical experiments Let D = (0, 1)2 be a spatial domain, U = [− 1 2, 1 2]N, and f(x) := x1 a fixed source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Let ξ: U → U, ξ(y) = ( 1 √ 6 sin(2πyj))j≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' We consider the PDE problem � −∇ · (aξ(x, y)∇uξ(x, y)) = f(x), x ∈ D, y ∈ U, uξ(x, y) = 0, x ∈ ∂D, y ∈ U, (6) equipped with the diffusion coefficient aξ(x, y) = 3 2 + � j≥1 ξ(yj)j−ϑ sin(jπx1) sin(jπx2), x ∈ D, y ∈ U, ϑ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The PDE (6) is spatially discretized using a first-order conforming finite element method with mesh size h = 2−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' We consider the dimension truncation error for the full PDE solution using the formula ∥uξ − uξ,s∥L2(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='L2(D)) ≈ � � [− 1 2 , 1 2 ]s′ ∥uξ,s′(·, y) − uξ,s(·, y)∥2 L2(D) dy � 1 2 , and we also consider the nonlinear quantity of interest (5), estimating the dimension truncation error using the formula ∥Gnl(uξ) − Gnl(uξ,s)∥L2(U) ≈ � � [− 1 2, 1 2]s′ |Gnl(uξ,s′(·, y)) − Gnl(uξ,s(·, y))|2 dy � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' In both cases, we choose s′ ≫ s and the high-dimensional integrals are approximated using a randomly shifted rank-1 lattice rule with 220 cubature nodes and a single random shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' As the integration lattice, we use in both cases an off-the-shelf rank-1 lattice rule [19, lattice-39101-1024-1048576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='3600] and use the same random shift for each value of ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' As the reference solution, we use the PDE solution corresponding to s′ = 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The numerical results for dimensions s ∈ {2k : k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' , 9} and decay rates ϑ ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='0} corresponding to the full PDE solution and the nonlinear quantity of interest are displayed in Figures 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The theoretical convergence rates in each case are −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='5, and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='5, respectively, and they are displayed alongside the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The convergence graphs corresponding to the full PDE solution in Figure 1 display an aliasing behavior between 10 ≤ s ≤ 100, which may be explained by the contributions of the finite element discretization error as well as the use of an “off-the-shelf” lattice rule (in contrast to a “tailored” lattice rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' This behavior appear to be exacerbated in the convergence graphs corresponding to the nonlinear quantity of interest in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Nonetheless, in all cases the theoretically anticipated convergence rates are easily observed in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' We remark that the convergence graphs corresponding to the affine and uniform model with ξ(y) := (yj)j≥1 are extremely similar to the results corresponding to the periodic model, and have thus been omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 9 Figure 1: The dimension truncation errors of the full PDE solution corresponding to a periodically parameterized input random field with decay parameters ϑ ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The expected dimension truncation error rates are −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='5, and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Figure 2: The dimension truncation errors of the nonlinear quantity of interest corresponding to a periodically parameterized input random field with decay parameters ϑ ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The expected dimension truncation error rates are −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='5, and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 6 Conclusions Unlike many studies which have considered the dimension truncation error rate within the context of high-dimensional numerical integration, we considered the L2 dimension truncation error rate for parametric Hilbert space valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Our theory covers both affine parametric as well as non-affine parametric problems with sufficiently smooth dependence on a sequence of bounded, parametric variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The main dimension trun- cation results presented in this work can be applied to nonlinear quantities of interest of parametric model problems, provided that they satisfy the conditions of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' 10 In addition, the Hilbert space can be chosen to be a finite element subspace, indicating that our dimension truncation results are also valid for conforming finite element approx- imations of parametric PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The L2 dimension truncation error rates considered in this work arise, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', in the study of high-dimensional function approximation of parametric PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' An example of such an approximation scheme is the kernel method over lattice point sets considered in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' The kernel method was analyzed in the context of the so-called periodic model, in which a countable number of independent random variables enter the input random field of the PDE as periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' Our second main result shows that the L2 dimension truncation error rate remains invariant under certain transformations of the parametric variables: especially, the L2 dimension truncation rate considered in this work holds for periodically parametrized model problems such as those studied in [13, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=' References [1] Bachmayr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', Cohen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} +page_content=', Dahmen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQf_DhA/content/2301.13693v1.pdf'} 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b/XNE3T4oBgHgl3EQfFwkF/content/tmp_files/2301.04307v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6ca9cd28f48d29e6e7693ee943daf041df1ffb0 --- /dev/null +++ b/XNE3T4oBgHgl3EQfFwkF/content/tmp_files/2301.04307v1.pdf.txt @@ -0,0 +1,1057 @@ +MNRAS 000, 1–?? (20XX) +Preprint 12 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Accretion of supersonic magnetized winds onto black holes +Miguel Gracia-Linares★1 and Francisco S. Guzmán†2 +1 Center for Gravitational Physics, Department of Physics. The University of Texas at Austin Austin, TX 78712, U. S. A. +2Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo. Edificio C-3, Cd. Universitaria, 58040 Morelia, Michoacán, México. +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We present the accretion of magnetized supersonic winds onto a rotating black hole in three dimensions. We select representative +spin-wind orientations in order to illustrate its effects on the evolution and morphology of the shock cone. The most important +finding in the magnetized case, unlike the purely hydrodynamical scenario, is the formation of rarified spots where the magnetic +field pressure dominates over the gas pressure. In these rarified spots we find the formation of eddies within the shock cone. +Key words: accretion – black hole physics – instabilities +1 INTRODUCTION +The Bondi-Hoyle-Littleton (BHL) or wind accretion occurs when a +compact object moves through a constant fluid which is considered +to be perfect and free of self-gravity, or conversely that a uniform +fluid moves toward the accretor (Bondi & Hoyle 1944; Bondi 1952). +This process has been studied analytically and numerically in both +Newtonian and relativistic regimes for example in (Fryxell et al. +1987; Matsuda et al. 1987; Shima et al. 1985; Sawada et al. 1989; +Matsuda et al. 1992, 1991) and in (Petrich et al. 1988; Font & Ibáñez +1998; Dönmez et al. 2011; Cruz-Osorio et al. 2012; Lora-Clavijo & +Guzmán 2013) respectively. +The BHL accretion by itself becomes more realistic as more ingre- +dients are added up to the wind and accretor models. Recent versions +of BHL analyses include the relativistic accretion a supersonic fluid +onto a spinning black hole in 3D (Gracia-Linares & Guzmán 2015), +the study of 2.5D Hydrodynamic BHL accretion onto black holes +including magnetic fields (Penner 2011; Takahashi & Ohsuga 2015), +the BHL accretion onto black holes including and the coupling be- +tween radiation and fluid (Zanotti et al. 2011; Park & Ricotti 2012). +In the Newtonian regime the study of magnetized BHL accretion +onto a neutron star can be found in (Toropina et al. 2012), also in +(Lee et al. 2014) the process of a magnetized plasma onto black hole +was studied using a Newtonian model of black hole. +Astrophysically motivated scenarios for the BHL accretion process +involve the properties of the gas around the black hole, including its +equation of state, possible radiation transport processes that even- +tually could affect the dynamics of the plasma and magnetic field +configurations. The velocity of the wind is another important param- +eter, directly related to the possible cause of the motion of the black +hole. For instance, it has been found using numerical simulations, +that the collision of two black holes with appropriate initial spins +and masses may produce final black holes that travel at very high +speeds, including velocities for supermassive black holes of the or- +der of 100-1000 km/s in early analyses (González et al. 2007), and +★ E-mail: mgracia@austin.utexas.edu +† E-mail: francisco.s.guzman@umich.mx +up to 15000 km/s (Sperhake et al. 2011) in more recent studies. As- +tronomical observations indicate that a candidate to be a wandering +black hole, whose velocity has been modeled with this process, the +object known as QSO 3C 186 (Chiaberge et al. 2017), traveling at +a speed of 2100km/s, which could be the result of the collision of +two black holes with appropriate initial orbital and spin parameters +(Lousto et al. 2017). Mechanisms that promote stellar mass black +holes to move in the interstellar space are the supernovae natal kicks, +for example as found in (Sahu et al. 2022), which have considerable +lower velocities. Even though it is possible to carry out simulations of +the BHL accretion process with these low speeds (e. g. González & +Guzmán (2018)), technically such simulations require a considerable +amount of resources due to the big numerical domain required for the +accretion regime to hold, where the spatial scale can be hundreds of +horizon radii for a supersonic scenario. In our analysis we use rather +high values of the velocity, which allows the use of a smaller nu- +merical domain, with appropriate numerical parameters for accurate +simulations that suffice to illustrate the effects of the magnetic field +on the wind in a spatial scale of a few horizon radii. Even in such +case, also very high velocity realistic scenarios exist, for example the +matter model using BHL accretion on black holes prior to mergers +in GW sources within the common envelope stage (Cruz-Osorio & +Rezzolla 2020). +Among the astrophysical applications of the BHL accretion we +find recent advances, that include the application of the shock-cone +flip-flop instability (Dönmez et al. 2011) and the shock cone vibra- +tions (Lora-Clavijo & Guzmán 2013) that occur during the BHL +accretion, as models of X-ray quasi periodic oscillators (QPOs); the +BHL has been studied in environments with non-trivial density gra- +dients (Lora-Clavijo et al. 2015) and in the presence of small rigid +bodies (Cruz-Osorio et al. 2017); BHL has been also studied in bi- +nary stars (Comerford et al. 2019), and has also been proposed as a +possible ignition mechanism of type Ia supernovae (Steigerwald & +Tejeda 2021); very recently the formation of jets in BHL processes +has been also presented (Kaaz et al. 2022), as well as the influence +of BHL in sources of gravitational waves (Cruz-Osorio & Rezzolla +2020). +The degree of applicability of models with more ingredients would +© 20XX The Authors +arXiv:2301.04307v1 [astro-ph.HE] 11 Jan 2023 + +2 +Gracia-Linares and & Guzmán +depend on the observational resolution of black hole horizon size +scale, for example using the Event Horizon Telescope array, which +has revealed high resolution images of plasma surrounding the super- +massive black hole at the centre of M87 (Event Horizon Telescope +Collaboration et al. 2019) and at the center of the Milky Way’s Sgr +A* (Akiyama et al. 2022). Other scenarios like the interesting moving +black hole associated to the quasar 3C 186 (Chiaberge et al. 2017), +that could be a kicked black hole moving through the galaxy medium +resulting from the merger of two black holes (Lousto et al. 2017), +would require a resolution currently out of reach due to the distance +from earth, although resolution is expected to always improve. +In order to contribute to the addition of ingredients modeling the +BHL process onto a spinning black hole, in this paper we study the +3D supersonic accretion of magnetized winds within the ideal mag- +netohydrodynamics approximation (MHD) and compare the general +results with the accretion of a purely Hydrodynamical gas (HD). In +this sense, the present paper is a follow up of (Gracia-Linares & +Guzmán 2015). We study the evolution of the MHD variables and +describe the differences between the accretion of a purely hydrody- +namical fluid and a magnetized plasma. In our analysis we assume +the black hole and wind to be initially immersed in a constant mag- +netic field, aligned with the direction of the axis of rotation of the +black hole. In order to investigate the potential properties of a general +case scenario, we choose three principal wind directions with respect +to the spin of the black hole: wind parallel to the axis of rotation, +diagonal wind and a wind perpendicular to the axis of rotation of the +black hole. Notice that the last two cases can only be studied in full +3D without symmetries. +The paper is organized as follows. In section 2 we describe the +ideal MHD equations modeling the magnetized fluid and the numer- +ical methods used. In section 3 we present the set of configurations +we experiment with and the main aspects we compare between the +HD and MHD scenarios. In 4 we describe the conclusions from +our analysis and in the appendix we show convergence tests of our +simulations. +2 THE WIND MODEL +2.1 Equations and numerical methods +The plasma is modeled with a magnetized fluid that obeys the ideal +MHD, which assumes infinite electric conductivity and the electric +field measured by a comoving observer set to zero. The stress-energy +tensor of such fluid is explicitly +𝑇 𝜇𝜈 = (𝜌ℎ + 𝑏2)𝑢𝜇𝑢𝜈 + +� +𝑝 + 𝑏2 +2 +� +𝑔𝜇𝜈 − 𝑏𝜇𝑏𝜈, +(1) +where 𝜌 is the rest-mass density, 𝑝 the pressure, 𝑏𝜇 the magnetic field +measured by a comoving observer, 𝑢𝜇 is the 4-velocity, ℎ ≡ 1+𝜖+𝑝/𝜌 +the specific enthalpy, 𝜖 the specific internal energy and 𝑔𝜇𝜈 are the +contravariant components of the 4-metric. +The equations for this matter field are those of the general rela- +tivistic magnetohydrodynamics (GRMHD). These can be written as +a flux conservative system that assumes a standard 3+1 decomposi- +tion of the space-time. The space-time metric described in Cartesian +coordinates (𝑡, 𝑥𝑖) is given by 𝑑𝑠2 = (−𝛼2 + 𝛽𝑖𝛽𝑖)𝑑𝑡2 + 2𝛽𝑖𝑑𝑡𝑑𝑥𝑖 + +𝛾𝑖 𝑗𝑑𝑥𝑖𝑑𝑥 𝑗, where 𝛼 is the lapse function and 𝛽𝑖 the components of the +shift vector associated to the 3+1 decomposition of the space-time, +and 𝛾𝑖 𝑗 are the components of the 3-metric of spatial hypersurfaces +used to foliate the space-time. In these terms, the GRMHD equations +according to the Valencia formulation are written as (Banyuls et al. +1997): +𝜕𝑡u + 𝜕𝑥𝑖F(𝑖) (u) = S(u), +(2) +where the vector u = {𝐷, 𝑆𝑖, 𝜏, 𝐵𝑘} contains the following conserved +variables, 𝐷 the generalized rest mass density of the fluid, 𝑆𝑖 the +momentum components along in each direction, 𝜏 the internal energy, +𝐵𝑘 the magnetic field measured by an eulerian observer, F(𝑖) (u) the +fluxes and S(u) a sources vector. In terms of the primitive variables +of the fluid elements, the conserved variables are defined by +𝐷 += +√𝛾𝜌𝑊 +𝑆𝑖 += +√𝛾[(𝜌ℎ + 𝑏2)𝑊2𝑣 𝑗 − 𝛼𝑏0𝑏 𝑗] +𝜏 += +√𝛾[(𝜌ℎ+𝑏2)𝑊2 − +� +𝑝 + 𝑏2 +2 +� +− 𝛼2(𝑏0)2 − 𝜌𝑊] +𝐵𝑖 += +√𝛾𝑊 +� +𝑏𝑖 − 𝛼𝑏0 +� +𝑣𝑖 − 𝛽𝑖 +𝛼 +�� +, +where 𝛾 is the determinant of the spatial 3-metric 𝛾𝑖 𝑗 of the three- +dimensional spatial slices which foliate the space-time, 𝑊 = (1 − +𝛾𝑖 𝑗𝑣𝑖𝑣 𝑗)−1/2 is the Lorentz factor and 𝑏0 = 𝑊𝐵𝑖𝑣𝑖/𝛼. As usual, +the system of equations is closed with an equation of state specified +below. In terms of primitive and conservative variables the fluxes +and sources are +F𝑖 (u) += +������� +� +(𝛼𝑣𝑖 − 𝛽𝑖)𝐷 +(𝛼𝑣𝑖 − 𝛽𝑖)𝑆𝑗 + 𝛼√𝛾 +� +𝑝 + 𝑏2 +2 +� +𝛿𝑖 +𝑗 − 𝛼√𝛾𝑏𝑗 𝐵𝑖/𝑊 +�𝛼𝑣𝑖 − 𝛽𝑖� 𝜏 + 𝛼√𝛾 +� +𝑝 + 𝑏2 +2 +� +𝑣𝑖 − 𝛼2√𝛾𝑏0𝐵𝑖/𝑊 +�𝛼𝑣𝑖 − 𝛽𝑖� 𝐵𝑘 − +� +𝛼𝑣𝑘 − 𝛽𝑘� +𝐵𝑖 +������� +� +, +S(u) += +���� +� +0 +𝑇 𝜇𝜈 �𝜕𝜇𝑔𝜈 𝑗 + Γ𝛿𝜇𝜈𝑔𝛿 𝑗 +� +𝛼�𝑇 𝜇0𝜕𝜇 ln 𝛼 − 𝑇 𝜇𝜈Γ0𝜇𝜈 +� +0 +���� +� +, +(3) +where 𝑔𝜇𝜈 are the covariant components of the 4-metric and Γ𝛿 𝜇𝜈 +the Christoffel symbols of the space-time. We solve the system of +equations (2-3) using the publically available GRHydro thorn (Mösta +et al. 2014), within the Cactus Einstein Toolkit (ETK) code (Löffler +et al. 2012). We use the high resolution shock capturing methods pro- +vided to solve the GRMHD equations. Specifically our simulations +use the HLLE numerical flux formula and the minmod reconstructor. +In order to preserve the magnetic field divergence near to zero, we +use the constraint transport method (Balsara & Spicer 1999) imple- +mented within the GRHydro thorn. For the integration in time we use +a fourth order Runge-Kutta method. What we added to the ETK is a +module that applies appropriate boundary conditions in the upstream +boundary, which is the part of the boundary from which we inject +the wind into the domain, where we set the density and velocity field +to their initial values during the evolution. On the other hand we im- +plement out-flux boundary conditions in the downstream boundary, +which is the part of the boundary through which the wind is expected +to leave the domain. These conditions are a key ingredient in the +accretion of winds in numerical domains that contain the accretion +sphere. In order to avoid divergences of the variables, the GRHydro +thorn implements an atmosphere that is triggered during the primi- +tive variables calculation for tiny or negative values of the density. +For that we use a floor density value of 10−12 in code units, and then +the pressure and internal energy are set to consistent values. In our +results the density never approaches such small values, including the +cases where rarefaction spots are formed. +MNRAS 000, 1–?? (20XX) + +MHD winds +3 +2.2 Description of space-time and wind +We describe the space-time metric of the black hole of mass 𝑀 and +spin S = 𝑎ˆ𝑧 using Kerr-Schild (KS) horizon penetrating coordinates: +𝑑𝑠2 += +� +𝜂𝜇𝜈 + +2𝑀𝑟3 +𝑟4 + 𝑎2𝑧2 𝑙𝜇𝑙𝜈 +� +𝑑𝑥𝜇𝑑𝑥𝜈, +𝑙𝜇 += +� +1, 𝑟𝑥 + 𝑎𝑦 +𝑟2 + 𝑎2 , 𝑟𝑦 − 𝑎𝑥 +𝑟2 + 𝑎2 , 𝑧 +𝑟 +� +, +𝑟 += +� +� +� +𝑟2∗ − 𝑎2 + +√︃ +(𝑟2∗ − 𝑎2)2 + 4𝑎2𝑧2 +2 +, +𝑟∗ += +√︃ +𝑥2 + 𝑦2 + 𝑧2, +(4) +where 𝜂𝜇𝜈 is the flat metric. +The wind is set initially as a spatially constant rest mass density +ideal gas 𝜌 = 1 × 10−6 [1/𝑀2], moving toward the black hole at a +given asymptotic supersonic velocity 𝑣2∞ = 𝑣𝑖𝑣𝑖. We assume the fluid +obeys a gamma-law equation of state 𝑝 = (Γ − 1)𝜌𝜖, and consider a +relativistic fluid with adiabatic index Γ = 4/3. We use the asymptotic +speed of sound 𝑐𝑠∞ = 0.05 in order to have a sufficiently slow wind +but still supersonic and to compare with previous hydrodynamical +results. The initial fluid pressure is written as 𝑝ini = 𝑐2s∞𝜌ini/(Γ − +𝑐2s∞Γ1), where Γ1 = Γ/(Γ − 1). In order to avoid negative and zero +values of the pressure we choose the sound speed such that 𝑐s∞ < +√ +Γ − 1. Finally, the initial specific internal energy 𝜖 is reconstructed +using the equation of state. +The magnetic field is defined initially to be constant and parallel +to the spin of the black hole B = 𝐵0 ˆ𝑧. We choose two representative +values of magnetic field strength 𝐵0,𝑠𝑡𝑟𝑜𝑛𝑔 = 1 × 10−5 [1/𝑀] and +𝐵0,𝑤𝑒𝑎𝑘 = 1 × 10−10 [1/𝑀]. After the initial time the magnetic +field evolves according to the GRMHD equations and responds to +the evolution of the plasma. +We focus on three representative scenarios, in which the wind is +assumed to have different direction with respect to the black hole +spin S. These three different orientations of the wind are represented +by ↑ ← ↗ in our tables and correspond to directions ˆ𝑧, ˆ𝑥 and ˆ𝑥 + ˆ𝑧 +respectively. The black hole spin is denoted by ⇑. +We also use the excision method inside the black hole horizon in +order to avoid the variables to interact with the black hole’s singu- +larity (Hawke et al. 2005). We performed all of our evolution runs +using an isotropic cubic grid Δ𝑥 = Δ𝑦 = Δ𝑧 with base resolution +Δ𝑥 = 0.5𝑀 and one refinement level with resolution Δ𝑥 = 0.25𝑀. +The base domain is set to [−30𝑀, 30𝑀]3 and is approximately twice +as big as a sphere of the accretion radius 𝑟𝑎𝑐𝑐 = 𝑀/(𝑐2𝑠∞ +𝑣2∞). This +is important because in case the domain is smaller than a sphere with +radius equal to the accretion radius, the flow could enter the wind +regime. +Continuing with the set up of the initial configuration, another +important property is the size scale. This is usually set by the accretion +radius defined by 𝑟𝑎𝑐𝑐 = 𝑟ℎ𝑜𝑟/(𝑣2∞ + 𝑐2𝑠∞), where 𝑟ℎ𝑜𝑟 is the radius +of an accretor, in our case the horizon radius of the black hole. The +accretion radius has the information of how fast or slow the wind +is and it is important because as studied in (Foglizzo et al. 2005), +the parameter that could trigger a possible flip-flop instability in the +Bondi-Hoyle process, is the relative size of the accretor with respect +to the accretion radius 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 𝑣2∞ + 𝑐2𝑠∞. Sometimes this is +called the accretor size (Foglizzo et al. 2005), but it is actually a size +relative to the accretion radius that indicates how fast or slow the +wind is. In our study we use two wind velocities corresponding to a +Figure 1. Snapshot on the 𝑦 = 0 plane of the density at time 𝑡 = 1000𝑀 when +the accretion is already stationary, for cases HD1 and MHD1 models ↑⇑ and +𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.065 shown in the top. We show the magnetic field lines in the +MHD1 case indicating how they bend toward the cone and after that continue +to their asymptotic vertical direction. In the bottom we show the 𝛽 parameter +and the shock cone superposed with a field of vectors indicating the Lorentz +force direction. The shock cone in the presence of magnetic field is slightly +wider than in the purely HD case and in the MHD case the maximum of the +gas density density (1.3 × 10−4) is smaller than the HD case (2.1 × 10−4). +slow case 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.065 and a fast case 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.25 with +asymptotic velocity 𝑣∞ = 0.25𝑐 and 0.5𝑐 respectively. +Finally, we set the dimensionless spin of the black hole to S = 0.8ˆ𝑧, +which is comparable to the one estimated by numerical simulations +of the QSO 3C186 quasar’s kicked black hole (Lousto et al. 2017). +3 RESULTS +With the above set of physical parameters, including two wind ve- +locities, two magnetic filed strengths, we performed a series of sim- +ulations for the wind orientations summarized in Table 1, where we +include the purely Hydrodynamical counterparts in order to compare +the impact of the magnetic field on the process. The simulations start +with the constant values of the wind variables except in the excision +region. During a transient stage the fluid and magnetic field interact +until they approach nearly stationary configurations, where the bow +shock if any and the shock cone are formed, and in the MHD cases +the magnetic field also stabilizes. This stationary stage is the one we +illustrate in the results discussed below. +The general properties of the morphology and dynamics of the +process once the evolution of the fluid has settled down to a nearly +stationary regime are presented in Figures 1, 2, 3 and 4, in geomet- +ric units, corresponding to the four first models in Table 1. In these +figures we show the rest mass density of the gas in the purely hydro- +dynamical scenario; the rest mass density of the plasma in the MHD +cases with the magnetic field lines superposed, indicating the distor- +tion due to the presence of the shock-cone, we remind that initially +the magnetic field lines are parallel to the 𝑧 axis; we also show the +value of 𝛽 = +2𝑝 +𝑏𝑖𝑏𝑖 , which reveals that there is a region 𝛽 < 1 where +the magnetic pressure dominates over the hydrodynamical pressure; +finally we show the Lorentz force field, indicating the direction in +which the plasma is being affected by this force. +Stability. Similarly to the purely hydrodynamical process, where +MNRAS 000, 1–?? (20XX) + +30-302.145e - 41.0e - 6W/ XZ/MHD1MHDI30-302.145e - 41.0e - 6W/ XZ/M1230-30W/ XZ/MMHDI0MHDI30-302.145e - 41.0e - 6W/ XZ/M4 +Gracia-Linares and & Guzmán +Figure 2. In the top we show a snapshot on the 𝑦 = 0 plane of the density +at 𝑡 = 1000𝑀, for the HD2 and MHD2 cases with orientation ↗⇑ and +𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.065. The first important difference between the HD and +MHD is that in the later, the shock-cone is detached from the black hole, +that is, the black hole is contained within the high density zone. The second +is that the shock cone in the presence of the magnetic field is wider than in +the pure HD case. Finally the third one is that in the MHD case there is a +low density region. In the MHD2 case we superpose the magnetic field lines, +which show asymptotically the vertical direction, however shows a complex +structure at the boundary of the shock cone. The structure of field lines would +be interesting to see in detail within the context of resistive MHD, because +there are some candidates to X-points that eventually could trigger magnetic +reconnection. In the bottom we show a plot of the 𝛽 parameter, which shows +that the magnetic field dominates in the region of the shock-cone. In the final +plot we superpose the density of the plasma with a field of arrows indicating +the direction of the Lorentz force. It can be seen that the arrows precisely +indicate that the Lorentz force pulls the plasma out of the rarified zone. +no Flip-Flop (FF) kind of instability was found, even within the +regime where it was predicted in the Newtonian theory (Gracia- +Linares & Guzmán 2015), in the case of MHD there was not such +instability either, even though some different dynamical features were +found. +The bow shock. Another important dependence of the orientation +of the wind is the bow shock. In Fig. 2 we show that for the MHD case +the bow shock is detached from the black hole surface, a condition +that usually triggers the shock cone instability in Newtonian systems +(Foglizzo et al. 2005) but has been shown to be inoffensive in the +relativistic case (Gracia-Linares & Guzmán 2015). +Shock cone angle. As reported in (Penner 2011) for the axisym- +metric case, the open angle of the shock cone is bigger for the MHD +than for the purely HD scenario. We confirm this result in the ax- +isymmetric cases and also in the full non-symmetric diagonal ↗⇑ +and horizontal ⇑← cases. +Effects on the magnetic field. We also study the effects of the +initially uniform magnetic field, due to the shock cone formation +process. The fact that the plasma piles up in a high density cone- +shaped region, indicates that some important effects may happen, +namely the magnitude of the magnetic field is expected to change +and the resulting currents from the process of formation will promote +Lorentz forces. First we diagnose the magnetic field amplification. +In Fig. 5 we show the magnetic field strength as a function of time, +measured at points where the plasma 𝛽 = +2𝑝 +𝑏𝑖𝑏𝑖 < 1, because there +the magnetic field dominates over the hydrodynamical pressure, for +the cases MHD1, MHD2, MHD3 and MHD4. In all these scenarios +the highest amplification occurs at points near the black hole horizon. +Notice that in the MHD2, MHD3 and MHD4 cases the magnetic field +increases approximately between one and two orders of magnitude +and in the case MHD1 the amplification is of one order of magnitude. +Rarefaction regions and Lorentz force. A second interesting im- +plication of the distortion of the magnetic field is the formation of +rarified zones not seen in the purely hydrodynamical case. Coinci- +dentally, as shown for the cases MHD2, MHD3 and MHD4 in Figures +2, 3, 4, these low density zones develop precisely where the mag- +netic pressure dominates over the fluid pressure 𝛽 < 1. In order to +investigate the possible reason for this we tracked the Lorentz force +𝜖𝑖 𝑗𝑘𝐽 𝑗 𝐵𝑘 where 𝐽 𝑗 is the current density. In these Figures we show a +snapshot at time 𝑡 = 1000𝑀 (after the process has become stationary) +of the rest mass density for the HD and MHD cases for comparison, +and show the direction in which this Lorentz force acts. We observe +that in the low density spots the direction of the Lorentz force points +in the appropriate direction as to move the plasma outwards. This +is an indication that -even if an approximation- this force is the re- +sponsible for the formation of these spots. These rarefaction zones +appear in different places depending on the direction of the wind, for +example in the diagonal MHD2 case the rarified zone is bigger in the +top half of the shock cone but in the horizontal MHD3 case the zone +is symmetric with respect to the 𝑧 = 0 plane. +Formation of eddies. Another interesting effect is the formation +of eddies in the shock cone. In Fig. 6 we present the formation of +eddies in two general cases, one is the diagonal case MHD2 ↗⇑, and +the second one is the horizontal wind MHD3 ⇑←. For both cases +we show the rest mass density on a plane perpendicular to the wind +and near to the black hole, together with its purely hydrodynamical +counterpart for comparison. We superpose the velocity field of the +plasma in order to have a clear idea of the motion in the low density +spots. +In the diagonal case HD2 and MHD2 we present the plot on +the plane 𝑥 + 𝑧 = 4, which is perpendicular to the wind in the +shock-cone region at a close distance from the black hole’s horizon. +This map is a perpendicular view of that in Fig. 2 and shows two +different rarified spots that do not appear in the HD2 case. The +most interesting part is that the velocity field indicates the plasma is +rotating approximately around the center of the big spot, whereas in +the purely hydrodynamical case the gas is simply moving toward the +center of the shock-cone. +In the horizontal wind case HD3 and MHD3 we show the density +and velocity field of the shock-cone also in a plane perpendicular to +the direction of the wind at a distance 2𝑀 from the center of the black +hole. In this case, the symmetry allows to notice two symmetric low +density spots. This is a perpendicular view from that in Fig. 3. In this +case it is clear that two eddies are formed in the rarified zones and +the plasma is rotating around. +Effect of the wind velocity. One may wonder whether the wind ve- +locity produces significant morphological changes. In the purely HD +regime we found in our previous paper (Gracia-Linares & Guzmán +2015) that the accretion rate of the system and the open angle of +the shock depends on the wind velocity, slower winds correspond to +wider shock cones and higher accretion rates. In this paper we ver- +ified the same happens for the magnetized case. An important new +feature appeared, this is the one related to the rarified zone within +the shock-cone. The density in the rarified zone is one order of mag- +nitude smaller in the fast case model MHD4 than in the slow case +model MHD1 as can be seen in Figs. 1 and 4. +Accretion rate. Another point to see is the influence of the magnetic +field on the accretion rate. We measured �𝑀 in a spherical surface +MNRAS 000, 1–?? (20XX) + +30-301.0e - 6W/ XZ/M1.61e - 4HD2MHD22.05e - 430-301.0e - 6W/ XZ/M2MHD230-30W/ XZ/M01MHD22.05e - 430-301.0e - 6W/ XZ/MMHD winds +5 +Figure 3. As in the previous case, we show in the top a snapshot of the +rest mass density on the 𝑦 = 0 plane at time 𝑡 = 1000𝑀 for the HD3 and +MHD3 models with 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.065 for the case ⇑←. Again a difference +is that in the MHD case the shock-cone is detached from the black hole and +the magnetic field lines shows an interesting distortion near the bow-shock. +Due to the bitant symmetry of this case, there are two symmetric rarified +regions. In the bottom we show that the region 𝛽 < 1 appears again within +the shock-cone region. In the bottom-right panel we show that the Lorentz +force points in the direction in which the plasma should move to produce the +low density spots. +Figure 4. This is the case of a fast wind. In the top we show a snapshot of +the rest mass density on the 𝑦 = 0 plane at time 𝑡 = 1000𝑀, for the HD4 +and MHD4 models with 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.25. Since this is a faster wind, in +both HD and MHD cases, the shock cone is attached to the black hole. In the +bottom we show the parameter 𝛽 which shows the magnetic field dominates in +the shock-cone region. In this case there is a central rarified zone. According +to the direction of the Lorentz force shown in the last panel, this force is the +responsible for the depletion of the low density regions. The low velocity +counterpart of the case is MHD1 in Figure 1. The density in the rarified zone +within the shock-cone of this MHD4 case is one order of magnitude smaller +than in the MHD1 case. This is an example of how the velocity of the wind +influences the properties of the system. +Table 1. Parameters of the 8 simulations used in this study. Four of them +involve MHD and the other ones are the purely HD equivalent counterparts. +In this Table the units of 𝑎 are [1/𝑀2], the velocity 𝑣∞ is in units of 𝑐 = 1 +and the units of the magnetic field 𝐵0 are [1/𝑀 ]. +Name +𝑎 = 0.8 +𝑣∞ = 0.25 +𝐵0 +Orientation +MHD1 +1 × 10−5 +↑⇑ +HD1 +0 +↑⇑ +MHD2 +1 × 10−5 +↗⇑ +HD2 +0 +↗⇑ +MHD3 +1 × 10−5 +⇑← +HD3 +0 +⇑← +MHD5 +1 × 10−10 +↑⇑ +MHD6 +1 × 10−10 +↗⇑ +MHD7 +1 × 10−10 +⇑← +Name +𝑎 = 0.8 +𝑣∞ = 0.5 +𝐵0 +Orientation +MHD4 +1 × 10−5 +↑⇑ +HD4 +0 +↑⇑ +MHD8 +1 × 10−10 +↑⇑ +Figure 5. Time series of the magnetic field strength measured at four different +points of the shock cone with a magnetic field 𝐵0 = 𝐵0,𝑠𝑡𝑟𝑜𝑛𝑔 for models +MHD1, MHD2, MHD3 and MHD4. For this we have chosen points located +where the magnetic pressure dominates. These points can be identified in +Fig. 1 for the MHD1 case, Fig. 2 for MHD2, Fig. 3 for MHD3 and Fig. 4 for +MHD4. The coordinates of the points where the magnetic field is measured +are indicated in each line for each of the cases. +MNRAS 000, 1–?? (20XX) + +MHD330-301.0e - 6W/ XZ/M2.89e - 42MHD330-30W/ XZ/M01MHD330-301.0e - 6W/ XZ/M2.89e - 4HD45.81e - 530-301.0e - 6W/ XZ/M5.81e - 5MHD430-301.0e - 6W/ XZ/M2MHD430-30W/ XZ/M01MHD4-15151.0e - 6W/ XZ/M5.81e - 5MHD1 +MHD2 +401 +A= (2.5,0,2.5) +A= (2.5,0,2.5) +301 +B=(5.0,0,5.0) +B= (5.0,0,5.0) +351- +C= (7.5,0,7.5) +C = (7.5,0, 7.5) +251 +D=(10.0,0,10.0) +D=(10.0,0,10.0) +301- +201 +(x10^-5) +(x10^-5) +251 +151 +201 +/BI +151 +101 +101- +51- +51- +0 +1002003004005006007008009001000 +1002003004005006007008009001000 +t/M +t/M +MHD3 +MHD4 +301 +451 +A=(5.0,0,5.0) +A=(2.5,0,2.5) +B = (7.5,0, 7.5) +B= (2.5,0,5.0) +401 +C= (10.0,0,10.0) +C=(2.5,0,7.5) +251- +D=(12.5,0,12.5) +351- +D= (2.5,0,10.0) +201 +301 +(x10^-5) +(x10^-5) +251 +151 +B +201- +101- +151- +101- +51- +51- +1+ +0 +100 +200 +300 +400 +0500 +600 +700800 +9001000 +0 +100200300 +4005006007008009001000 +t/M +t/MHD330-301.0e - 6W/ XZ/M2.89e - 46 +Gracia-Linares and & Guzmán +Figure 6. Snapshot of the rest mass density 𝜌 and velocity field 𝑣𝑖 for models +HD2, MHD2, HD3 and MHD3. For models HD2 and MHD2 we show the +results on the plane 𝑥 + 𝑧 = 4, which is a plane perpendicular to the wind +direction. For models HD3 and MHD3 we show the results on the plane +𝑥 = −2 which is a plane also perpendicular to the wind. The velocity field +in both cases shows the formation of eddies precisely in the rarified zones of +the shock-cone. It is useful to compare the shape and location of the rarified +zones with the perpendicular view in Figures 2 and 3. +located near the black hole’s horizon, at 𝑟 = 2𝑀 after the shock has +been formed and the evolution regime becomes stationary. In Table +2 we show the values of the accretion rate for all the models at a +stationary stage. We first observe that the horizontal wind (models +HD3 and MHD3) has the highest accretion rate for the velocity used +here, approximately between 10% to 20% above the other cases. +We found that the accretion rate of the MHD as compared with +the purely hydrodynamical counterparts is within a 3% of difference. +This table also shows that the direction of the wind influences more +the accretion rate than the fact of having pure HD or MHD. Besides +the direction of the wind, another factor that influences the accretion +rate is the wind velocity which can be seen from the comparison of +models MHD1 and MHD4, with significant differences of 100%. +The weak magnetic field case 𝐵0 = 𝐵0,𝑤𝑒𝑎𝑘. Unlike the strong +magnetic field case, the general properties for 𝐵0 = 𝐵0,𝑤𝑒𝑎𝑘 +(MHD5, MHD6, MHD7 and MHD8) are very similar to the purely +hydrodynamical counterparts. In Figures 7 and 8 we show the rest +mass density of these models and the respective HD cases. Contrary +to the strong magnetic field case, we do not observe a noticeable +change in the shock cone morphology. We attribute this behavior to +the fact that for this magnetic field the magnetic field pressure is of +order 10−20 and the plasma 𝛽 ≫ 1, even after the magnetic field is +amplified. Thus there is no significant impact on the gas dynamics +as the one observed in the strong field cases MHD1, MHD2, MHD3 +and MHD4. In this sense this regime works as a correspondence case +between MHD and HD, although the structure of magnetic field lines +holds. +Effects on the magnetic field. In Figure 9 we show the amplification +of the magnetic field, which is increased by one order of magnitude. +The initial and amplified values of the magnetic field are not sufficient +to compete with the gas pressure, and this is the reason why there are +not low density spots, first because the plasma 𝛽 is of the order of +1010 and second because the Lorentz force is ten orders of magnitude +smaller than in the 𝐵0,𝑠𝑡𝑟𝑜𝑛𝑔 case. As shown in Figures 7 and 8 this +magnetic field is too small to produce any relevant change. In general, +Table 2. Accretion rate values ( �𝑀) for all the cases. The perpendicular case +to the axis of rotation of the black hole is the configuration with the highest +�𝑀. Moreover +�𝑀 does not change significantly between the MHD and HD +cases. +Model +Orientation +�𝑀 +MHD1 +↑⇑ +0.667 × 10−3 +HD1 +↑⇑ +0.682 × 10−3 +MHD2 +↗⇑ +0.673 × 10−3 +HD2 +↗⇑ +0.637 × 10−3 +MHD3 +→⇑ +0.781 × 10−3 +HD3 +→⇑ +0.774 × 10−3 +Model +Orientation +�𝑀 +MHD4 +↑⇑ +0.315 × 10−3 +HD4 +↑⇑ +0.333 × 10−3 +MHD5 +↑⇑ +0.679 × 10−3 +MHD6 +↗⇑ +0.635 × 10−3 +MHD7 +→⇑ +0.769 × 10−3 +MHD8 +↑⇑ +0.331 × 10−3 +even though the magnetic field lines distort, purely hydrodynamics +gas evolution rules the process. +Accretion rate. Based on the previous description, for this scenario +with a weak magnetic field, it is expected the accretion rate to be +even more similar to that of the HD counterparts. We find that the +differences in accretion rate are within 0.5%. +Attractor behavior. The stationarity of the flow, morphology and +magnetic field lines distribution is recovered when the wind density +is varied with sinusoidal fluctuations of amplitude 10% of 𝜌ini and +injected over a nearly arbitrary time window as long as 𝜌ini is recov- +ered. This indicates that the configurations resist inhomogeneities, +which are expected to happen in real scenarios. More formally, it +would be interesting to find a parametrization of the basin of at- +traction of these nearly-stationary accretion configurations, and how +rapidly they approach stationarity in terms of Lyapunov exponents. +4 CONCLUSIONS AND DISCUSSION +We present the accretion of a supersonic magnetized wind onto a +black hole in 3D. For this we selected a set of different wind orien- +tations with respect to the spin of the black hole. In all the cases we +compared the results with the purely hydrodynamical counterpart. +We studied two different values of the magnetic field strength, +strong and weak cases. We found that in the strong field case there +are zones within the shock cone in which the magnetic field pressure +dominates 𝛽 < 1 and produces low density spots within the shock +cone. We also show the formation of eddies consisting of plasma +rotating around the low density spots within the shock cone near the +black hole horizon. Using an approximate calculation of the Lorentz +force, we found that this force can be the responsible for the depletion +of the low density regions. The weak field case on the other hand, was +such that the gas pressure dominates over the magnetic field and the +plasma 𝛽 ≫ 1, and neither low density spots nor eddies are formed, +MNRAS 000, 1–?? (20XX) + +-1515HD23.66e - 4Y/MVx2 + z2 / M1.0e - 6W/ XMHD21.0e - 6W/ X-15153.66e - 4Y/MVx2 + z2 / M1.0e - 6-1515Y/MHD3Z/M2.89e - 41.0e - 6-1515Y/MZ/M2.89e - 4MHD3MHD winds +7 +Figure 7. Snapshot of the rest mass density 𝜌 during the stationary regime, +for models HD1, HD2, and the MHD counterparts with the weak magnetic +field strength 𝐵0 = 𝐵0,𝑤𝑒𝑎𝑘, specifically MHD5 and MHD6. These models +exhibit the distortion of the magnetic field lines likewise in the strong field +case. However we did not find any drastic change in the morphology and +magnitude of the rest mass density that could for example trigger the formation +of rarified zones. +Figure 8. Snapshot of the rest mass density 𝜌 during the stationary regime, +for models HD3, HD4, and the MHD counterparts for the weak magnetic +field 𝐵0 = 𝐵0,𝑤𝑒𝑎𝑘, specifically MHD7 and MHD8. Again, we notice the +distortion of the magnetic field lines but the field is not strong enough to +produce zones of magnetic field domination that eventually could produce +rarified zones. +although the magnetic field lines follow a similar pattern as in the +strong field case. +We find that the accretion rate is not significantly different between +the HD and MHD cases. Instead, the wind velocity and orientation +is more important for the accretion rate. +We are sure that with the insight obtained with the general sce- +narios presented here, the infrastructure developed, specifically the +implementation of upstream and downstream boundary conditions, +can be applied to astrophysical systems, like wandering black holes of +type QSO 3C186 (Chiaberge et al. 2017) and supernovae natal kicks +like that in (Sahu et al. 2022). The various properties of the process, +Figure 9. Time series of the magnetic field strength measured at four different +points of the shock cone for 𝐵0 = 𝐵0,𝑤𝑒𝑎𝑘 and models MHD5, MHD6, +MHD7 and MHD8. For this we have chosen points located where the magnetic +pressure dominates. These points can be identified in Fig. 7 for the MHD5 +and MHD6 cases, and Fig. 8 for MHD7 and MHD8. The coordinates where +the magnetic field is measured are indicated for each case. +including the morphology of the shock cone, the formation of rarified +zones and eddies are expected to change for different parameters of +the wind, including the asymptotic velocity and equation of state of +the gas. It is possible now to construct a catalog of simulations with +astrophysical parameters to be contrasted with observations. We also +expect that our set up, applied to specific scenarios will be useful +for example in the study of the cases like the high speed black hole +prior to merger, during the common envelop face in (Cruz-Osorio & +Rezzolla 2020). +Concerning the observability of the processes detailed in this pa- +per, a direct observation does not seem to be affordable at the mo- +ment, considering the candidates need a higher resolution than those +used by the EHT. Then it is not expected a direct observation of +the shock-cone itself, and differences with the presence of magnetic +fields should be even finer. However it can be expected that the for- +mation process of the shock-cone will reveal the intrinsic properties +of the system, like black hole velocity and mass, perhaps spin, and +the gas density and equation of state as illustrated in (González & +Guzmán 2018) for the case of a wandering black hole, along with +a correlation of vibrational modes of the shock-cone as indicated +in (Lora-Clavijo & Guzmán 2013). Vibrations, that will depend on +each combination of parameters of wind inclination angle and mag- +netic field strength, that we showed here that have local fingerprints, +should be analyzed as done for the spin-less, hydrodynamical model +in (Lora-Clavijo & Guzmán 2013). The various scenarios, with the +appropriate vibration modes would compose the catalog of possible +scenarios to be correlated with observations on candidates like QSO +3C186. On aother case, the binary black hole merger, the first steps +have been already explored and signatures are expected together with +gravitational wave signals (Cruz-Osorio & Rezzolla 2020). +MNRAS 000, 1–?? (20XX) + +30-302.145e - 41.0e - 6W/ XZ/MHD1HD45.81e - 530-301.0e - 6W/ XZ/M30-302.145e - 41.0e - 6W/ XZ/MMHD530-301.0e - 6W/ XZ/MMHD61.61e - 430-301.0e - 6W/ XZ/M2.89e - 4MHD730-301.0e - 6W/ XZ/M5.81e - 5MHD8MHD5 +MHD6 +6.5 +10.0 +A=(2.5,0,2.5) +A= (2.5,0,2.5) +6.0 +B= (5.0,0,5.0) +9.0 +B = (5.0, 0, 5.0) +C = (7.5,0, 7.5) +C=(7.5,0,7.5) +5.5 +D=(10.0,0,10.0) +8.0 +D = (10.0,0,10.0) +5.0- +7.0 +(x10^-10) +4.5 +(x10^-10) +6.0- +4.0 +3.5 +5.0 +IBI +B/ +3.0 +4.0 +2.5 +3.0- +2.0 +2.0- +1.5 +1.0 +1.0 +1002003004005006007008009001000 +0 +1002003004005006007008009001000 +t/M +t/M +MHD7 +MHD8 +A= (5.0,0,5.0) +41 +23.5 +A= (2.5,0,2.5) +B= (7.5,0, 7.5) +B= (2.5,0,5.0) +21 +..C (10.0,0,10.0) +C= (2.5,0,7.5) +D=(12.5,0,12.5) +D = (2.5, 0, 10.0) +18.5 +31- +16 +(x10^-10) +(X10A-10) +13.5 +21 +B +IBI +11 +8.5 +11- +6 +3.5 +0 +100 200 300 400 +5006007008009001000 +0 +100200300400 +5006007008009001000 +t/M +t/M30-301.0e - 6W/ XZ/M1.61e - 4HD2HD330-301.0e - 6W/ XZ/M2.89e - 48 +Gracia-Linares and & Guzmán +ACKNOWLEDGEMENTS +MGL is supported by NSF grant PHY-1550461, PHY-2207780, +PHY-2114582, and the Mexican National Council of Science and +Technology (CONACyT) CVU 391996. FSG acknowledges support +from grant CIC-UMSNH-4.9. The runs were carried out in the Big +Mamma cluster of the Laboratorio de Inteligencia Artificial y Super- +cómputo, IFM-UMSNH. +DATA AVAILABILITY +The data underlying this article will be shared on reasonable request +to the corresponding author. +REFERENCES +Akiyama K., et al., 2022, ApJ, 930, L12 +Balsara D. S., Spicer D. S., 1999, Journal of Computational Physics, 149, 270 +Banyuls F., Font J. A., Ibáñez J. M., Martí J. M., Miralles J. A., 1997, ApJ, +476, 221 +Bondi H., 1952, MNRAS, 112, 195 +Bondi H., Hoyle F., 1944, MNRAS, 104, 273 +Chiaberge M., et al., 2017, A&A, 600, A57 +Comerford T. A. F., Izzard R. G., Booth R. A., Rosotti G., 2019, Monthly +Notices of the Royal Astronomical Society, 490, 5196 +Cruz-Osorio A., Rezzolla L., 2020, The Astrophysical Journal, 894, 147 +Cruz-Osorio A., Lora-Clavijo F. D., Guzmán F. S., 2012, MNRAS, 426, 732 +Cruz-Osorio A., Sánchez-Salcedo F. J., Lora-Clavijo F. 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In order to support the validity of our results we present a +self-convergence test for one of the scenarios described in this paper. +We show the order of convergence 𝑄 for the density along the 𝑧 axis +outside of the black hole horizon, where the high density shock-cone +forms, at two different times 1000𝑀 and 2000𝑀 for the MHD4 case. +For this test we calculated three numerical solutions 𝜌1, 𝜌2 and +𝜌3 using respectively the resolutions Δ𝑥1 = 0.5𝑀, Δ𝑥2 = Δ𝑥1/2 +and Δ𝑥3 = Δ𝑥2/2. We then evaluate 𝑄 = +log (𝜌1−𝜌2)−log (𝜌2−𝜌3) +log 2 +and show its value in Figure A1. The result is that 𝑄 takes values +between 1 and 2, which is the expected order of convergence of +the high resolution shock capturing methods using the HLLE flux +formula, with a linear reconstructor in the presence of shocks. This +shows that the simulations for the evolution of the wind, and for +the parameters in this paper, are carried out within the convergence +regime of the numerical methods used. +MNRAS 000, 1–?? (20XX) + +MHD winds +9 +Figure A1. Order of convergence 𝑄 of the rest mass density measured along +the 𝑧 axis, on the downstream zone, at two different times, 𝑡 = 1000𝑀 (top) +and 𝑡 = 2000𝑀 (bottom) for the case MHD4 along the shock cone. +MNRAS 000, 1–?? (20XX) + +4 +3.5 +3 +2.5 +Q +2 +1.5 +0.5 +0 +4 +6 +8 +10 +12 +14 +16 +z/M +4 +3.5 +3 +2.5 +Q +2 +1.5 +80000090 +R& +0.5 +0 +4 +6 +8 +10 +12 +14 +16 +z/M \ No newline at end of file diff --git a/XNE3T4oBgHgl3EQfFwkF/content/tmp_files/load_file.txt b/XNE3T4oBgHgl3EQfFwkF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b46456684c37be0d70981cdb61d36c774055c7b7 --- /dev/null +++ b/XNE3T4oBgHgl3EQfFwkF/content/tmp_files/load_file.txt @@ -0,0 +1,719 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf,len=718 +page_content='MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' (20XX) Preprint 12 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0 Accretion of supersonic magnetized winds onto black holes Miguel Gracia-Linares★1 and Francisco S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Guzmán†2 1 Center for Gravitational Physics, Department of Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The University of Texas at Austin Austin, TX 78712, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Edificio C-3, Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Universitaria, 58040 Morelia, Michoacán, México.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We present the accretion of magnetized supersonic winds onto a rotating black hole in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We select representative spin-wind orientations in order to illustrate its effects on the evolution and morphology of the shock cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The most important finding in the magnetized case, unlike the purely hydrodynamical scenario, is the formation of rarified spots where the magnetic field pressure dominates over the gas pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In these rarified spots we find the formation of eddies within the shock cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Key words: accretion – black hole physics – instabilities 1 INTRODUCTION The Bondi-Hoyle-Littleton (BHL) or wind accretion occurs when a compact object moves through a constant fluid which is considered to be perfect and free of self-gravity, or conversely that a uniform fluid moves toward the accretor (Bondi & Hoyle 1944;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Bondi 1952).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This process has been studied analytically and numerically in both Newtonian and relativistic regimes for example in (Fryxell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Matsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Shima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Sawada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Matsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 1992, 1991) and in (Petrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Font & Ibáñez 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Dönmez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Cruz-Osorio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Lora-Clavijo & Guzmán 2013) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The BHL accretion by itself becomes more realistic as more ingre- dients are added up to the wind and accretor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Recent versions of BHL analyses include the relativistic accretion a supersonic fluid onto a spinning black hole in 3D (Gracia-Linares & Guzmán 2015), the study of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5D Hydrodynamic BHL accretion onto black holes including magnetic fields (Penner 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Takahashi & Ohsuga 2015), the BHL accretion onto black holes including and the coupling be- tween radiation and fluid (Zanotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Park & Ricotti 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the Newtonian regime the study of magnetized BHL accretion onto a neutron star can be found in (Toropina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2012), also in (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2014) the process of a magnetized plasma onto black hole was studied using a Newtonian model of black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Astrophysically motivated scenarios for the BHL accretion process involve the properties of the gas around the black hole, including its equation of state, possible radiation transport processes that even- tually could affect the dynamics of the plasma and magnetic field configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The velocity of the wind is another important param- eter, directly related to the possible cause of the motion of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' For instance, it has been found using numerical simulations, that the collision of two black holes with appropriate initial spins and masses may produce final black holes that travel at very high speeds, including velocities for supermassive black holes of the or- der of 100-1000 km/s in early analyses (González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2007), and ★ E-mail: mgracia@austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='edu † E-mail: francisco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='guzman@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='mx up to 15000 km/s (Sperhake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2011) in more recent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' As- tronomical observations indicate that a candidate to be a wandering black hole, whose velocity has been modeled with this process, the object known as QSO 3C 186 (Chiaberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2017), traveling at a speed of 2100km/s, which could be the result of the collision of two black holes with appropriate initial orbital and spin parameters (Lousto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Mechanisms that promote stellar mass black holes to move in the interstellar space are the supernovae natal kicks, for example as found in (Sahu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2022), which have considerable lower velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Even though it is possible to carry out simulations of the BHL accretion process with these low speeds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' González & Guzmán (2018)), technically such simulations require a considerable amount of resources due to the big numerical domain required for the accretion regime to hold, where the spatial scale can be hundreds of horizon radii for a supersonic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In our analysis we use rather high values of the velocity, which allows the use of a smaller nu- merical domain, with appropriate numerical parameters for accurate simulations that suffice to illustrate the effects of the magnetic field on the wind in a spatial scale of a few horizon radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Even in such case, also very high velocity realistic scenarios exist, for example the matter model using BHL accretion on black holes prior to mergers in GW sources within the common envelope stage (Cruz-Osorio & Rezzolla 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Among the astrophysical applications of the BHL accretion we find recent advances, that include the application of the shock-cone flip-flop instability (Dönmez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2011) and the shock cone vibra- tions (Lora-Clavijo & Guzmán 2013) that occur during the BHL accretion, as models of X-ray quasi periodic oscillators (QPOs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' the BHL has been studied in environments with non-trivial density gra- dients (Lora-Clavijo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2015) and in the presence of small rigid bodies (Cruz-Osorio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' BHL has been also studied in bi- nary stars (Comerford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2019), and has also been proposed as a possible ignition mechanism of type Ia supernovae (Steigerwald & Tejeda 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' very recently the formation of jets in BHL processes has been also presented (Kaaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2022), as well as the influence of BHL in sources of gravitational waves (Cruz-Osorio & Rezzolla 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The degree of applicability of models with more ingredients would © 20XX The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='04307v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='HE] 11 Jan 2023 2 Gracia-Linares and & Guzmán depend on the observational resolution of black hole horizon size scale, for example using the Event Horizon Telescope array, which has revealed high resolution images of plasma surrounding the super- massive black hole at the centre of M87 (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2019) and at the center of the Milky Way’s Sgr A* (Akiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Other scenarios like the interesting moving black hole associated to the quasar 3C 186 (Chiaberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2017), that could be a kicked black hole moving through the galaxy medium resulting from the merger of two black holes (Lousto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2017), would require a resolution currently out of reach due to the distance from earth, although resolution is expected to always improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In order to contribute to the addition of ingredients modeling the BHL process onto a spinning black hole, in this paper we study the 3D supersonic accretion of magnetized winds within the ideal mag- netohydrodynamics approximation (MHD) and compare the general results with the accretion of a purely Hydrodynamical gas (HD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In this sense, the present paper is a follow up of (Gracia-Linares & Guzmán 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We study the evolution of the MHD variables and describe the differences between the accretion of a purely hydrody- namical fluid and a magnetized plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In our analysis we assume the black hole and wind to be initially immersed in a constant mag- netic field, aligned with the direction of the axis of rotation of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In order to investigate the potential properties of a general case scenario, we choose three principal wind directions with respect to the spin of the black hole: wind parallel to the axis of rotation, diagonal wind and a wind perpendicular to the axis of rotation of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Notice that the last two cases can only be studied in full 3D without symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In section 2 we describe the ideal MHD equations modeling the magnetized fluid and the numer- ical methods used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In section 3 we present the set of configurations we experiment with and the main aspects we compare between the HD and MHD scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In 4 we describe the conclusions from our analysis and in the appendix we show convergence tests of our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2 THE WIND MODEL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='1 Equations and numerical methods The plasma is modeled with a magnetized fluid that obeys the ideal MHD, which assumes infinite electric conductivity and the electric field measured by a comoving observer set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The stress-energy tensor of such fluid is explicitly 𝑇 𝜇𝜈 = (𝜌ℎ + 𝑏2)𝑢𝜇𝑢𝜈 + � 𝑝 + 𝑏2 2 � 𝑔𝜇𝜈 − 𝑏𝜇𝑏𝜈, (1) where 𝜌 is the rest-mass density, 𝑝 the pressure, 𝑏𝜇 the magnetic field measured by a comoving observer, 𝑢𝜇 is the 4-velocity, ℎ ≡ 1+𝜖+𝑝/𝜌 the specific enthalpy, 𝜖 the specific internal energy and 𝑔𝜇𝜈 are the contravariant components of the 4-metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The equations for this matter field are those of the general rela- tivistic magnetohydrodynamics (GRMHD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' These can be written as a flux conservative system that assumes a standard 3+1 decomposi- tion of the space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The space-time metric described in Cartesian coordinates (𝑡, 𝑥𝑖) is given by 𝑑𝑠2 = (−𝛼2 + 𝛽𝑖𝛽𝑖)𝑑𝑡2 + 2𝛽𝑖𝑑𝑡𝑑𝑥𝑖 + 𝛾𝑖 𝑗𝑑𝑥𝑖𝑑𝑥 𝑗, where 𝛼 is the lapse function and 𝛽𝑖 the components of the shift vector associated to the 3+1 decomposition of the space-time, and 𝛾𝑖 𝑗 are the components of the 3-metric of spatial hypersurfaces used to foliate the space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In these terms, the GRMHD equations according to the Valencia formulation are written as (Banyuls et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 1997): 𝜕𝑡u + 𝜕𝑥𝑖F(𝑖) (u) = S(u), (2) where the vector u = {𝐷, 𝑆𝑖, 𝜏, 𝐵𝑘} contains the following conserved variables, 𝐷 the generalized rest mass density of the fluid, 𝑆𝑖 the momentum components along in each direction, 𝜏 the internal energy, 𝐵𝑘 the magnetic field measured by an eulerian observer, F(𝑖) (u) the fluxes and S(u) a sources vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In terms of the primitive variables of the fluid elements, the conserved variables are defined by 𝐷 = √𝛾𝜌𝑊 𝑆𝑖 = √𝛾[(𝜌ℎ + 𝑏2)𝑊2𝑣 𝑗 − 𝛼𝑏0𝑏 𝑗] 𝜏 = √𝛾[(𝜌ℎ+𝑏2)𝑊2 − � 𝑝 + 𝑏2 2 � − 𝛼2(𝑏0)2 − 𝜌𝑊] 𝐵𝑖 = √𝛾𝑊 � 𝑏𝑖 − 𝛼𝑏0 � 𝑣𝑖 − 𝛽𝑖 𝛼 �� , where 𝛾 is the determinant of the spatial 3-metric 𝛾𝑖 𝑗 of the three- dimensional spatial slices which foliate the space-time, 𝑊 = (1 − 𝛾𝑖 𝑗𝑣𝑖𝑣 𝑗)−1/2 is the Lorentz factor and 𝑏0 = 𝑊𝐵𝑖𝑣𝑖/𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' As usual, the system of equations is closed with an equation of state specified below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In terms of primitive and conservative variables the fluxes and sources are F𝑖 (u) = ������� � (𝛼𝑣𝑖 − 𝛽𝑖)𝐷 (𝛼𝑣𝑖 − 𝛽𝑖)𝑆𝑗 + 𝛼√𝛾 � 𝑝 + 𝑏2 2 � 𝛿𝑖 𝑗 − 𝛼√𝛾𝑏𝑗 𝐵𝑖/𝑊 �𝛼𝑣𝑖 − 𝛽𝑖� 𝜏 + 𝛼√𝛾 � 𝑝 + 𝑏2 2 � 𝑣𝑖 − 𝛼2√𝛾𝑏0𝐵𝑖/𝑊 �𝛼𝑣𝑖 − 𝛽𝑖� 𝐵𝑘 − � 𝛼𝑣𝑘 − 𝛽𝑘� 𝐵𝑖 ������� � , S(u) = ���� � 0 𝑇 𝜇𝜈 �𝜕𝜇𝑔𝜈 𝑗 + Γ𝛿𝜇𝜈𝑔𝛿 𝑗 � 𝛼�𝑇 𝜇0𝜕𝜇 ln 𝛼 − 𝑇 𝜇𝜈Γ0𝜇𝜈 � 0 ���� � , (3) where 𝑔𝜇𝜈 are the covariant components of the 4-metric and Γ𝛿 𝜇𝜈 the Christoffel symbols of the space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We solve the system of equations (2-3) using the publically available GRHydro thorn (Mösta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2014), within the Cactus Einstein Toolkit (ETK) code (Löffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We use the high resolution shock capturing methods pro- vided to solve the GRMHD equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Specifically our simulations use the HLLE numerical flux formula and the minmod reconstructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In order to preserve the magnetic field divergence near to zero, we use the constraint transport method (Balsara & Spicer 1999) imple- mented within the GRHydro thorn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' For the integration in time we use a fourth order Runge-Kutta method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' What we added to the ETK is a module that applies appropriate boundary conditions in the upstream boundary, which is the part of the boundary from which we inject the wind into the domain, where we set the density and velocity field to their initial values during the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' On the other hand we im- plement out-flux boundary conditions in the downstream boundary, which is the part of the boundary through which the wind is expected to leave the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' These conditions are a key ingredient in the accretion of winds in numerical domains that contain the accretion sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In order to avoid divergences of the variables, the GRHydro thorn implements an atmosphere that is triggered during the primi- tive variables calculation for tiny or negative values of the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' For that we use a floor density value of 10−12 in code units, and then the pressure and internal energy are set to consistent values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In our results the density never approaches such small values, including the cases where rarefaction spots are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' (20XX) MHD winds 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='2 Description of space-time and wind We describe the space-time metric of the black hole of mass 𝑀 and spin S = 𝑎ˆ𝑧 using Kerr-Schild (KS) horizon penetrating coordinates: 𝑑𝑠2 = � 𝜂𝜇𝜈 + 2𝑀𝑟3 𝑟4 + 𝑎2𝑧2 𝑙𝜇𝑙𝜈 � 𝑑𝑥𝜇𝑑𝑥𝜈, 𝑙𝜇 = � 1, 𝑟𝑥 + 𝑎𝑦 𝑟2 + 𝑎2 , 𝑟𝑦 − 𝑎𝑥 𝑟2 + 𝑎2 , 𝑧 𝑟 � , 𝑟 = � � � 𝑟2∗ − 𝑎2 + √︃ (𝑟2∗ − 𝑎2)2 + 4𝑎2𝑧2 2 , 𝑟∗ = √︃ 𝑥2 + 𝑦2 + 𝑧2, (4) where 𝜂𝜇𝜈 is the flat metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The wind is set initially as a spatially constant rest mass density ideal gas 𝜌 = 1 × 10−6 [1/𝑀2], moving toward the black hole at a given asymptotic supersonic velocity 𝑣2∞ = 𝑣𝑖𝑣𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We assume the fluid obeys a gamma-law equation of state 𝑝 = (Γ − 1)𝜌𝜖, and consider a relativistic fluid with adiabatic index Γ = 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We use the asymptotic speed of sound 𝑐𝑠∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='05 in order to have a sufficiently slow wind but still supersonic and to compare with previous hydrodynamical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The initial fluid pressure is written as 𝑝ini = 𝑐2s∞𝜌ini/(Γ − 𝑐2s∞Γ1), where Γ1 = Γ/(Γ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In order to avoid negative and zero values of the pressure we choose the sound speed such that 𝑐s∞ < √ Γ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Finally, the initial specific internal energy 𝜖 is reconstructed using the equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The magnetic field is defined initially to be constant and parallel to the spin of the black hole B = 𝐵0 ˆ𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We choose two representative values of magnetic field strength 𝐵0,𝑠𝑡𝑟𝑜𝑛𝑔 = 1 × 10−5 [1/𝑀] and 𝐵0,𝑤𝑒𝑎𝑘 = 1 × 10−10 [1/𝑀].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' After the initial time the magnetic field evolves according to the GRMHD equations and responds to the evolution of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We focus on three representative scenarios, in which the wind is assumed to have different direction with respect to the black hole spin S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' These three different orientations of the wind are represented by ↑ ← ↗ in our tables and correspond to directions ˆ𝑧, ˆ𝑥 and ˆ𝑥 + ˆ𝑧 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The black hole spin is denoted by ⇑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We also use the excision method inside the black hole horizon in order to avoid the variables to interact with the black hole’s singu- larity (Hawke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We performed all of our evolution runs using an isotropic cubic grid Δ𝑥 = Δ𝑦 = Δ𝑧 with base resolution Δ𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5𝑀 and one refinement level with resolution Δ𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='25𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The base domain is set to [−30𝑀, 30𝑀]3 and is approximately twice as big as a sphere of the accretion radius 𝑟𝑎𝑐𝑐 = 𝑀/(𝑐2𝑠∞ +𝑣2∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This is important because in case the domain is smaller than a sphere with radius equal to the accretion radius, the flow could enter the wind regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Continuing with the set up of the initial configuration, another important property is the size scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This is usually set by the accretion radius defined by 𝑟𝑎𝑐𝑐 = 𝑟ℎ𝑜𝑟/(𝑣2∞ + 𝑐2𝑠∞), where 𝑟ℎ𝑜𝑟 is the radius of an accretor, in our case the horizon radius of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The accretion radius has the information of how fast or slow the wind is and it is important because as studied in (Foglizzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2005), the parameter that could trigger a possible flip-flop instability in the Bondi-Hoyle process, is the relative size of the accretor with respect to the accretion radius 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 𝑣2∞ + 𝑐2𝑠∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Sometimes this is called the accretor size (Foglizzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2005), but it is actually a size relative to the accretion radius that indicates how fast or slow the wind is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In our study we use two wind velocities corresponding to a Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Snapshot on the 𝑦 = 0 plane of the density at time 𝑡 = 1000𝑀 when the accretion is already stationary, for cases HD1 and MHD1 models ↑⇑ and 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='065 shown in the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We show the magnetic field lines in the MHD1 case indicating how they bend toward the cone and after that continue to their asymptotic vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the bottom we show the 𝛽 parameter and the shock cone superposed with a field of vectors indicating the Lorentz force direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The shock cone in the presence of magnetic field is slightly wider than in the purely HD case and in the MHD case the maximum of the gas density density (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='3 × 10−4) is smaller than the HD case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='1 × 10−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' slow case 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='065 and a fast case 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='25 with asymptotic velocity 𝑣∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='25𝑐 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5𝑐 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Finally, we set the dimensionless spin of the black hole to S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='8ˆ𝑧, which is comparable to the one estimated by numerical simulations of the QSO 3C186 quasar’s kicked black hole (Lousto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 3 RESULTS With the above set of physical parameters, including two wind ve- locities, two magnetic filed strengths, we performed a series of sim- ulations for the wind orientations summarized in Table 1, where we include the purely Hydrodynamical counterparts in order to compare the impact of the magnetic field on the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The simulations start with the constant values of the wind variables except in the excision region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' During a transient stage the fluid and magnetic field interact until they approach nearly stationary configurations, where the bow shock if any and the shock cone are formed, and in the MHD cases the magnetic field also stabilizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This stationary stage is the one we illustrate in the results discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The general properties of the morphology and dynamics of the process once the evolution of the fluid has settled down to a nearly stationary regime are presented in Figures 1, 2, 3 and 4, in geomet- ric units, corresponding to the four first models in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In these figures we show the rest mass density of the gas in the purely hydro- dynamical scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' the rest mass density of the plasma in the MHD cases with the magnetic field lines superposed, indicating the distor- tion due to the presence of the shock-cone, we remind that initially the magnetic field lines are parallel to the 𝑧 axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' we also show the value of 𝛽 = 2𝑝 𝑏𝑖𝑏𝑖 , which reveals that there is a region 𝛽 < 1 where the magnetic pressure dominates over the hydrodynamical pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' finally we show the Lorentz force field, indicating the direction in which the plasma is being affected by this force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Similarly to the purely hydrodynamical process, where MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' (20XX) 30-302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='145e - 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/MHD1MHDI30-302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='145e - 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M1230-30W/ XZ/MMHDI0MHDI30-302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='145e - 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M4 Gracia-Linares and & Guzmán Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the top we show a snapshot on the 𝑦 = 0 plane of the density at 𝑡 = 1000𝑀, for the HD2 and MHD2 cases with orientation ↗⇑ and 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The first important difference between the HD and MHD is that in the later, the shock-cone is detached from the black hole, that is, the black hole is contained within the high density zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The second is that the shock cone in the presence of the magnetic field is wider than in the pure HD case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Finally the third one is that in the MHD case there is a low density region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the MHD2 case we superpose the magnetic field lines, which show asymptotically the vertical direction, however shows a complex structure at the boundary of the shock cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The structure of field lines would be interesting to see in detail within the context of resistive MHD, because there are some candidates to X-points that eventually could trigger magnetic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the bottom we show a plot of the 𝛽 parameter, which shows that the magnetic field dominates in the region of the shock-cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the final plot we superpose the density of the plasma with a field of arrows indicating the direction of the Lorentz force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' It can be seen that the arrows precisely indicate that the Lorentz force pulls the plasma out of the rarified zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' no Flip-Flop (FF) kind of instability was found, even within the regime where it was predicted in the Newtonian theory (Gracia- Linares & Guzmán 2015), in the case of MHD there was not such instability either, even though some different dynamical features were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The bow shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Another important dependence of the orientation of the wind is the bow shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2 we show that for the MHD case the bow shock is detached from the black hole surface, a condition that usually triggers the shock cone instability in Newtonian systems (Foglizzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2005) but has been shown to be inoffensive in the relativistic case (Gracia-Linares & Guzmán 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Shock cone angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' As reported in (Penner 2011) for the axisym- metric case, the open angle of the shock cone is bigger for the MHD than for the purely HD scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We confirm this result in the ax- isymmetric cases and also in the full non-symmetric diagonal ↗⇑ and horizontal ⇑← cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Effects on the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We also study the effects of the initially uniform magnetic field, due to the shock cone formation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The fact that the plasma piles up in a high density cone- shaped region, indicates that some important effects may happen, namely the magnitude of the magnetic field is expected to change and the resulting currents from the process of formation will promote Lorentz forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' First we diagnose the magnetic field amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 5 we show the magnetic field strength as a function of time, measured at points where the plasma 𝛽 = 2𝑝 𝑏𝑖𝑏𝑖 < 1, because there the magnetic field dominates over the hydrodynamical pressure, for the cases MHD1, MHD2, MHD3 and MHD4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In all these scenarios the highest amplification occurs at points near the black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Notice that in the MHD2, MHD3 and MHD4 cases the magnetic field increases approximately between one and two orders of magnitude and in the case MHD1 the amplification is of one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Rarefaction regions and Lorentz force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' A second interesting im- plication of the distortion of the magnetic field is the formation of rarified zones not seen in the purely hydrodynamical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Coinci- dentally, as shown for the cases MHD2, MHD3 and MHD4 in Figures 2, 3, 4, these low density zones develop precisely where the mag- netic pressure dominates over the fluid pressure 𝛽 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In order to investigate the possible reason for this we tracked the Lorentz force 𝜖𝑖 𝑗𝑘𝐽 𝑗 𝐵𝑘 where 𝐽 𝑗 is the current density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In these Figures we show a snapshot at time 𝑡 = 1000𝑀 (after the process has become stationary) of the rest mass density for the HD and MHD cases for comparison, and show the direction in which this Lorentz force acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We observe that in the low density spots the direction of the Lorentz force points in the appropriate direction as to move the plasma outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This is an indication that -even if an approximation- this force is the re- sponsible for the formation of these spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' These rarefaction zones appear in different places depending on the direction of the wind, for example in the diagonal MHD2 case the rarified zone is bigger in the top half of the shock cone but in the horizontal MHD3 case the zone is symmetric with respect to the 𝑧 = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Formation of eddies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Another interesting effect is the formation of eddies in the shock cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 6 we present the formation of eddies in two general cases, one is the diagonal case MHD2 ↗⇑, and the second one is the horizontal wind MHD3 ⇑←.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' For both cases we show the rest mass density on a plane perpendicular to the wind and near to the black hole, together with its purely hydrodynamical counterpart for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We superpose the velocity field of the plasma in order to have a clear idea of the motion in the low density spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the diagonal case HD2 and MHD2 we present the plot on the plane 𝑥 + 𝑧 = 4, which is perpendicular to the wind in the shock-cone region at a close distance from the black hole’s horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This map is a perpendicular view of that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2 and shows two different rarified spots that do not appear in the HD2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The most interesting part is that the velocity field indicates the plasma is rotating approximately around the center of the big spot, whereas in the purely hydrodynamical case the gas is simply moving toward the center of the shock-cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the horizontal wind case HD3 and MHD3 we show the density and velocity field of the shock-cone also in a plane perpendicular to the direction of the wind at a distance 2𝑀 from the center of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In this case, the symmetry allows to notice two symmetric low density spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This is a perpendicular view from that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In this case it is clear that two eddies are formed in the rarified zones and the plasma is rotating around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Effect of the wind velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' One may wonder whether the wind ve- locity produces significant morphological changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the purely HD regime we found in our previous paper (Gracia-Linares & Guzmán 2015) that the accretion rate of the system and the open angle of the shock depends on the wind velocity, slower winds correspond to wider shock cones and higher accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In this paper we ver- ified the same happens for the magnetized case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' An important new feature appeared, this is the one related to the rarified zone within the shock-cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The density in the rarified zone is one order of mag- nitude smaller in the fast case model MHD4 than in the slow case model MHD1 as can be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Another point to see is the influence of the magnetic field on the accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We measured �𝑀 in a spherical surface MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' (20XX) 30-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='61e - 4HD2MHD22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='05e - 430-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M2MHD230-30W/ XZ/M01MHD22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='05e - 430-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/MMHD winds 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' As in the previous case, we show in the top a snapshot of the rest mass density on the 𝑦 = 0 plane at time 𝑡 = 1000𝑀 for the HD3 and MHD3 models with 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='065 for the case ⇑←.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Again a difference is that in the MHD case the shock-cone is detached from the black hole and the magnetic field lines shows an interesting distortion near the bow-shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Due to the bitant symmetry of this case, there are two symmetric rarified regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the bottom we show that the region 𝛽 < 1 appears again within the shock-cone region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the bottom-right panel we show that the Lorentz force points in the direction in which the plasma should move to produce the low density spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This is the case of a fast wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the top we show a snapshot of the rest mass density on the 𝑦 = 0 plane at time 𝑡 = 1000𝑀, for the HD4 and MHD4 models with 𝑟ℎ𝑜𝑟/𝑟𝑎𝑐𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Since this is a faster wind, in both HD and MHD cases, the shock cone is attached to the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In the bottom we show the parameter 𝛽 which shows the magnetic field dominates in the shock-cone region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In this case there is a central rarified zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' According to the direction of the Lorentz force shown in the last panel, this force is the responsible for the depletion of the low density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The low velocity counterpart of the case is MHD1 in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The density in the rarified zone within the shock-cone of this MHD4 case is one order of magnitude smaller than in the MHD1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This is an example of how the velocity of the wind influences the properties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Parameters of the 8 simulations used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Four of them involve MHD and the other ones are the purely HD equivalent counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In this Table the units of 𝑎 are [1/𝑀2], the velocity 𝑣∞ is in units of 𝑐 = 1 and the units of the magnetic field 𝐵0 are [1/𝑀 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Name 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='8 𝑣∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='25 𝐵0 Orientation MHD1 1 × 10−5 ↑⇑ HD1 0 ↑⇑ MHD2 1 × 10−5 ↗⇑ HD2 0 ↗⇑ MHD3 1 × 10−5 ⇑← HD3 0 ⇑← MHD5 1 × 10−10 ↑⇑ MHD6 1 × 10−10 ↗⇑ MHD7 1 × 10−10 ⇑← Name 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='8 𝑣∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5 𝐵0 Orientation MHD4 1 × 10−5 ↑⇑ HD4 0 ↑⇑ MHD8 1 × 10−10 ↑⇑ Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Time series of the magnetic field strength measured at four different points of the shock cone with a magnetic field 𝐵0 = 𝐵0,𝑠𝑡𝑟𝑜𝑛𝑔 for models MHD1, MHD2, MHD3 and MHD4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' For this we have chosen points located where the magnetic pressure dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' These points can be identified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 1 for the MHD1 case, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2 for MHD2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 3 for MHD3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 4 for MHD4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The coordinates of the points where the magnetic field is measured are indicated in each line for each of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' (20XX) MHD330-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='89e - 42MHD330-30W/ XZ/M01MHD330-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='89e - 4HD45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='81e - 530-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='81e - 5MHD430-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M2MHD430-30W/ XZ/M01MHD4-15151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='81e - 5MHD1 MHD2 401 A= (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5) A= (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5) 301 B=(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0,0,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0) B= (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0,0,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0) 351- C= (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5) C = (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5) 251 D=(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0,0,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0) D=(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0,0,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0) 301- 201 (x10^-5) (x10^-5) 251 151 201 /BI 151 101 101- 51- 51- 0 1002003004005006007008009001000 1002003004005006007008009001000 t/M t/M MHD3 MHD4 301 451 A=(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0,0,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0) A=(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5) B = (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5) B= (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0) 401 C= (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0,0,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0) C=(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5) 251- D=(12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5) 351- D= (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0) 201 301 (x10^-5) (x10^-5) 251 151 B 201- 101- 151- 101- 51- 51- 1+ 0 100 200 300 400 0500 600 700800 9001000 0 100200300 4005006007008009001000 t/M t/MHD330-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='89e - 46 Gracia-Linares and & Guzmán Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Snapshot of the rest mass density 𝜌 and velocity field 𝑣𝑖 for models HD2, MHD2, HD3 and MHD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' For models HD2 and MHD2 we show the results on the plane 𝑥 + 𝑧 = 4, which is a plane perpendicular to the wind direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' For models HD3 and MHD3 we show the results on the plane 𝑥 = −2 which is a plane also perpendicular to the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The velocity field in both cases shows the formation of eddies precisely in the rarified zones of the shock-cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' It is useful to compare the shape and location of the rarified zones with the perpendicular view in Figures 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' located near the black hole’s horizon, at 𝑟 = 2𝑀 after the shock has been formed and the evolution regime becomes stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In Table 2 we show the values of the accretion rate for all the models at a stationary stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We first observe that the horizontal wind (models HD3 and MHD3) has the highest accretion rate for the velocity used here, approximately between 10% to 20% above the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We found that the accretion rate of the MHD as compared with the purely hydrodynamical counterparts is within a 3% of difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This table also shows that the direction of the wind influences more the accretion rate than the fact of having pure HD or MHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Besides the direction of the wind, another factor that influences the accretion rate is the wind velocity which can be seen from the comparison of models MHD1 and MHD4, with significant differences of 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The weak magnetic field case 𝐵0 = 𝐵0,𝑤𝑒𝑎𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Unlike the strong magnetic field case, the general properties for 𝐵0 = 𝐵0,𝑤𝑒𝑎𝑘 (MHD5, MHD6, MHD7 and MHD8) are very similar to the purely hydrodynamical counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In Figures 7 and 8 we show the rest mass density of these models and the respective HD cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Contrary to the strong magnetic field case, we do not observe a noticeable change in the shock cone morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We attribute this behavior to the fact that for this magnetic field the magnetic field pressure is of order 10−20 and the plasma 𝛽 ≫ 1, even after the magnetic field is amplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Thus there is no significant impact on the gas dynamics as the one observed in the strong field cases MHD1, MHD2, MHD3 and MHD4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In this sense this regime works as a correspondence case between MHD and HD, although the structure of magnetic field lines holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Effects on the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In Figure 9 we show the amplification of the magnetic field, which is increased by one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The initial and amplified values of the magnetic field are not sufficient to compete with the gas pressure, and this is the reason why there are not low density spots, first because the plasma 𝛽 is of the order of 1010 and second because the Lorentz force is ten orders of magnitude smaller than in the 𝐵0,𝑠𝑡𝑟𝑜𝑛𝑔 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' As shown in Figures 7 and 8 this magnetic field is too small to produce any relevant change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In general, Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Accretion rate values ( �𝑀) for all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The perpendicular case to the axis of rotation of the black hole is the configuration with the highest �𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Moreover �𝑀 does not change significantly between the MHD and HD cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Model Orientation �𝑀 MHD1 ↑⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='667 × 10−3 HD1 ↑⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='682 × 10−3 MHD2 ↗⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='673 × 10−3 HD2 ↗⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='637 × 10−3 MHD3 →⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='781 × 10−3 HD3 →⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='774 × 10−3 Model Orientation �𝑀 MHD4 ↑⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='315 × 10−3 HD4 ↑⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='333 × 10−3 MHD5 ↑⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='679 × 10−3 MHD6 ↗⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='635 × 10−3 MHD7 →⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='769 × 10−3 MHD8 ↑⇑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='331 × 10−3 even though the magnetic field lines distort, purely hydrodynamics gas evolution rules the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Based on the previous description, for this scenario with a weak magnetic field, it is expected the accretion rate to be even more similar to that of the HD counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We find that the differences in accretion rate are within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Attractor behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The stationarity of the flow, morphology and magnetic field lines distribution is recovered when the wind density is varied with sinusoidal fluctuations of amplitude 10% of 𝜌ini and injected over a nearly arbitrary time window as long as 𝜌ini is recov- ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This indicates that the configurations resist inhomogeneities, which are expected to happen in real scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' More formally, it would be interesting to find a parametrization of the basin of at- traction of these nearly-stationary accretion configurations, and how rapidly they approach stationarity in terms of Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 4 CONCLUSIONS AND DISCUSSION We present the accretion of a supersonic magnetized wind onto a black hole in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' For this we selected a set of different wind orien- tations with respect to the spin of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In all the cases we compared the results with the purely hydrodynamical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We studied two different values of the magnetic field strength, strong and weak cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We found that in the strong field case there are zones within the shock cone in which the magnetic field pressure dominates 𝛽 < 1 and produces low density spots within the shock cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We also show the formation of eddies consisting of plasma rotating around the low density spots within the shock cone near the black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Using an approximate calculation of the Lorentz force, we found that this force can be the responsible for the depletion of the low density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The weak field case on the other hand, was such that the gas pressure dominates over the magnetic field and the plasma 𝛽 ≫ 1, and neither low density spots nor eddies are formed, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' (20XX) 1515HD23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='66e - 4Y/MVx2 + z2 / M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XMHD21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ X-15153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='66e - 4Y/MVx2 + z2 / M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6-1515Y/MHD3Z/M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='89e - 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6-1515Y/MZ/M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='89e - 4MHD3MHD winds 7 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Snapshot of the rest mass density 𝜌 during the stationary regime, for models HD1, HD2, and the MHD counterparts with the weak magnetic field strength 𝐵0 = 𝐵0,𝑤𝑒𝑎𝑘, specifically MHD5 and MHD6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' These models exhibit the distortion of the magnetic field lines likewise in the strong field case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' However we did not find any drastic change in the morphology and magnitude of the rest mass density that could for example trigger the formation of rarified zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Snapshot of the rest mass density 𝜌 during the stationary regime, for models HD3, HD4, and the MHD counterparts for the weak magnetic field 𝐵0 = 𝐵0,𝑤𝑒𝑎𝑘, specifically MHD7 and MHD8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Again, we notice the distortion of the magnetic field lines but the field is not strong enough to produce zones of magnetic field domination that eventually could produce rarified zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' although the magnetic field lines follow a similar pattern as in the strong field case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We find that the accretion rate is not significantly different between the HD and MHD cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Instead, the wind velocity and orientation is more important for the accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We are sure that with the insight obtained with the general sce- narios presented here, the infrastructure developed, specifically the implementation of upstream and downstream boundary conditions, can be applied to astrophysical systems, like wandering black holes of type QSO 3C186 (Chiaberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2017) and supernovae natal kicks like that in (Sahu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The various properties of the process, Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Time series of the magnetic field strength measured at four different points of the shock cone for 𝐵0 = 𝐵0,𝑤𝑒𝑎𝑘 and models MHD5, MHD6, MHD7 and MHD8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' For this we have chosen points located where the magnetic pressure dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' These points can be identified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 7 for the MHD5 and MHD6 cases, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 8 for MHD7 and MHD8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The coordinates where the magnetic field is measured are indicated for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' including the morphology of the shock cone, the formation of rarified zones and eddies are expected to change for different parameters of the wind, including the asymptotic velocity and equation of state of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' It is possible now to construct a catalog of simulations with astrophysical parameters to be contrasted with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We also expect that our set up, applied to specific scenarios will be useful for example in the study of the cases like the high speed black hole prior to merger, during the common envelop face in (Cruz-Osorio & Rezzolla 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Concerning the observability of the processes detailed in this pa- per, a direct observation does not seem to be affordable at the mo- ment, considering the candidates need a higher resolution than those used by the EHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Then it is not expected a direct observation of the shock-cone itself, and differences with the presence of magnetic fields should be even finer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' However it can be expected that the for- mation process of the shock-cone will reveal the intrinsic properties of the system, like black hole velocity and mass, perhaps spin, and the gas density and equation of state as illustrated in (González & Guzmán 2018) for the case of a wandering black hole, along with a correlation of vibrational modes of the shock-cone as indicated in (Lora-Clavijo & Guzmán 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Vibrations, that will depend on each combination of parameters of wind inclination angle and mag- netic field strength, that we showed here that have local fingerprints, should be analyzed as done for the spin-less, hydrodynamical model in (Lora-Clavijo & Guzmán 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The various scenarios, with the appropriate vibration modes would compose the catalog of possible scenarios to be correlated with observations on candidates like QSO 3C186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' On aother case, the binary black hole merger, the first steps have been already explored and signatures are expected together with gravitational wave signals (Cruz-Osorio & Rezzolla 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' (20XX) 30-302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='145e - 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/MHD1HD45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='81e - 530-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M30-302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='145e - 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/MMHD530-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/MMHD61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='61e - 430-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='89e - 4MHD730-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='81e - 5MHD8MHD5 MHD6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0 A=(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5) A= (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5,0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5) 6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='61e - 4HD2HD330-301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='0e - 6W/ XZ/M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='89e - 48 Gracia-Linares and & Guzmán ACKNOWLEDGEMENTS MGL is supported by NSF grant PHY-1550461, PHY-2207780, PHY-2114582, and the Mexican National Council of Science and Technology (CONACyT) CVU 391996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' FSG acknowledges support from grant CIC-UMSNH-4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=', 2015, PASJ, 67, 60 Toropina O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=', Romanova M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=', Lovelace R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=', 2012, MNRAS, 420, 810 Zanotti O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=', Roedig C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=', Rezzolla L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=', Del Zanna L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=', 2011, MNRAS, 417, 2899 APPENDIX A: CONVERGENCE The implementation of the relativistic hydrodynamics GRHydro thorn, both in special and general relativity of the ETK has been tested severely for particular test scenarios (Löffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Mösta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' In order to support the validity of our results we present a self-convergence test for one of the scenarios described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We show the order of convergence 𝑄 for the density along the 𝑧 axis outside of the black hole horizon, where the high density shock-cone forms, at two different times 1000𝑀 and 2000𝑀 for the MHD4 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' For this test we calculated three numerical solutions 𝜌1, 𝜌2 and 𝜌3 using respectively the resolutions Δ𝑥1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5𝑀, Δ𝑥2 = Δ𝑥1/2 and Δ𝑥3 = Δ𝑥2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' We then evaluate 𝑄 = log (𝜌1−𝜌2)−log (𝜌2−𝜌3) log 2 and show its value in Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' The result is that 𝑄 takes values between 1 and 2, which is the expected order of convergence of the high resolution shock capturing methods using the HLLE flux formula, with a linear reconstructor in the presence of shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' This shows that the simulations for the evolution of the wind, and for the parameters in this paper, are carried out within the convergence regime of the numerical methods used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' (20XX) MHD winds 9 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' Order of convergence 𝑄 of the rest mass density measured along the 𝑧 axis, on the downstream zone, at two different times, 𝑡 = 1000𝑀 (top) and 𝑡 = 2000𝑀 (bottom) for the case MHD4 along the shock cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content=' (20XX) 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5 Q 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5 0 4 6 8 10 12 14 16 z/M 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5 Q 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5 80000090 R& 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} +page_content='5 0 4 6 8 10 12 14 16 z/M' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQfFwkF/content/2301.04307v1.pdf'} diff --git a/YNAyT4oBgHgl3EQf9fo9/content/tmp_files/2301.00874v1.pdf.txt b/YNAyT4oBgHgl3EQf9fo9/content/tmp_files/2301.00874v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe50c2bd45722f7237cc40a6425017ed0914f49c --- /dev/null +++ b/YNAyT4oBgHgl3EQf9fo9/content/tmp_files/2301.00874v1.pdf.txt @@ -0,0 +1,2122 @@ +A multiple scattering formulation for elastic wave propagation in space-time +modulated metamaterials +Xingbo Pua, Alessandro Marzania,∗, Antonio Palermoa,∗ +aDepartment of Civil, Chemical, Environmental and Materials Engineering, University of Bologna, 40136 Bologna, Italy +Abstract +Space-time modulation of material parameters offers new possibilities for manipulating elastic wave prop- +agation by exploiting time-reversal symmetry breaking. Here we propose and validate a general framework +based on the multiple scattering theory to model space-time modulated elastic metamaterials, namely elastic +waveguides equipped with modulated resonators. The formulation allows to consider an arbitrary distribution +of resonators with a generic space-time modulation profile and compute the wavefield within and outside the +resonators’ region. Additionally, under appropriate assumptions, the same framework can be exploited to pre- +dict the waveguide dispersion relation. We demonstrate the capabilities of our formulation by revisiting the +dynamics of two representative space-time modulated systems, e.g. the non-reciprocal propagation of (i) flexural +waves along a metabeam and (ii) surface acoustic waves along a metasurface. Given its flexibility, the proposed +method can pave the way towards the design of novel devices able to realize unidirectional transport of elastic +energy for vibration isolation, signal processing and energy harvesting purposes. +Keywords: +Space-time modulation, Non-reciprocity, Metamaterials, Metasurfaces, One-way mode conversion +1. Introduction +In the last decade, the research on active (or activated) materials has fueled the discovery of novel dy- +namic functionalities to design devices for vibrations and waves control [1, 2]. Activated materials are often +characterized by constitutive properties that are modulated in space and time according to an external energy +source. The study of such space-time modulated materials was originally pioneered in optics [3] and, shortly +afterward, extended to acoustics [4] and elasticity [1]. Elastic waves propagating in these space-time varying +media are of particular interest since the modulation can create a directional bias that breaks the time-reversal +symmetry. Breaking reciprocity allows to realize rich and unconventional phenomena, including, but not limited +to, unidirectional wave propagation, adiabatic energy pumping [5, 6], frequency conversion [7]. These effects +can be leveraged to design novel devices such as acoustic rectifiers [8], circulators [9], and topological insulators +[10], which can find applications in acoustic communication, signal processing, energy harvesting and vibration +isolation [11, 12, 13]. +In the context of elastodynamics, space-time modulation can be achieved following two strategies. +The +first one relies on a bias directly introduced in the waveguide, as a modulation of the elastic and/or mass +properties, so to obtain a modulated phononic crystal [14, 15, 16]. +The second option utilizes space-time +modulated mechanical oscillators attached to a non-modulated waveguide [17, 18, 19] to obtain a modulated +elastic metamaterial. Both approaches proved to be technically feasible by a series of experimental works where +∗Corresponding authors +Email addresses: alessandro.marzani@unibo.it (Alessandro Marzani), antonio.palermo6@unibo.it (Antonio Palermo) +arXiv:2301.00874v1 [physics.app-ph] 2 Jan 2023 + +programmable electric components were used to modulate the media/oscillators [20, 21, 22, 7, 23]. Nonetheless, +modulated metamaterials, compared to their phononic counterpart, are easier to realize, since only the resonant +elements need to be modulated, and support non-reciprocal effects at sub-wavelength scales. +Besides the numerous examples of modulated waveguides [24, 25], most of the conducted studies rely on the +use of numerical simulations, typically developed via finite element (FE) or finite difference (FD) algorithms, to +describe the expected non-reciprocal effects. Nonetheless, numerical simulations are always bounded by their +computational cost which inherently limits the development of design and optimization studies. Computation- +ally inexpensive analytical tools for modulated media are thus desirable, not only to reduce the computational +burden but also to gain a deeper understanding of non-reciprocal effects. Currently, analytical formulations for +time-modulated systems are mainly used to predict the dispersion relations of both discrete [26, 27, 28] and +continuous media [15, 17, 20, 21, 22, 18, 19]. +Although knowledge of the dispersion relations provides physical insights into the existence of directional +band gaps, evidence of non-reciprocal phenomena can be found only by computing transient or steady-state +responses across finite modulated systems. To the best of our knowledge, analytical methods for the computation +of wavefields and transmission/reflection coefficients are currently limited to one-dimensional (1D) problems +[29, 30, 20, 16]. Additionally, there is no unified framework that enables the computation of both the dispersion +relation and the wavefield of a generic modulated system. +To fill this gap, we here propose a generalized multiple scattering formulation able to model the dynamic +response of space-time modulated resonators coupled to a generic elastic waveguide. As observed in experiments, +space-time modulated resonators can generate scattered fields at lower and higher harmonics with respect to +the excitation frequency [20]. To capture this response, we first describe the coupling between the vibrating +resonators and the waveguide motion with an ad-hoc impedance operator able to account for the expected +additional harmonics. Then, we compute the scattered fields in the waveguide by means of Green’s functions. +Finally, we set our multiple scattering scheme to couple the incident and scattered fields and compute the +related unknown amplitudes by imposing proper boundary conditions at each resonator base. The proposed +formulation allows us to investigate the dynamic of an arbitrary number N of resonators with an arbitrary +spatial-temporal modulation profile, since all the space-time varying oscillators can be described individually. +Additionally, by introducing appropriate assumptions, the same formulation can be used to derive the related +dispersion equations. +The details of the methodology and its modeling capabilities are discussed in what follows. In particular, in +Section 2 we describe the proposed general multiple scattering formulation for the computation of the wavefield +and the dispersion relation of waveguides coupled with space-time-modulated resonators. In Section 3, we apply +the formulation to model flexural waves in a beam and Rayleigh waves on a substrate, both coupled with an +array of modulated surface resonators. For both scenarios we show the capability of the formulation to predict +non-reciprocal guided waves. Finally, we derive conclusions and outlook of the work in Section 4. +2. Theoretical formulation +2.1. Statement of the problem +We propose a general analytical framework to model a cluster of space-time-modulated oscillators attached +to a given elastic waveguide (Fig. 1). The formulation includes the following three steps: (i) the definition of +2 + +the elastic force exerted on the waveguide by a time-modulated resonator when excited by a base motion; (ii) +the use of Green’s functions to describe the scattered wavefields generated by the resonators in the waveguide; +(iii) the construction of a multiple scattering formulation to couple the waveguide with an arbitrary number of +time-modulated resonators. The approach allows computing the lower- and higher-order scattered harmonics, +generated by the collective response of the time-modulated resonators, and responsible for the non-reciprocal +wave motion in the waveguide. +First, we present the framework in its most general form, i.e. considering a finite array of time-modulated +oscillators with mechanical properties obeying the same modulation period Tm and arbitrarily arranged over the +waveguide surface. Then, we show how to derive the waveguide dispersion relation by introducing appropriate +assumptions, e.g., considering an infinite array of identical resonators regularly arranged along the elastic +support. + +Incidence +Scattering +Fig. 1. Schematics of space-time modulated resonators laying over an elastic waveguide. +2.2. Elastic force of a time-modulated resonator +Let us recall the dynamics of the generic nth resonator attached to the waveguide surface at the location rn +(see Fig. 1). The resonator has a mass mn, damping coefficient cn, and time-modulated spring stiffness kn(t): +kn(t) = kn(t + Tm), +(1) +where Tm is the modulation time period. The governing equation of the nth resonator motion reads: +mn +∂2Wn(t) +∂t2 ++ cn +�∂Wn(t) +∂t +− ∂wn(t) +∂t +� ++ kn(t)[Wn(t) − wn(t)] = 0, +(2) +in which Wn(t) = W(rn, t) denotes the mass vertical displacement while wn(t) = w(rn, t) is the vertical motion +at the resonator base. Accordingly, the point force Fn(t) = F(rn, t) exerted by the resonator onto the waveguide +3 + +surface reads: +Fn(t) = −mn +∂2Wn(t) +∂t2 +. +(3) +Since the modulated stiffness kn(t) in Eq. (1) is time-periodic, we express it in Fourier series form as: +kn(t) = +∞ +� +j=−∞ +ˆk(j) +n eijωmt, +j ∈ Z, +(4) +in which i = √−1 is the imaginary unit, ωm = 2π/Tm is the modulation frequency, and where the Fourier +coefficients are defined as: +ˆk(j) +n += ωm +2π +� +π +ωm +−π +ωm +kn(t)e−ijωmt dt. +(5) +As we will see in the next section, the motion along the waveguide excited by a harmonic (eiωt) incident +field, contains several lower- and higher-order harmonics generated by the scattering of the time-modulated +mechanical resonators. +As a result, the vertical motion at the resonator base, namely the motion at the +waveguide surface, can be written as [29]: +wn(t) = +∞ +� +h=−∞ +ˆw(h) +n ei(ω+hωm)t, +h ∈ Z, +(6) +so that the solution of Eq. (2) is sought in the form [21, 20, 16]: +Wn(t) = +∞ +� +h=−∞ +ˆW (h) +n ei(ω+hωm)t, +h ∈ Z. +(7) +Substituting Eqs. (4), (6) and (7) into Eq. (2), yields: +∞ +� +h=−∞ +[−mn(ω + hωm)2 + icn(ω + hωm)] ˆW (h) +n eihωmt + +∞ +� +h=−∞ +∞ +� +j=−∞ +ˆk(j) +n +ˆW (h) +n ei(j+h)ωmt += +∞ +� +h=−∞ +icn(ω + hωm) ˆw(h) +n eihωmt + +∞ +� +h=−∞ +∞ +� +j=−∞ +ˆk(j) +n ˆw(h) +n ei(j+h)ωmt. +(8) +Exploiting the orthogonality of harmonic functions, we simplify Eq. (8) by multiplying it for ωme−ipωmt/(2π), +and integrating it from −π/ωm to π/ωm, to obtain: +[−mn(ω + pωm)2 + icn(ω + pωm)] ˆW (p) +n ++ +∞ +� +j=−∞ +ˆk(j) +n +ˆW (p−j) +n += icn(ω + pωm) ˆw(p) +n ++ +∞ +� +j=−∞ +ˆk(j) +n ˆw(p−j) +n +, +p ∈ Z. +(9) +By truncating the orders from −P to P, Eq. (9) can be reorganized in matrix form as: +Mn ˆ +Wn = Qn ˆwn, +(10) +4 + +with: +Mn = +� +���������� +ˆm(−P ) +n +ˆk(−1) +n +ˆk(−2) +n +· · · +ˆk(−2P ) +n +ˆk(1) +n +ˆm(−P +1) +n +ˆk(−1) +n +· · · +ˆk(−2P +1) +n +ˆk(2) +n +ˆk(1) +n +ˆm(−P +2) +n +· · · +ˆk(−2P +2) +n +... +... +... +... +... +ˆk(2P ) +n +ˆk(2P −1) +n +ˆk(2P −2) +n +· · · +ˆm(P ) +n +� +���������� +, Qn = +� +���������� +ˆq(−P ) +n +ˆk(−1) +n +ˆk(−2) +n +· · · +ˆk(−2P ) +n +ˆk(1) +n +ˆq(−P +1) +n +ˆk(−1) +n +· · · +ˆk(−2P +1) +n +ˆk(2) +n +ˆk(1) +n +ˆq(−P +2) +n +· · · +ˆk(−2P +2) +n +... +... +... +... +... +ˆk(2P ) +n +ˆk(2P −1) +n +ˆk(2P −2) +n +· · · +ˆq(P ) +n +� +���������� +, +ˆ +Wn = [ ˆW (−P ) +n +, ˆW (−P +1) +n +, · · · , ˆW (P −1) +n +, ˆW (P ) +n +]T , +ˆwn = [ ˆw(−P ) +n +, ˆw(−P +1) +n +, · · · , ˆw(P −1) +n +, ˆw(P ) +n +]T , +(11) +in which ˆm(j) +n += ˆk(0) +n +− mn(ω + jωm)2 + icn(ω + jωm), and ˆq(j) +n += ˆk(0) +n ++ icn(ω + jωm). +The vertical force at the base of the resonator can thus be obtained by substituting Eq. (7) into Eq. (3): +Fn(t) = −mn +∂2 +∂t2 +∞ +� +h=−∞ +ˆW (h) +n ei(ω+hωm)t = mn +∞ +� +h=−∞ +(ω+hωm)2 ˆW (h) +n ei(ω+hωm)t = +∞ +� +h=−∞ +ˆF (h) +n ei(ω+hωm)t, +h ∈ Z, +(12) +where the ˆF (h) +n +coefficients from h = −P to h = P, collected in the vector ˆFn, read: +ˆFn = Dn ˆ +Wn = DnM−1 +n Qn ˆwn =: Zn ˆwn, +(13) +with: +ˆFn = +� +������� +ˆF (−P ) +n +ˆF (−P +1) +n +... +ˆF (P ) +n +� +������� +, Dn = +� +������� +mn(ω − Pωm)2 +0 +· · · +0 +0 +mn[ω + (−P + 1)ωm]2 +· · · +0 +... +... +... +... +0 +0 +· · · +mn(ω + Pωm)2 +� +������� +. +(14) +In Eq. +(13), the matrix Zn is the impedance operator which relates the resonator base motion to the +resonator base force. It can be observed that the force exerted by each modulated resonator on the elastic +substrate comprises multiple harmonics (ω + hωm). In the next section, we discuss how these forces generate +the related multiple scattered wavefields. +2.3. Elastic wave field of a finite cluster of modulated resonators +We now consider an arbitrary distribution of N space-time modulated resonators arranged on top of a +given elastic waveguide. We assume that the resonators have an identical stiffness modulation frequency ωm. +When an incident wave field u0 = [u0, v0, w0] impinges the bases of such resonators, it triggers their vibrations +which, in turn, generate scattered waves in the waveguide. Following a standard multiple scattering description +[31, 32, 33], the total wave field u = (u, v, w) at the generic position r along the waveguide can be expressed as +5 + +the summation of the incident and scattered wave fields of the N resonators: +u(r, t) = u0(r, t) + +N +� +n=1 +Fn(t)Gu(r − rn), +(15a) +v(r, t) = v0(r, t) + +N +� +n=1 +Fn(t)Gv(r − rn), +(15b) +w(r, t) = w0(r, t) + +N +� +n=1 +Fn(t)Gw(r − rn), +(15c) +where Gu, Gv, Gw are the related Green’s functions in terms of displacements along x, y, z. As in the previous +section, we express the displacements of Eqs. (15a), (15b), (15c) accounting for the multiple harmonics: +∞ +� +h=−∞ +ˆu(h)(r)ei(ω+hωm)t = +∞ +� +h=−∞ +ˆu(h) +0 (r)ei(ω+hωm)t + +N +� +n=1 +∞ +� +h=−∞ +ˆF (h) +n +ˆG(h) +u (r − rn, ω + hωm)ei(ω+hωm)t, +h ∈ Z. +(16a) +∞ +� +h=−∞ +ˆv(h)(r)ei(ω+hωm)t = +∞ +� +h=−∞ +ˆv(h) +0 (r)ei(ω+hωm)t + +N +� +n=1 +∞ +� +h=−∞ +ˆF (h) +n +ˆG(h) +v (r − rn, ω + hωm)ei(ω+hωm)t, +h ∈ Z. +(16b) +∞ +� +h=−∞ +ˆw(h)(r)ei(ω+hωm)t = +∞ +� +h=−∞ +ˆw(h) +0 (r)ei(ω+hωm)t + +N +� +n=1 +∞ +� +h=−∞ +ˆF (h) +n +ˆG(h) +w (r − rn, ω + hωm)ei(ω+hωm)t, +h ∈ Z. +(16c) +Truncating the harmonic terms from h = −P to h = P, Eqs. (16a), (16b), (16c) can be rewritten as: +ˆu(r) = ˆu0(r) + +N +� +n=1 +ˆGu(r − rn)ˆFn, +(17a) +ˆv(r) = ˆv0(r) + +N +� +n=1 +ˆGv(r − rn)ˆFn, +(17b) +ˆw(r) = ˆw0(r) + +N +� +n=1 +ˆGw(r − rn)ˆFn, +(17c) +with: +6 + +ˆϕ(r) = +� +������� +ˆϕ(−P )(r) +ˆϕ(−P +1)(r) +... +ˆϕ(P )(r) +� +������� +, ˆϕ0(r) = +� +������� +ˆϕ(−P ) +0 +(r) +ˆϕ(−P +1) +0 +(r) +... +ˆϕ(P ) +0 +(r) +� +������� +, ϕ = u, v, w. +ˆGϕ(r − rn) = +� +������� +ˆG(−P ) +ϕ +(r − rn, ω − Pωm) +0 +· · · +0 +0 +ˆG(−P +1) +ϕ +(r − rn, ω − Pωm + ωm) +· · · +0 +... +... +... +... +0 +0 +· · · +ˆG(P ) +ϕ (r − rn, ω + Pωm) +� +������� +, +and where ˆϕ0 has non zero components only for the incident field ϕ0 = u0, v0, w0: +ˆϕ(j) +0 += +� +� +� +ϕ0 +j = 0 +0 +j ̸= 0 +, +j ∈ [−P, −P + 1, ..., P] +(18) +Note that in Eqs. +(17a), (17b), (17c) the total displacement components ˆu, ˆv, ˆw and the elastic force +coefficients ˆFn are unknown. Nonetheless, following a classical multiple scattering approach, we can obtain the +coefficients ˆFn by setting boundary conditions at resonator bases. In particular, we substitute Eq. (13) into +Eq. (17c) and specify it at the resonator location rm: +Z−1 +m ˆFm = ˆw0(rm) + +N +� +n=1 +ˆGw(rm − rn)ˆFn, +n, m = 1, ..., N. +(19) +Eq. (19) leads to a system of m = N equations that we can recast in matrix form as: +AX = B, +(20) +with: +A = +� +������� +Z−1 +1 +− ˆGw(0) +− ˆGw(r1 − r2) +· · · +− ˆGw(r1 − rN) +− ˆGw(r2 − r1) +Z−1 +2 +− ˆGw(0) +· · · +− ˆGw(r2 − rN) +... +... +... +... +− ˆGw(rN − r1) +− ˆGw(rN − r2) +· · · +Z−1 +N − ˆGw(0) +� +������� +, X = +� +������� +ˆF1 +ˆF2 +... +ˆFN +� +������� +, B = +� +������� +ˆw0(r1) +ˆw0(r2) +... +ˆw0(rN) +� +������� +. +(21) +It follows that for a given incident wave field ˆw0, the vector X of the force amplitudes ˆFn can be computed +as X = A−1B, and the total wave field in the waveguide evaluated by using Eqs. (15a), (15b), (15c). +In addition, we will show in the following subsection that Eq. (20), under appropriate assumptions, allows +to derive the dispersion relation of time-modulated waveguides. +2.4. Dispersion relation +We here consider an infinite array of equally spaced resonators, arranged atop an elastic waveguide (see Fig. +2) at lattice distance a. We restrict our interest to oscillators with identical mass and with spring constant +7 + +modulated in time and space with a wave-like modulation of period Tm = 2π/ωm and wavelength λm = 2π/κm, +whose general form reads: +k(x, t) = k(x + λm, t + Tm). +(22) +As before, we express k(x, t) in a Fourier series form: +k(x, t) = +∞ +� +j=−∞ +ˆk(j)eij(ωmt−κmx), +j ∈ Z, +(23) +where the Fourier coefficients are computed as: +ˆk(j) = κm +2π +ωm +2π +� +π +κm +−π +κm +� +π +ωm +−π +ωm +k(x, t)e−ij(ωmt−κmx) dxdt. +(24) +As discussed in [29, 19], a stable response of the modulated system requires each modulation amplitude +ˆk(j)(j ̸= 0) to be sufficiently small with respect to the static stiffness ˆk(0). Under this assumption, for the +assumed infinite (N → ∞) periodic array of identical resonators, the scattering Eqs. (19) are the same at any +location xm and satisfy: +Z−1ˆF = +N +� +n=−N +ˆGw(xn)ˆF(xn) = +∞ +� +n=−∞ +ˆGw(xn)ˆF(xn) +(25) +where xm has been conveniently set equal to 0. Following the effective medium approach [34, 35, 36], namely +considering the lattice spacing a much smaller than the characteristic wavelength, we approximate the discrete +point force as an average line load. As a result, the total vertical displacement at the generic resonator base +can be computed as: +Z−1ˆF = 1 +a +∞ +� +n=−∞ +� xn+a/2 +xn−a/2 +ˆGw(x)ˆF(x) dx = 1 +a +� ∞ +−∞ +ˆGw(x)ˆF(x) dx. +(26) +Due to the space-time modulation of the resonator properties, we can express the force in Eq. (12) in the +form: +F(x, t) = +∞ +� +h=−∞ +ˆF (h)e−i(κ+hκm)x+i(ω+hωm)t, +h ∈ Z. +(27) +Substituting Eq. (27) into Eq. (26) and truncating the orders from h = −P to h = P we obtain: +8 + +aZ−1 +� +������� +ˆF (−P ) +ˆF (−P +1) +... +ˆF (P ) +� +������� += +� ∞ +−∞ +� +������� +ˆG(−P ) +w +(x, ω − Pωm) +0 +· · · +0 +0 +ˆG(−P +1) +w +(x, ω − Pωm + ωm) +· · · +0 +... +... +... +... +0 +0 +· · · +ˆG(P ) +w (x, ω + Pωm) +� +������� +� +������� +ˆF (−P )e−i(κ−P κm)x +ˆF (−P +1)e−i(κ−P κm+κm)x +... +ˆF (P )e−i(κ+P κm)x +� +������� +dx. +(28) +Some minor algebra yields the following system of homogeneous equations: +� +� +� +� +� +� +� +� +aZ−1 − +� +������� +˜G(κ − Pκm, ω − Pωm) +0 +· · · +0 +0 +˜G(κ − Pκm + κm, ω − Pωm + ωm) +· · · +0 +... +... +... +... +0 +0 +· · · +˜G(κ + Pκm, ω + Pωm) +� +������� +� +� +� +� +� +� +� +� +� +������� +ˆF (−P ) +ˆF (−P +1) +... +ˆF (P ) +� +������� += 0, +(29) +in which ˜G(κ + Pκm, ω + Pωm) is Pth order Green’s function in space-domain which is obtained as: +� ∞ +−∞ +ˆG(P ) +w (x, ω + Pωm)e−i(κ+P κm)x dx = ˜G(κ + Pκm, ω + Pωm). +(30) +Non-trivial solutions of Eq. (29) provide the dispersion relation: +˜C(κ, ω) := +������������ +aZ−1 − +� +������� +˜G(κ − Pκm, ω − Pωm) +0 +· · · +0 +0 +˜G(κ − Pκm + κm, ω − Pωm + ωm) +· · · +0 +... +... +... +... +0 +0 +· · · +˜G(κ + Pκm, ω + Pωm) +� +������� +������������ += 0. +(31) +9 + + +Fig. 2. Schematic of wave propagation in space-time modulated (a) metabeam and (b) metasurface. +3. Examples and applications +To show the potential of our formulation, we consider two space-time-modulated waveguides that have been +thoroughly discussed in previous works [17, 20, 18, 19]. We begin our investigation by considering an Euler +beam coupled with an array of modulated resonators. For this example, we validate our approach against the +results of Transfer Matrix Method (TMM). For the sake of completeness we report in Appendix +A the full +derivation of the adopted TMM [20]. As a second example, we consider a 2D elastic half-space coupled to +modulated resonators. For this configuration, given the absence of closed-form formulations, we compare our +findings vs. those obtained via finite element simulations, as in Ref. [19]. +3.1. Modeling non-reciprocal flexural waves in a space-time modulated beam +We consider an Euler-Bernoulli beam equipped with an array of undamped resonators, see Fig. 2a, modulated +in a wave-like fashion according to the relationship [27, 17, 20, 26]: +k(t, x) = k0 + ka cos(ωmt − κmx), +(32) +where k0 denotes the static stiffness, ka the amplitude of the modulation, ωm the modulation angular frequency, +κm the modulation wavenumber. At any location xn, the modulated stiffness is time-periodic and its non-zero +Fourier coefficients read: +ˆk(0) +n += k0, +ˆk(−1) +n += 1 +2kaeiκmxn, +ˆk(1) +n += 1 +2kae−iκmxn, +(33) +as ˆk(j) +n += 0 for |j| > 1. For the numerical example, we consider the mechanical parameters originally adopted in +Ref. [17], i.e., a resonator mass m0 = ρAa, where ρ is the mass density of the beam and A is the cross-section +area of the beam. Similarly, the modulation frequency is set as ωm = 0.25ω0 and modulation amplitude as +ka = 0.2k0, in which ω0 is the resonance frequency of resonators and k0 = m0ω2 +0 is the unmodulated stiffness; +the modulation wavenumber is κm = 1.25κ0, where κ0 = +4� +k0/(aD), D the bending stiffness of the beam. +10 + +3.1.1. Dispersion relation +According to the Euler–Bernoulli beam theory, the Pth order governing equation under the action of a +harmonic point force can be written as: +D∂4w +∂x4 + ρA∂2w +∂t2 = δ (x) eiωP t, +P ∈ Z. +(34) +We Fourier transform Eq. (34) along the x direction, and obtain the transformed Pth order Green’s function +in space-domain as: +˜G(κP , ωP ) = +1 +Dκ4 +P − ρAω2 +P +, +(35) +where the shifted frequency and wavenumber are defined as: +ωP = ω + Pωm, +κP = κ + Pκm, +P ∈ Z. +(36) +First, by setting P = 0 we get the non-modulated impedance parameter Z from Eq. (13) as: +Z = m0ω2 +0ω2 +ω2 +0 − ω2 . +(37) +Substituting Eqs. (35) and (37) into Eq. (31) we obtain immediately the dispersion relation of a non-modulated +metabeam: +C(κ, ω) := Dκ4 − +� +ρA + m0 +a +1 +1 − ω2/ω2 +0 +� +ω2 = 0. +(38) +This dispersion equation is identical to the one obtained in Refs. [17, 20] and matches the dispersion curve +provided by FE simulations, see Appendix B for details. +In the presence of modulation, scattered waves are expected when the phase matching condition is satisfied, +i.e., C(κ, ω) = C(κP , ωP ) [27]. As an example, in Fig. 3a we show the original (C) and the two shifted dispersion +curves (C±1) for P = ±1, respectively. +The phase matching condition is met at the intersections between +the original curve and the shifted ones, namely at six magenta points of the pairs (A), (B) and (C). The +asymmetric distribution of these intersections suggests the breaking of time-reversal symmetry which, in turn, +leads to direction-dependent phenomena within the metabeam [20]. +We now predict the dispersion properties of the modulated metabeam. To do so, we substitute Eq. (35) +into Eq. (31) by truncating waves to the first order (P = 1), which yields: +˜C(κ, ω) := +��������� +aZ−1 − +� +���� +1/[D(κ − κm)4 − ρA(ω − ωm)2] +0 +0 +0 +1/[Dκ4 − ρAω2] +0 +0 +0 +1/[D(κ + κm)4 − ρA(ω + ωm)2] +� +���� +��������� += 0, +(39) +11 + +with the impedance operator: +Z = m0 +� +���� +(ω − ωm)2 +0 +0 +0 +ω2 +0 +0 +0 +(ω + ωm)2 +� +���� +� +���� +k0 − m0(ω − ωm)2 +0.5ka +0 +0.5ka +k0 − m0ω2 +0.5ka +0 +0.5ka +k0 − m0(ω + ωm)2 +� +���� +−1 � +���� +k0 +0.5ka +0 +0.5ka +k0 +0.5ka +0 +0.5ka +k0 +� +���� . +(40) +We remark that the coupled dispersion relation in Eq. (39) holds only near the above-mentioned intersections +in Fig. 3a [28]. Thus, we compute and plot the coupled dispersion in the range of ±0.1κ and ±0.1ω around each +crossing point, as shown in Fig. 3b (red circular markers). For comparison, we also provide the unmodulated +dispersion curve (solution of Eq. (38)) and its shifted analogs on the same figure. As discussed in Ref. [17], +in the vicinity of pair B no directional band gap is generated, since both modes have positive group velocities. +Conversely, for contra-directional branches such as pairs A and C, the repulsion effect can lead to narrow band +directional gaps, for instance, around A. Within these gaps, waves are hindered only when propagating along +the specific direction (dictated by the sign of the related wavenumber); conversely, they are fully transmitted +when propagating along the opposite direction [27]. +This directional wave-filtering is usually accompanied +by the generation of lower/higher-order waves at the phase-matched frequencies, thus resulting in a reflection +combined with a frequency conversion [17]. Evidence of these effects is provided in the next section where the +steady-state solution of waves propagating along a finite modulated metabeam is computed. +3.1.2. Steady-state solutions +To evidence the non-reciprocal behavior predicted by the dispersion analysis, we utilize Eq. (17c) to compute +the steady-state response of a finite metabeam. In particular, we are interested in verifying the non-reciprocal +reflection/transmission in the directional band gap at pair A in Fig. 3. As an example, an array of 50 resonators +is considered for these investigations. The response is recorded at locations xr and xt, and later used to compute +the reflection and transmission values, respectively. In both scenarios, the harmonic point source eiωt and the +receiver are located at distances of ds = 600a and dr = 300a from the closest oscillator. +According to the formulation discussed in Section 2, the impedance operators Z1 to ZN are obtained from +Eq. (13) while the Pth order Green’s function in Eq. (20) is obtained by applying the inverse Fourier transform +to Eq. (35) as: +ˆG(P ) +w (x, ω + Pωm) = +−1 +4Dβ3 +P +(e−βP |x| + i e−iβP |x|), +(41) +where the Pth order wavenumber for flexural waves reads: +βP = +4 +� +ρA(ω + Pωm)2 +D +. +(42) +Substituting Eq. (41) into Eq. (20) we obtain the elastic force coefficients ˆFn, which are inserted into Eq. +(17c) for the calculation of the displacement components ˆw(x) in the beam. +We begin our investigation considering a right-propagating (κ > 0) flexural wave at frequency ω = 1.66ω0, +i.e., the intersection at pair A in Fig. 3a. The reflection coefficient, normalized with respect to the incident +wave, |wr/w0|, is displayed in Fig. 3c, considering the scattered waves truncated at P = ±5 order. +12 + +(c) +(d) +(a) +(b) +(A) +(B) +(C) +Receiver +Receiver +Fundamental mode +Incident ++3rd +-3rd ++1st ++5th +-5th +-1st +Fundamental mode +Incident ++3rd +-3rd ++1st ++5th +-5th +-1st +Fig. 3. (a) Dispersion curve of a non-modulated metabeam (black dashed lines) and its shifted analogs for P = −1 and P = 1, +respectively. (b) Dispersion curves (circular red markers) of a modulated metabeam in proximity of the phase matching pairs. +Normalized reflection and transmission coefficients for flexural waves propagating at ω = 1.66ω0 inside the directional band gap +(pair A) for (c) a right and (d) a left traveling incident wave (star marker), respectively. For comparison, results obtained by the +transfer matrix method (TMM) are also provided (blue solid lines). +13 + +As expected, right-propagating incident waves at ω = 1.66ω0 (dashed line in Fig. 3c) undergo a strong +reflection with different frequency contents, including the largest component at the first-order harmonic (1.66ω0− +ωm) and non-negligible components at the second (1.66ω0 − 2ωm) and third-order harmonic (1.66ω0 − 3ωm). +The amplitude of other higher-order harmonics is negligible. Conversely, the left-propagating wave (opposite +to the modulation direction) with the same frequency ω = 1.66ω0 can travel through the resonators almost +undisturbed, as shown by the normalized transmission |wt/w0| in Fig. 3d. To verify the predictions provided by +our approach, we compute the same transmission and reflection coefficients using the transfer matrix method. +The results, marked by solid lines in Figs. 3c,d, are in excellent agreement with our analytical solutions (see +more details on the transfer matrix method in Appendix A). +3.2. Modeling non-reciprocal Rayleigh wave propagation in a space-time modulated metasurface +We now consider the propagation of Rayleigh waves across a cluster of modulated resonators. +Such a +problem has been recently investigated with the aid of FE numerical simulations [18, 19]. Our purpose is to +show the capability of the proposed analytical formulation to reproduce both the non-reciprocal dispersion and +the reflection/transmission coefficients in this complex configuration. +For our example, we consider the parameters recently used in Ref. [19]: a half-space with cL/cT = 2, a +resonator with mass ratio m0ω0/(ρacT ) = 0.15, the modulation frequency ωm/ω0 = 0.25, and the modulation +wavenumber κm/κr = 2.5, in which κr = ω0/cT . +3.2.1. Dispersion relation +Let us briefly recall the Green’s function for a 2D isotropic elastic half-space actuated by a harmonic vertical +load acting at the surface. For this configuration, the equilibrium equation can be formulated as a boundary +value problem: +c2 +L∇(∇ · u) − c2 +T ∇ × (∇ × u) = ∂2u +∂t2 , +for z < 0, +(43a) +τzx(x, 0) = 0, +σz(x, 0) = δ(x)eiωP t, +(43b) +in which cL and cT denote the longitudinal (L) and transverse (T) wave velocities, and τzx, σz represent the +shear and normal stresses, respectively; u is the displacement field with components u and w; δ(x) is the Dirac +delta function. +In analogy with the metabeam problem, we Fourier transform the equilibrium Eqs. (43a) and (43b) along +the x direction, and obtain the transformed Pth order Green’s function at z = 0 as: +˜G(κP , ωP ) = +1 +ρc4 +T +ω2 +P +� +κ2 +P − ω2 +P +c2 +L +4κ2 +P +�� +κ2 +P − ω2 +P +c2 +T +� � +κ2 +P − ω2 +P +c2 +L +� +− +� +2κ2 +P − ω2 +P +c2 +T +�2 , +(44) +where ρ is the density of the substrate, and the shifted frequency ωP and wavenumber κP are defined in Eq. +(36). Substituting Eqs. (37) and (44) into Eq. (31) and setting P = 0, we obtain immediately the dispersion +14 + +relation for Rayleigh waves existing in a non-modulated metasurface: +C(κ, ω) := +� +2κ2 − ω2 +c2 +T +�2 +− 4κ2 +�� +κ2 − ω2 +c2 +T +� � +κ2 − ω2 +c2 +L +� +− +m0ω4� +κ2 − ω2 +c2 +L +ρac4 +T (ω2/ω2 +0 − 1) = 0. +(45) +This dispersion equation is identical to the one obtained in Refs. [35, 36], and matches the numerical dispersion +curve computed via FEM, see Appendix B. +As for the metabeam scenario, we first plot the unmodulated C(κ, ω) and the shifted C(κP , ωP ) dispersion +curves for P = ±1, Fig. 4a. Again, phase matching points (e.g., pairs A to E) are found when C(κ, ω) = +C(κP , ωP ) is met. We predict the dispersion properties of the modulated metasurface around these points using +Eq. (31). To this purpose, we substitute Eq. (44) into Eq. (31) and truncate the expansion to the first order, +using the impedance operator Z computed according to Eq. (38). +We display the modulated dispersion relation in the range of ±0.1κ and ±0.1ω around each intersection in +Fig. 4b (red circular markers). As an example, the intersection between contra-directional branches gives rise +to the locking pair C which results in a directional band gap. Harmonic waves propagating with wavenumber- +frequency falling within the directional gap (1.21κr, 1.185ω0) are reflected by the metasurface as a propagating +mode at the phase-matched frequency-wavenumber pair (1.21κr−κm, 1.185ω0−ωm). Conversely, such reflection +by conversion does not occur for waves propagating along the opposite direction at the same frequency 1.185ω0, +confirming the non-reciprocity due to the broken time-reversal symmetry [19]. Again, clear evidence of these +effects predicted by the dispersion curve is provided in the next section by computing the steady-state solutions +of Rayleigh waves propagating along a finite modulated metasurface. +3.2.2. Steady-state solutions +To shown the non-reciprocal Rayleigh wave propagation discussed above, we use Eq. (17c) to compute the +steady-state response of a finite metasurface. The impedance operators Z1 to ZN are computed from Eq. (13), +while the Green’s function in Eq. (20) is obtained by applying the inverse Fourier transform to Eq. (44), +yielding the Pth order wave field (Green’s function) at z = 0: +ˆG(P ) +w (x, 0, ωP ) = +1 +2πρc4 +T +� ∞ +−∞ +ω2 +P +� +κ2 +P − ω2 +P +c2 +L +4κ2 +P +�� +κ2 +P − ω2 +P +c2 +T +� � +κ2 +P − ω2 +P +c2 +L +� +− +� +2κ2 +P − ω2 +P +c2 +T +�2 eiκP x dκP . +(46) +We note that unlike the Green’s function of an Euler beam, Eq. (41), the one for a 2D elastic substrate is +divergent at the origin, Eq. (46). To avoid any convergence issue, we introduce a small footprint of length ℓs +for each resonator so that the associated Green’s functions are [33]: +ˆG(P ) +w (x, z, ωP ) = +1 +πρc2 +T +� ∞ +−∞ +sin(κP ℓs/2) +κP +2κ2 +P βLeβT z − βL(2κ2 +P − ω2 +P +c2 +T )eβLz +4κ2 +P βLβT − +� +2κ2 +P − ω2 +P +c2 +T +�2 +eiκP x dκP , +(47a) +for the vertical displacement components and +ˆG(P ) +u +(x, z, ωP ) = +i +πρc2 +T +� ∞ +−∞ +sin(κP ℓs/2) +2βLβT eβT z − (2κ2 +P − ω2 +P +c2 +T )eβLz +4κ2 +P βLβT − +� +2κ2 +P − ω2 +P +c2 +T +�2 +eiκP x dκP , +(47b) +15 + +for the horizontal ones, where: +βL = +� +κ2 +P − ω2 +P +c2 +L +, +βT = +� +κ2 +P − ω2 +P +c2 +T +. +(48) +Substituting Eq. (47a) into Eq. (20) we obtain the elastic force coefficients ˆFn, which are used in Eqs. +(17a), (17c) to compute the wave field ˆu(x, z), ˆw(x, z) in the substrate. +For our example, we compute the steady-state response at locations (xr, 0) and (xt, 0) on the substrate +surface considering an array of 100 resonators with footprint width ℓs = a/20, where the harmonic point source +and the receiver are located at distances of ds = 600a and dr = 300a from the closest resonator. For a right-going +(κ > 0) incident Rayleigh wave (dashed line) at ω = 1.185ω0 the reflected field, shown in Fig. 4c, confirms a +back-scattering at the coupled frequency ω = 1.185ω0−ωm. Conversely, the left-going (κ < 0) incident Rayleigh +wave (dashed line) at the same frequency ω = 1.185ω0 propagates without reflection or frequency conversion +phenomena (see Fig. 4d). +We now resort to the FEM to verify our analytical solutions. To this purpose, we build a 2D plane-strain +model in a commercial FE software (COMSOL Multiphysics). The reader can find the details of the numerical +model in Appendix +C. Specifically, we compute the transient response of the system actuated by a vertical +tone-burst-shaped force having central frequency ω = 1.185ω0, and analyze the vertical displacement field w (see +the insets of Fig. 4). The corresponding frequency spectra (solid line) computed through the Fourier transform +(FFT) of the record time-domain data at the receiver are displayed in Figs. 4c,d. The reader can appreciate +how the numerical results match the analytical solutions. +Finally, we inspect the steady-state response at the “veering pair” (intersection between two co-directional +branches) [19] E in Fig. 4, where we expect the Rayleigh wave to be transmitted and converted from one +harmonic to another [18, 19]. To evidence such a conversion, we utilize the same model (see Fig. 5) excited by +a right-going incident Rayleigh wave at frequency ω = 0.734ω0. The results obtained with both the analytical +solutions and the FE model are collected in Fig. 5b. As expected, the modulated metasurface can convert the +incident wave (ω = 0.734ω0) into a transmitted wave with a different frequency content, e.g., the phase matched +first-order harmonic at ω = 0.734ω0 + ωm. +To better appreciate this effect, we compute the total wave field +� +ℜ(u)2 + ℜ(w)2, using Eqs. (17a), (17c), +(47a), (47b), in the domain x = [550, 750]a, z = [−150, 0]a. The total wave field, shown in Fig. 5c, can be +decomposed by Eqs. (17a), (17c) into the incident field at ω = 0.734ω0, Fig. 5d, and scattered wave fields: the +fundamental mode at ω = 0.734ω0 in Fig. 5e and the first-order harmonic at ω = 0.734ω0 + ωm in Fig. 5f. +Both scattering fields exhibit a clear asymmetry, with the right-hand side having a greater amplitude than the +left-hand side, a clear feature of the forward scattering behavior at veering pairs of the modulated metasurface. +16 + +(A) +(B) +(C) +(D) +(E) +(a) +(b) +(c) +(d) +Receiver +Receiver +Fundamental mode ++3rd ++1st ++5th +-3rd +-5th +-1st +Incident +Incident +Fundamental mode ++3rd ++1st ++5th +-3rd +-5th +-1st +Fig. 4. Analytical modeling of a metasurface. (a) Dispersion relation of a non-modulated metasurface (black solid lines) and its +shifted analogs for P = −1 and P = 1, respectively. (b) Dispersion relation (circular markers) of a modulated metasurface in the +vicinity of phase matching pairs. Steady-state solutions of Rayleigh wave propagation at ω = 1.185ω0 inside the narrow directional +band gap (pair C) for (c) a right and (d) a left traveling incident wave (star marker), respectively. +17 + +Receiver +disp. +max. +0 ++1st +Incident +Fundamental mode +-3rd +-5th +-1st ++3rd ++5th +(a) +(e) +(b) +(f) +(c) +(d) +Fig. 5. Harmonic responses of a modulated metasurface. (a) Schematic for right-propagating Rayleigh waves. (b) Steady-state +solutions of Rayleigh wave propagation at ω = 0.734ω0 (pair E in Fig. 4). The total wave field, free field, fundamental scattered +field (ω = 0.734ω0), and the first-order scattered field (ω = 0.734ω0 + ωm) excited by the source at ω = 0.734ω0 are shown in (c), +(d), (e) and (f), respectively. +18 + +4. Conclusion +We developed a multiple scattering formulation to model the interaction of a given incident field with a +cluster of space-time-modulated resonators located at the surface of a given elastic waveguide. The effect of +time-varying resonators is modeled by means of impedance operators, able to account for lower- and higher- +order harmonics generated by the modulated oscillators. The vertical motion of resonators, actuated by the +incident field, generates scattered fields in the waveguide, which are characterized via ad-hoc Green’s functions. +The unknown amplitudes of scattered fields are then obtained from a multiple scattering scheme by ensuring +the continuity of displacement at the footprint of resonators. +We have demonstrated the capabilities and accuracy of our framework by computing both the dispersion +relation and wave field of flexural and Rayleigh waves propagating along modulated beams and substrates, +respectively. +Our approach has several advantages compared to currently available methods for studying elastic waves +along space-time-modulated metamaterials. First, it allows to investigate an arbitrary number of resonators +with no restriction on their spatial configuration and modulation profile, apart from their common modulation +period Tm. Second, it enables the analytical treatment of non-reciprocal wave propagation in higher dimensional +systems (2D and 3D), thus overcoming the limitation of currently available analytical methods (e.g, the transfer +matrix method) valid only for 1D wave propagation problems [29]. Third, our method is able to reduce the +computational cost with respect to classical numerical schemes since it does not require the discretization of +the entire system. This feature is particularly appealing for modeling wave propagation in higher dimensional +systems and will prove its value for future design and optimization studies. Fourth, it advances the knowledge +of multiple scattering theory which has demonstrated its superior capabilities in modeling the interaction of +oscillators with elastic flexural and surface acoustic waves [31, 32, 33, 37]. +Overall, we anticipate the proposed formulation will serve as a powerful tool to explore various modulation +profiles on elastic waveguides and to guide future experiments on space-time-modulated systems. Since the +framework developed in this work is general, we also expect an extension into acoustics and electromagnetism, +thus supporting the development of nonreciprocal devices for both acoustics and optics. +CRediT authorship contribution statement +Xingbo Pu: Conceptualization, Methodology, Investigation, Formal analysis, Validation, Software, Writing - +original draft. Antonio Palermo: Conceptualization, Software, Formal analysis, Writing - review & editing, +Supervision. +Alessandro Marzani: Conceptualization, Writing - review & editing, Supervision, Funding +acquisition. +Declaration of competing interest +The authors declare that they have no conflict of interest. +Acknowledgments +This project has received funding from the European Union’s Horizon 2020 research and innovation programme +under the Marie Sk�lodowska Curie grant agreement No 813424. +19 + +Appendix A. Details on the transfer matrix method (TMM) +In this Appendix, we provide the details of the transfer matrix method for the modulated beam (see Fig. +A.1) [20]. According to Euler beam theory, the pth order displacement in the nth cell can be expressed as: +ˆw(p) +n (x) = [eiβ(p)(x−xn), e−iβ(p)(x−xn), eβ(p)(x−xn), e−β(p)(x−xn)][A(p) +n , B(p) +n , C(p) +n , D(p) +n ]T := L(p) +n (x)U(p) +n , +(A.1) +in which β(p) = +4� +ρA(ω + pωm)2/D is the pth order wavenumber. + +Fig. A.1. Schematic of transfer matrix method: (a) the nth cell, (b) the global system. +By truncating the orders from p = −P to p = P, the displacement can be expressed in matrix form +ˆwn(x) = Ln(x)An, with: +ˆwn(x) = +� +������� +ˆw(−P )(x) +ˆw(−P +1)(x) +... +ˆw(P )(x) +� +������� +, Ln(x) = +� +������� +L(−P ) +n +(x) +0 +· · · +0 +0 +L(−P +1) +n +(x) +· · · +0 +... +... +... +... +0 +0 +· · · +L(P ) +n (x) +� +������� +, An = +� +������� +U(−P ) +n +U(−P +1) +n +... +U(P ) +n +� +������� +. +(A.2) +Hence, the vertical force ˆFn in Eq. (13) can be written as: +ˆFn = DnM−1 +n Qn ˆwn(xn) = DnM−1 +n QnLn(xn)An. +(A.3) +For an arbitrary pth order, the continuities of the displacement, slope, bending moment and shear force at +xn yield: +ˆw(p) +n−1(xn) = ˆw(p) +n (xn), +(A.4a) +∂ +∂x ˆw(p) +n−1(xn) = ∂ +∂x ˆw(p) +n (xn), +(A.4b) +20 + +D ∂2 +∂x2 ˆw(p) +n−1(xn) = D ∂2 +∂x2 ˆw(p) +n (xn), +(A.4c) +D ∂3 +∂x3 ˆw(p) +n−1(xn) = D ∂3 +∂x3 ˆw(p) +n (xn) − ˆF (p) +n . +(A.4d) +Substituting Eq. (A.1) into Eqs. (A.4a)-(A.4d) yields: +α(p) +n−1U(p) +n−1 = ζ(p) +n U(p) +n ++ γ(p) +n , +(A.5) +with the coefficients: +α(p) +n−1 = +� +������� +eiβ(p)ℓn +e−iβ(p)ℓn +eβ(p)ℓn +e−β(p)ℓn +ieiβ(p)ℓn +−ie−iβ(p)ℓn +eβ(p)ℓn +−e−β(p)ℓn +−eiβ(p)ℓn +−e−iβ(p)ℓn +eβ(p)ℓn +e−β(p)ℓn +−ieiβ(p)ℓn +ie−iβ(p)ℓn +eβ(p)ℓn +−e−β(p)ℓn +� +������� +, ζ(p) +n += +� +������� +1 +1 +1 +1 +i +−i +1 +−1 +−1 +−1 +1 +1 +−i +i +1 +−1 +� +������� +, γ(p) +n += +� +������� +0 +0 +0 +ˆF (p) +n χ(p) +� +������� +, +(A.6) +where ℓn = xn − xn−1, and χ(p) = −1/[(β(p))3D]. Similarly, by truncating the orders from −P to P, the +displacements can be expressed in matrix form: +αAn−1 = ζAn + γ ˆFn, +(A.7) +with: +α = +� +������� +α(−P ) +n−1 +α(−P +1) +n−1 +... +α(P ) +n−1 +� +������� +, ζ = +� +������� +ζ(−P ) +n +ζ(−P +1) +n +... +ζ(P ) +n +� +������� +, γ = +� +���������������������������������� +0 +0 +· · · +0 +0 +0 +· · · +0 +0 +0 +· · · +0 +χ(−P ) +0 +· · · +0 +0 +0 +· · · +0 +0 +0 +· · · +0 +0 +0 +· · · +0 +0 +χ(−P +1) +· · · +0 +... +... +... +... +0 +0 +· · · +0 +0 +0 +· · · +0 +0 +0 +· · · +0 +0 +0 +· · · +χ(P ) +� +���������������������������������� +(A.8) +Combining Eq. (A.3) and Eq. (A.7) we obtain: +αAn−1 = ζAn + γDnM−1 +n QnLn(xn)An, +(A.9) +21 + +from which we obtain the local transfer matrix relating An−1 to An: +TnAn−1 = An, +(A.10) +where: +Tn = [ζ + γDnM−1 +n QnLn(xn)]−1α. +(A.11) +Therefore, for an infinite beam coupled with N resonators, the global equation is expressed as: +T A0 = AN, +(A.12) +where the global transfer matrix T reads: +T = TNTN−1 · · · T1. +(A.13) +After some algebraic operations, Eq. (A.12) can be further written as: +� +� M11 +M12 +M21 +M22 +� +� +� +� IL +RL +� +� = +� +� TR +IR +� +� , +(A.14) +with coefficients: +M = PT PT , +(A.15a) +IL = [B(−P ) +0 +, D(−P ) +0 +, B(−P +1) +0 +, D(−P +1) +0 +, · · · , B(P ) +0 +, D(P ) +0 +]T , +(A.15b) +RL = [A(−P ) +0 +, C(−P ) +0 +, A(−P +1) +0 +, C(−P +1) +0 +, · · · , A(P ) +0 +, C(P ) +0 +]T , +(A.15c) +TR = [B(−P ) +N +, D(−P ) +N +, B(−P +1) +N +, D(−P +1) +N +, · · · , B(P ) +N , D(P ) +N ]T , +(A.15d) +IR = [A(−P ) +N +, C(−P ) +N +, A(−P +1) +N +, C(−P +1) +N +, · · · , A(P ) +N , C(P ) +N ]T , +(A.15e) +22 + +in which P is an elementary matrix which reads: +P = +� +������������������������� +0 +1 +0 +0 +0 +0 +· · · +0 +0 +0 +0 +0 +1 +0 +0 +· · · +0 +0 +0 +0 +0 +0 +0 +1 +· · · +0 +0 +... +... +... +... +... +... +... +... +... +0 +0 +0 +0 +0 +0 +· · · +0 +1 +1 +0 +0 +0 +0 +0 +· · · +0 +0 +0 +0 +1 +0 +0 +0 +· · · +0 +0 +0 +0 +0 +0 +1 +0 +· · · +0 +0 +... +... +... +... +... +... +... +... +... +0 +0 +0 +0 +0 +0 +· · · +1 +0 +� +������������������������� +. +(A.16) +Eq. (A.14) can be further transformed to: +ψout = Sψin, +(A.17) +where: +ψout = +� +� RL +TR +� +� , +ψin = +� +� IL +IR +� +� , +S = +� +� +−M−1 +22 M21 +M−1 +22 +M11 − M12M−1 +22 M21 +M12M−1 +22 +� +� . +(A.18) +With Eq. (A.17) we can compute both the transmission and reflection coefficients directly. It is worth +mentioning that, due to the presence of exponential amplification terms in α(p) +n−1 in Eq. (A.5), the transfer +matrix method may encounter numerical divergence in some occasions, e.g., when considering a large number +of oscillators or large values of resonators spacing. Such a limitation can be well addressed by the multiple +scattering formulation proposed in this work. +Appendix B. Validation of non-modulated dispersion equation +In this Appendix, we validate the analytical dispersion equation of non-modulated metamaterials via the +finite element method (FEM). To do so, we build 2D FE models (unit cells) using 2D elasticity in COMSOL +Multiphysics. In particular, the Euler beam is modeled by a 2D plane-stress FE model with dimensions a × hb +(Fig. B.1a), while the half-space is modeled by a 2D plane-strain FE model with the height ℓz = 4πcT /ω0 (Fig. +B.1b). To model the linear spring, we use a truss model with the unit cross-sectional area and unit height whose +equivalent Young modulus satisfies Et = m0ω2 +0. Additionally, the resonator mass is modeled by a point mass +model with mass m0. To simulate the dynamics of an infinite array of periodic resonators, we impose a pair +of Floquet periodic boundary conditions on the vertical substrate edges. In Fig. B.1b, a clamped boundary +condition is enforced at the bottom edge to avoid rigid motions. +For the metabeam, the parameters used in this work are set as: mass density ρ = 2700 kg/m3, Young +modulus E = 69 GPa, Poisson ratio ν = 0.33, lattice constant a = 0.04 m, beam thickness hb = 0.002 m, beam +width bw = 0.03 m, resonance frequency of oscillators ω0 = 80π rad/s, and damping coefficient of oscillators +c = 0. For the metasurface, the parameters used are: mass density ρ = 2700 kg/m3, Young modulus E = 69 +23 + +GPa, Poisson ratio ν = 0.33, lattice constant a = 0.3 m, resonance frequency of oscillators ω0 = 200π rad/s, and +damping coefficient of oscillators c = 0. The numerical dispersion curves are obtained by solving the eigenvalue +problem for given wave number varying between k = [0, π/a]. The comparison between the analytical dispersion +curves computed by Eqs. (38), (45) and FE simulations for non-modulated metabeam and metasurface is shown +in Fig. B.1c and Fig. B.1d, respectively. Excellent agreement between them is observed. +Point mass +Truss +(c) +(b) +(a) +(d) +Fig. B.1. Comparison of non-modulated (original) dispersion curves between the analytical solution and the FE solution. (a, b) +Schematic of unit cells used for the FE simulation. The dispersion curves for (c) flexural waves in a metabeam, and (d) Rayleigh +waves in a metasurface. +Appendix C. Details on the FE model for transient simulations +In this Appendix, we provide the details of the 2D plane-strain FE model, implemented in COMSOL +Multiphysics, used to verify our analytical solutions in Section 3.2.2. The FE model consists of an array of +resonators and a substrate with width ℓx = 32λ0 and depth ℓx = 8λ0, where λ0 = 2πcT /ω0 (see Figs. C.1a,b). +As in Appendix B, the resonator is modeled by a point mass m0, while the spring is modeled by a truss element +with unit length and cross-sectional area whose Young modulus reads Et = k0+ka cos(ωmt−κmx). To minimize +reflections from the domain borders, we add low-reflecting boundary conditions around the substrate (denoted +by the dashed lines). The substrate is discretized using a fine mesh (λ0/10) of quadratic serendipity elements, +which allows to obtain convergent results at the frequency of interest. +We perform numerical simulations in the time domain. A narrow tone-burst signal of the form F0(t) = +A0[H(t)−H(t−2πN/ω)] sin(ωt)[1−cos(ωt/N)] is used to generate Rayleigh waves, where H(t) is the Heaviside +function. In the numerical example, the amplitude is set as A0 = 1, the central frequency is ω = 1.185ω0, and +the number of cycles is N = 60. We display the signal and its Fourier spectrum in Figs. C.1c,d. +24 + + +Receiver +(a) +(c) +(d) +Receiver +(b) +Fig. C.1. 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Boechler, J. Eliason, A. Kumar, A. Maznev, K. Nelson, N. Fang, Interaction of a contact resonance of +microspheres with surface acoustic waves, Physical Review Letters 111 (3) (2013) 036103. doi:10.1103/ +PhysRevLett.111.036103. +[36] A. Maznev, On the effective medium model of the interaction of rayleigh waves with mass–spring oscillators +on the surface, Wave Motion 115 (2022) 103074. doi:10.1016/j.wavemoti.2022.103074. +[37] X. Pu, A. Palermo, A. Marzani, Topological edge states of quasiperiodic elastic metasurfaces, Mechanical +Systems and Signal Processing 181 (2022) 109478. doi:10.1016/j.ymssp.2022.109478. +28 + diff --git a/YNAyT4oBgHgl3EQf9fo9/content/tmp_files/load_file.txt b/YNAyT4oBgHgl3EQf9fo9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f28a8e6ec66971a3cac44dea374e05325478ec4c --- /dev/null +++ b/YNAyT4oBgHgl3EQf9fo9/content/tmp_files/load_file.txt @@ -0,0 +1,1056 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf,len=1055 +page_content='A multiple scattering formulation for elastic wave propagation in space-time modulated metamaterials Xingbo Pua, Alessandro Marzania,∗, Antonio Palermoa,∗ aDepartment of Civil, Chemical, Environmental and Materials Engineering, University of Bologna, 40136 Bologna, Italy Abstract Space-time modulation of material parameters offers new possibilities for manipulating elastic wave prop- agation by exploiting time-reversal symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Here we propose and validate a general framework based on the multiple scattering theory to model space-time modulated elastic metamaterials, namely elastic waveguides equipped with modulated resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The formulation allows to consider an arbitrary distribution of resonators with a generic space-time modulation profile and compute the wavefield within and outside the resonators’ region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Additionally, under appropriate assumptions, the same framework can be exploited to pre- dict the waveguide dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We demonstrate the capabilities of our formulation by revisiting the dynamics of two representative space-time modulated systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' the non-reciprocal propagation of (i) flexural waves along a metabeam and (ii) surface acoustic waves along a metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Given its flexibility, the proposed method can pave the way towards the design of novel devices able to realize unidirectional transport of elastic energy for vibration isolation, signal processing and energy harvesting purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Keywords: Space-time modulation, Non-reciprocity, Metamaterials, Metasurfaces, One-way mode conversion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Introduction In the last decade, the research on active (or activated) materials has fueled the discovery of novel dy- namic functionalities to design devices for vibrations and waves control [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Activated materials are often characterized by constitutive properties that are modulated in space and time according to an external energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The study of such space-time modulated materials was originally pioneered in optics [3] and, shortly afterward, extended to acoustics [4] and elasticity [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Elastic waves propagating in these space-time varying media are of particular interest since the modulation can create a directional bias that breaks the time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Breaking reciprocity allows to realize rich and unconventional phenomena, including, but not limited to, unidirectional wave propagation, adiabatic energy pumping [5, 6], frequency conversion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' These effects can be leveraged to design novel devices such as acoustic rectifiers [8], circulators [9], and topological insulators [10], which can find applications in acoustic communication, signal processing, energy harvesting and vibration isolation [11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In the context of elastodynamics, space-time modulation can be achieved following two strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The first one relies on a bias directly introduced in the waveguide, as a modulation of the elastic and/or mass properties, so to obtain a modulated phononic crystal [14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The second option utilizes space-time modulated mechanical oscillators attached to a non-modulated waveguide [17, 18, 19] to obtain a modulated elastic metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Both approaches proved to be technically feasible by a series of experimental works where ∗Corresponding authors Email addresses: alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='marzani@unibo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='it (Alessandro Marzani), antonio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='palermo6@unibo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='it (Antonio Palermo) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='00874v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='app-ph] 2 Jan 2023 programmable electric components were used to modulate the media/oscillators [20, 21, 22, 7, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Nonetheless, modulated metamaterials, compared to their phononic counterpart, are easier to realize, since only the resonant elements need to be modulated, and support non-reciprocal effects at sub-wavelength scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Besides the numerous examples of modulated waveguides [24, 25], most of the conducted studies rely on the use of numerical simulations, typically developed via finite element (FE) or finite difference (FD) algorithms, to describe the expected non-reciprocal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Nonetheless, numerical simulations are always bounded by their computational cost which inherently limits the development of design and optimization studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Computation- ally inexpensive analytical tools for modulated media are thus desirable, not only to reduce the computational burden but also to gain a deeper understanding of non-reciprocal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Currently, analytical formulations for time-modulated systems are mainly used to predict the dispersion relations of both discrete [26, 27, 28] and continuous media [15, 17, 20, 21, 22, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Although knowledge of the dispersion relations provides physical insights into the existence of directional band gaps, evidence of non-reciprocal phenomena can be found only by computing transient or steady-state responses across finite modulated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To the best of our knowledge, analytical methods for the computation of wavefields and transmission/reflection coefficients are currently limited to one-dimensional (1D) problems [29, 30, 20, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Additionally, there is no unified framework that enables the computation of both the dispersion relation and the wavefield of a generic modulated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To fill this gap, we here propose a generalized multiple scattering formulation able to model the dynamic response of space-time modulated resonators coupled to a generic elastic waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As observed in experiments, space-time modulated resonators can generate scattered fields at lower and higher harmonics with respect to the excitation frequency [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To capture this response, we first describe the coupling between the vibrating resonators and the waveguide motion with an ad-hoc impedance operator able to account for the expected additional harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Then, we compute the scattered fields in the waveguide by means of Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Finally, we set our multiple scattering scheme to couple the incident and scattered fields and compute the related unknown amplitudes by imposing proper boundary conditions at each resonator base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The proposed formulation allows us to investigate the dynamic of an arbitrary number N of resonators with an arbitrary spatial-temporal modulation profile, since all the space-time varying oscillators can be described individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Additionally, by introducing appropriate assumptions, the same formulation can be used to derive the related dispersion equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The details of the methodology and its modeling capabilities are discussed in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In particular, in Section 2 we describe the proposed general multiple scattering formulation for the computation of the wavefield and the dispersion relation of waveguides coupled with space-time-modulated resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In Section 3, we apply the formulation to model flexural waves in a beam and Rayleigh waves on a substrate, both coupled with an array of modulated surface resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For both scenarios we show the capability of the formulation to predict non-reciprocal guided waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Finally, we derive conclusions and outlook of the work in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Theoretical formulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Statement of the problem We propose a general analytical framework to model a cluster of space-time-modulated oscillators attached to a given elastic waveguide (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The formulation includes the following three steps: (i) the definition of 2 the elastic force exerted on the waveguide by a time-modulated resonator when excited by a base motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (ii) the use of Green’s functions to describe the scattered wavefields generated by the resonators in the waveguide;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (iii) the construction of a multiple scattering formulation to couple the waveguide with an arbitrary number of time-modulated resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The approach allows computing the lower- and higher-order scattered harmonics, generated by the collective response of the time-modulated resonators, and responsible for the non-reciprocal wave motion in the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' First, we present the framework in its most general form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' considering a finite array of time-modulated oscillators with mechanical properties obeying the same modulation period Tm and arbitrarily arranged over the waveguide surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Then, we show how to derive the waveguide dispersion relation by introducing appropriate assumptions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=', considering an infinite array of identical resonators regularly arranged along the elastic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Incidence Scattering Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Schematics of space-time modulated resonators laying over an elastic waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Elastic force of a time-modulated resonator Let us recall the dynamics of the generic nth resonator attached to the waveguide surface at the location rn (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The resonator has a mass mn, damping coefficient cn, and time-modulated spring stiffness kn(t): kn(t) = kn(t + Tm), (1) where Tm is the modulation time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The governing equation of the nth resonator motion reads: mn ∂2Wn(t) ∂t2 + cn �∂Wn(t) ∂t − ∂wn(t) ∂t � + kn(t)[Wn(t) − wn(t)] = 0, (2) in which Wn(t) = W(rn, t) denotes the mass vertical displacement while wn(t) = w(rn, t) is the vertical motion at the resonator base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Accordingly, the point force Fn(t) = F(rn, t) exerted by the resonator onto the waveguide 3 surface reads: Fn(t) = −mn ∂2Wn(t) ∂t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (3) Since the modulated stiffness kn(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (1) is time-periodic, we express it in Fourier series form as: kn(t) = ∞ � j=−∞ ˆk(j) n eijωmt, j ∈ Z, (4) in which i = √−1 is the imaginary unit, ωm = 2π/Tm is the modulation frequency, and where the Fourier coefficients are defined as: ˆk(j) n = ωm 2π � π ωm −π ωm kn(t)e−ijωmt dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (5) As we will see in the next section, the motion along the waveguide excited by a harmonic (eiωt) incident field, contains several lower- and higher-order harmonics generated by the scattering of the time-modulated mechanical resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As a result, the vertical motion at the resonator base, namely the motion at the waveguide surface, can be written as [29]: wn(t) = ∞ � h=−∞ ˆw(h) n ei(ω+hωm)t, h ∈ Z, (6) so that the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (2) is sought in the form [21, 20, 16]: Wn(t) = ∞ � h=−∞ ˆW (h) n ei(ω+hωm)t, h ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (7) Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (4), (6) and (7) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (2), yields: ∞ � h=−∞ [−mn(ω + hωm)2 + icn(ω + hωm)] ˆW (h) n eihωmt + ∞ � h=−∞ ∞ � j=−∞ ˆk(j) n ˆW (h) n ei(j+h)ωmt = ∞ � h=−∞ icn(ω + hωm) ˆw(h) n eihωmt + ∞ � h=−∞ ∞ � j=−∞ ˆk(j) n ˆw(h) n ei(j+h)ωmt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (8) Exploiting the orthogonality of harmonic functions, we simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (8) by multiplying it for ωme−ipωmt/(2π), and integrating it from −π/ωm to π/ωm, to obtain: [−mn(ω + pωm)2 + icn(ω + pωm)] ˆW (p) n + ∞ � j=−∞ ˆk(j) n ˆW (p−j) n = icn(ω + pωm) ˆw(p) n + ∞ � j=−∞ ˆk(j) n ˆw(p−j) n , p ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (9) By truncating the orders from −P to P, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (9) can be reorganized in matrix form as: Mn ˆ Wn = Qn ˆwn, (10) 4 with: Mn = � ���������� ˆm(−P ) n ˆk(−1) n ˆk(−2) n · · ˆk(−2P ) n ˆk(1) n ˆm(−P +1) n ˆk(−1) n · · ˆk(−2P +1) n ˆk(2) n ˆk(1) n ˆm(−P +2) n · · ˆk(−2P +2) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆk(2P ) n ˆk(2P −1) n ˆk(2P −2) n · · ˆm(P ) n � ���������� , Qn = � ���������� ˆq(−P ) n ˆk(−1) n ˆk(−2) n · · ˆk(−2P ) n ˆk(1) n ˆq(−P +1) n ˆk(−1) n · · ˆk(−2P +1) n ˆk(2) n ˆk(1) n ˆq(−P +2) n · · ˆk(−2P +2) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆk(2P ) n ˆk(2P −1) n ˆk(2P −2) n · · ˆq(P ) n � ���������� , ˆ Wn = [ ˆW (−P ) n , ˆW (−P +1) n , · · · , ˆW (P −1) n , ˆW (P ) n ]T , ˆwn = [ ˆw(−P ) n , ˆw(−P +1) n , · · · , ˆw(P −1) n , ˆw(P ) n ]T , (11) in which ˆm(j) n = ˆk(0) n − mn(ω + jωm)2 + icn(ω + jωm), and ˆq(j) n = ˆk(0) n + icn(ω + jωm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The vertical force at the base of the resonator can thus be obtained by substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (7) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (3): Fn(t) = −mn ∂2 ∂t2 ∞ � h=−∞ ˆW (h) n ei(ω+hωm)t = mn ∞ � h=−∞ (ω+hωm)2 ˆW (h) n ei(ω+hωm)t = ∞ � h=−∞ ˆF (h) n ei(ω+hωm)t, h ∈ Z, (12) where the ˆF (h) n coefficients from h = −P to h = P, collected in the vector ˆFn, read: ˆFn = Dn ˆ Wn = DnM−1 n Qn ˆwn =: Zn ˆwn, (13) with: ˆFn = � ������� ˆF (−P ) n ˆF (−P +1) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆF (P ) n � ������� , Dn = � ������� mn(ω − Pωm)2 0 · · 0 0 mn[ω + (−P + 1)ωm]2 · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 0 0 · · mn(ω + Pωm)2 � ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (14) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (13), the matrix Zn is the impedance operator which relates the resonator base motion to the resonator base force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' It can be observed that the force exerted by each modulated resonator on the elastic substrate comprises multiple harmonics (ω + hωm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In the next section, we discuss how these forces generate the related multiple scattered wavefields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Elastic wave field of a finite cluster of modulated resonators We now consider an arbitrary distribution of N space-time modulated resonators arranged on top of a given elastic waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We assume that the resonators have an identical stiffness modulation frequency ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' When an incident wave field u0 = [u0, v0, w0] impinges the bases of such resonators, it triggers their vibrations which, in turn, generate scattered waves in the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Following a standard multiple scattering description [31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' the total wave field u = (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' w) at the generic position r along the waveguide can be expressed as 5 the summation of the incident and scattered wave fields of the N resonators: u(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' t) = u0(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' t) + N � n=1 Fn(t)Gu(r − rn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (15a) v(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' t) = v0(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' t) + N � n=1 Fn(t)Gv(r − rn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (15b) w(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' t) = w0(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' t) + N � n=1 Fn(t)Gw(r − rn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (15c) where Gu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Gv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Gw are the related Green’s functions in terms of displacements along x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As in the previous section, we express the displacements of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (15a), (15b), (15c) accounting for the multiple harmonics: ∞ � h=−∞ ˆu(h)(r)ei(ω+hωm)t = ∞ � h=−∞ ˆu(h) 0 (r)ei(ω+hωm)t + N � n=1 ∞ � h=−∞ ˆF (h) n ˆG(h) u (r − rn, ω + hωm)ei(ω+hωm)t, h ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (16a) ∞ � h=−∞ ˆv(h)(r)ei(ω+hωm)t = ∞ � h=−∞ ˆv(h) 0 (r)ei(ω+hωm)t + N � n=1 ∞ � h=−∞ ˆF (h) n ˆG(h) v (r − rn, ω + hωm)ei(ω+hωm)t, h ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (16b) ∞ � h=−∞ ˆw(h)(r)ei(ω+hωm)t = ∞ � h=−∞ ˆw(h) 0 (r)ei(ω+hωm)t + N � n=1 ∞ � h=−∞ ˆF (h) n ˆG(h) w (r − rn, ω + hωm)ei(ω+hωm)t, h ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (16c) Truncating the harmonic terms from h = −P to h = P, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (16a), (16b), (16c) can be rewritten as: ˆu(r) = ˆu0(r) + N � n=1 ˆGu(r − rn)ˆFn, (17a) ˆv(r) = ˆv0(r) + N � n=1 ˆGv(r − rn)ˆFn, (17b) ˆw(r) = ˆw0(r) + N � n=1 ˆGw(r − rn)ˆFn, (17c) with: 6 ˆϕ(r) = � ������� ˆϕ(−P )(r) ˆϕ(−P +1)(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆϕ(P )(r) � ������� , ˆϕ0(r) = � ������� ˆϕ(−P ) 0 (r) ˆϕ(−P +1) 0 (r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆϕ(P ) 0 (r) � ������� , ϕ = u, v, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆGϕ(r − rn) = � ������� ˆG(−P ) ϕ (r − rn, ω − Pωm) 0 · · 0 0 ˆG(−P +1) ϕ (r − rn, ω − Pωm + ωm) · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 0 0 · · ˆG(P ) ϕ (r − rn, ω + Pωm) � ������� , and where ˆϕ0 has non zero components only for the incident field ϕ0 = u0, v0, w0: ˆϕ(j) 0 = � � � ϕ0 j = 0 0 j ̸= 0 , j ∈ [−P, −P + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=', P] (18) Note that in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (17a), (17b), (17c) the total displacement components ˆu, ˆv, ˆw and the elastic force coefficients ˆFn are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Nonetheless, following a classical multiple scattering approach, we can obtain the coefficients ˆFn by setting boundary conditions at resonator bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In particular, we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (13) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (17c) and specify it at the resonator location rm: Z−1 m ˆFm = ˆw0(rm) + N � n=1 ˆGw(rm − rn)ˆFn, n, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (19) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (19) leads to a system of m = N equations that we can recast in matrix form as: AX = B, (20) with: A = � ������� Z−1 1 − ˆGw(0) − ˆGw(r1 − r2) · · − ˆGw(r1 − rN) − ˆGw(r2 − r1) Z−1 2 − ˆGw(0) · · − ˆGw(r2 − rN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' − ˆGw(rN − r1) − ˆGw(rN − r2) · · Z−1 N − ˆGw(0) � ������� , X = � ������� ˆF1 ˆF2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆFN � ������� , B = � ������� ˆw0(r1) ˆw0(r2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆw0(rN) � ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (21) It follows that for a given incident wave field ˆw0, the vector X of the force amplitudes ˆFn can be computed as X = A−1B, and the total wave field in the waveguide evaluated by using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (15a), (15b), (15c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In addition, we will show in the following subsection that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (20), under appropriate assumptions, allows to derive the dispersion relation of time-modulated waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Dispersion relation We here consider an infinite array of equally spaced resonators, arranged atop an elastic waveguide (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 2) at lattice distance a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We restrict our interest to oscillators with identical mass and with spring constant 7 modulated in time and space with a wave-like modulation of period Tm = 2π/ωm and wavelength λm = 2π/κm, whose general form reads: k(x, t) = k(x + λm, t + Tm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (22) As before, we express k(x, t) in a Fourier series form: k(x, t) = ∞ � j=−∞ ˆk(j)eij(ωmt−κmx), j ∈ Z, (23) where the Fourier coefficients are computed as: ˆk(j) = κm 2π ωm 2π � π κm −π κm � π ωm −π ωm k(x, t)e−ij(ωmt−κmx) dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (24) As discussed in [29, 19], a stable response of the modulated system requires each modulation amplitude ˆk(j)(j ̸= 0) to be sufficiently small with respect to the static stiffness ˆk(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Under this assumption, for the assumed infinite (N → ∞) periodic array of identical resonators, the scattering Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (19) are the same at any location xm and satisfy: Z−1ˆF = N � n=−N ˆGw(xn)ˆF(xn) = ∞ � n=−∞ ˆGw(xn)ˆF(xn) (25) where xm has been conveniently set equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Following the effective medium approach [34, 35, 36], namely considering the lattice spacing a much smaller than the characteristic wavelength, we approximate the discrete point force as an average line load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As a result, the total vertical displacement at the generic resonator base can be computed as: Z−1ˆF = 1 a ∞ � n=−∞ � xn+a/2 xn−a/2 ˆGw(x)ˆF(x) dx = 1 a � ∞ −∞ ˆGw(x)ˆF(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (26) Due to the space-time modulation of the resonator properties, we can express the force in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (12) in the form: F(x, t) = ∞ � h=−∞ ˆF (h)e−i(κ+hκm)x+i(ω+hωm)t, h ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (27) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (27) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (26) and truncating the orders from h = −P to h = P we obtain: 8 aZ−1 � ������� ˆF (−P ) ˆF (−P +1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆF (P ) � ������� = � ∞ −∞ � ������� ˆG(−P ) w (x, ω − Pωm) 0 · · 0 0 ˆG(−P +1) w (x, ω − Pωm + ωm) · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 0 0 · · ˆG(P ) w (x, ω + Pωm) � ������� � ������� ˆF (−P )e−i(κ−P κm)x ˆF (−P +1)e−i(κ−P κm+κm)x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆF (P )e−i(κ+P κm)x � ������� dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (28) Some minor algebra yields the following system of homogeneous equations: � � � � � � � � aZ−1 − � ������� ˜G(κ − Pκm, ω − Pωm) 0 · · 0 0 ˜G(κ − Pκm + κm, ω − Pωm + ωm) · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 0 0 · · ˜G(κ + Pκm, ω + Pωm) � ������� � � � � � � � � � ������� ˆF (−P ) ˆF (−P +1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆF (P ) � ������� = 0, (29) in which ˜G(κ + Pκm, ω + Pωm) is Pth order Green’s function in space-domain which is obtained as: � ∞ −∞ ˆG(P ) w (x, ω + Pωm)e−i(κ+P κm)x dx = ˜G(κ + Pκm, ω + Pωm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (30) Non-trivial solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (29) provide the dispersion relation: ˜C(κ, ω) := ������������ aZ−1 − � ������� ˜G(κ − Pκm, ω − Pωm) 0 · · 0 0 ˜G(κ − Pκm + κm, ω − Pωm + ωm) · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 0 0 · · ˜G(κ + Pκm, ω + Pωm) � ������� ������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (31) 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Schematic of wave propagation in space-time modulated (a) metabeam and (b) metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Examples and applications To show the potential of our formulation, we consider two space-time-modulated waveguides that have been thoroughly discussed in previous works [17, 20, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We begin our investigation by considering an Euler beam coupled with an array of modulated resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For this example, we validate our approach against the results of Transfer Matrix Method (TMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For the sake of completeness we report in Appendix A the full derivation of the adopted TMM [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As a second example, we consider a 2D elastic half-space coupled to modulated resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For this configuration, given the absence of closed-form formulations, we compare our findings vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' those obtained via finite element simulations, as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Modeling non-reciprocal flexural waves in a space-time modulated beam We consider an Euler-Bernoulli beam equipped with an array of undamped resonators, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 2a, modulated in a wave-like fashion according to the relationship [27, 17, 20, 26]: k(t, x) = k0 + ka cos(ωmt − κmx), (32) where k0 denotes the static stiffness, ka the amplitude of the modulation, ωm the modulation angular frequency, κm the modulation wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' At any location xn, the modulated stiffness is time-periodic and its non-zero Fourier coefficients read: ˆk(0) n = k0, ˆk(−1) n = 1 2kaeiκmxn, ˆk(1) n = 1 2kae−iκmxn, (33) as ˆk(j) n = 0 for |j| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For the numerical example, we consider the mechanical parameters originally adopted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' [17], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=', a resonator mass m0 = ρAa, where ρ is the mass density of the beam and A is the cross-section area of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Similarly, the modulation frequency is set as ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='25ω0 and modulation amplitude as ka = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='2k0, in which ω0 is the resonance frequency of resonators and k0 = m0ω2 0 is the unmodulated stiffness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' the modulation wavenumber is κm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='25κ0, where κ0 = 4� k0/(aD), D the bending stiffness of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Dispersion relation According to the Euler–Bernoulli beam theory, the Pth order governing equation under the action of a harmonic point force can be written as: D∂4w ∂x4 + ρA∂2w ∂t2 = δ (x) eiωP t, P ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (34) We Fourier transform Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (34) along the x direction, and obtain the transformed Pth order Green’s function in space-domain as: ˜G(κP , ωP ) = 1 Dκ4 P − ρAω2 P , (35) where the shifted frequency and wavenumber are defined as: ωP = ω + Pωm, κP = κ + Pκm, P ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (36) First, by setting P = 0 we get the non-modulated impedance parameter Z from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (13) as: Z = m0ω2 0ω2 ω2 0 − ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (37) Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (35) and (37) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (31) we obtain immediately the dispersion relation of a non-modulated metabeam: C(κ, ω) := Dκ4 − � ρA + m0 a 1 1 − ω2/ω2 0 � ω2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (38) This dispersion equation is identical to the one obtained in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' [17, 20] and matches the dispersion curve provided by FE simulations, see Appendix B for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In the presence of modulation, scattered waves are expected when the phase matching condition is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=', C(κ, ω) = C(κP , ωP ) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As an example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3a we show the original (C) and the two shifted dispersion curves (C±1) for P = ±1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The phase matching condition is met at the intersections between the original curve and the shifted ones, namely at six magenta points of the pairs (A), (B) and (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The asymmetric distribution of these intersections suggests the breaking of time-reversal symmetry which, in turn, leads to direction-dependent phenomena within the metabeam [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We now predict the dispersion properties of the modulated metabeam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To do so, we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (35) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (31) by truncating waves to the first order (P = 1), which yields: ˜C(κ, ω) := ��������� aZ−1 − � ���� 1/[D(κ − κm)4 − ρA(ω − ωm)2] 0 0 0 1/[Dκ4 − ρAω2] 0 0 0 1/[D(κ + κm)4 − ρA(ω + ωm)2] � ���� ��������� = 0, (39) 11 with the impedance operator: Z = m0 � ���� (ω − ωm)2 0 0 0 ω2 0 0 0 (ω + ωm)2 � ���� � ���� k0 − m0(ω − ωm)2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='5ka 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='5ka k0 − m0ω2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='5ka 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='5ka k0 − m0(ω + ωm)2 � ���� −1 � ���� k0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='5ka 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='5ka k0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='5ka 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='5ka k0 � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (40) We remark that the coupled dispersion relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (39) holds only near the above-mentioned intersections in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3a [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Thus, we compute and plot the coupled dispersion in the range of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1κ and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1ω around each crossing point, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3b (red circular markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For comparison, we also provide the unmodulated dispersion curve (solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (38)) and its shifted analogs on the same figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' [17], in the vicinity of pair B no directional band gap is generated, since both modes have positive group velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Conversely, for contra-directional branches such as pairs A and C, the repulsion effect can lead to narrow band directional gaps, for instance, around A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Within these gaps, waves are hindered only when propagating along the specific direction (dictated by the sign of the related wavenumber);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' conversely, they are fully transmitted when propagating along the opposite direction [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' This directional wave-filtering is usually accompanied by the generation of lower/higher-order waves at the phase-matched frequencies, thus resulting in a reflection combined with a frequency conversion [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Evidence of these effects is provided in the next section where the steady-state solution of waves propagating along a finite modulated metabeam is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Steady-state solutions To evidence the non-reciprocal behavior predicted by the dispersion analysis, we utilize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (17c) to compute the steady-state response of a finite metabeam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In particular, we are interested in verifying the non-reciprocal reflection/transmission in the directional band gap at pair A in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As an example, an array of 50 resonators is considered for these investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The response is recorded at locations xr and xt, and later used to compute the reflection and transmission values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In both scenarios, the harmonic point source eiωt and the receiver are located at distances of ds = 600a and dr = 300a from the closest oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' According to the formulation discussed in Section 2, the impedance operators Z1 to ZN are obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (13) while the Pth order Green’s function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (20) is obtained by applying the inverse Fourier transform to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (35) as: ˆG(P ) w (x, ω + Pωm) = −1 4Dβ3 P (e−βP |x| + i e−iβP |x|), (41) where the Pth order wavenumber for flexural waves reads: βP = 4 � ρA(ω + Pωm)2 D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (42) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (41) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (20) we obtain the elastic force coefficients ˆFn, which are inserted into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (17c) for the calculation of the displacement components ˆw(x) in the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We begin our investigation considering a right-propagating (κ > 0) flexural wave at frequency ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='66ω0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=', the intersection at pair A in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The reflection coefficient, normalized with respect to the incident wave, |wr/w0|, is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3c, considering the scattered waves truncated at P = ±5 order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 12 (c) (d) (a) (b) (A) (B) (C) Receiver Receiver Fundamental mode Incident +3rd 3rd +1st +5th 5th 1st Fundamental mode Incident +3rd 3rd +1st +5th 5th 1st Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (a) Dispersion curve of a non-modulated metabeam (black dashed lines) and its shifted analogs for P = −1 and P = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (b) Dispersion curves (circular red markers) of a modulated metabeam in proximity of the phase matching pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Normalized reflection and transmission coefficients for flexural waves propagating at ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='66ω0 inside the directional band gap (pair A) for (c) a right and (d) a left traveling incident wave (star marker), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For comparison, results obtained by the transfer matrix method (TMM) are also provided (blue solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 13 As expected, right-propagating incident waves at ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='66ω0 (dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3c) undergo a strong reflection with different frequency contents, including the largest component at the first-order harmonic (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='66ω0− ωm) and non-negligible components at the second (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='66ω0 − 2ωm) and third-order harmonic (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='66ω0 − 3ωm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The amplitude of other higher-order harmonics is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Conversely, the left-propagating wave (opposite to the modulation direction) with the same frequency ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='66ω0 can travel through the resonators almost undisturbed, as shown by the normalized transmission |wt/w0| in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To verify the predictions provided by our approach, we compute the same transmission and reflection coefficients using the transfer matrix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The results, marked by solid lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3c,d, are in excellent agreement with our analytical solutions (see more details on the transfer matrix method in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Modeling non-reciprocal Rayleigh wave propagation in a space-time modulated metasurface We now consider the propagation of Rayleigh waves across a cluster of modulated resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Such a problem has been recently investigated with the aid of FE numerical simulations [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Our purpose is to show the capability of the proposed analytical formulation to reproduce both the non-reciprocal dispersion and the reflection/transmission coefficients in this complex configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For our example, we consider the parameters recently used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' [19]: a half-space with cL/cT = 2, a resonator with mass ratio m0ω0/(ρacT ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='15, the modulation frequency ωm/ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='25, and the modulation wavenumber κm/κr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='5, in which κr = ω0/cT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Dispersion relation Let us briefly recall the Green’s function for a 2D isotropic elastic half-space actuated by a harmonic vertical load acting at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For this configuration, the equilibrium equation can be formulated as a boundary value problem: c2 L∇(∇ · u) − c2 T ∇ × (∇ × u) = ∂2u ∂t2 , for z < 0, (43a) τzx(x, 0) = 0, σz(x, 0) = δ(x)eiωP t, (43b) in which cL and cT denote the longitudinal (L) and transverse (T) wave velocities, and τzx, σz represent the shear and normal stresses, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' u is the displacement field with components u and w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' δ(x) is the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In analogy with the metabeam problem, we Fourier transform the equilibrium Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (43a) and (43b) along the x direction, and obtain the transformed Pth order Green’s function at z = 0 as: ˜G(κP , ωP ) = 1 ρc4 T ω2 P � κ2 P − ω2 P c2 L 4κ2 P �� κ2 P − ω2 P c2 T � � κ2 P − ω2 P c2 L � − � 2κ2 P − ω2 P c2 T �2 , (44) where ρ is the density of the substrate, and the shifted frequency ωP and wavenumber κP are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (37) and (44) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (31) and setting P = 0, we obtain immediately the dispersion 14 relation for Rayleigh waves existing in a non-modulated metasurface: C(κ, ω) := � 2κ2 − ω2 c2 T �2 − 4κ2 �� κ2 − ω2 c2 T � � κ2 − ω2 c2 L � − m0ω4� κ2 − ω2 c2 L ρac4 T (ω2/ω2 0 − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (45) This dispersion equation is identical to the one obtained in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' [35, 36], and matches the numerical dispersion curve computed via FEM, see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As for the metabeam scenario, we first plot the unmodulated C(κ, ω) and the shifted C(κP , ωP ) dispersion curves for P = ±1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Again, phase matching points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=', pairs A to E) are found when C(κ, ω) = C(κP , ωP ) is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We predict the dispersion properties of the modulated metasurface around these points using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To this purpose, we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (44) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (31) and truncate the expansion to the first order, using the impedance operator Z computed according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We display the modulated dispersion relation in the range of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1κ and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1ω around each intersection in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 4b (red circular markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As an example, the intersection between contra-directional branches gives rise to the locking pair C which results in a directional band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Harmonic waves propagating with wavenumber- frequency falling within the directional gap (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='21κr, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='185ω0) are reflected by the metasurface as a propagating mode at the phase-matched frequency-wavenumber pair (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='21κr−κm, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='185ω0−ωm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Conversely, such reflection by conversion does not occur for waves propagating along the opposite direction at the same frequency 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='185ω0, confirming the non-reciprocity due to the broken time-reversal symmetry [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Again, clear evidence of these effects predicted by the dispersion curve is provided in the next section by computing the steady-state solutions of Rayleigh waves propagating along a finite modulated metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Steady-state solutions To shown the non-reciprocal Rayleigh wave propagation discussed above, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (17c) to compute the steady-state response of a finite metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The impedance operators Z1 to ZN are computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (13), while the Green’s function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (20) is obtained by applying the inverse Fourier transform to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (44), yielding the Pth order wave field (Green’s function) at z = 0: ˆG(P ) w (x, 0, ωP ) = 1 2πρc4 T � ∞ −∞ ω2 P � κ2 P − ω2 P c2 L 4κ2 P �� κ2 P − ω2 P c2 T � � κ2 P − ω2 P c2 L � − � 2κ2 P − ω2 P c2 T �2 eiκP x dκP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (46) We note that unlike the Green’s function of an Euler beam, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (41), the one for a 2D elastic substrate is divergent at the origin, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To avoid any convergence issue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' we introduce a small footprint of length ℓs for each resonator so that the associated Green’s functions are [33]: ˆG(P ) w (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ωP ) = 1 πρc2 T � ∞ −∞ sin(κP ℓs/2) κP 2κ2 P βLeβT z − βL(2κ2 P − ω2 P c2 T )eβLz 4κ2 P βLβT − � 2κ2 P − ω2 P c2 T �2 eiκP x dκP ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (47a) for the vertical displacement components and ˆG(P ) u (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ωP ) = i πρc2 T � ∞ −∞ sin(κP ℓs/2) 2βLβT eβT z − (2κ2 P − ω2 P c2 T )eβLz 4κ2 P βLβT − � 2κ2 P − ω2 P c2 T �2 eiκP x dκP ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (47b) 15 for the horizontal ones,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' where: βL = � κ2 P − ω2 P c2 L ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' βT = � κ2 P − ω2 P c2 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (48) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (47a) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (20) we obtain the elastic force coefficients ˆFn, which are used in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (17a), (17c) to compute the wave field ˆu(x, z), ˆw(x, z) in the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For our example, we compute the steady-state response at locations (xr, 0) and (xt, 0) on the substrate surface considering an array of 100 resonators with footprint width ℓs = a/20, where the harmonic point source and the receiver are located at distances of ds = 600a and dr = 300a from the closest resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For a right-going (κ > 0) incident Rayleigh wave (dashed line) at ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='185ω0 the reflected field, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 4c, confirms a back-scattering at the coupled frequency ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='185ω0−ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Conversely, the left-going (κ < 0) incident Rayleigh wave (dashed line) at the same frequency ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='185ω0 propagates without reflection or frequency conversion phenomena (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We now resort to the FEM to verify our analytical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To this purpose, we build a 2D plane-strain model in a commercial FE software (COMSOL Multiphysics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The reader can find the details of the numerical model in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Specifically, we compute the transient response of the system actuated by a vertical tone-burst-shaped force having central frequency ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='185ω0, and analyze the vertical displacement field w (see the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The corresponding frequency spectra (solid line) computed through the Fourier transform (FFT) of the record time-domain data at the receiver are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 4c,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The reader can appreciate how the numerical results match the analytical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Finally, we inspect the steady-state response at the “veering pair” (intersection between two co-directional branches) [19] E in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 4, where we expect the Rayleigh wave to be transmitted and converted from one harmonic to another [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To evidence such a conversion, we utilize the same model (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 5) excited by a right-going incident Rayleigh wave at frequency ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='734ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The results obtained with both the analytical solutions and the FE model are collected in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As expected, the modulated metasurface can convert the incident wave (ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='734ω0) into a transmitted wave with a different frequency content, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=', the phase matched first-order harmonic at ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='734ω0 + ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To better appreciate this effect, we compute the total wave field � ℜ(u)2 + ℜ(w)2, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (17a), (17c), (47a), (47b), in the domain x = [550, 750]a, z = [−150, 0]a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The total wave field, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 5c, can be decomposed by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (17a), (17c) into the incident field at ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='734ω0, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 5d, and scattered wave fields: the fundamental mode at ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='734ω0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 5e and the first-order harmonic at ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='734ω0 + ωm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 5f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Both scattering fields exhibit a clear asymmetry, with the right-hand side having a greater amplitude than the left-hand side, a clear feature of the forward scattering behavior at veering pairs of the modulated metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 16 (A) (B) (C) (D) (E) (a) (b) (c) (d) Receiver Receiver Fundamental mode +3rd +1st +5th 3rd 5th 1st Incident Incident Fundamental mode +3rd +1st +5th 3rd 5th 1st Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Analytical modeling of a metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (a) Dispersion relation of a non-modulated metasurface (black solid lines) and its shifted analogs for P = −1 and P = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (b) Dispersion relation (circular markers) of a modulated metasurface in the vicinity of phase matching pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Steady-state solutions of Rayleigh wave propagation at ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='185ω0 inside the narrow directional band gap (pair C) for (c) a right and (d) a left traveling incident wave (star marker), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 17 Receiver disp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 0 +1st Incident Fundamental mode 3rd 5th 1st +3rd +5th (a) (e) (b) (f) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Harmonic responses of a modulated metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (a) Schematic for right-propagating Rayleigh waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (b) Steady-state solutions of Rayleigh wave propagation at ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='734ω0 (pair E in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The total wave field, free field, fundamental scattered field (ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='734ω0), and the first-order scattered field (ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='734ω0 + ωm) excited by the source at ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='734ω0 are shown in (c), (d), (e) and (f), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Conclusion We developed a multiple scattering formulation to model the interaction of a given incident field with a cluster of space-time-modulated resonators located at the surface of a given elastic waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The effect of time-varying resonators is modeled by means of impedance operators, able to account for lower- and higher- order harmonics generated by the modulated oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The vertical motion of resonators, actuated by the incident field, generates scattered fields in the waveguide, which are characterized via ad-hoc Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The unknown amplitudes of scattered fields are then obtained from a multiple scattering scheme by ensuring the continuity of displacement at the footprint of resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We have demonstrated the capabilities and accuracy of our framework by computing both the dispersion relation and wave field of flexural and Rayleigh waves propagating along modulated beams and substrates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Our approach has several advantages compared to currently available methods for studying elastic waves along space-time-modulated metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' First, it allows to investigate an arbitrary number of resonators with no restriction on their spatial configuration and modulation profile, apart from their common modulation period Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Second, it enables the analytical treatment of non-reciprocal wave propagation in higher dimensional systems (2D and 3D), thus overcoming the limitation of currently available analytical methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='g, the transfer matrix method) valid only for 1D wave propagation problems [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Third, our method is able to reduce the computational cost with respect to classical numerical schemes since it does not require the discretization of the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' This feature is particularly appealing for modeling wave propagation in higher dimensional systems and will prove its value for future design and optimization studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Fourth, it advances the knowledge of multiple scattering theory which has demonstrated its superior capabilities in modeling the interaction of oscillators with elastic flexural and surface acoustic waves [31, 32, 33, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Overall, we anticipate the proposed formulation will serve as a powerful tool to explore various modulation profiles on elastic waveguides and to guide future experiments on space-time-modulated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Since the framework developed in this work is general, we also expect an extension into acoustics and electromagnetism, thus supporting the development of nonreciprocal devices for both acoustics and optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' CRediT authorship contribution statement Xingbo Pu: Conceptualization, Methodology, Investigation, Formal analysis, Validation, Software, Writing - original draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Antonio Palermo: Conceptualization, Software, Formal analysis, Writing - review & editing, Supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Alessandro Marzani: Conceptualization, Writing - review & editing, Supervision, Funding acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Declaration of competing interest The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Acknowledgments This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk�lodowska Curie grant agreement No 813424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 19 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Details on the transfer matrix method (TMM) In this Appendix, we provide the details of the transfer matrix method for the modulated beam (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' According to Euler beam theory, the pth order displacement in the nth cell can be expressed as: ˆw(p) n (x) = [eiβ(p)(x−xn), e−iβ(p)(x−xn), eβ(p)(x−xn), e−β(p)(x−xn)][A(p) n , B(p) n , C(p) n , D(p) n ]T := L(p) n (x)U(p) n , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1) in which β(p) = 4� ρA(ω + pωm)2/D is the pth order wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Schematic of transfer matrix method: (a) the nth cell, (b) the global system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' By truncating the orders from p = −P to p = P, the displacement can be expressed in matrix form ˆwn(x) = Ln(x)An, with: ˆwn(x) = � ������� ˆw(−P )(x) ˆw(−P +1)(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ˆw(P )(x) � ������� , Ln(x) = � ������� L(−P ) n (x) 0 · · 0 0 L(−P +1) n (x) · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 0 0 · · L(P ) n (x) � ������� , An = � ������� U(−P ) n U(−P +1) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' U(P ) n � ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='2) Hence, the vertical force ˆFn in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (13) can be written as: ˆFn = DnM−1 n Qn ˆwn(xn) = DnM−1 n QnLn(xn)An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='3) For an arbitrary pth order, the continuities of the displacement, slope, bending moment and shear force at xn yield: ˆw(p) n−1(xn) = ˆw(p) n (xn), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='4a) ∂ ∂x ˆw(p) n−1(xn) = ∂ ∂x ˆw(p) n (xn), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='4b) 20 D ∂2 ∂x2 ˆw(p) n−1(xn) = D ∂2 ∂x2 ˆw(p) n (xn), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='4c) D ∂3 ∂x3 ˆw(p) n−1(xn) = D ∂3 ∂x3 ˆw(p) n (xn) − ˆF (p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='4d) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='4a)-(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='4d) yields: α(p) n−1U(p) n−1 = ζ(p) n U(p) n + γ(p) n , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='5) with the coefficients: α(p) n−1 = � ������� eiβ(p)ℓn e−iβ(p)ℓn eβ(p)ℓn e−β(p)ℓn ieiβ(p)ℓn −ie−iβ(p)ℓn eβ(p)ℓn −e−β(p)ℓn −eiβ(p)ℓn −e−iβ(p)ℓn eβ(p)ℓn e−β(p)ℓn −ieiβ(p)ℓn ie−iβ(p)ℓn eβ(p)ℓn −e−β(p)ℓn � ������� , ζ(p) n = � ������� 1 1 1 1 i −i 1 −1 −1 −1 1 1 −i i 1 −1 � ������� , γ(p) n = � ������� 0 0 0 ˆF (p) n χ(p) � ������� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='6) where ℓn = xn − xn−1, and χ(p) = −1/[(β(p))3D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Similarly, by truncating the orders from −P to P, the displacements can be expressed in matrix form: αAn−1 = ζAn + γ ˆFn, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='7) with: α = � ������� α(−P ) n−1 α(−P +1) n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' α(P ) n−1 � ������� , ζ = � ������� ζ(−P ) n ζ(−P +1) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' ζ(P ) n � ������� , γ = � ���������������������������������� 0 0 · · 0 0 0 · · 0 0 0 · · 0 χ(−P ) 0 · · 0 0 0 · · 0 0 0 · · 0 0 0 · · 0 0 χ(−P +1) · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 0 0 · · 0 0 0 · · 0 0 0 · · 0 0 0 · · χ(P ) � ���������������������������������� (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='8) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='7) we obtain: αAn−1 = ζAn + γDnM−1 n QnLn(xn)An, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='9) 21 from which we obtain the local transfer matrix relating An−1 to An: TnAn−1 = An, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='10) where: Tn = [ζ + γDnM−1 n QnLn(xn)]−1α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='11) Therefore, for an infinite beam coupled with N resonators, the global equation is expressed as: T A0 = AN, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='12) where the global transfer matrix T reads: T = TNTN−1 · · · T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='13) After some algebraic operations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='12) can be further written as: � � M11 M12 M21 M22 � � � � IL RL � � = � � TR IR � � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='14) with coefficients: M = PT PT , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='15a) IL = [B(−P ) 0 , D(−P ) 0 , B(−P +1) 0 , D(−P +1) 0 , · · · , B(P ) 0 , D(P ) 0 ]T , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='15b) RL = [A(−P ) 0 , C(−P ) 0 , A(−P +1) 0 , C(−P +1) 0 , · · · , A(P ) 0 , C(P ) 0 ]T , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='15c) TR = [B(−P ) N , D(−P ) N , B(−P +1) N , D(−P +1) N , · · · , B(P ) N , D(P ) N ]T , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='15d) IR = [A(−P ) N , C(−P ) N , A(−P +1) N , C(−P +1) N , · · · , A(P ) N , C(P ) N ]T , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='15e) 22 in which P is an elementary matrix which reads: P = � ������������������������� 0 1 0 0 0 0 · · 0 0 0 0 0 1 0 0 · · 0 0 0 0 0 0 0 1 · · 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 0 0 0 0 0 0 · · 0 1 1 0 0 0 0 0 · · 0 0 0 0 1 0 0 0 · · 0 0 0 0 0 0 1 0 · · 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 0 0 0 0 0 0 · · 1 0 � ������������������������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='16) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='14) can be further transformed to: ψout = Sψin, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='17) where: ψout = � � RL TR � � , ψin = � � IL IR � � , S = � � −M−1 22 M21 M−1 22 M11 − M12M−1 22 M21 M12M−1 22 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='18) With Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='17) we can compute both the transmission and reflection coefficients directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' It is worth mentioning that, due to the presence of exponential amplification terms in α(p) n−1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='5), the transfer matrix method may encounter numerical divergence in some occasions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=', when considering a large number of oscillators or large values of resonators spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Such a limitation can be well addressed by the multiple scattering formulation proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Validation of non-modulated dispersion equation In this Appendix, we validate the analytical dispersion equation of non-modulated metamaterials via the finite element method (FEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To do so, we build 2D FE models (unit cells) using 2D elasticity in COMSOL Multiphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In particular, the Euler beam is modeled by a 2D plane-stress FE model with dimensions a × hb (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1a), while the half-space is modeled by a 2D plane-strain FE model with the height ℓz = 4πcT /ω0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To model the linear spring, we use a truss model with the unit cross-sectional area and unit height whose equivalent Young modulus satisfies Et = m0ω2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Additionally, the resonator mass is modeled by a point mass model with mass m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To simulate the dynamics of an infinite array of periodic resonators, we impose a pair of Floquet periodic boundary conditions on the vertical substrate edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1b, a clamped boundary condition is enforced at the bottom edge to avoid rigid motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For the metabeam, the parameters used in this work are set as: mass density ρ = 2700 kg/m3, Young modulus E = 69 GPa, Poisson ratio ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='33, lattice constant a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='04 m, beam thickness hb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='002 m, beam width bw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='03 m, resonance frequency of oscillators ω0 = 80π rad/s, and damping coefficient of oscillators c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' For the metasurface, the parameters used are: mass density ρ = 2700 kg/m3, Young modulus E = 69 23 GPa, Poisson ratio ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='33, lattice constant a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='3 m, resonance frequency of oscillators ω0 = 200π rad/s, and damping coefficient of oscillators c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The numerical dispersion curves are obtained by solving the eigenvalue problem for given wave number varying between k = [0, π/a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The comparison between the analytical dispersion curves computed by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (38), (45) and FE simulations for non-modulated metabeam and metasurface is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Excellent agreement between them is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Point mass Truss (c) (b) (a) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Comparison of non-modulated (original) dispersion curves between the analytical solution and the FE solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' (a, b) Schematic of unit cells used for the FE simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The dispersion curves for (c) flexural waves in a metabeam, and (d) Rayleigh waves in a metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' Details on the FE model for transient simulations In this Appendix, we provide the details of the 2D plane-strain FE model, implemented in COMSOL Multiphysics, used to verify our analytical solutions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The FE model consists of an array of resonators and a substrate with width ℓx = 32λ0 and depth ℓx = 8λ0, where λ0 = 2πcT /ω0 (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' As in Appendix B, the resonator is modeled by a point mass m0, while the spring is modeled by a truss element with unit length and cross-sectional area whose Young modulus reads Et = k0+ka cos(ωmt−κmx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' To minimize reflections from the domain borders, we add low-reflecting boundary conditions around the substrate (denoted by the dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' The substrate is discretized using a fine mesh (λ0/10) of quadratic serendipity elements, which allows to obtain convergent results at the frequency of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We perform numerical simulations in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' A narrow tone-burst signal of the form F0(t) = A0[H(t)−H(t−2πN/ω)] sin(ωt)[1−cos(ωt/N)] is used to generate Rayleigh waves, where H(t) is the Heaviside function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' In the numerical example, the amplitude is set as A0 = 1, the central frequency is ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='185ω0, and the number of cycles is N = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' We display the signal and its Fourier spectrum in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 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Topological edge states of quasiperiodic elastic metasurfaces, Mechanical Systems and Signal Processing 181 (2022) 109478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='ymssp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content='109478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQf9fo9/content/2301.00874v1.pdf'} diff --git a/YtE2T4oBgHgl3EQfZAd1/content/tmp_files/2301.03860v1.pdf.txt b/YtE2T4oBgHgl3EQfZAd1/content/tmp_files/2301.03860v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f788ced49603180a9049f4247c11b7f487aabef --- /dev/null +++ b/YtE2T4oBgHgl3EQfZAd1/content/tmp_files/2301.03860v1.pdf.txt @@ -0,0 +1,1765 @@ +arXiv:2301.03860v1 [physics.plasm-ph] 10 Jan 2023 +Nonlinear interactions of ion acoustic waves explored using fast imaging +decompositions +Simon Vincent1,2, Vincent Dolique1, and Nicolas Plihon1 +1 Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France +2 École Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC), CH-1015 Lausanne, +Switzerland +(Dated: 11 January 2023) +Fast camera imaging is used to study ion acoustic waves propagating azimuthally in a magnetized plasma column. The +high speed image sequences are analyzed using Proper Orthogonal Decomposition and 2D Fourier Transform, allowing +to evaluate the assets and differences of both decomposition techniques. The spatio-temporal features of the waves are +extracted from the high speed images, and highlight energy exchanges between modes. Growth rates of the modes are +extracted from the reconstructed temporal evolution of the modes, revealing the influence of ion-neutral collisions as +pressure increases. Finally, the nonlinear interactions between modes are extracted using bicoherence computations, +and show the importance of interactions between modes with azimuthal wave numbers m, m − 1 and −1, with m an +integer. +I. +INTRODUCTION +The propagation of ion sound waves or ion acoustic waves +is ubiquitous in plasmas and their non-linear interactions, pos- +sibly leading to ion acoustic turbulence, is a widespread en- +ergizing process in plasma physics. The nonlinear evolution +of ion acoustic waves (IAW) generically leads to instabilities +and the development of non-linear structures. For instance, +IAW have long been observed in the solar wind, and related +to the anisotropy of the electron distribution function1. In this +context, IAW are driven unstable when the ratio of the elec- +tron to ion temperature is larger than unity, as observed by +the Helios spacecraft2, and very recently for oblique IAW by +Parker Solar Probe3. Heating of energetic particles from ion +acoustic turbulence was also proposed in the context of po- +lar aurorae4. The non-linear evolution of ion acoustic waves +into strongly non-linear structures such as solitons5 or double +layers has been reported in electro-positive plasmas6 or elec- +tronegative plasmas7, for which two branches of ion acoustic +waves exist8. In the context of bounded plasmas, IAW ex- +cited in sheaths may affect particle transport at low pressure9 +or lead to strong ion heating10 when the ratio of the electron +to ion temperature is larger than unity. IAW may also be use- +ful tools to probe sheath criteria in multiple ion plasmas11–13. +Technological plasmas may also trigger IAW, that, in return, +affect their operation, as reported for Hollow Cathodes14,15, +Hall thrusters16 and diverging magnetic nozzle thrusters17. +In this article, we report on the observation of localized ion +acoustic waves in a magnetized plasma column using high +speed camera imaging. Our observations thus shed new light +on the ion acoustic activity that has been previously reported +in similar configurations18–24. We do not investigate the ori- +gin of the IAW from parametric instability or waves interac- +tions here, as was done in these previous investigations, but +we analyse the spatio-temporal characteristics of the IAW us- +ing mode decomposition from high-speed imaging. The IAW +nonlinear interactions are quantitatively highlighted by means +of bicoherence computations. +The article is organized as follows. The experimental set- +up is introduced in Sec. II, the analysis of fast camera mea- +surements by mode decomposition techniques is presented in +Sec. III. In particular, we highlight the differences and com- +plementarities of two different mode decompositions, namely +Proper Orthogonal Decomposition and 2D Fourier Transform. +In section IV the waves observed by camera imaging are iden- +tified to be IAW from the waves phase velocities. Finally +the non-linear modes interactions are characterized and their +nonlinear aspect is exhibited in section V and conclusions are +drawn in section VI. +II. +EXPERIMENTAL SET-UP AND DIAGNOSTICS +A. +Experimental set-up +The experimental set-up25 consists in a 20 cm diameter, +1 m long stainless steel cylindrical chamber containing an ar- +gon plasma generated by a 1 kW, 13.56 MHz radio frequency +source. The coordinates will be referenced using a cartesian +coordinate system, where z denotes the axial direction (see +Fig. 1). A cylindrical coordinate system (r,θ,z) will be used +for the reconstruction of rotating modes. The plasma base +pressure p0 in the chamber is regulated at a fixed value, be- +tween 0.8 mTorr and 2 mTorr by steps of 0.1 mTorr. The +plasma is created by an inductive source around a 11 cm diam- +eter borosilicate tube connected at one end of the chamber (at +z = 0 cm). Three coils placed along the steel cylinder generate +an axial magnetic field that confines the plasma (see Fig. 1). +This magnetic field is not perfectly homogeneous along z. For +a current in the coils set here at 100 A, it has an averaged +amplitude along the z-axis of B = 170 G. +Radial profiles of the plasma density n, the electron tem- +perature Te and the plasma potential Φp are performed using +a 5-tips probe26,27 and an emissive probe respectively. The +probes were inserted radially at z = 49 cm (see blue dashed +line in Fig. 1). To keep the whole apparatus in a steady ther- +mal state, the operation of the plasma is pulsed: the plasma +is sustained over typically 5 seconds, during which data are +acquired, with a repetition period of typically 30 s. The ex- +periment is fully automated to allow high repeatability and + +2 +0.8 m + + +B +RF source +Coils +0.12 m + +Pump +inlet +0.2 m +z +y +x +0.11 m + + + + + + + + + + + + +-5 +0 +5 +-5 +0 +5 +0 +100 +200 +300 +-5 +0 +5 +-5 +0 +5 +0 +5 +10 +15 +θ +FIG. 1. Left: sketch of the experimental set-up. The dashed blue +line indicates where probe profiles are measured. Right: mean im- +age of the light intensity collected by the camera (up), and standard +deviation of the fluctuations around this average value (bottom), for +p0 = 1 mTorr. +reproducibility of the plasma. The level of shot to shot repro- +ducibility was ±0.6% for the ion saturation current of a Lang- +muir probe, with a standard deviation of 0.2% (estimated from +a series of 40 shots at the plasma column center). Radial scans +of the plasma parameters were performed sequentially: each +spatial point has been acquired during one plasma-pulse, and +the probe is translated between two pulses, from the center of +the plasma column (r = 0 cm) to its edge (r = 10 cm). +The results presented in this article mainly rely on high- +speed imaging of the plasma emitted light, performed through +a DN 200 borosilicate window closing the chamber at z = +80 cm (opposite to the source tube). A Phantom v2511 cam- +era is placed along the z-axis, 3.5 m away from this window, +and the light intensity naturally radiated by the plasma Icam is +captured at 200 kfps with a resolution of 256 × 256 px. A fil- +ter around 750±5 nm is used in order to restrict the collected +light to a single ArI spectral line. Examples of the mean inten- +sity ⟨Icam⟩ and fluctuation standard deviation σ(˜Icam) images +are shown in Fig. 1. Note that the plasma column is not per- +fectly axisymmetric, and the fluctuations are of the order of +10% of the mean amplitude. Note also that the depth of field +of the optical set-up being of the order of the length of the +chamber (with a camera objective aperture set at f/4, and a +focal length of 135 mm), the light intensity recorded by the +camera is actually the result of an integration along the z-axis. +Due to the magnetic field ripple and to the parallax, the di- +rect comparison of the probes’ measurements (at z = 49 cm) +and the camera images (where light is integrated along z), is +not relevant. We thus introduce a distorted space (x∗, y∗, z) +in which the camera lines of sight are parallel (see Ref.27 for +more details). The camera images are hence observed in the +plane (x∗, y∗), that may also be referenced as (r∗,θ) in a polar +coordinate system. +B. +Radial profiles of the plasma parameters +The top row of Fig. 2 displays the radial profiles of the +plasma density, electron temperature and plasma potential as +0 +2 +4 +6 +8 +10 +2 +3 +4 +5 +0 +2 +4 +6 +8 +10 +4 +5 +6 +7 +8 +1016 +0 +2 +4 +6 +8 +10 +0.08 +0.1 +0.12 +0.14 +0.16 +0 +2 +4 +6 +8 +10 +0.1 +0.15 +0.2 +0.25 +0.3 +103 +104 +105 +10-6 +10-4 +10-2 +100 +0 +2 +4 +6 +8 +10 +-4 +-3 +-2 +-1 +0 +0 +2 +4 +6 +8 +10 +4 +6 +8 +10 +12 +1017 +FIG. 2. +Radial profiles of the density, electron temperature, and +plasma potential mean values (top row) and fluctuations (middle row) +at p0 = 1 mTorr. Plasma parameters measurements were made with +5-tips (n, Te) and emissive (Φp) probes. Bottom: Power Spectral den- +sity of the plasma density and light intensity fluctuations (computed +from the average value of a 5×5 pixel box on the images). +a function of r for a pressure p0 = 1 mTorr. As previously +introduced, these profiles cannot be directly compared to the +images in the (r∗,θ) plane. However, assuming axisymme- +try and invariance of the plasma parameter along magnetic +field lines, a synthetic integration process detailed in Ref27 al- +lows to map the probe measurements along r (at z = 49 cm) +to the images expressed along r∗, enabling quantitative com- +parison. The density is approximately constant in a core re- +gion of the plasma for r ≤ 4 cm (r∗ ≤ 3 cm) and then de- +creases towards the edge. A clear peak can be seen around +r = 4.5 cm (r∗ ∼ 3 cm) for the electron temperature, produced +by the higher ionization rate of the RF inductive source close +to the wall at r = 5.5 cm, z ∼ −10 cm. This higher temperature +is also responsible for the higher light emission observed on +the images at r∗ ∼ 3 cm in Fig.1. Finally, the plasma potential +decreases from the center to r ∼ 4 cm (i.e. r∗ ∼ 3 cm), and +presents a strong positive gradient at the edge of the plasma +column. This is responsible for an ⃗E × ⃗B drift that drives +plasma rotation in the −⃗eθ direction, discussed later in this +work. +The radial profiles of the fluctuations of the plasma param- +eters are shown in the middle panel of Fig. 2. The fluctuations +are peaked at the edge of the plasma column at r ∼ 5.5 cm +(i.e. r∗ ∼ 4 cm). +Filtered light fluctuations recorded by fast camera are usu- +ally considered to be a proxy for density fluctuations28–30. For +the magnetic field value of B = 170 G reported in this ar- +ticle, simultaneous probe and camera measurements showed +the fluctuations of density ˜n and light intensity ˜Icam at the + +3 +probe location to have very similar spectra, as shown in the +bottom panel of Fig. 2. However, as it was shown in our pre- +vious work27, for magnetic field values in the range 100 to +700 G, light intensity naturally radiated by low temperature +plasmas are also highly correlated to the electron temperature. +In Sec. IV, the comparison of experimental phase velocities +with the theoretical ion acoustic speed nonetheless requires +to assume ˜Icam to be a reasonable proxy for ˜n. We therefore +underline that this is a rather strong assumption, and that for +further quantitative comparison ˜Icam should be interpreted as a +combination of both ˜n and ˜Te - a task beyond the scope of this +article. The strong spectral component observed at 5.6 kHz +is identified as a Kelvin-Helmholtz mode31. In the present +work, we will focus on fluctuations observed between 50 and +70 kHz, which are unambiguously identified as ion acoustic +waves. +III. +IMAGE ANALYSIS +The first and most natural tool that comes to mind for the +images analysis is the Fourier decomposition. This is used +later on; we prefer here to start the analysis by using an alter- +native method, the Proper Orthogonal Decomposition (POD). +Note that before performing these decomposition, we +choose to normalize each pixel by its fluctuations mean, as +was done in similar conditions32. +This choice greatly en- +hances the contrast, allowing to nicely extract the relative am- +plitudes of the modes (especially in the regions of low light +intensity), but at the cost of losing information on the abso- +lute amplitude of the modes. +Note finally that in the following the light fluctuations +recorded by the camera ˜Icam will simply be denoted I for eas- +ier readability. +A. +Proper Orthogonal Decomposition +The POD consists in extracting the spatial structures that +are dominant throughout time in a given data set I(r∗,t), +where r∗ denotes space. This is done by computing the eigen- +modes Ψi of the spatial autocorrelation of the time-averaged +field ⟨I⟩(r∗). These so-called spatial modes Ψi then define +an orthonormal basis onto which the original data can be pro- +jected. This can be written: +I(r∗,t) = ∑ +i +σi ai(t)ψi(r∗) +with σi ai(t) being the time evolution of the data projected on +the spatial modes Ψi. Here ai and Ψi are of norm unity; the +amplitude of the various components of the decomposition are +thus given by the values of σi. +One of the most interesting aspects of this decomposition +is that it is done without any a priori on the shape of the Ψi +structures: they simply come out from the computation pro- +cess, as natural modes, contrary to Fourier analysis, which +projects the data onto predefined spatial and temporal struc- +tures. Hence POD might allow the emergence of structures +with physical significance that are not well described by mere +Fourier modes. Thanks moreover to its simplicity of imple- +mentation and computational speed when performed onto a +discrete set of data, as is explained later, POD becomes a very +attractive and efficient analysis tool for experimentalists, and +has grown very popular in the last decades for the analysis of +data from experiments or from numerical simulations. Note +that depending on the field, this technique is also referred to +as Karhunen-Loève decomposition (as a reference to the orig- +inal mathematical theorem) or principal component analysis. +The set of spatial modes (Ψi) has the property of being the +optimal basis for approximating the data I33: for any N, the +norm of the projection of I onto (Ψi)1,N, which reads ∑N +1 σ2 +i , is +higher than the projection onto any other basis than one might +choose. For a given value of N, the spatial modes (Ψi)1,N can +then be interpreted as the vectors that are the best suited to +reproduce the information carried by I, in the most efficient +way. Applied to physical data, this property is even more in- +teresting if the σ2 +i have a clear physical meaning. The use +of POD on experimental data has been initiated in fluid dy- +namics, for the analysis of turbulent velocity fields34. In this +context the norm ∑N +1 σ2 +i of the projected data represents a ki- +netic energy, and the modes Ψi may then be interpreted as +the most important flow structures in terms of kinetic energy. +POD has then been applied to spatio-temporal measurements +of plasma fluctuations in tokamaks, either measured by sets of +Langmuir probes35–37 or by means of soft x-ray emission38. It +has also been applied to camera imaging data of plasma natu- +rally radiated light to exhibit spiral shaped structure generated +by a m = 2 instability in a linear device39 and in a tokamak +to highlight plasma response to resonant magnetic perturba- +tions40. More recently, the technique was used to characterize +instabilities in the plume of a Hall thruster from fast imaging +data41. Following this study, POD has been applied to decom- +pose the camera imaging data of a plasma plume produced +by a high-current hollow cathode42. Unfortunately, in the lat- +ter cases, the physical interpretation of the Ψi vectors is not +as straightforward as in fluid mechanics, since the extracted +modes result from the decomposition of light intensity fields, +which depends in a non trivial way on the plasma parameters. +And even by considering as a crude approximation ˜Icam ∝ ˜n, +not much can be said on the norm ∑N +1 σ2 +i in terms of physical +significance. This does not mean the amplitude of the modes +extracted from plasma emitted light is void of meaning, but +simply that one has to be careful before thinking of it as a +precise energy estimation. In this article POD decomposition +is thus discussed in a purely qualitative way. Finally, POD +does not require any symmetry, which is a significant advan- +tage over Fourier decomposition for instance. In the case of +complex geometries, POD can be an efficient alternative for +capturing the physical structures in the data. +In practice, it can be shown that a direct extraction of the +spatial modes Ψi associated to their temporal evolution ai, is +in fact achieved by applying a mere singular value decompo- +sition (SVD) to the matrix containing the data I, rearranged +in such a way that one dimension of the matrix represents +space, and the other time. This way of computing a POD, +also referred to as bi-orthogonal decomposition43 is the one + +4 +-5 +0 +5 +-5 +0 +5 +0 +50 +100 +-1 +0 +1 +100 +102 +10-5 +10-3 +10-1 +-5 +0 +5 +-5 +0 +5 +0 +50 +100 +-1 +0 +1 +100 +102 +10-5 +10-3 +10-1 +-5 +0 +5 +-5 +0 +5 +0 +50 +100 +-1 +0 +1 +100 +102 +10-5 +10-3 +10-1 +-5 +0 +5 +-5 +0 +5 +0 +50 +100 +-1 +0 +1 +100 +102 +10-5 +10-3 +10-1 +-5 +0 +5 +-5 +0 +5 +0 +50 +100 +-1 +0 +1 +100 +102 +10-5 +10-3 +10-1 +-5 +0 +5 +-5 +0 +5 +0 +50 +100 +-1 +0 +1 +100 +102 +10-5 +10-3 +10-1 +-5 +0 +5 +-5 +0 +5 +-1 +-0.5 +0 +0.5 +1 +0 +50 +100 +-1 +0 +1 +100 +102 +10-5 +10-3 +10-1 +0 +0.05 +0.1 +-0.2 +0 +0.2 +a) +c) +1 +5 +10 +15 +0 +0.5 +1 +b) +FIG. 3. Proper Orthogonal Decomposition modes (a) and singular values (b) of light intensity normalized fluctuations, for p0 = 1.3 mTorr. c) +Zoom on the evolution of a1 and a2. See text for details. +implemented here. Each image (of p pixels) of a video con- +taining q frames is rearranged to form a matrix I of size p×q +to which a singular value decomposition is applied I = ΨΣAt. +Ψ and A are orthonormal matrices of respective sizes p × p +and q × q, and Σ is a matrix of the same size as I. The matrix +Σ only contains diagonal elements, which are the decomposi- +tion’s singular values σi: +space + + + + + + + + + + + + + + + + + + + +(I)i j + + +time +� �� � += + + +... +... +... +Ψ1 +Ψ2 +Ψq +... +... +... + + + + +σ1 +... +σq + + + + +··· a1 ··· +··· a2 ··· +··· ap ··· + + +Figure 3 a) shows the result of a POD applied to a 100 ms +time series of intensity fluctuations (i.e. +20000 images) +recorded at a pressure p0 = 1.3 mTorr. The spatial modes +Ψi(r∗) are displayed in the top row, the time series of the am- +plitudes ai(t) in the the middle row, and the corresponding +power spectral density S(f) in the bottom row. The interpreta- +tion of the decomposition requires to consider pairs of modes, +such as IPOD +1,2 (r∗,t) = σ1a1(t)Ψ1(r∗)+ σ2a2(t)Ψ2(r∗), which +yields rotating azimuthal waves of the type e−iωt−imθ (shown +later in subsection III C), since the spatial modes Ψ1 and Ψ2 +are shifted by a quarter wavelength and the temporal modes +a1 and a2 are in quadrature (see Fig. 3 c))44. In the example +shown in Fig. 3, the modes (Ψ1, a1) and (Ψ2, a2) correspond +to a m = −5 rotating azimuthal wave, the modes (Ψ3, a3) and +(Ψ4, a4) correspond to a m = −6 rotating azimuthal wave, +and the modes (Ψ6, a6) and (Ψ7, a7) correspond to a m = −4 +rotating azimuthal wave. +The rotation frequencies of the m-modes can be deduced +from the spectra of the temporal signals ai, shown in the +bottom row of Fig. 3 a). +A very clear peak at frequency +f = 55.6 kHz is observed for the modes 1&2, capturing the +m = −5 azimuthal wave. The m = −6 wave (POD modes +3&4) has a f = 65.1 kHz frequency, and the m = −4 wave +(POD modes 6&7) has a f = 45.2 kHz frequency. Section IV +shows that these modes are ion acoustic waves. +The singular values σi of the modes are plotted in Fig. 3 +b). The amplitudes of σ1 and σ2 are nearly identical (within +0.3%), and more than twice larger than the other singular val- +ues, showing that the dynamics is dominated by a m = −5 +rotating mode. The time series ai(t) show sudden changes +in amplitude, for instance at times t ∼ 3.8 ms, 29.5 ms and +65.2 ms, corresponding to an energy exchange between modes +m = −6 and m = −5, that will be investigated in Section V. +The relatively intense POD mode (Ψ5, a5), with a strong spec- +tral component at f = 5.6 kHz, was identified as a m = 3 +Kelvin-Helmoltz mode31, not discussed here. The POD analy- +sis presented here was straightforward to implement, and pro- +vide a very efficient way of extracting global features captured +in a video sample. Now we present the results of a Fourier +analysis performed on the same data that complements the +POD analysis. +B. +2D Fourier Transform +The 2D Fourier Transform (2D-FT) of a two variables func- +tion f(x,t) reads: +ˆf (k,ω) = +�� f(x,t)e−i(k·x+ωt)dxdt. 2D- +FT is classically used to decompose the spatio-temporal sig- +nals collected by azimuthally distributed probe arrays into az- +imuthal modes45,46. Following many studies using camera +imaging and performed in the context of linear plasma de- +vices29,47–49 and plasma thrusters50,51, the 2D-FT is here per- +formed on virtual rings at various radii r∗. For a given value +of the radius r∗, a time series I(θ,t)|r∗ is extracted from the +camera images. For each angle θn = 2πn +Nθ with n ∈ [0 : Nθ −1], +the value of the pixel at position (r∗, θn) is extracted. The +angle resolution of Nθ = 700 is chosen here, such that no in- + +*米5 +-10 +-5 +0 +5 +10 +0 +20 +40 +60 +80 +100 +22 +24 +26 +28 +30 +32 +Power spectrum +FIG. 4. Power spectrum of the raw light intensity fluctuations from +camera imaging, taken on a corona of radius r∗ = 3.3 cm for p0 = +1.3 mTorr. Dispersion relations are plotted from experimental fits of +the power spectrum maxima (red) and from the ion acoustic speed +(black) (see section IV). +terpolation is needed in the processes of converting either a +ring of pixels in the image space (x∗, y∗) into a vector along +the θ direction, nor in the inverse process, when reconstruct- +ing images in the (x∗, y∗) plane from the mode decomposition +results. Since the images are 2π periodic in the θ direction, +the wave-vectors read m/r∗, with m an integer, and the 2D-FT +of I(θ,t)|r∗ is computed as +ˆIr∗(m, f) = +�� +I(θ,t)|r∗e−i(mθ+2π ft)dθdt +with f the frequency. +The resulting 2D power spectrum +Sr∗(m, f) = | ˆIr∗(m, f)|2 displays the amplitudes of light inten- +sity fluctuations as a function of the spatial mode m and the +frequency f, at a given radius r∗. Note that at radius r∗ ∼ +3.5 cm on the images, the intensity I(θ,t)|r∗ is reconstructed +from a corona of ∼ 400 pixels, which ensures a very good pre- +cision in the extraction of the first modes m up to m ∼ 20. The +power spectrum at r∗ = 3.3 cm and p0 = 1.3 mTorr is shown +in Fig. 4. The observations are similar to those drawn from +the POD analysis : the dominant mode is an m = −5 mode +whose frequency is peaked at 55.6 kHz, and the other impor- +tant modes are an m = −6 mode peaked at 65.2 kHz and an +m = −4 mode peaked at 44.9 kHz. Note that this 2D power +spectrum provides a dependence of the dominant frequency +on the mode number, and can therefore be seen as an experi- +mental dispersion relation. This is used in section IV for the +identification of the modes. +The full spatio-temporal evolution of any given m mode can +also be extracted by 2D-FT. To this end, the 2D Fourier Trans- +form is computed for radii r∗ covering the full image (here +2D-FT are computed for r∗ = 2i px, i ∈ [1 : 64], instead of +r∗ = [1 : 128] px, to limit memory storage and increase compu- +tational speed). For given m and r∗ values, the inverse Fourier +Transform of ˆIr∗(m, f) is computed, resulting in the spatio- +temporal signal of the m mode, at radius r∗. Performing this +0 +0.5 +1 +0 +20 +40 +60 +80 +100 +4 +5 +Average amplitude +Radial location +t0 +FIG. 5. Time evolution of m-mode average amplitudes extracted by +2D-FT, for p0 = 1.3 mTorr. +inverse computation for all radii previously mentioned leads +to the full spatio-temporal reconstruction of the m mode. Ex- +amples of snapshots of such reconstructed 2D-FT modes are +shown and commented in subsection IIIC. +The average amplitude of the reconstructed modes is then +computed along time, providing a global picture of the modes +dynamics. +The global m-modes time evolution for p0 = +1,3 mTorr is plotted in Fig. 5 (top). As already observed on +the time signals of the POD in Fig. 3, clear exchange events +can be observed involving modes m = −5 and m = −6. The +m = −4 mode is seen to follow the dynamics of m = −5 while +the m = −7 mode follows the dynamics of m = −6 mode, +a feature that was not detected by the POD analysis. From +the reconstructed signals of the individual m-modes, the in- +stantaneous mean radial profile is computed by an integration +over θ, allowing for the computation of the radial location +r∗ +max where the wave amplitude is maximal. Figure 5 (bottom) +shows r∗ +max for the m = −5 and m = −6 modes, that are highly +correlated to the global dynamics of the modes. Again this +could not be deduced from POD, since spatial modes struc- +ture are deduced from a time-averaged analysis. Figure 5 is +further discussed in section V. Now let us compare the results +obtained from both POD and 2D-FT analysis. +C. +Comparison between POD and 2D Fourier Transform +Figure 6 shows snapshots of the m = −5 and m = −6 2D- +FT reconstructed modes, as well as the corresponding modes +reconstructed from the POD analysis IPOD +1,2 +and IPOD +3,4 , for the +experiment achieved at p0 = 1.3 mTorr. The snapshots are +shown every 0.06 ms following t0 = 77.27 ms, marking the +beginning of an energy exchange between modes m = −5 and +m = −6 (see Fig. 5 (top)). The time interval 0.06 ms repre- +sents slightly more than 3 wave periods for the m = −5 mode, +and nearly 4 wave periods for the m = −6 mode. The spa- + +6 ++c +-c +0 ++c +-c +0 ++c +-c +0 ++c +-c +0 +c = 1 +c = 0.72 +c = 0.80 +c = 0.38 +c = 0.42 +c = 0.66 +c = 0.78 +c = 1 +m = -5 +I1,2 +m = -6 +POD +I3,4 +POD +c = 1 +c = 0.75 +c = 0.73 +c = 0.61 +c = 0.39 +c = 0.53 +c = 0.61 +c = 1 +-5 +0 +5 +-5 +0 +5 +-5 +0 +5 +-5 +0 +5 +-5 +0 +5 +0 +0 +0 +0 ++ ++ ++ +FIG. 6. Evolution of 2D-FT modes m = −5, m = −6 and POD re- +constructed modes IPOD +1,2 +and IPOD +3,4 , during the exchange event high- +lighted by the dashed-black box in Fig. 5, for p0 = 1.3 mTorr. +tial shape of the 2D-FT modes varies significantly. On the +contrary the shapes of the POD reconstructed modes remains +almost unchanged. This is actually expected since each of +these signals is merely composed of the linear combination +of two spatial fields. Note also that the spatial structures Ψi +were extracted from the time-averaged data field (see subsec- +tion III A): the reconstructed mode are unable to account for +spatially localized variations. +Let us now compare the spatial structures provided by POD +and 2D-FT. Figure 7 (top) shows time-average radial profiles +of the modes for both decompositions. The profiles are com- +puted by an integration along θ, averaged over the 20000 im- +ages. Fig. 7 (bottom) shows the azimuthal profiles taken at a +given time and for r∗ = 4 cm. The comparison between POD +modes (1&2) and the m = −5 2D-FT mode shows an almost +perfect match (note that the match slightly decreases when the +amplitude of the m = −5 mode strongly decreases). The com- +parison between POD modes IPOD +3,4 +and the m = −6 2D-FT +mode give similar results, although with a lower agreement +on the outward part (r∗ ≥ 4 cm) of the radial profiles. An +overall good match is observed between the lower amplitude +POD modes (6&7) and the m = −4 2D-FT mode radial pro- +files. The instantaneous azimuthal profile are not identical, +with a phase shift up to ∼ π/8 depending on the frame. These +results show that, in the context of data having 2-π periodic- +ity, POD and 2D-FT decompositions share several common +features, while they do provide exactly the same knowledge. +Note finally that for the computations performed here with a +number of images N = 20000, the POD is twice faster than +the 2D-FT (even though it was taken into account for the 2D- +FT only 20 mode reconstructions, and half of the images pix- +0 +2 +4 +6 +0 +0.5 +1 +0 +2 +4 +6 +0 +0.5 +1 +0 +2 +4 +6 +0 +0.5 +1 +0 +/2 +3 /2 +2 +-1 +0 +1 +0 +/2 +3 /2 +2 +-1 +0 +1 +0 +/2 +3 /2 +2 +-1 +0 +1 +FIG. 7. Comparison of (top) radial and (bottom) azimuthal profiles of +the POD and 2D-FT modes, at p0 = 1.3 mTorr, for the (left) m = −5, +(center) m = −6 and (right) m = −4 modes. +els as mentioned in subsection III B). Then when only taking +N = 2000, the POD is more than 12 times faster than the 2D- +FT. +The main strengths of both POD and 2D-FT techniques are +summarized: +• POD is fast and easy. It is extremely simple to imple- +ment, and it provides quick and direct results on the +spatio-temporal dynamics of a dataset. +• POD is flexible. It does not rely on any particular shape +of the physical structure at play, nor on a specific loca- +tion in the images analysed. It will therefore be partic- +ularly well suited to study for instance non-linearly sat- +urated modes exhibiting a complex spatial or temporal +pattern. Note however that if the results can be particu- +larly insightful, they might also be difficult to interpret +(and in some cases even unusable). +• 2D-FT is explicit, hence robust. Projecting the data onto +a predefined set of wave modes (here for instance of the +form e−iωt−imθ) prevents the emergence of unexpected +structures, but it provides the results with a well identi- +fied physical meaning. +• 2D-FT is exhaustive for linear mode analysis. Since +it provides the full spatio-temporal evolution of linear +wave, 2D-FT is particularly attractive to study their +dynamics, exhibit the corresponding dispersion rela- +tions, or use for instance the phase correlations between +modes to study weakly non-linear interactions (see the +use of bicoherence in section V). +Both techniques can provide insightful and complementary +results. A recent preprint, reporting on the specific compar- +ison between POD and 2D-FT applied to Hall thruster cam- +era imaging52, concludes similarly. Applied to the present +datasets, POD shows that the dominant physical structures are +m-modes of the form e−iωt−imθ. This indicates that the 2D-FT +as implemented here, is an appropriate numerical tool for the +mode decomposition. Hence POD does not constitute a strong + +*****7 +gain for further analysis here. In the following, for the iden- +tification of the waves and the in-depth study of their weakly +non-linear interactions, we will use the results from the 2D-FT +decomposition. +IV. +WAVES IDENTIFICATION +The azimuthal waves detected by both POD and 2D-FT are +now unambiguously identified as ion acoustic waves. A series +of high-speed imaging acquisitions was performed for pres- +sures p0 in the range [0.8;2] mTorr by steps of 0.1 mTorr. +For each value of the pressure, the radius r∗ +max at which the +wave amplitude is maximal is deduced from the time-average +of the raw images. The experimental phase velocity vφ is de- +termined by a linear fit of the most energetic modes observed +on the spectrum Sr∗max(f,m) as f(m) = vφm/(2πr∗ +max). A typ- +ical linear fit is shown in Fig. 4 for p0 = 1.3 mTorr. The ex- +perimental phase velocities vexp +φ +are displayed in Fig. 8 as red +dots. The errorbars are estimated by the combination of the +uncertainties on the fit on S(m, f), and on the evaluation of +r∗ +max (r∗ +max(t) fluctuates around its mean value with a standard +deviation of ∼ 3%, see Fig. 5 (bottom)). +These experimental phase velocities are compared to the +theoretical ion acoustic speed cs = +� +eTe/mi, with e the ele- +mentary charge and mi the ion mass. The computation of the +latter requires careful estimates of Te where the phase velocity +is measured on the high-speed images. Note that at z = 49 cm, +where the probe measurement is performed, the radial posi- +tion that is best representative of what is seen at r∗ = 3.3 cm +on the images is in fact at r = 5 cm (see Appendix A and for +a detailed explanation see27). A detailed pressure scan of the +electron temperature Te was performed with the 5-tips probe +at a radius r = 4 cm, and from a finely resolved radial scan +at p0 = 1 mTorr27,31, we have Te(5 cm) ≈ Te(4 cm)+ 0.2 eV. +Therefore, from the measured values Te(4 cm), Te(5 cm) is +evaluated to lie in the range [Te(4 cm);Te(4 cm)+0.5] eV. The +resulting theoretical ion acoustic speeds cs(p0) are shown in +Fig. 8 (gray area). +The experimental phase velocities follow the trend of +cs(p0), with values shifted down by approximately 700 m/s. +This is well explained by a Doppler shift due to the plasma +column rotation. +The plasma column indeed rotates, as +was reported previously53, where the electric drift +⃗E ×⃗B +B2 += +−∇rφp/B⃗eθ was shown to overcome the diamagnetic drift +−Ti +n +⃗∇n ×⃗B +B2 +. Two damping mechanisms also need to be ac- +counted for: ion-neutral friction and effective friction due to +ionization. The ion-neutral collision frequency reads νin = +nnσinvth,i, with vth,i = +� +eTi(eV)/mi, nn being the neutral den- +sity and Ti the ion temperature. We consider nn ≈ p0/kBTn +with Tn(p0) = 350 K, σin = 1.6 × 1018 m−2 from experi- +mental cross sections54, and Ti ≈ 0.2 eV using previous LIF +measurements. The effective friction due to the ionization +originates from ions created with a temperature much lower +than the surrounding Ti and depends upon the ionization +frequency νiz, computed as νiz = nnKiz,0T 0.59 +e +exp(−εiz/Te), +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2 +2.5 +3 +3.5 +4 +FIG. 8. Comparison between the experimental phase velocity (red +dots), the ion sound velocity cs (black curve), and its Doppler shifted +values (green curve). +with Kiz,0 = 2.34 × 10−14 m3/s and εiz = 17.44 eV, and Te +in eV55. +A global damping factor K is then given31,53 as +K = 1 + +�νin + νiz +ωci +�2 +, with ωci the ion cyclotron frequency. +This finally gives a background azimuthal rotation of the ions +as vi0,θ ≈ −(1/K)∂rφ(r)/B. +The rotation velocity vi0,θ is estimated from the experimen- +tal profiles φp(r) shown in Fig. 2 (measured at p0 = 1 mTorr, +and assuming variations with pressure within ±20%). The +estimated values of nn and Ti are considered to be bounded +within ±10%, and Te is estimated from T r=4cm +e +(p0) as ex- +plained above. The results for the estimate of cs + vi0,θ are +shown in Fig. 8 (green curve). In spite of all the approxima- +tions made, the comparison between experimental phase ve- +locities and the Doppler shifted values of cs provides a very +satisfactory agreement. This allows us to identify with great +confidence the azimuthal waves observed at B = 170 G as ion +acoustic waves. An interesting feature is that the ion acoustic +waves travel in the positive θ direction, i.e. opposite to the +E × B drive. +We stress here that adding the ion background velocity to +the classical ion acoustic wave speed is a crude approxima- +tion, deemed sufficient here for the purpose of wave identifica- +tion. However, a careful calculation would require to compute +a complete dispersion relation from the governing equations, +which couple in a complex way and prescribe direct analytical +computation. Indeed the effect of an ion background velocity +on the ion acoustic phase velocity is likely to be coupled with +other effects such as electron magnetization or friction with +the neutrals, leading to computations well beyond the scope +of this article. +Interestingly, we observed that the ion acoustic waves are +only observed over a narrow range of magnetic field values. +For B = 80 G no clear wave emerges from the fluctuations of +the plasma density or emitted light intensity; on the other hand +for B ≥ 300 G low frequency waves develop31. + +8 +7� +�� +-2.5 +-2 +-1.5 +-1 +-0.5 +0�� +1.1 +1.3 +0.2 +0.3 +��� +0.5 +FIG. 9. Left: time evolution of m-modes average amplitudes, ex- +tracted by 2D-FT. Left: fit of the 2D-FT m = −5 mode growth rate at +an exchange event with m = −6 mode, for p0 = 1.3 mTorr. The fit is +done on a selected interval of the raw data (light blue). The red curve +is the result of a filter. Right: Evolution of the growth time scale of +m = −5 mode, evaluated during exchange events, as a function of the +pressure p0. +V. +MODES DYNAMICS AND INTERACTIONS +The spatio-temporal dynamics and the non-linear nature +of the energy exchanges between the ion acoustic modes, as +clearly shown in Fig. 5, are now described. +A. +Growth rates of ion acoustic modes +The time series shown in Fig. 6 is taken around the ex- +change event highlighted at t0 in Fig. 5. At time t0 the am- +plitude of the m = −6 mode is close to its maximum, while +the amplitude of the m = −5 mode is close to its minimum. +At time t0 + 18 ms, the amplitude of the m = −6 mode has +decreased close to its minimum value, and the m = −5 mode +dominates. Figure 5 (bottom) shows that the radial position of +the dominant mode (either the m = −5 or the m = −6 mode) +is indeed very stable. On the other hand, the radial position of +the low amplitude mode strongly fluctuates around its equi- +librium value (with standard deviations around 0.6 cm for the +m = −5 mode and ∼ 0.5 cm for the m = −6 mode). +The exchange events observed for p0 = 1.3 mTorr between +modes m = −5 and m = −6 (Figures 3 and 5) are similarly ob- +served at p0 = 1.1 mTorr and p0 = 0.9 mTorr. The timescales +of the exchange events are now determined at these three val- +ues of the pressure. This is done by fitting the mode amplitude +Am as exponentially growing: Am ∝ exp(t/τ). Figure 9 (left) +shows a typical fit around t0: the green part shows the interval +over which the raw signal (blue) is fitted; a low-pass filtered +signal is shown for clarity (red). Figure 9 (right) shows the +resulting values of τ found for the m = −5 mode. The growth- +time τ significantly increases with the pressure, its value dou- +bles from p0 = 0.9 mTorr to 1.3 mTorr. This is interpreted as +being the result of an increased friction from the neutrals at +higher pressure. Note that this observation of a decrease of +the ion acoustic wave growth rate with increasing pressure is +consistent with theoretical predictions9. +0 +10 +20 +30 +10-3 +10-2 +10-1 +FIG. 10. Residence time PDF, obtained for a neutral pressure p0 = +1.30 mTorr. +B. +Residence time distribution +The statistics of the transitions between the m = −5 and +m = −6 modes were obtained in a new set of experiments, per- +formed at a lower sampling frequency (20 kfps)56 over longer +times (2 seconds). This allows to extract the time evolution +of the modes average amplitude, extracted by 2D-FT. The +probability distribution function of the residence time of the +m = −4, m = −5 and m = −6 modes are shown in Fig. 10, +for a total duration of 4 seconds (i.e. more than one thou- +sand transitions between modes). The distributions are com- +patible with an exponential distribution, which implies that +the transition events are not correlated. Such distributions of +residence times or waiting times are ubiquitous to transitions +observed in aerodynamics57, turbulent flows58,59 or convec- +tion60, to the waiting time between reversals in dynamo ex- +periments61, or the turbulent dynamics of the scrape-off layer +in tokamaks62,63. +For all modes, the probability distribution function is com- +patible with a functional fit of the form e−t/τ, with τ = 5.4 ms +for m = −4, and τ = 6.0 ms for m = −5 and τ = 3.2 ms for +m = −6. As already observed in Fig. 5, the m = −4 mode +is tied to the m = −5 mode, resulting in similar pdf. Fig- +ure 5 also shows that the system is more often dominated by +a m = −5 mode, which results in an exponential pdf with a +larger characteristic time for the m = −5 mode as compared +to the m = −6 mode. High speed imaging of the dynamics of +the plasma allows to probe long-time statistics of the waves +dynamics. It opens the possibility to probe the evolution of +the characteristic residence time as a function of the control +parameters (for instance pressure), possibly shedding light to +the physical processes leading to exchange events. Note that +the dominant mode (and the associated characteristic time) +was observed to strongly evolve with pressure (data not shown +and beyond the scope of this article). + +9 +4 +� +50 +55 +60 +65 +0 +5 +10 +15 +0 +0.1 +0.2 +0.3 +� + +0 +20 + +60 +8 +105 +106 +10 + +a) +b) +c) +  +50 +55 +60 +65 +0 +5 +10 +15 +0 +0.1 +0.2 +0.3 + +FIG. 11. Maps of the threshold b2 +0 (a) and bicoherence b2 (b), cor- +responding to the three-wave interaction (m = −5) + (m = −1) ↔ +(m = −6). c) Frequency power spectra of modes m = −1, m = −5 +and m = −6 from 2D-FT computed at r∗ = 3.3 cm. B = 170 G and +p0 = 1.3 mTorr. +C. +Non-linear behaviour +In order to further assess the non-linear nature of the dy- +namics between the dominant ion acoustic modes, the bico- +herence b2(fm=−5, fm=−1) is computed for the three wave in- +teraction (m = −5) + (m = −1) ↔ (m = −6), and shown in +Fig 11. Note that to increase statistics, the 2D-FT at all radii +1 ≤ r∗ ≤ 5 around the wave maximal amplitude are used. +More details on the bicoherence computations are provided +in appendix B. The threshold map shown in Fig 11 a) was +computed using a basic surrogate technique where the phases +of the 2D-FT signal are randomly mixed. This yields bico- +herence values for signals without any preferential phase re- +lations, from which a threshold value of max(b2 +0) = 0.12 is +estimated. Figure 11 b) shows the map of b2(fm=−5, fm=−1) +with fm=−5 and fm=−1 the frequencies of modes m = −5 +and m = −1 respectively. For the sake of visibility, the ar- +eas of bicoherence high values are highlighted by gray con- +tours (defined at 40% of the maximum value of a Gaussian +filtered b2 map). Most of the bicoherence highest values lie +around the diagonal fm=−5 + fm=−1 = 65 kHz, that is the +dominant frequency of the m = −6 mode. This reveals the +strong non-linear behaviour of the (m = −6, f ∼ 65 kHz) +mode component, which interacts with m = −5 and m = −1 +modes via continuous sets of frequencies. The points dis- +played as red dots in Fig. 11 b) are also enlarged for clar- +ity: they correspond to b2 ≳ 0.36, i.e. more than three times +the threshold value. The points for which fm=−1 = 0 kHz, +and fm=−5 ∈ [64.6;65.4] kHz correspond to frequency com- +ponents of the m = −5 mode being fed by the high amplitude +of the (m = −6, f ∼ 65 kHz) component. Note that these +interactions are not the dominant process characterizing the +energy exchanges detailed in subsection V B, since they only +involves frequency components of the m = −5 mode around +65 kHz, with a low energy. The point at fm=−5 = 55.4 kHz +and fm=−1 = 11.2 kHz however corresponds to the interac- +tion: +(m = −5, f = 55.4) + (m = −1, f = 11.2) ↔ (m = −6, f = 66.7) +which involves the dominant frequency components of the +m = −5 and m = −6 modes. The very high bicoherence value +at this location (b2 = 0.39) definitively establishes the non- +linearity of the interactions between the ion acoustic modes +(m = −5, f = 55.4 kHz) and (m = −6, f = 66.7 kHz), at the +origin of the transitions observed in Fig. 5. +Finally, Fig. 11 c) shows the frequency spectra of the m = +−6, m = −5 and m = −1 modes involved in the three-wave +interactions described above. These spectra correspond to 1D +cuts along the frequency axis of the 2D-FT spectrum shown +in Fig. 4. These spectra clearly display the non-linear feeding +of the m = −5 mode by the high amplitude m = −6 mode +around 65 kHz. The non-linear feeding of modes m = −1 +and m = −6 by the high amplitude m = −5 mode around 55 +kHz is also visible. The component (m = −6, f = 55 kHz) is +then non-linearly interacting with (m = −5, f = 44) kHz and +(m = −1, f = 11 kHz), as can be deduced by the high values +of b2(fm=−5 ∼ 44, fm=−1 ∼ 11) from Fig. 11 b). +The computation of other bicoherence maps (not shown +here) reveals additional non-linear behaviours. The bicoher- +ence computation of the (m = −6)+ (m = −1) ↔ (m = −7) +coupling unambiguously shows that (m = −6, f = 65.2 kHz) +non-linearly interacts with (m = −7, f = 74.0 kHz) via an +(m = −1, f = 8.8 kHz) mode component (with b2 = 0.36 > +3max(b2 +0)). Similarly, bicoherence computation of the (m = +−4) + (m = −1) ↔ (m = −5) coupling highlights that (m = +−4, f = 44.2 kHz) and (m = −5, f = 55.4 kHz) modes non- +linearly interact via (m = −1, f = 11.2 kHz) (with b2 = +0.38 > 3max(b2 +0)). As a last example, the bicoherence map + +10 +for the interaction (m = −4) + (m = −2) ↔ (m = −6) does +not exhibit high values indicating the absence of non-linear +interaction between the corresponding ion acoustic modes. It +however reveals that the frequency components (f = 55 kHz) +of (m = −4) and (m = −6) modes (resulting from the spread +of the m = −5 mode, visible in Fig. 4) are non-linearly linked +via the (m = −2, f = 0 kHz) mode component. +Thanks to the rich spatio-temporal information provided by +camera imaging and to the use of bicoherence, the weakly +non-linear interactions are clearly highlighted. In particular +the existence of three-wave interactions between ion acoustic +modes m = −p, m = −p − 1 and m = −1, for p ∈ [4,5,6], is +demonstrated. +VI. +CONCLUSION +We have presented the first report of temporally and spa- +tially entirely resolved ion acoustic waves in a magnetized +plasma column. The ion acoustic waves were observed by +means of fast camera imaging in a low temperature argon +plasma column, with dominant azimuthal mode numbers m = +−4, m = −5 and m = −6 depending on the neutral pressure +that was varied from 0.8 mTorr to 2 mTorr. +Two image analysis techniques, namely proper orthogo- +nal decomposition (POD) and 2D Fourier transform (2D-FT), +were presented and thoroughly compared. These tools are +found to be complementary. POD is easy to implement and +adaptable to any type of data, and useful to provide a fast +overview of the underlying dynamics of a given dataset. This +helps focusing in a second time on a more precise and targeted +analysis, that 2D-FT can then provide, yielding detailed and +unambiguous information. +Using 2D-FT analysis of high speed images, the ion acous- +tic waves were found to rotate in opposite direction to the +global E ×B drift of the plasma column, with a phase velocity +Doppler shifted by this actual electric drift velocity. +The dynamics of the dominant ion acoustic modes was +then explored using the 2D-FT decomposition. Growth rates, +which extraction was made possible by the camera high tem- +poral resolution, were found to decrease as pressure increases, +following previous numerical predictions. A detailed analysis +was then carried out in the particular case of p0 = 1.3 mTorr. +At this pressure the exchange dynamics between dominant +modes m = −5 and m = −6 was shown to be of a bistable +nature. More generally the weakly non-linear nature of the +m = −p and m = −p − 1 mode interaction (p ∈ [4,5,6]), in- +volved in a three-wave interaction with a m = −1 mode, was +demonstrated by means of bicoherence computation. +Finally we emphasize that, except from probe measure- +ments that were needed for the wave identification, all the re- +sults that were presented exclusively rely on fast camera imag- +ing measurements. This work can therefore be considered as +a case study demonstrating the very powerful capabilities of +fast camera imaging as a plasma diagnostics, notably for the +exploration of complex waves dynamics. +ACKNOWLEDGEMENTS +This work was partly supported by the French National +Research Agency under Contract No. ANR-13-JS04–0003- +01. We acknowledge support from the CNRS for the acquisi- +tion of the high-speed camera and useful discussions with V. +Désangles and G. Bousselin and warmly thank P. Borgnat for +advises on surrogate techniques. +AUTHOR DECLARATIONS +Conflict of Interest +The authors have no conflicts to disclose. +DATA AVAILABILITY +The data that support the findings of this study are available +from the corresponding author upon reasonable request. +Appendix A: Radial scale: camera imaging v.s. probe +The magnetic field ripple and parallax in our experimen- +tal set-up leads the camera lines of sight to cross regions of +different plasma parameters. The light recorded by camera, +resulting of an integration process along these lines of sight, +cannot be directly compared to probe measurements that are +performed at z = L2. +A transformation is implemented, modeling the integration +along the camera lines of sight of any plasma parameter that +is measured at z = L2. The details of this transformation are +provided in Ref.27. +Figure 12 shows the result of this artificial integration pro- +cess, applied to a test profile peaked at r = 5 cm (blue curve). +The resulting profile (red curve), expressed along the camera +imaging coordinate r∗, shows that what is seen on the cam- +era images at r∗ = 3.3 cm mainly corresponds to the plasma +parameter evolution that is located at r = 5 cm on the axis +z = L2. +0 +2 +4 +6 +8 +10 +0 +0.5 +1 +FIG. 12. Camera lines of sight integration process, applied to a test +profile measured at z = L2. + +11 +Appendix B: Bicoherence and confidence level +Bicoherence is a spectral analysis tool that is commonly +used in physics, for the detection of non-linear three waves +interactions. Bicoherence computation essentially consists in +extracting the frequency components of one of several signals, +and comparing their phases. The signal decomposition at the +basis of a bicoherence analysis can be done by Fourier trans- +form64,65 as it is the case in this work, or based on a wavelet +approach66,67. In this appendix, we first remind the basic prin- +ciple of bicoherence, and then explained how bicoherence is +computed in the particular case of camera images. Then we +provide the definition of a clear and mathematically meaning- +ful threshold, that is often lacking when bicoherence is used +onto experimental data in plasma physics. +Evaluate three wave interactions by bicoherence +Let us consider three signals x, y and z with correspond- +ing Fourier transforms ˆx, ˆy and ˆz. The cross bispectrum of +x, y and z is defined as a function of frequencies (f1, f2) +as : B(f1, f2) = ˆx(f1)ˆy(f2)ˆz∗(f1 + f2). If the frequency com- +ponents f1 and f2 of x and y respectively (with phases φx +1 +and φy +2) are involved in a three-wave interaction with the fre- +quency component f1 + f2 of z (with a phase φz +1+2) the phase +difference between these signals is a constant. Computing +the bispectrum onto successive reduced parts δt of the sig- +nals is therefore a way of measuring this phase locking, since +Bδt(f1, f2) ∝ exp−i(φx +1+φy +2−φz +1+2). The bicoherence is defined +by the normalized average over a statistically significant num- +ber of such bispectrum computations: +b2(f1, f2) = +|⟨ˆx(f1).ˆy(f2).ˆz∗(f1 + f2)⟩δt|2 +⟨|ˆx(f1).ˆy(f2)|2⟩δt⟨|ˆz(f1 + f2)|2⟩δt +If the signal frequency components previously mentionned +are perfectly uncorrelated, b2 corresponds to the average of +random complex numbers, and tends to cancel out. If those +frequency components are on the contrary perfectly phase +locked, the computations of Bδt(f1, f2) have a constant value +and b2 = 1. In the case of experimental data, neither case is +realistic, and a threshold value b2 +0 above which the bicoher- +ence can be considered significant needs to be defined (see +last paragraph of this appendix). +Bicoherence on camera images +With camera images that provide 2D spatio-temporal sig- +nals, bicoherence can be computed between the frequency +components of distinct modes m. Bicoherence allows to probe +the phases of signal components for given set of wave vector +and frequency (m, f). This analysis is applied on the present +camera images, following the work of Ref.46. For a given +radius r∗, let us denote the 2D Fourier decomposition of the +light intensity: +A(t,θ) = ∑ +n,p +a(fn,mp)ei(2π fnt−mpθ+φn,p) +The spectrum associated with a single mp mode is a part of +this decomposition: +ˆAmp(fn) = a(fn,mp)eiφn,p +Similarly to computations achieved for 1D signals, the +bispectrum is defined as a statistical averaging, over parts +of lengths δt of the signal. +In order to improve the sta- +tistical averaging here, the sum is also done over the sig- +nals from various radii r∗. +This double averaging pro- +cess is denoted ⟨.⟩r∗,δt. +The bispectrum between compo- +nents (m1,f1) and (m2,f2) is then defined as Bm1,m2(f1, f2) = +ˆAm1(f1). ˆAm2(f2). ˆAm1+m2(f1 + f2)∗, and the bicoherence is +computed as: +b2 +m1,m2( f1, f2) = +|⟨ ˆAm1( f1). ˆAm2( f2). ˆA∗ +m1+m2( f1 + f2)⟩r∗,δt |2 +⟨| ˆAm1( f1). ˆAm2( f2)|2⟩r∗,δt ⟨| ˆAm1+m2( f1 + f2)|2⟩r∗,δt +The bicoherence as it is implemented in our code takes +mode numbers m1 and m2 as an entry and explores all possible +three-wave interactions (m1, f1) + (m2, f2) ↔ (m1 + m2, f1 + +f2) in terms of frequencies f1 and f2. The operation is fixed +as an addition, and the result is in a form of a 2D map of +b2 +m1,m2(f1, f2), with [f1, f2] ∈ [0,Fs/2]2, Fs being the data sam- +pling frequency. Here for simplicity, the bicoherence applied +to camera images is simply denoted b2. +Definition of a threshold +The phase correlation between any set of experimental sig- +nals is likely to be imperfect or partial, leading to 0 < b2 < 1. +Moreover the absolute values of the bicoherence are relative +to each set of signals investigated: a general threshold value is +not relevant. A method to systematically determine the level +above which the value of b2 becomes physically meaningful, +that depends on each bicoherence computation, is therefore +needed. +A possible method consists in the creation of an artificial +set of signals, sharing the same characteristics than the orig- +inal signals, but without any preferential relation between its +frequency components. 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Slow dynamics in a turbulent von kármán +swirling flow. Phys. Rev. Lett., 99:054101, 2007. +60E. Brown and G. Ahlers. Rotations and cessations of the large-scale circula- +tion in turbulent rayleigh–bénard convection. Journal of Fluid Mechanics, +568:351–386, 2006. +61R. Monchaux, M. Berhanu, S. Aumaître, A. Chiffaudel, F. Daviaud, +B. Dubrulle, F. Ravelet, St. Fauve, N. Mordant, F. Pétrélis, M. Bourgoin, +P. Odier, J.-F. Pinton, N. Plihon, and R. Volk. The von kármán sodium ex- +periment: Turbulent dynamical dynamos. Physics of Fluids, 21(3):035108, +2009. +62O.E. Garcia, J. Horacek, and R.A. Pitts. Intermittent fluctuations in the +TCV scrape-off layer. Nuclear Fusion, 55(6):062002, 2015. +63A. Theodorsen, O. E. Garcia, R. Kube, B. LaBombard, and J. L. Terry. Uni- +versality of poisson-driven plasma fluctuations in the alcator c-mod scrape- +off layer. Physics of Plasmas, 25(12):122309, 2018. +64Ch. P. Ritz, E. J. Powers, T. L. Rhodes, R. D. Bengtson, K. W. Gentle, Hong +Lin, P. E. Phillips, and A. J. Wootton. Advanced plasma fluctuation analysis +techniques and their impact on fusion research. Rev. Sci. Instrum., 59:1739, +1988. +65S.-I. Itoh, K. Itoh, Y. Nagashima, and Y. Kosuga. On the application of +cross bispectrum and cross bicoherence. +Plasma and Fusion Research, +12:1101003–1101003, 2017. +66B. Ph. van Milligen, E. Sanchez, T. Estrada, C. Hidalgo, B. Brafias, B. Car- +reras, and L. Garda. Wavelet bicoherence: A new turbulence analysis tool. +Phys. Plasmas, 2:3017, 1995. +67S. Oldenbürger, F. Brochard, and G. Bonhomme. Investigation of mode +coupling in a magnetized plasma column using fast imaging. Physics of +Plasmas, 18:032307, 2011. +68Kin L. Siu and Ki H. Chon. On the efficacy of the combined use of the +cross-bicoherence with surrogate data technique to statistically quantify +the presence of nonlinear interactions. Annals of Biomedical Engineering, +37:1839–1848, 2009. + diff --git a/YtE2T4oBgHgl3EQfZAd1/content/tmp_files/load_file.txt b/YtE2T4oBgHgl3EQfZAd1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3af90ea9547b1aa4ac1b7f106f0ee75f2e3b6c8 --- /dev/null +++ b/YtE2T4oBgHgl3EQfZAd1/content/tmp_files/load_file.txt @@ -0,0 +1,1251 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf,len=1250 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='03860v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='plasm-ph] 10 Jan 2023 Nonlinear interactions of ion acoustic waves explored using fast imaging decompositions Simon Vincent1,2, Vincent Dolique1, and Nicolas Plihon1 1 Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France 2 École Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC), CH-1015 Lausanne, Switzerland (Dated: 11 January 2023) Fast camera imaging is used to study ion acoustic waves propagating azimuthally in a magnetized plasma column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The high speed image sequences are analyzed using Proper Orthogonal Decomposition and 2D Fourier Transform, allowing to evaluate the assets and differences of both decomposition techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The spatio-temporal features of the waves are extracted from the high speed images, and highlight energy exchanges between modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Growth rates of the modes are extracted from the reconstructed temporal evolution of the modes, revealing the influence of ion-neutral collisions as pressure increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Finally, the nonlinear interactions between modes are extracted using bicoherence computations, and show the importance of interactions between modes with azimuthal wave numbers m, m − 1 and −1, with m an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' INTRODUCTION The propagation of ion sound waves or ion acoustic waves is ubiquitous in plasmas and their non-linear interactions, pos- sibly leading to ion acoustic turbulence, is a widespread en- ergizing process in plasma physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The nonlinear evolution of ion acoustic waves (IAW) generically leads to instabilities and the development of non-linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For instance, IAW have long been observed in the solar wind, and related to the anisotropy of the electron distribution function1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In this context, IAW are driven unstable when the ratio of the elec- tron to ion temperature is larger than unity, as observed by the Helios spacecraft2, and very recently for oblique IAW by Parker Solar Probe3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Heating of energetic particles from ion acoustic turbulence was also proposed in the context of po- lar aurorae4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The non-linear evolution of ion acoustic waves into strongly non-linear structures such as solitons5 or double layers has been reported in electro-positive plasmas6 or elec- tronegative plasmas7, for which two branches of ion acoustic waves exist8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In the context of bounded plasmas, IAW ex- cited in sheaths may affect particle transport at low pressure9 or lead to strong ion heating10 when the ratio of the electron to ion temperature is larger than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' IAW may also be use- ful tools to probe sheath criteria in multiple ion plasmas11–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Technological plasmas may also trigger IAW, that, in return, affect their operation, as reported for Hollow Cathodes14,15, Hall thrusters16 and diverging magnetic nozzle thrusters17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In this article, we report on the observation of localized ion acoustic waves in a magnetized plasma column using high speed camera imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Our observations thus shed new light on the ion acoustic activity that has been previously reported in similar configurations18–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' We do not investigate the ori- gin of the IAW from parametric instability or waves interac- tions here, as was done in these previous investigations, but we analyse the spatio-temporal characteristics of the IAW us- ing mode decomposition from high-speed imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The IAW nonlinear interactions are quantitatively highlighted by means of bicoherence computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The experimental set- up is introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' II, the analysis of fast camera mea- surements by mode decomposition techniques is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In particular, we highlight the differences and com- plementarities of two different mode decompositions, namely Proper Orthogonal Decomposition and 2D Fourier Transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In section IV the waves observed by camera imaging are iden- tified to be IAW from the waves phase velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Finally the non-linear modes interactions are characterized and their nonlinear aspect is exhibited in section V and conclusions are drawn in section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' EXPERIMENTAL SET-UP AND DIAGNOSTICS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Experimental set-up The experimental set-up25 consists in a 20 cm diameter, 1 m long stainless steel cylindrical chamber containing an ar- gon plasma generated by a 1 kW, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='56 MHz radio frequency source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The coordinates will be referenced using a cartesian coordinate system, where z denotes the axial direction (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A cylindrical coordinate system (r,θ,z) will be used for the reconstruction of rotating modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The plasma base pressure p0 in the chamber is regulated at a fixed value, be- tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='8 mTorr and 2 mTorr by steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='1 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The plasma is created by an inductive source around a 11 cm diam- eter borosilicate tube connected at one end of the chamber (at z = 0 cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Three coils placed along the steel cylinder generate an axial magnetic field that confines the plasma (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This magnetic field is not perfectly homogeneous along z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For a current in the coils set here at 100 A, it has an averaged amplitude along the z-axis of B = 170 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Radial profiles of the plasma density n, the electron tem- perature Te and the plasma potential Φp are performed using a 5-tips probe26,27 and an emissive probe respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The probes were inserted radially at z = 49 cm (see blue dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' To keep the whole apparatus in a steady ther- mal state, the operation of the plasma is pulsed: the plasma is sustained over typically 5 seconds, during which data are acquired, with a repetition period of typically 30 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The ex- periment is fully automated to allow high repeatability and 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='8 m B RF source Coils 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='12 m Pump inlet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 m z y x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='11 m 5 0 5 5 0 5 0 100 200 300 5 0 5 5 0 5 0 5 10 15 θ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Left: sketch of the experimental set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The dashed blue line indicates where probe profiles are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Right: mean im- age of the light intensity collected by the camera (up), and standard deviation of the fluctuations around this average value (bottom), for p0 = 1 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' reproducibility of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The level of shot to shot repro- ducibility was ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='6% for the ion saturation current of a Lang- muir probe, with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2% (estimated from a series of 40 shots at the plasma column center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Radial scans of the plasma parameters were performed sequentially: each spatial point has been acquired during one plasma-pulse, and the probe is translated between two pulses, from the center of the plasma column (r = 0 cm) to its edge (r = 10 cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The results presented in this article mainly rely on high- speed imaging of the plasma emitted light, performed through a DN 200 borosilicate window closing the chamber at z = 80 cm (opposite to the source tube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A Phantom v2511 cam- era is placed along the z-axis, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 m away from this window, and the light intensity naturally radiated by the plasma Icam is captured at 200 kfps with a resolution of 256 × 256 px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A fil- ter around 750±5 nm is used in order to restrict the collected light to a single ArI spectral line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Examples of the mean inten- sity ⟨Icam⟩ and fluctuation standard deviation σ(˜Icam) images are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note that the plasma column is not per- fectly axisymmetric, and the fluctuations are of the order of 10% of the mean amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note also that the depth of field of the optical set-up being of the order of the length of the chamber (with a camera objective aperture set at f/4, and a focal length of 135 mm), the light intensity recorded by the camera is actually the result of an integration along the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Due to the magnetic field ripple and to the parallax, the di- rect comparison of the probes’ measurements (at z = 49 cm) and the camera images (where light is integrated along z), is not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' We thus introduce a distorted space (x∗, y∗, z) in which the camera lines of sight are parallel (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='27 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The camera images are hence observed in the plane (x∗, y∗), that may also be referenced as (r∗,θ) in a polar coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Radial profiles of the plasma parameters The top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 2 displays the radial profiles of the plasma density, electron temperature and plasma potential as 0 2 4 6 8 10 2 3 4 5 0 2 4 6 8 10 4 5 6 7 8 1016 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='16 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 103 104 105 10-6 10-4 10-2 100 0 2 4 6 8 10 4 3 2 1 0 0 2 4 6 8 10 4 6 8 10 12 1017 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Radial profiles of the density, electron temperature, and plasma potential mean values (top row) and fluctuations (middle row) at p0 = 1 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Plasma parameters measurements were made with 5-tips (n, Te) and emissive (Φp) probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Bottom: Power Spectral den- sity of the plasma density and light intensity fluctuations (computed from the average value of a 5×5 pixel box on the images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' a function of r for a pressure p0 = 1 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' As previously introduced, these profiles cannot be directly compared to the images in the (r∗,θ) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' However, assuming axisymme- try and invariance of the plasma parameter along magnetic field lines, a synthetic integration process detailed in Ref27 al- lows to map the probe measurements along r (at z = 49 cm) to the images expressed along r∗, enabling quantitative com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The density is approximately constant in a core re- gion of the plasma for r ≤ 4 cm (r∗ ≤ 3 cm) and then de- creases towards the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A clear peak can be seen around r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 cm (r∗ ∼ 3 cm) for the electron temperature, produced by the higher ionization rate of the RF inductive source close to the wall at r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 cm, z ∼ −10 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This higher temperature is also responsible for the higher light emission observed on the images at r∗ ∼ 3 cm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Finally, the plasma potential decreases from the center to r ∼ 4 cm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' r∗ ∼ 3 cm), and presents a strong positive gradient at the edge of the plasma column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This is responsible for an ⃗E × ⃗B drift that drives plasma rotation in the −⃗eθ direction, discussed later in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The radial profiles of the fluctuations of the plasma param- eters are shown in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The fluctuations are peaked at the edge of the plasma column at r ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 cm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' r∗ ∼ 4 cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Filtered light fluctuations recorded by fast camera are usu- ally considered to be a proxy for density fluctuations28–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For the magnetic field value of B = 170 G reported in this ar- ticle, simultaneous probe and camera measurements showed the fluctuations of density ˜n and light intensity ˜Icam at the 3 probe location to have very similar spectra, as shown in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' However, as it was shown in our pre- vious work27, for magnetic field values in the range 100 to 700 G, light intensity naturally radiated by low temperature plasmas are also highly correlated to the electron temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' IV, the comparison of experimental phase velocities with the theoretical ion acoustic speed nonetheless requires to assume ˜Icam to be a reasonable proxy for ˜n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' We therefore underline that this is a rather strong assumption, and that for further quantitative comparison ˜Icam should be interpreted as a combination of both ˜n and ˜Te - a task beyond the scope of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The strong spectral component observed at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='6 kHz is identified as a Kelvin-Helmholtz mode31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In the present work, we will focus on fluctuations observed between 50 and 70 kHz, which are unambiguously identified as ion acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' IMAGE ANALYSIS The first and most natural tool that comes to mind for the images analysis is the Fourier decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This is used later on;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' we prefer here to start the analysis by using an alter- native method, the Proper Orthogonal Decomposition (POD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note that before performing these decomposition, we choose to normalize each pixel by its fluctuations mean, as was done in similar conditions32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This choice greatly en- hances the contrast, allowing to nicely extract the relative am- plitudes of the modes (especially in the regions of low light intensity), but at the cost of losing information on the abso- lute amplitude of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note finally that in the following the light fluctuations recorded by the camera ˜Icam will simply be denoted I for eas- ier readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Proper Orthogonal Decomposition The POD consists in extracting the spatial structures that are dominant throughout time in a given data set I(r∗,t), where r∗ denotes space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This is done by computing the eigen- modes Ψi of the spatial autocorrelation of the time-averaged field ⟨I⟩(r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' These so-called spatial modes Ψi then define an orthonormal basis onto which the original data can be pro- jected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This can be written: I(r∗,t) = ∑ i σi ai(t)ψi(r∗) with σi ai(t) being the time evolution of the data projected on the spatial modes Ψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Here ai and Ψi are of norm unity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' the amplitude of the various components of the decomposition are thus given by the values of σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' One of the most interesting aspects of this decomposition is that it is done without any a priori on the shape of the Ψi structures: they simply come out from the computation pro- cess, as natural modes, contrary to Fourier analysis, which projects the data onto predefined spatial and temporal struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Hence POD might allow the emergence of structures with physical significance that are not well described by mere Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Thanks moreover to its simplicity of imple- mentation and computational speed when performed onto a discrete set of data, as is explained later, POD becomes a very attractive and efficient analysis tool for experimentalists, and has grown very popular in the last decades for the analysis of data from experiments or from numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note that depending on the field, this technique is also referred to as Karhunen-Loève decomposition (as a reference to the orig- inal mathematical theorem) or principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The set of spatial modes (Ψi) has the property of being the optimal basis for approximating the data I33: for any N, the norm of the projection of I onto (Ψi)1,N, which reads ∑N 1 σ2 i , is higher than the projection onto any other basis than one might choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For a given value of N, the spatial modes (Ψi)1,N can then be interpreted as the vectors that are the best suited to reproduce the information carried by I, in the most efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Applied to physical data, this property is even more in- teresting if the σ2 i have a clear physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The use of POD on experimental data has been initiated in fluid dy- namics, for the analysis of turbulent velocity fields34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In this context the norm ∑N 1 σ2 i of the projected data represents a ki- netic energy, and the modes Ψi may then be interpreted as the most important flow structures in terms of kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' POD has then been applied to spatio-temporal measurements of plasma fluctuations in tokamaks, either measured by sets of Langmuir probes35–37 or by means of soft x-ray emission38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' It has also been applied to camera imaging data of plasma natu- rally radiated light to exhibit spiral shaped structure generated by a m = 2 instability in a linear device39 and in a tokamak to highlight plasma response to resonant magnetic perturba- tions40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' More recently, the technique was used to characterize instabilities in the plume of a Hall thruster from fast imaging data41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Following this study, POD has been applied to decom- pose the camera imaging data of a plasma plume produced by a high-current hollow cathode42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Unfortunately, in the lat- ter cases, the physical interpretation of the Ψi vectors is not as straightforward as in fluid mechanics, since the extracted modes result from the decomposition of light intensity fields, which depends in a non trivial way on the plasma parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' And even by considering as a crude approximation ˜Icam ∝ ˜n, not much can be said on the norm ∑N 1 σ2 i in terms of physical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This does not mean the amplitude of the modes extracted from plasma emitted light is void of meaning, but simply that one has to be careful before thinking of it as a precise energy estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In this article POD decomposition is thus discussed in a purely qualitative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Finally, POD does not require any symmetry, which is a significant advan- tage over Fourier decomposition for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In the case of complex geometries, POD can be an efficient alternative for capturing the physical structures in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In practice, it can be shown that a direct extraction of the spatial modes Ψi associated to their temporal evolution ai, is in fact achieved by applying a mere singular value decompo- sition (SVD) to the matrix containing the data I, rearranged in such a way that one dimension of the matrix represents space, and the other time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This way of computing a POD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' also referred to as bi-orthogonal decomposition43 is the one 4 5 0 5 5 0 5 0 50 100 1 0 1 100 102 10-5 10-3 10-1 5 0 5 5 0 5 0 50 100 1 0 1 100 102 10-5 10-3 10-1 5 0 5 5 0 5 0 50 100 1 0 1 100 102 10-5 10-3 10-1 5 0 5 5 0 5 0 50 100 1 0 1 100 102 10-5 10-3 10-1 5 0 5 5 0 5 0 50 100 1 0 1 100 102 10-5 10-3 10-1 5 0 5 5 0 5 0 50 100 1 0 1 100 102 10-5 10-3 10-1 5 0 5 5 0 5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 1 0 50 100 1 0 1 100 102 10-5 10-3 10-1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 a) c) 1 5 10 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 1 b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Proper Orthogonal Decomposition modes (a) and singular values (b) of light intensity normalized fluctuations, for p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' c) Zoom on the evolution of a1 and a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' implemented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Each image (of p pixels) of a video con- taining q frames is rearranged to form a matrix I of size p×q to which a singular value decomposition is applied I = ΨΣAt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Ψ and A are orthonormal matrices of respective sizes p × p and q × q, and Σ is a matrix of the same size as I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The matrix Σ only contains diagonal elements, which are the decomposi- tion’s singular values σi: space \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 (I)i j \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb time � �� � = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Ψ1 Ψ2 Ψq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' σq \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb \uf8ee \uf8ef\uf8f0 ··· a1 ··· ··· a2 ··· ··· ap ··· \uf8f9 \uf8fa\uf8fb Figure 3 a) shows the result of a POD applied to a 100 ms time series of intensity fluctuations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 20000 images) recorded at a pressure p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The spatial modes Ψi(r∗) are displayed in the top row, the time series of the am- plitudes ai(t) in the the middle row, and the corresponding power spectral density S(f) in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The interpreta- tion of the decomposition requires to consider pairs of modes, such as IPOD 1,2 (r∗,t) = σ1a1(t)Ψ1(r∗)+ σ2a2(t)Ψ2(r∗), which yields rotating azimuthal waves of the type e−iωt−imθ (shown later in subsection III C), since the spatial modes Ψ1 and Ψ2 are shifted by a quarter wavelength and the temporal modes a1 and a2 are in quadrature (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 3 c))44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 3, the modes (Ψ1, a1) and (Ψ2, a2) correspond to a m = −5 rotating azimuthal wave, the modes (Ψ3, a3) and (Ψ4, a4) correspond to a m = −6 rotating azimuthal wave, and the modes (Ψ6, a6) and (Ψ7, a7) correspond to a m = −4 rotating azimuthal wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The rotation frequencies of the m-modes can be deduced from the spectra of the temporal signals ai, shown in the bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 3 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A very clear peak at frequency f = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='6 kHz is observed for the modes 1&2, capturing the m = −5 azimuthal wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The m = −6 wave (POD modes 3&4) has a f = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='1 kHz frequency, and the m = −4 wave (POD modes 6&7) has a f = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 kHz frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Section IV shows that these modes are ion acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The singular values σi of the modes are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 3 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The amplitudes of σ1 and σ2 are nearly identical (within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3%), and more than twice larger than the other singular val- ues, showing that the dynamics is dominated by a m = −5 rotating mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The time series ai(t) show sudden changes in amplitude, for instance at times t ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='8 ms, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 ms and 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 ms, corresponding to an energy exchange between modes m = −6 and m = −5, that will be investigated in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The relatively intense POD mode (Ψ5, a5), with a strong spec- tral component at f = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='6 kHz, was identified as a m = 3 Kelvin-Helmoltz mode31, not discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The POD analy- sis presented here was straightforward to implement, and pro- vide a very efficient way of extracting global features captured in a video sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Now we present the results of a Fourier analysis performed on the same data that complements the POD analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 2D Fourier Transform The 2D Fourier Transform (2D-FT) of a two variables func- tion f(x,t) reads: ˆf (k,ω) = �� f(x,t)e−i(k·x+ωt)dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 2D- FT is classically used to decompose the spatio-temporal sig- nals collected by azimuthally distributed probe arrays into az- imuthal modes45,46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Following many studies using camera imaging and performed in the context of linear plasma de- vices29,47–49 and plasma thrusters50,51, the 2D-FT is here per- formed on virtual rings at various radii r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For a given value of the radius r∗, a time series I(θ,t)|r∗ is extracted from the camera images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For each angle θn = 2πn Nθ with n ∈ [0 : Nθ −1], the value of the pixel at position (r∗, θn) is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The angle resolution of Nθ = 700 is chosen here, such that no in- 米5 10 5 0 5 10 0 20 40 60 80 100 22 24 26 28 30 32 Power spectrum FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Power spectrum of the raw light intensity fluctuations from camera imaging, taken on a corona of radius r∗ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 cm for p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Dispersion relations are plotted from experimental fits of the power spectrum maxima (red) and from the ion acoustic speed (black) (see section IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' terpolation is needed in the processes of converting either a ring of pixels in the image space (x∗, y∗) into a vector along the θ direction, nor in the inverse process, when reconstruct- ing images in the (x∗, y∗) plane from the mode decomposition results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Since the images are 2π periodic in the θ direction, the wave-vectors read m/r∗, with m an integer, and the 2D-FT of I(θ,t)|r∗ is computed as ˆIr∗(m, f) = �� I(θ,t)|r∗e−i(mθ+2π ft)dθdt with f the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The resulting 2D power spectrum Sr∗(m, f) = | ˆIr∗(m, f)|2 displays the amplitudes of light inten- sity fluctuations as a function of the spatial mode m and the frequency f, at a given radius r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note that at radius r∗ ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 cm on the images, the intensity I(θ,t)|r∗ is reconstructed from a corona of ∼ 400 pixels, which ensures a very good pre- cision in the extraction of the first modes m up to m ∼ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The power spectrum at r∗ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 cm and p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The observations are similar to those drawn from the POD analysis : the dominant mode is an m = −5 mode whose frequency is peaked at 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='6 kHz, and the other impor- tant modes are an m = −6 mode peaked at 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 kHz and an m = −4 mode peaked at 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='9 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note that this 2D power spectrum provides a dependence of the dominant frequency on the mode number, and can therefore be seen as an experi- mental dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This is used in section IV for the identification of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The full spatio-temporal evolution of any given m mode can also be extracted by 2D-FT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' To this end, the 2D Fourier Trans- form is computed for radii r∗ covering the full image (here 2D-FT are computed for r∗ = 2i px, i ∈ [1 : 64], instead of r∗ = [1 : 128] px, to limit memory storage and increase compu- tational speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For given m and r∗ values, the inverse Fourier Transform of ˆIr∗(m, f) is computed, resulting in the spatio- temporal signal of the m mode, at radius r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Performing this 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 1 0 20 40 60 80 100 4 5 Average amplitude Radial location t0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Time evolution of m-mode average amplitudes extracted by 2D-FT, for p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' inverse computation for all radii previously mentioned leads to the full spatio-temporal reconstruction of the m mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Ex- amples of snapshots of such reconstructed 2D-FT modes are shown and commented in subsection IIIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The average amplitude of the reconstructed modes is then computed along time, providing a global picture of the modes dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The global m-modes time evolution for p0 = 1,3 mTorr is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 5 (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' As already observed on the time signals of the POD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 3, clear exchange events can be observed involving modes m = −5 and m = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The m = −4 mode is seen to follow the dynamics of m = −5 while the m = −7 mode follows the dynamics of m = −6 mode, a feature that was not detected by the POD analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' From the reconstructed signals of the individual m-modes, the in- stantaneous mean radial profile is computed by an integration over θ, allowing for the computation of the radial location r∗ max where the wave amplitude is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Figure 5 (bottom) shows r∗ max for the m = −5 and m = −6 modes, that are highly correlated to the global dynamics of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Again this could not be deduced from POD, since spatial modes struc- ture are deduced from a time-averaged analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Figure 5 is further discussed in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Now let us compare the results obtained from both POD and 2D-FT analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Comparison between POD and 2D Fourier Transform Figure 6 shows snapshots of the m = −5 and m = −6 2D- FT reconstructed modes, as well as the corresponding modes reconstructed from the POD analysis IPOD 1,2 and IPOD 3,4 , for the experiment achieved at p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The snapshots are shown every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='06 ms following t0 = 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='27 ms, marking the beginning of an energy exchange between modes m = −5 and m = −6 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 5 (top)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The time interval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='06 ms repre- sents slightly more than 3 wave periods for the m = −5 mode, and nearly 4 wave periods for the m = −6 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The spa- 6 +c c 0 +c c 0 +c c 0 +c c 0 c = 1 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='72 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='80 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='38 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='42 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='66 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='78 c = 1 m = -5 I1,2 m = -6 POD I3,4 POD c = 1 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='75 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='73 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='61 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='39 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='53 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='61 c = 1 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 0 0 0 0 + + + FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Evolution of 2D-FT modes m = −5, m = −6 and POD re- constructed modes IPOD 1,2 and IPOD 3,4 , during the exchange event high- lighted by the dashed-black box in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 5, for p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' tial shape of the 2D-FT modes varies significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' On the contrary the shapes of the POD reconstructed modes remains almost unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This is actually expected since each of these signals is merely composed of the linear combination of two spatial fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note also that the spatial structures Ψi were extracted from the time-averaged data field (see subsec- tion III A): the reconstructed mode are unable to account for spatially localized variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Let us now compare the spatial structures provided by POD and 2D-FT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Figure 7 (top) shows time-average radial profiles of the modes for both decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The profiles are com- puted by an integration along θ, averaged over the 20000 im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 7 (bottom) shows the azimuthal profiles taken at a given time and for r∗ = 4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The comparison between POD modes (1&2) and the m = −5 2D-FT mode shows an almost perfect match (note that the match slightly decreases when the amplitude of the m = −5 mode strongly decreases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The com- parison between POD modes IPOD 3,4 and the m = −6 2D-FT mode give similar results, although with a lower agreement on the outward part (r∗ ≥ 4 cm) of the radial profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' An overall good match is observed between the lower amplitude POD modes (6&7) and the m = −4 2D-FT mode radial pro- files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The instantaneous azimuthal profile are not identical, with a phase shift up to ∼ π/8 depending on the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' These results show that, in the context of data having 2-π periodic- ity, POD and 2D-FT decompositions share several common features, while they do provide exactly the same knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note finally that for the computations performed here with a number of images N = 20000, the POD is twice faster than the 2D-FT (even though it was taken into account for the 2D- FT only 20 mode reconstructions, and half of the images pix- 0 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 1 0 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 1 0 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 1 0 /2 3 /2 2 1 0 1 0 /2 3 /2 2 1 0 1 0 /2 3 /2 2 1 0 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Comparison of (top) radial and (bottom) azimuthal profiles of the POD and 2D-FT modes, at p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr, for the (left) m = −5, (center) m = −6 and (right) m = −4 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' els as mentioned in subsection III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Then when only taking N = 2000, the POD is more than 12 times faster than the 2D- FT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The main strengths of both POD and 2D-FT techniques are summarized: POD is fast and easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' It is extremely simple to imple- ment, and it provides quick and direct results on the spatio-temporal dynamics of a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' POD is flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' It does not rely on any particular shape of the physical structure at play, nor on a specific loca- tion in the images analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' It will therefore be partic- ularly well suited to study for instance non-linearly sat- urated modes exhibiting a complex spatial or temporal pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note however that if the results can be particu- larly insightful, they might also be difficult to interpret (and in some cases even unusable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 2D-FT is explicit, hence robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Projecting the data onto a predefined set of wave modes (here for instance of the form e−iωt−imθ) prevents the emergence of unexpected structures, but it provides the results with a well identi- fied physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 2D-FT is exhaustive for linear mode analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Since it provides the full spatio-temporal evolution of linear wave, 2D-FT is particularly attractive to study their dynamics, exhibit the corresponding dispersion rela- tions, or use for instance the phase correlations between modes to study weakly non-linear interactions (see the use of bicoherence in section V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Both techniques can provide insightful and complementary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A recent preprint, reporting on the specific compar- ison between POD and 2D-FT applied to Hall thruster cam- era imaging52, concludes similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Applied to the present datasets, POD shows that the dominant physical structures are m-modes of the form e−iωt−imθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This indicates that the 2D-FT as implemented here, is an appropriate numerical tool for the mode decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Hence POD does not constitute a strong *****7 gain for further analysis here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In the following, for the iden- tification of the waves and the in-depth study of their weakly non-linear interactions, we will use the results from the 2D-FT decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' WAVES IDENTIFICATION The azimuthal waves detected by both POD and 2D-FT are now unambiguously identified as ion acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A series of high-speed imaging acquisitions was performed for pres- sures p0 in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2] mTorr by steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='1 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For each value of the pressure, the radius r∗ max at which the wave amplitude is maximal is deduced from the time-average of the raw images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The experimental phase velocity vφ is de- termined by a linear fit of the most energetic modes observed on the spectrum Sr∗max(f,m) as f(m) = vφm/(2πr∗ max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A typ- ical linear fit is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 4 for p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The ex- perimental phase velocities vexp φ are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 8 as red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The errorbars are estimated by the combination of the uncertainties on the fit on S(m, f), and on the evaluation of r∗ max (r∗ max(t) fluctuates around its mean value with a standard deviation of ∼ 3%, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 5 (bottom)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' These experimental phase velocities are compared to the theoretical ion acoustic speed cs = � eTe/mi, with e the ele- mentary charge and mi the ion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The computation of the latter requires careful estimates of Te where the phase velocity is measured on the high-speed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note that at z = 49 cm, where the probe measurement is performed, the radial posi- tion that is best representative of what is seen at r∗ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 cm on the images is in fact at r = 5 cm (see Appendix A and for a detailed explanation see27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A detailed pressure scan of the electron temperature Te was performed with the 5-tips probe at a radius r = 4 cm, and from a finely resolved radial scan at p0 = 1 mTorr27,31, we have Te(5 cm) ≈ Te(4 cm)+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Therefore, from the measured values Te(4 cm), Te(5 cm) is evaluated to lie in the range [Te(4 cm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='Te(4 cm)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5] eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The resulting theoretical ion acoustic speeds cs(p0) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 8 (gray area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The experimental phase velocities follow the trend of cs(p0), with values shifted down by approximately 700 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This is well explained by a Doppler shift due to the plasma column rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The plasma column indeed rotates, as was reported previously53, where the electric drift ⃗E ×⃗B B2 = −∇rφp/B⃗eθ was shown to overcome the diamagnetic drift −Ti n ⃗∇n ×⃗B B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Two damping mechanisms also need to be ac- counted for: ion-neutral friction and effective friction due to ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The ion-neutral collision frequency reads νin = nnσinvth,i, with vth,i = � eTi(eV)/mi, nn being the neutral den- sity and Ti the ion temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' We consider nn ≈ p0/kBTn with Tn(p0) = 350 K, σin = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='6 × 1018 m−2 from experi- mental cross sections54, and Ti ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 eV using previous LIF measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The effective friction due to the ionization originates from ions created with a temperature much lower than the surrounding Ti and depends upon the ionization frequency νiz, computed as νiz = nnKiz,0T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='59 e exp(−εiz/Te), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='8 2 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Comparison between the experimental phase velocity (red dots), the ion sound velocity cs (black curve), and its Doppler shifted values (green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' with Kiz,0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='34 × 10−14 m3/s and εiz = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='44 eV, and Te in eV55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A global damping factor K is then given31,53 as K = 1 + �νin + νiz ωci �2 , with ωci the ion cyclotron frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This finally gives a background azimuthal rotation of the ions as vi0,θ ≈ −(1/K)∂rφ(r)/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The rotation velocity vi0,θ is estimated from the experimen- tal profiles φp(r) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 2 (measured at p0 = 1 mTorr, and assuming variations with pressure within ±20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The estimated values of nn and Ti are considered to be bounded within ±10%, and Te is estimated from T r=4cm e (p0) as ex- plained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The results for the estimate of cs + vi0,θ are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 8 (green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In spite of all the approxima- tions made, the comparison between experimental phase ve- locities and the Doppler shifted values of cs provides a very satisfactory agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This allows us to identify with great confidence the azimuthal waves observed at B = 170 G as ion acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' An interesting feature is that the ion acoustic waves travel in the positive θ direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' opposite to the E × B drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' We stress here that adding the ion background velocity to the classical ion acoustic wave speed is a crude approxima- tion, deemed sufficient here for the purpose of wave identifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' However, a careful calculation would require to compute a complete dispersion relation from the governing equations, which couple in a complex way and prescribe direct analytical computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Indeed the effect of an ion background velocity on the ion acoustic phase velocity is likely to be coupled with other effects such as electron magnetization or friction with the neutrals, leading to computations well beyond the scope of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Interestingly, we observed that the ion acoustic waves are only observed over a narrow range of magnetic field values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For B = 80 G no clear wave emerges from the fluctuations of the plasma density or emitted light intensity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' on the other hand for B ≥ 300 G low frequency waves develop31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 8 7� �� 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 0�� 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 ��� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Left: time evolution of m-modes average amplitudes, ex- tracted by 2D-FT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Left: fit of the 2D-FT m = −5 mode growth rate at an exchange event with m = −6 mode, for p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The fit is done on a selected interval of the raw data (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The red curve is the result of a filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Right: Evolution of the growth time scale of m = −5 mode, evaluated during exchange events, as a function of the pressure p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' MODES DYNAMICS AND INTERACTIONS The spatio-temporal dynamics and the non-linear nature of the energy exchanges between the ion acoustic modes, as clearly shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 5, are now described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Growth rates of ion acoustic modes The time series shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 6 is taken around the ex- change event highlighted at t0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' At time t0 the am- plitude of the m = −6 mode is close to its maximum, while the amplitude of the m = −5 mode is close to its minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' At time t0 + 18 ms, the amplitude of the m = −6 mode has decreased close to its minimum value, and the m = −5 mode dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Figure 5 (bottom) shows that the radial position of the dominant mode (either the m = −5 or the m = −6 mode) is indeed very stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' On the other hand, the radial position of the low amplitude mode strongly fluctuates around its equi- librium value (with standard deviations around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='6 cm for the m = −5 mode and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 cm for the m = −6 mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The exchange events observed for p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr between modes m = −5 and m = −6 (Figures 3 and 5) are similarly ob- served at p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='1 mTorr and p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='9 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The timescales of the exchange events are now determined at these three val- ues of the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This is done by fitting the mode amplitude Am as exponentially growing: Am ∝ exp(t/τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Figure 9 (left) shows a typical fit around t0: the green part shows the interval over which the raw signal (blue) is fitted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' a low-pass filtered signal is shown for clarity (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Figure 9 (right) shows the resulting values of τ found for the m = −5 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The growth- time τ significantly increases with the pressure, its value dou- bles from p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='9 mTorr to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This is interpreted as being the result of an increased friction from the neutrals at higher pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note that this observation of a decrease of the ion acoustic wave growth rate with increasing pressure is consistent with theoretical predictions9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 0 10 20 30 10-3 10-2 10-1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Residence time PDF, obtained for a neutral pressure p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='30 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Residence time distribution The statistics of the transitions between the m = −5 and m = −6 modes were obtained in a new set of experiments, per- formed at a lower sampling frequency (20 kfps)56 over longer times (2 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This allows to extract the time evolution of the modes average amplitude, extracted by 2D-FT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The probability distribution function of the residence time of the m = −4, m = −5 and m = −6 modes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 10, for a total duration of 4 seconds (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' more than one thou- sand transitions between modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The distributions are com- patible with an exponential distribution, which implies that the transition events are not correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Such distributions of residence times or waiting times are ubiquitous to transitions observed in aerodynamics57, turbulent flows58,59 or convec- tion60, to the waiting time between reversals in dynamo ex- periments61, or the turbulent dynamics of the scrape-off layer in tokamaks62,63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For all modes, the probability distribution function is com- patible with a functional fit of the form e−t/τ, with τ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='4 ms for m = −4, and τ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='0 ms for m = −5 and τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 ms for m = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' As already observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 5, the m = −4 mode is tied to the m = −5 mode, resulting in similar pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Fig- ure 5 also shows that the system is more often dominated by a m = −5 mode, which results in an exponential pdf with a larger characteristic time for the m = −5 mode as compared to the m = −6 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' High speed imaging of the dynamics of the plasma allows to probe long-time statistics of the waves dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' It opens the possibility to probe the evolution of the characteristic residence time as a function of the control parameters (for instance pressure), possibly shedding light to the physical processes leading to exchange events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note that the dominant mode (and the associated characteristic time) was observed to strongly evolve with pressure (data not shown and beyond the scope of this article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 9 4 � 50 55 60 65 0 5 10 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 � 0 20 60 8\x0e 105 106 10 \x0f a) b) c) \x10 \x11 50 55 60 65 0 5 10 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 \x12\x13\x14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Maps of the threshold b2 0 (a) and bicoherence b2 (b), cor- responding to the three-wave interaction (m = −5) + (m = −1) ↔ (m = −6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' c) Frequency power spectra of modes m = −1, m = −5 and m = −6 from 2D-FT computed at r∗ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' B = 170 G and p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Non-linear behaviour In order to further assess the non-linear nature of the dy- namics between the dominant ion acoustic modes, the bico- herence b2(fm=−5, fm=−1) is computed for the three wave in- teraction (m = −5) + (m = −1) ↔ (m = −6), and shown in Fig 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note that to increase statistics, the 2D-FT at all radii 1 ≤ r∗ ≤ 5 around the wave maximal amplitude are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' More details on the bicoherence computations are provided in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The threshold map shown in Fig 11 a) was computed using a basic surrogate technique where the phases of the 2D-FT signal are randomly mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This yields bico- herence values for signals without any preferential phase re- lations, from which a threshold value of max(b2 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='12 is estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Figure 11 b) shows the map of b2(fm=−5, fm=−1) with fm=−5 and fm=−1 the frequencies of modes m = −5 and m = −1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For the sake of visibility, the ar- eas of bicoherence high values are highlighted by gray con- tours (defined at 40% of the maximum value of a Gaussian filtered b2 map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Most of the bicoherence highest values lie around the diagonal fm=−5 + fm=−1 = 65 kHz, that is the dominant frequency of the m = −6 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This reveals the strong non-linear behaviour of the (m = −6, f ∼ 65 kHz) mode component, which interacts with m = −5 and m = −1 modes via continuous sets of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The points dis- played as red dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 11 b) are also enlarged for clar- ity: they correspond to b2 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='36, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' more than three times the threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The points for which fm=−1 = 0 kHz, and fm=−5 ∈ [64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='4] kHz correspond to frequency com- ponents of the m = −5 mode being fed by the high amplitude of the (m = −6, f ∼ 65 kHz) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Note that these interactions are not the dominant process characterizing the energy exchanges detailed in subsection V B, since they only involves frequency components of the m = −5 mode around 65 kHz, with a low energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The point at fm=−5 = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='4 kHz and fm=−1 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 kHz however corresponds to the interac- tion: (m = −5, f = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='4) + (m = −1, f = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2) ↔ (m = −6, f = 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='7) which involves the dominant frequency components of the m = −5 and m = −6 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The very high bicoherence value at this location (b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='39) definitively establishes the non- linearity of the interactions between the ion acoustic modes (m = −5, f = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='4 kHz) and (m = −6, f = 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='7 kHz), at the origin of the transitions observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 11 c) shows the frequency spectra of the m = −6, m = −5 and m = −1 modes involved in the three-wave interactions described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' These spectra correspond to 1D cuts along the frequency axis of the 2D-FT spectrum shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' These spectra clearly display the non-linear feeding of the m = −5 mode by the high amplitude m = −6 mode around 65 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The non-linear feeding of modes m = −1 and m = −6 by the high amplitude m = −5 mode around 55 kHz is also visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The component (m = −6, f = 55 kHz) is then non-linearly interacting with (m = −5, f = 44) kHz and (m = −1, f = 11 kHz), as can be deduced by the high values of b2(fm=−5 ∼ 44, fm=−1 ∼ 11) from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 11 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The computation of other bicoherence maps (not shown here) reveals additional non-linear behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The bicoher- ence computation of the (m = −6)+ (m = −1) ↔ (m = −7) coupling unambiguously shows that (m = −6, f = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 kHz) non-linearly interacts with (m = −7, f = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='0 kHz) via an (m = −1, f = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='8 kHz) mode component (with b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='36 > 3max(b2 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Similarly, bicoherence computation of the (m = −4) + (m = −1) ↔ (m = −5) coupling highlights that (m = −4, f = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 kHz) and (m = −5, f = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='4 kHz) modes non- linearly interact via (m = −1, f = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='2 kHz) (with b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='38 > 3max(b2 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' As a last example, the bicoherence map 10 for the interaction (m = −4) + (m = −2) ↔ (m = −6) does not exhibit high values indicating the absence of non-linear interaction between the corresponding ion acoustic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' It however reveals that the frequency components (f = 55 kHz) of (m = −4) and (m = −6) modes (resulting from the spread of the m = −5 mode, visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 4) are non-linearly linked via the (m = −2, f = 0 kHz) mode component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Thanks to the rich spatio-temporal information provided by camera imaging and to the use of bicoherence, the weakly non-linear interactions are clearly highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In particular the existence of three-wave interactions between ion acoustic modes m = −p, m = −p − 1 and m = −1, for p ∈ [4,5,6], is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' CONCLUSION We have presented the first report of temporally and spa- tially entirely resolved ion acoustic waves in a magnetized plasma column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The ion acoustic waves were observed by means of fast camera imaging in a low temperature argon plasma column, with dominant azimuthal mode numbers m = −4, m = −5 and m = −6 depending on the neutral pressure that was varied from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='8 mTorr to 2 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Two image analysis techniques, namely proper orthogo- nal decomposition (POD) and 2D Fourier transform (2D-FT), were presented and thoroughly compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' These tools are found to be complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' POD is easy to implement and adaptable to any type of data, and useful to provide a fast overview of the underlying dynamics of a given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This helps focusing in a second time on a more precise and targeted analysis, that 2D-FT can then provide, yielding detailed and unambiguous information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Using 2D-FT analysis of high speed images, the ion acous- tic waves were found to rotate in opposite direction to the global E ×B drift of the plasma column, with a phase velocity Doppler shifted by this actual electric drift velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The dynamics of the dominant ion acoustic modes was then explored using the 2D-FT decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Growth rates, which extraction was made possible by the camera high tem- poral resolution, were found to decrease as pressure increases, following previous numerical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A detailed analysis was then carried out in the particular case of p0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' At this pressure the exchange dynamics between dominant modes m = −5 and m = −6 was shown to be of a bistable nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' More generally the weakly non-linear nature of the m = −p and m = −p − 1 mode interaction (p ∈ [4,5,6]), in- volved in a three-wave interaction with a m = −1 mode, was demonstrated by means of bicoherence computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Finally we emphasize that, except from probe measure- ments that were needed for the wave identification, all the re- sults that were presented exclusively rely on fast camera imag- ing measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This work can therefore be considered as a case study demonstrating the very powerful capabilities of fast camera imaging as a plasma diagnostics, notably for the exploration of complex waves dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work was partly supported by the French National Research Agency under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' ANR-13-JS04–0003- 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' We acknowledge support from the CNRS for the acquisi- tion of the high-speed camera and useful discussions with V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Désangles and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Bousselin and warmly thank P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Borgnat for advises on surrogate techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Appendix A: Radial scale: camera imaging v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' probe The magnetic field ripple and parallax in our experimen- tal set-up leads the camera lines of sight to cross regions of different plasma parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The light recorded by camera, resulting of an integration process along these lines of sight, cannot be directly compared to probe measurements that are performed at z = L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A transformation is implemented, modeling the integration along the camera lines of sight of any plasma parameter that is measured at z = L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The details of this transformation are provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Figure 12 shows the result of this artificial integration pro- cess, applied to a test profile peaked at r = 5 cm (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The resulting profile (red curve), expressed along the camera imaging coordinate r∗, shows that what is seen on the cam- era images at r∗ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='3 cm mainly corresponds to the plasma parameter evolution that is located at r = 5 cm on the axis z = L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='5 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Camera lines of sight integration process, applied to a test profile measured at z = L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 11 Appendix B: Bicoherence and confidence level Bicoherence is a spectral analysis tool that is commonly used in physics, for the detection of non-linear three waves interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Bicoherence computation essentially consists in extracting the frequency components of one of several signals, and comparing their phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The signal decomposition at the basis of a bicoherence analysis can be done by Fourier trans- form64,65 as it is the case in this work, or based on a wavelet approach66,67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In this appendix, we first remind the basic prin- ciple of bicoherence, and then explained how bicoherence is computed in the particular case of camera images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Then we provide the definition of a clear and mathematically meaning- ful threshold, that is often lacking when bicoherence is used onto experimental data in plasma physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Evaluate three wave interactions by bicoherence Let us consider three signals x, y and z with correspond- ing Fourier transforms ˆx, ˆy and ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The cross bispectrum of x, y and z is defined as a function of frequencies (f1, f2) as : B(f1, f2) = ˆx(f1)ˆy(f2)ˆz∗(f1 + f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' If the frequency com- ponents f1 and f2 of x and y respectively (with phases φx 1 and φy 2) are involved in a three-wave interaction with the fre- quency component f1 + f2 of z (with a phase φz 1+2) the phase difference between these signals is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Computing the bispectrum onto successive reduced parts δt of the sig- nals is therefore a way of measuring this phase locking, since Bδt(f1, f2) ∝ exp−i(φx 1+φy 2−φz 1+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The bicoherence is defined by the normalized average over a statistically significant num- ber of such bispectrum computations: b2(f1, f2) = |⟨ˆx(f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='ˆy(f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='ˆz∗(f1 + f2)⟩δt|2 ⟨|ˆx(f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='ˆy(f2)|2⟩δt⟨|ˆz(f1 + f2)|2⟩δt If the signal frequency components previously mentionned are perfectly uncorrelated, b2 corresponds to the average of random complex numbers, and tends to cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' If those frequency components are on the contrary perfectly phase locked, the computations of Bδt(f1, f2) have a constant value and b2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In the case of experimental data, neither case is realistic, and a threshold value b2 0 above which the bicoher- ence can be considered significant needs to be defined (see last paragraph of this appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Bicoherence on camera images With camera images that provide 2D spatio-temporal sig- nals, bicoherence can be computed between the frequency components of distinct modes m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Bicoherence allows to probe the phases of signal components for given set of wave vector and frequency (m, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This analysis is applied on the present camera images, following the work of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' For a given radius r∗, let us denote the 2D Fourier decomposition of the light intensity: A(t,θ) = ∑ n,p a(fn,mp)ei(2π fnt−mpθ+φn,p) The spectrum associated with a single mp mode is a part of this decomposition: ˆAmp(fn) = a(fn,mp)eiφn,p Similarly to computations achieved for 1D signals, the bispectrum is defined as a statistical averaging, over parts of lengths δt of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' In order to improve the sta- tistical averaging here, the sum is also done over the sig- nals from various radii r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This double averaging pro- cess is denoted ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content='⟩r∗,δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The bispectrum between compo- nents (m1,f1) and (m2,f2) is then defined as Bm1,m2(f1, f2) = ˆAm1(f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' ˆAm2(f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' ˆAm1+m2(f1 + f2)∗, and the bicoherence is computed as: b2 m1,m2( f1, f2) = |⟨ ˆAm1( f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' ˆAm2( f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' ˆA∗ m1+m2( f1 + f2)⟩r∗,δt |2 ⟨| ˆAm1( f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' ˆAm2( f2)|2⟩r∗,δt ⟨| ˆAm1+m2( f1 + f2)|2⟩r∗,δt The bicoherence as it is implemented in our code takes mode numbers m1 and m2 as an entry and explores all possible three-wave interactions (m1, f1) + (m2, f2) ↔ (m1 + m2, f1 + f2) in terms of frequencies f1 and f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The operation is fixed as an addition, and the result is in a form of a 2D map of b2 m1,m2(f1, f2), with [f1, f2] ∈ [0,Fs/2]2, Fs being the data sam- pling frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Here for simplicity, the bicoherence applied to camera images is simply denoted b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Definition of a threshold The phase correlation between any set of experimental sig- nals is likely to be imperfect or partial, leading to 0 < b2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Moreover the absolute values of the bicoherence are relative to each set of signals investigated: a general threshold value is not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A method to systematically determine the level above which the value of b2 becomes physically meaningful, that depends on each bicoherence computation, is therefore needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A possible method consists in the creation of an artificial set of signals, sharing the same characteristics than the orig- inal signals, but without any preferential relation between its frequency components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The bicoherence of this artificial set of signals is then computed, providing a lower limit for the values of b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' This type of method is called surrogate tech- nique68, and can be very sophisticated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Here we use a very basic version of the surrogate techniques: the phases of each 2D-FT spectra are randomly mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The bicoherence compu- tation applied to this modified data defines a threshold map b2 0(f1, f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Then for simplicity we take the maximal value max(b2 0) and define it as a global threshold value for the real bicoherence computation b2(f1, f2) of this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 1D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Ion acoustic wave studies near the presheath/sheath boundary in a weakly collisional argon/xenon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Plasma Sources Science and Technology, 17:015012, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 13A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Hala and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Hershkowitz.' 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shear control of a weakly magnetized plasma column using current injection by emissive electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Journal of Plasma Physics, 87:905870308, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 54A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Phelps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' The application of scattering cross sections to ion flux models in discharge sheaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Journal of Applied Physics, 76:747, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 55M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Lieberman and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Lichtenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Principles of Plasma discharges and materials processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' John Wiley and Sons, 2 edition, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 56Note that with the lower sampling frequency of 20 kfps, the extracted IAW modes with frequencies ∼ 70 kHz can not be resolved temporally, which prevent the distinction between modes +m and −m for a given integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' However it is observed that in the same conditions, with higher sampling frequency acquisition, the (m=+p) mode amplitude is negligible in front of the (m=-p) amplitude for p = [4,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' We therefore use at Fs = 20 kfps the sum of the amplitudes of extracted modes +p and -p, to estimate the amplitude of mode (m = -p), for p = [4,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 57A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Gayout, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Bourgoin, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Plihon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Rare event-triggered 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Investigation of mode coupling in a magnetized plasma column using fast imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Physics of Plasmas, 18:032307, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' 68Kin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Siu and Ki H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Chon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' On the efficacy of the combined use of the cross-bicoherence with surrogate data technique to statistically quantify the presence of nonlinear interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} +page_content=' Annals of Biomedical Engineering, 37:1839–1848, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfZAd1/content/2301.03860v1.pdf'} diff --git a/_NE2T4oBgHgl3EQfQwY9/content/tmp_files/2301.03773v1.pdf.txt b/_NE2T4oBgHgl3EQfQwY9/content/tmp_files/2301.03773v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6ad62a5a91bea5eab43366483bfb6e00785434f4 --- /dev/null +++ b/_NE2T4oBgHgl3EQfQwY9/content/tmp_files/2301.03773v1.pdf.txt @@ -0,0 +1,2057 @@ +SiFall: Practical Online Fall Detection with RF Sensing +Sijie Ji +sijie001@e.ntu.edu.sg +Nanyang Technological University +Singapore, Singapore +Yaxiong Xie +yaxiongx@buffalo.edu +University at Buffalo +Buffalo, New York +Mo Li +limo@ntu.edu.sg +Nanyang Technological University +Singapore, Singapore +ABSTRACT +Falls are one of the leading causes of death in the elderly people +aged 65 and above. In order to prevent death by sending prompt +fall detection alarms, non-invasive radio-frequency (RF) based fall +detection has attracted significant attention, due to its wide cover- +age and privacy preserving nature. Existing RF-based fall detection +systems process fall as an activity classification problem and as- +sume that human falls introduce reproducible patterns to the RF +signals. We, however, argue that the fall is essentially an accident, +hence, its impact is uncontrollable and unforeseeable. We propose +to solve the fall detection problem in a fundamentally different +manner. Instead of directly identifying the human falls which are +difficult to quantify, we recognize the normal repeatable human +activities and then identify the fall as abnormal activities out of the +normal activity distribution. We implement our idea and build a +prototype based on commercial Wi-Fi. We conduct extensive ex- +periments with 16 human subjects. The experiment results show +that our system can achieve high fall detection accuracy and adapt +to different environments for real-time fall detection. +CCS CONCEPTS +• Human-centered computing → Ubiquitous and mobile com- +puting systems and tools; • Computer systems organization +→ Real-time systems; • Applied computing → Health care in- +formation systems; • Computing methodologies → Machine +learning. +KEYWORDS +Self-supervised Learning, Wireless Sensing, Real-time System, Adap- +tive Segmentation, Fall Detection, Device-free +ACM Reference Format: +Sijie Ji, Yaxiong Xie, and Mo Li. 2022. SiFall: Practical Online Fall Detection +with RF Sensing. In ACM Conference on Embedded Networked Sensor Systems +(SenSys ’22), November 6–9, 2022, Boston, MA, USA. ACM, Boston, MA, USA, +15 pages. https://doi.org/10.1145/3560905.3568517 +1 +INTRODUCTION +Fall is an important global public health issue [37]. Every year there +are approximately 37.3 million fall-related injuries that require +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +SenSys ’22, November 6–9, 2022, Boston, MA, USA +© 2022 Association for Computing Machinery. +ACM ISBN 978-1-4503-9886-2/22/11...$15.00 +https://doi.org/10.1145/3560905.3568517 +Figure 1: Diversity of falls. +medical attention and directly cost $34 billion [7]. Clinical reports +show that timely treatment (<1 hour) can prevent deaths from fatal +falls [58]. Therefore, an effective fall detection system is necessary +to facilitate timely treatment and benefit the current aging society +where more and more elderly people are living alone [12]. +Existing fall detection solutions can be classified into two cate- +gories: wearable-based solutions and device-free solutions. Medical +research has reported that wearable-based solutions do not work +well in practice due to the burden of carrying and charging those +devices from time to time [16]. In contrast, device-free solutions in- +cluding computer vision (CV) based, acoustic-based, and RF-based +are more user-friendly. Among them, the CV-based solutions cannot +work under dim light conditions, occlusions and often compromises +user privacy. The acoustic-based solutions are limited by its sensing +range (<4.5m) [56] and possibly subject to restriction by ambient +loudness (<40dB SPL) [33]. However, RF-based solutions are not +constrained by the above and also cost-effective as they take the +advantage of existing ubiquitous communication infrastructures +such as WiFi APs. +Existing RF-based fall detection systems [38, 51, 53, 57] consider +falls as a type of normal human activity and applies traditional +human activity recognition method to identify the falls out of simi- +lar activities such as sitting, sleeping and jumping. Generally, the +solution consists of off-line training and on-line inference. During +the off-line training, the system builds up a model based on feature +engineering [38, 57] or machine learning [51, 53], to separate the +falls from other human activities. The RF signals are collected for +training purposes when the human being performs a set of pre- +defined activities, such as falling, sitting and jumping. The system +then applies the trained model to identify falls from the received +signals. All existing solutions implicitly assume that human falls in- +troduce reproducible patterns to the RF signals which can be captured +by the trained model and used to differentiate the falls from other +activities. +In this paper, we revisit such a problem and argue that the sig- +nal patterns introduced by human falls are full of randomness and +consequently hard to be fully captured by trained templates. Our +key intuition is that the human fall, by its nature, is an accident +arXiv:2301.03773v1 [cs.HC] 10 Jan 2023 + +external factor +level-change +Slip +Stumble +Fall on the floor +Lying on the bed +internalfactor +α change +Lost consciousness +Lost balance +Fall on the floor +Sit to the chairSenSys ’22, November 6–9, 2022, Boston, MA, USA +Sijie Ji, Yaxiong Xie, and Mo Li +that is unforeseeable and the human reaction is highly uncon- +trollable, introducing highly dynamic disturbance to the wireless +signals. Specifically, as depicted in Figure 1, there are diverse causes +of human falls, such as a stumble, a slip, loss of consciousness, loss +of balance, a sudden fright, etc, which may result in randomness, +e.g., a stumble or a slip may result in displacement of the human +body. In contrast, a person stays at the same place if he loses his +consciousness. In addition, the free range of movement in the joints +of human body brings in another level of randomness when the +human being cannot properly control his behavior during the falls. +Extracting representative features of the human falls becomes im- +practical because of such uncontrolled randomness. Even collecting +adequate data is challenging because one person can hardly repeat +real and uncontrollable falls. +With the above observation, in this paper we handle the fall +detection problem in a fundamentally different manner. Instead of +seeking features to characterize the unforeseeable and uncontrol- +lable human falls, we turn to solving an easier problem: recogniz- +ing normal repeatable human activities including but not limited +to jumping, sitting, and walking. We formulate fall detection as +adaptive anomaly detection and identity an abnormal activity that +cannot be classified as any of the known activities as a fall. Our +hypothesis is that after an adequate time period of training, a self- +supervised learning process will eventually perfect the model to dif- +ferentiate uncontrolled falls from other repeated controlled human +activities. To prune the search space and speed up the convergence +of the model training, we apply analyzable signal processing to +early filter out non-fall human activities with distinguishable sig- +nal features. Our observation suggests that falls change the status +of the human body in a short period of time and thus introduce +high frequency components to the signal variations. We feed the +identified suspicious fall-like activities to a deep neural network +called FallNet to recognize the true falls. Specifically, the FallNet +trains an auto-encoder [22] to learn a compressed representation +of normal fall-like activities. When used for inference, the auto- +decoder is only able to accurately reconstruct the normal fall-like +activities but not real human falls. The FallNet, therefore, identifies +the activities that result in large reconstruction error as falls. After +deployment, the FallNet is continuously updated using the freshly +collected data in a self-supervised manner, so it evolves to adapt +to the local propagation environment and the particular human +subjects that the system monitors. We expect that the FallNet will +eventually perfect its detection accuracy and false alarm rate over +time. +To realize our idea, we implement a Self-supervised Incremental +learning Fall detection system, SiFall. To the best of our knowledge, +SiFall is the first RF-based fall detection system that can work in +real time for online fall detection on a daily basis across different +human, different environments and different types of activities. +SiFall possesses the following three advantages: +• SiFall works with daily human activities in runtime - WiFi +CSI samples are dynamically processed, segmented, and dis- +criminated to detect ongoing "falls". +• SiFall’s self-learning process can adapt to the variation of +human subjects, environment, and types of falls. The core +anomaly detection model of SiFall evolves during its use; +• SiFall separates the signal processing from its machine learn- +ing model, which is designed to be lightweight and may +easily be accommodated at the edge devices. +The developed SiFall prototype has been comprehensively eval- +uated with a total amount of over 92 hours of test data collected +from 16 human subjects of different ages and genders. During our +experimental evaluation, SiFall is able to achieve 98.3% accuracy in +a real-world setting with extensive movements. During a continu- +ous three-day adoption in a normal living environment, SiFall is +able to detect 94.1% falls with only one false alarm in the end. +2 +CHALLENGES AND OPPORTUNITIES +This section first discusses the challenges in developing a practical +RF-based fall detection system and then presents the key observa- +tions and opportunities. +2.1 +Challenges +2.1.1 +Fall Ambiguity. There is no uniform quantitative definition +of "fall" in medicine, biology, or physics. According to the World +Health Organization (WHO), fall is a subjective term, which is +measured by the level of discomfort in the human body after a +person accidentally lies on the ground or other low level [37]. As a +result, it is hard to identify a quantified signal template to feature +the "fall" when performing RF sensing. Besides, the orientation and +the reflection surface of the human body may impact the reflected +RF signal which leads to inter-activity similarities (e.g., falling v.s. +lying down) [1, 31]. Other factors including deployment layout +and individual difference may also contribute to the ambiguity in +defining and quantifying the "fall" in the RF signal space. +2.1.2 +Data Scarcity. Recent advances in deep learning allow learn- +ing powerful discriminative models from a number of represen- +tative samples [14], which may bypass the difficulty in defining +precise signal templates of "falls". However, since the "falls" are +high exceptional human activities that often occur uncontrollably, +it is extremely difficult to obtain sufficient repeatable real-life data +samples containing different types of falls, leading to a data scarcity +issue. Most existing fall detection studies depend on learning from +artificial fall samples collected from the laboratory environment +and thus may have gaps in detecting real falls that take place in +daily life. The lack of fall data may also result in class distribution +skews where the learned model is biased towards the majority types +of falls and may have poor predictive performance for other types +of falls. As long as the types of falls are not sufficiently emulated, +the learned model may be unreliable with poor generalizability. +2.1.3 +Unstructured Input Signal. Human motions, even of the same +type, may last for different durations of time, and as a result, the +relevant RF signals are unstructured and of different lengths. The +processing of variable-length input signals is very different from +processing fixed-length data samples in many machine learning +models. Real-time processing of variable-length sequences is par- +ticularly difficult because data structurization techniques like se- +quence padding or dynamic template mapping can hardly be applied +in real time [50]. In addition, real-time segmentation of the RF sig- +nals from consecutive activities is also challenging, the inaccuracy + +SiFall: Practical Online Fall Detection with RF Sensing +SenSys ’22, November 6–9, 2022, Boston, MA, USA +(a) STFT segments from the same testing subject +#Person 1 +(b) STFT segments from different testing subjects +#Person 8, 9, and 16 +Figure 2: STFT segments across activities and testing subjects. +of which may lead to inconsistency of features in the machine learn- +ing model. Most existing fall detection solutions assume pre-defined +fixed-length RF signal input. +2.2 +Opportunities +While the practical challenges suggest extreme difficulties in learn- +ing the RF templates of human falls, we observe that there is an +opportunity on the other hand to categorize human daily activities +as they are usually repeatable and there exist plenty daily data +samples for training a model to describe them. To showcase such +an observation, Figure 2b visualizes the extracted WiFi signal fea- +tures after short time Fourier transformation (STFT) across various +human activities (details in §3.2). Figure 2a depicts the STFT seg- +ments collected from the same testing human subject and Figure 2b +depicts those collected from three different human subjects. It is +obvious to see that the daily human activities give very consistent +STFT patterns, e.g., the kneeling and sitting patterns in Figure 2a. +Even across different human subjects the patterns of the same daily +activities remain consistent, e.g., the kneeling and sitting patterns +in Figure 2b. The "falls" however appear highly varied and non- +repeatable across the types, e.g., the "stop fall", "walk fall", and "slow +fall" (details in §4.2), as well as the testing human subjects. The +above observations suggest that it is reliable to train a model to +accurately describe the normal daily activities and as an oppor- +tunity to identify "falls" as abnormal outlier output from such a +model. As there are plenty of daily activities to see when the sys- +tem is deployed in reality, a self-supervised learning scheme may +continuously perfect the trained model with improved accuracy in +distinguishing the falls from normal daily activities. +3 +SYSTEM DESIGN +A desired fall detection system should have the following charac- +teristics: (i) it must work in real-time and detect falls with run-time +data input; (ii) it must be able to evolve itself without involving +human efforts to label the data samples; (iii) it must adapt to envi- +ronment and different users. In this section, we present the design +of SiFall, a system that accommodates the above design consider- +ations. We begin with the system overview followed by fall-like +activity segmentation and the design of FallNet. +3.1 +Overview +SiFall consists of a front-end to process RF signals and a back-end +server to train the neural network model and detect the fall, as +shown in Figure 3. +SiFall’s front-end collects WiFi channel state information (CSI) +measurements, denoises the CSI and extracts the dynamic compo- +nent of the CSI to obtain an approximate RF-signal description of +human movement. Finally, a lightweight algorithm is used to quan- +tify the motion intensity and segment the RF signals accordingly +(§3.2). In the end, SiFall applies short-time Fourier transformation +(STFT) to derive the time-frequency spectrum of each piece of +segmented RF signal clip and supplies the STFT spectrum to the +back-end server for fall detection. The purpose of the front-end +signal processing is two folded: to early rule out normal activities +that possess clear daily activity features, and to present segmented +RF signals with data cleansing. Typical daily human movements +without high-frequency components are expected to be filtered out +to narrow down the learning space of the back-end neural network +model. +In the back-end server, a self-evolving deep neural network called +FallNet takes the segmented RF signal as input and identify the +falls from the normal fall-like activities. The FallNet is designed +based on the auto-encoder framework to do the self-supervised +learning where the encoder learns a nonlinear mapping from the +unstructured RF-signal space to uniformed compact latent feature +space and thus addresses challenge from the unstructured input +signal. The decoder learns the mapping from the latent space back +to the RF-signal space with the goal of reconstructing the original + +60 +SlowFall +WalkFall +StopFall +Kneel +Sit +40 +30 +20 +10 +0 +123456 +12345 +2345 +1 +12345 +60 +SlowFall +WalkFall +StopFall +Kneel +Sit +40 +30 +20 +10 +0 +123456 +2345 +60 +SlowFall +WalkFall +StopFall +Kneel +Sit +40 +30 +20 +10 +3456 +1 +234 +5 +3 +2 +3 +Time (s) +Time (s) +Time (s) +Time (s) +Time (s)60 +Kneel +Sit +Fall +50 +#Person16 +4 +30 +20 +10 +0 +3 +4 +5123456 +60 +Kneel +Sit +Fall +50 +#Person8 +40 +30 +20 +10 +0 +2 +23456 +60 +Kneel +Sit +Fall +50 +#Person9 +40 +30 +20 +10 +0 +3 +、 +3 +6 +Time (s) +Time (s) +Time (s)SenSys ’22, November 6–9, 2022, Boston, MA, USA +Sijie Ji, Yaxiong Xie, and Mo Li +Figure 3: Overview of SiFall +RF-signal as closely as possible. After training with a large num- +ber of repeated regular human activities, the FallNet establishes a +Gaussian mixture distribution of normal human activities in the +latent space (§3.3) and thus is capable of accurately recognizing +and recovering the RF-signal clips of normal activities. When de- +ployed, the FallNet classifies the fall-like RF-signal clips that can be +well reconstructed as normal daily activities and those RF-signal +clips that cannot be reconstructed as falls. The FallNet is continu- +ously updated with RF-signal clips of repeatedly-appearing normal +human activities fed from the front-end (§3.4). +3.2 +RF signal Segmentation +3.2.1 +CSI Extraction and Denoising. The received WiFi signal can +be modeled as: +𝑌 (𝑓 ,𝑡) = 𝐻 (𝑓 ,𝑡) × 𝑋 (𝑓 ,𝑡) +(1) +where 𝑋 (𝑓 ,𝑡) represents the signals carried at subcarrier fre- +quency 𝑓 and time point 𝑡 and 𝐻 (𝑓 ,𝑡) denotes the CSI value at +𝑓 . The CSI describes how the RF signals are transformed by the +current wireless channel - the amplitude attenuation and phase +rotation of different frequency components due to multipath re- +flection, diffraction, and scattering by objects in the environment. +On top of that, RF chipset processing at WiFi transceivers may +introduce additional distortion and noises [59, 60]. Therefore, we +perform necessary data cleaning to eliminate the impact of the +hardware imperfections. +The CSI 𝐻 consists of a static part induced by ambient environ- +ment 𝐻𝑠 and a dynamic part related to human movement 𝐻𝑑. CSI is +also subject to WiFi hardware distortion 𝐻ℎ. Therefore, we model +the overall CSI as: +𝐻 (𝑓 ,𝑡) = (𝐻𝑠 (𝑓 ,𝑡) + 𝐻𝑑 (𝑓 ,𝑡)) · 𝐻ℎ (𝑓 ,𝑡) += (𝐻𝑠 (𝑓 ,𝑡) + 𝐻𝑑 (𝑓 ,𝑡)) · 𝜀1 (𝑡) 𝑒𝜀2(𝑓 ,𝑡)+𝜀3(𝑡)+𝜀4 +(2) +where 𝜀1(𝑡) is the amplitude scaling caused by automatic gain +control (AGC), 𝜀2(𝑓 ,𝑡) represents the phase offset introduced by the +combination of packet detection delay (PDD), sampling frequency +offset (SFO) and sampling time offset (STO), 𝜀3(𝑡) is the phase offset +caused by the carrier frequency offset (CFO), and 𝜀4 is the initial +Figure 4: Static CSI Amplitude (upper left), Dynamic CSI Am- +plitude (upper right) and the Extract Channel Dynamic 𝑆(𝑡), +gray is the ground truth static. +phase offset of the radio chains. We utilize relatively clean CSI +amplitude and mitigate the impact of noisy CSI phase by calculating +the conjugate multiplication of CSI as ˆ𝐻 (𝑓 ,𝑡) for each subcarrier: +ˆ𝐻 (𝑓 ,𝑡) ≡ 𝐻 (𝑓 ,𝑡) 𝐻 (𝑓 ,𝑡) = 𝜀2 +1 (𝑡) |𝐻𝑠 (𝑓 ,𝑡) + 𝐻𝑑 (𝑓 ,𝑡)|2 , +(3) +The resulting ˆ𝐻 (𝑓 ,𝑡) is still affected by the amplitude scaling 𝜀1(𝑡) +that AGC introduces. To visualize the impact of 𝜀1(𝑡), we collect +CSI measurements from a static environment and calculate CSI +amplitude across subcarriers in Figure 4 (upper left), from which +we see that the CSI amplitude curves across subcarriers are similar +but not identical. The reason is that the amplitude scaling factor +𝜀1(𝑡) is time-varying but consistent across subcarriers. We note that, +because of the amplitude scaling factor, the CSI amplitude of a single +subcarrier ˆ𝐻 (𝑓 ,𝑡) is time-varying even when the environment is +static and thus cannot capture the dynamics introduced by the +human motion. +3.2.2 +Capturing Channel Dynamics. We use the variations of the +CSI amplitude curve to capture the channel dynamics introduced by +human motion. To illustrate the intuition, we plot the CSI amplitude +when the human is moving in Figure 4 (upper right), from which +we see that the shape of amplitude curve varies significantly in +non-static environment. We use cosine similarity to quantify the +similarity between consecutive CSI measurements: +𝑆(𝑡𝑛) = ⟨ ˆ𝐻 (𝑡𝑛), ˆ𝐻 (𝑡𝑛−1)⟩ +| ˆ𝐻 (𝑡𝑛)|| ˆ𝐻 (𝑡𝑛−1)| +(4) +where ˆ𝐻 (𝑡𝑛) = [ ˆ𝐻 (𝑓1,𝑡𝑛), · · · ˆ𝐻 (𝑓𝑀,𝑡𝑛)] represents the CSI ampli- +tude vector of all 𝑀 subcarriers sampled at 𝑛-th time point. We plot +the calculated S(t) for CSI collected from both static and non-static +environment in Figure 4 (bottom), from which we see the variation +of the 𝑆(𝑡) accurately captures the dynamics of the wireless chan- +nels, because the normalization operation to compute similarity +essentially removes the effect of AGC and thus 𝜀1(𝑡) is removed. +We note that, the similarity 𝑆(𝑡) is affected by CSI sampled at two +time point, so its value may also vary when the sampling interval + +Detecting&Segmenting +Self-learning +Signal Preprocessing +FallNet +Model Updating +MovementDetection +pe(x) +Fall-like Segmentation +FALLALARM70 +Amplitude(dB) +65 +static +dynamic +60 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +Subcarrier +Subcarrier +0.9 +S +0.8 +0 +1 +2 +3 +4 +Time(s)SiFall: Practical Online Fall Detection with RF Sensing +SenSys ’22, November 6–9, 2022, Boston, MA, USA +(a) Spectrum +(b) 𝑎(𝑡) +Figure 5: The STFT spectrum and the corresponding acceler- +ation of channel dynamic. +varies, adding another unpredictable factor. In our implementation, +we introduce a reference vector �𝑟 = [1, . . . , 1] and derive 𝑆(𝑡) as +the similarity between the ˆ𝐻 (𝑡𝑛) and the reference vector. We use +the variance of S(t) across 0.1s and above a threshold Γ to detect +the human movement. +3.2.3 +Segmenting Fall-like Activities. To forge efficient online de- +tection and relatively consistent feature extraction, we propose a +heuristic algorithm to segment fall-like activities from continuous +monitored RF signals. The key observation is that a fall and fall-like +activity (e.g. Sit, Jump and Squat) usually comes to a full pause +at the end of the motion before transitioning to next movement, +which may be due to the direction change of the movement (vertical +to horizontal). Similar observation has been reported in previous +studies with WiFi [53] and RFID [10] signals as well. Therefore, +we segment 𝑆(𝑡) in a backtracking manner from an observation of +motion pause, which is easier to capture than the actual start of +an activity. Meanwhile, as the channel dynamic is caused by the +human movements, we derive an approximate acceleration descrip- +tor 𝑎 to help further filter out daily movements accompanied by +a pause with low-intensity (e.g. walk and stop). In addition, we +assume the RF signals collected after a fall are also useful and thus +a greedy algorithm is used to keep monitoring the 𝑆(𝑡) to window +the entire fall-like activity. +The approximate 𝑎 is computed by using the relationship [43]: +𝑎(𝑡) = d2 +dt2 𝑆(𝑡) = 𝜆 d +dt 𝑓𝐷 (𝑡) +(5) +where 𝜆 is wave-length of the subcarrier wave, 𝑓𝐷 (𝑡) is the Doppler +frequency shift. We approximate d +d𝑡 𝑓𝐷 (𝑡) by computing STFT of +𝑆(𝑡) as STFT is used to capture the frequency component in a small +time duration and the frequency component change is caused by the +relative movement between transceivers and the reflecting human +body. Denote the STFT spectrum as S ∈ R𝐹×𝑇 , where 𝐹 is the fix +frequency bins and 𝑇 is the number of time bins. For each time +bins, we have a vector of approximate d +d𝑡 𝑓𝐷 (𝑡) denote as �𝑣, �𝑣 ∈ R𝐹 . +We search 𝑚𝑎𝑥(𝑣) as the function of indices of frequency bins that +exceed the noise floor via dynamic programming such that: +max(v) = argmax +𝑓1,···,𝑓𝑇 +𝑇 +∑︁ +𝑖=1 +S𝑖,𝑓𝑖, +s.t. |𝑓𝑖 − 𝑓𝑖−1| <= 1;𝑖 = 2, · · · ,𝑇 . +(6) +𝑚𝑎𝑥(𝑣) ≜ d +d𝑡 𝑓𝐷 (𝑡) so 𝑎(𝑡) may be obtained by calculating 𝜆𝑚𝑎𝑥(𝑣). +We consider 𝑎(𝑡) > Θ = 2.5 indicates a potential fall-like activity as +the human normal acceleration in walking is less than 2.5𝑚/𝑠2 [63]. +Figure 5a is the STFT spectrum derived from 𝑆(𝑡) contained in Fig- +ure 4 and Figure 5b is the 𝑚𝑎𝑥(𝑣) derived from the STFT contained +in Figure 5a. +When applied in real time, once the variance of 𝑆(𝑡) is estimated +below Γ, suggesting a pause after a move, SiFall records the time +as 𝑡𝑒𝑛𝑑 and then searches if there exists 𝑎(𝑡) > 2.5 in the past +five seconds (𝑡 ∈ [𝑡𝑒𝑛𝑑 − 5,𝑡𝑒𝑛𝑑]) and records the 𝑚𝑎𝑥(𝑎) and its +corresponding time as 𝑡𝑚𝑎𝑥. +An online greedy change point detection algorithm [28] is ap- +plied to continuously update 𝑡𝑒𝑛𝑑 for one second afterwards to +obtain 𝑡∗ +𝑒𝑛𝑑: +C +� +𝑆(𝑡𝑒𝑛𝑑 : 𝑡∗ +𝑒𝑛𝑑) +� ++ 𝛽 < C +� +𝑆(𝑡𝑒𝑛𝑑 : 𝑡∗ +𝑒𝑛𝑑+1) +� +(7) +where C stands for the error of the linear regression and 𝛽 is a +penalty value. The rationale behind is that the RF signals collected +after the fall-like activity may also contain useful information for +identifying the fall. +In the end, SiFall extracts 𝑆(𝑡) between +� +𝑡𝑚𝑎𝑥 − 3𝑠,𝑡∗ +𝑒𝑛𝑑 +� +and per- +forms STFT on 𝑆(𝑡) to obtain the fall-like segments. The additional +three-second time before 𝑡𝑚𝑎𝑥 is used to include as complete fall- +like activity as possible because we would rather contain redundant +signal data as compared to missing any possible important data. +Note that the lengths of STFT segments and their corresponding +spectrums are variable because the time between 𝑡𝑚𝑎𝑥 and 𝑡∗ +𝑒𝑛𝑑 +depends on the duration of the captured activity. Following that, +the STFT segment of the fall-like activity is supplied to the neural +network in the back-end for affirmative fall detection. Algorithm 1 +defines the whole backtracking segmentation process. +Algorithm 1: Fall-like Segmentation Algorithm +Input: 𝑆(𝑡); Threshold: Θ, Γ; Penalty: 𝛽; 𝑓 𝑠 +if movstd(S(t),fs/10) < Γ; then +record 𝑡 as 𝑡𝑒𝑛𝑑, S=STFT([S(t-5fs),...,S(t)]); +if 𝑚𝑎𝑥(𝑣) > Θ then +record 𝑡𝑚𝑎𝑥; +while 𝑡 < 𝑡𝑒𝑛𝑑+𝑓 𝑠 do +err(𝑡)=C (𝑡𝑒𝑛𝑑 : 𝑡); +if err(t) > err(t-1) + 𝛽 then +𝑡∗ +𝑒𝑛𝑑=𝑡 − 1 +continue; +temp = [𝑆(𝑡𝑚𝑎𝑥 − 3𝑓 𝑠),𝑆(𝑡∗ +𝑒𝑛𝑑)]; +seg = STFT(temp); +3.3 +FallNet Design +The difficulty now lies in identifying ongoing falls from those RF +clips of fall-like activities. This section elaborates on the design +of FallNet, which is able to further identify falls from the RF clips +of fall-like activities. Specifically, we learn the complicated distri- +bution of normal fall-like activities by a variational auto-encoder + +80 +Fall +60 +Freguency +Jump +40 +Squat +20 +0 +0.5 +1.5 +2 +2.5 +3 +3.5 +Time (s)10 +max(v) +Acceleration(m/s +smooth +7.5 +5 +2.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Time (s)SenSys ’22, November 6–9, 2022, Boston, MA, USA +Sijie Ji, Yaxiong Xie, and Mo Li +������ �� � � ������� � ��� +�� +� +� +�� ��� +σ +���� +μ +ɛ +iteration +constrain +� +��� ��� +��� � � �� � +Conv + IN + LeakyReLU +Pooling +Uppooling +Pooling Indices +encoder +decoder +Figure 6: The FallNet Architecture: the encoder, decoder and bottleneck layer. +based FallNet. When used for inference, the FallNet is only able +to accurately reconstruct the normal fall-like activities but not +real human falls. We, therefore, identify the activities that result +in large reconstruction error as falls. We first construct the core +encoder-decoder architecture of FallNet, which does not rely on +data annotation and is able to accept unstructured input data. Then, +we elaborate on some special designs of FallNet to cope with partic- +ular issues. Finally, we import the variational inference technique +to FallNet to make it more generalizable. +3.3.1 +FallNet Architecture. We design FallNet based on autoen- +coder architecture which is a well-known deep learning framework +to compress data without labels. The ability to compress data shows +its high ability to understand the intrinsic relationship between the +compressed data and the original data, hence, a trained encoder +is also widely used as a feature extractor. We train the encoder- +decoder only based on the fall-like STFT segments collected from +daily activities so the FallNet learns the representative features of +daily activities. When used for inference, the encoder-decoder is +able to fully reconstruct the signals of those repeatedly seen normal +activities. +Encoder. The input to the network is the signal clip of the 𝑖th +activity 𝑥𝑖 ∈ R𝐹×𝑇 (𝑖)×𝐶 from a total number of 𝑁 activities, where +the 𝐹 is a chosen frequency resolution of the STFT image,𝑇 (𝑖) is the +time duration of the activity, which might vary across activities, and +𝐶 is the number of spatial streams(between Tx and Rx antennas). +Therefore, the complete information of the three domains, i.e., time, +frequency and spatial, are fed into the FallNet. The encoder of +the FallNet learns a nonlinear transformation FE : X → Z that +maps the original data space X ⊆ R𝑚(𝑖) with variable dimensions +and inconsistency to a compact latent feature space Z ⊆ R𝑛 with +uniform dimension. 𝑚(𝑖) denotes the flattened dimension of 𝑥𝑖 and +𝑛 represents the dimension of the latent space of features that are +most representative to describe the activities such that: +z = FE (x, ΘE) +where ΘE is a set of parameters of the encoder. As the encoder +learns the most representative features and automatically filters +out the redundancy, it works well with the STFT segments, which +may be longer than the actual activities. +The ability of the encoder to project variable-length data space +into a uniform latent feature space is owing to our fully convolu- +tional network structure design of the building blocks. The convo- +lution operation itself intrinsically can cope with input of varying +lengths, although many people don’t notice this because the con- +volution operation is usually used to process images that are of +same length. The convolution operator in fact works on local tensor +regions and depends only on relative spatial coordinates determi- +nated by the convolution kernel size [19] (refer to Appendix B for +more details). As a result, when using the "same padding" [19] in a +convolution layer, for an input with dimension 𝐹 × 𝑇 (𝑖) × 𝐶, the +output will be with the dimension of 𝐹 ×𝑇 (𝑖) × 𝐶′, where the only +change is the channel dimension 𝐶′, depending on the number of +convolution filters. In particular, the encoder of FallNet consists of +five building blocks of decreased size that are stacked together. Each +building block consists of two convolution layers with instance nor- +malization (IN) [52], an activation function of LeakyReLU [61], and +a max-pooling layer. All convolution layers fix the convolution +filter size to 3 which simulates a larger filter while keeping the +benefits of smaller filter sizes in order to reduce the computational +overhead [48]. IN is used to cope with the antenna imbalance is- +sue. LeakyReLU is the activation function to bring in non-linearity +ability of the network and it can avoid the dying ReLU problem. +Max-pooling is used to achieve translation in-variance over small +spatial shifts in the input tensor [49]. The max-pooling layer will +decrease the size of the input to half so that the final output size +of each building block is 𝐹/2 ×𝑇 (𝑖)/2 × 𝐶′. At the end of the five +building blocks, we first average pooling the feature values along +the time dimension with an index to record its dimension. Note that +𝐶′ is determinated by the number of convolution filters which is +controlled by us and the 𝐹 is fixed, a fully connected layer hence can +be used to conduct channel-wise linear transformation to map the +tensor to a fixed-length vector 𝑧 with 𝑛 dimension that represents +the extracted features. +Decoder. The decoder learns to reconstruct the input signal 𝑥𝑖 +from the output 𝑧 of the encoder, such that +ˆx = FD (z, ΘD) +where ˆx is the reconstructed signal, and ΘD is a set of parameters +of the decoder. To reconstruct ˆ𝑥, the decoder needs up-sampling + +EZSiFall: Practical Online Fall Detection with RF Sensing +SenSys ’22, November 6–9, 2022, Boston, MA, USA +Figure 7: Up-pooling diagram. +oprations to map 𝑧 back to the size𝑚(𝑖) of the original input smaple +𝑥(𝑖). In consequence, the decoder and the encoder are symmetric +with the same number of building blocks, except that the max- +pooling layers at the encoder is replaced by up-pooling layers at the +decoder. As up-pooling [6] utilizes the 2-bit indices stored during +max-pooling operation in the encoding phase and up-samples the +feature map by filling the values directly to the index position and +zero-padding the remaining positions. It avoids parameter learning +to reduce the computation overhead. Figure 7 illustrates the up- +pooling operations.Another small detail is that the decoder first +uses the record index from the previous average pooling operation +to zero-pad the 𝑧 back to the dimension before the fully connected +layer, then goes through the five identical building blocks of the +decoder. +Consequently, the goal of the FallNet is to learn the parameter +sets of encoder and decoder satisfying: +� +ˆΘE, ˆΘD +� += arg min +ΘE,ΘD +E𝑥∼X +� +∥x − FD (FE (x, ΘE) , ΘD) ∥2� +It is worth noting that this learning process only needs the input +sample 𝑥 and does not require any labelled data, therefore, it can +benefit from substantial and easily accessible RF samples of daily +activities. +3.3.2 +Coping with Antenna Imbalance. CSI collcted from different +antennas may have different amplitudes, which lead to the imbal- +ance of the power of STFT spectrums. The removal of AGC impact +in the CSI denoising phase further amplifies this issue. As the 𝐶 +channels of input tensor corresponds to different Tx-Rx antenna +streams, the FallNet adopts IN that normalizes the antenna streams +with learnable affine parameters 𝛾 , 𝛽 to cope with the antenna +imbalance: +IN𝛾,𝛽 (𝑋) ≡ 𝛾 � +𝑋 + 𝛽, where � +𝑋 = +𝑋 − 𝜇 +√ +𝜎2 + 𝜖 +where 𝜇 and 𝜎2 are computed across spatial dimensions indepen- +dently for each channel so that every spectrum has the same range +of values. 𝜖 is a small constant added for numerical stability. Noted +that the FallNet removes the commonly adopted Batch Normaliza- +tion (BN), as the data samples in our case are generated online and +may follow different distributions. IN has the same characteristics +as BN does, which helps the entire neural network to alleviate gra- +dient saturation and accelerate convergence [24] (refer to Appendix +A for more details). +3.3.3 +Coping with RF Data Scarcity. Although the training of the +FallNet is free from data annotation, making it possible to con- +tinuously learn from daily fall-like activities, it is not realistic to +enumerate all possible fall-like activities. Besides, some types of +activities may be relatively dominant owing to specific user activity +patterns. As a result, FallNet may be prone to be overfitted. To +make the FallNet resistant to such overfitting and be generalized to +function properly, instead of using a vector 𝑧 with 𝑛 dimension to +represent the learned fall-like activity features, we adopt a bottle- +neck layer with stochastic sampling operation to make the FallNet +become probabilistic. +The reason for doing this is based on our observation (Figure 2) +that fall-like activities of the same type are similar, though not +identical. By introducing this prior knowledge, we can construct +the obtained samples with certain distributions and assume that +the same type of activities come from the corresponding distribu- +tion to obtain more general sample characteristics. We, therefore, +import such prior knowledge into the network, allowing the neu- +ral network to learn more generalizable features from the limited +data. In particular, we assume that each of the 𝑛 features of the RF +signals follows a normal distribution due to different body shapes +or orientations. Refer to Figure 2b to see that the same actions +performed by a single person or multiple persons have similarity +due to the kinematic consistency. Thus, in the feature space, sam- +ples from each normal activity group 𝐴𝑐 are supposed to follow +an 𝑛-dimensional Gaussian distribution as the activities from the +same group (e.g., sit, bow, or jump) are repeated and controlled. We +denote a certain activity group as 𝐴𝑐 with number of 𝑗 (𝑐) samples. +Ideally, all normal samples from different daily activities together +form a mixture distribution of Gaussian. +With such prior knowledge, we therefore impose the constraint +to FallNet’s learning process and force it to learn a mixture Gaussian +distribution over the latent feature space, rather than learning a vec- +tor of feature representations 𝑧 that may be over-fitted with limited +data samples. To this end, we modify the output of the encoder from +𝑧 to two vectors 𝜇𝑐 and 𝜎𝑐 that represents mean and variance of +the activity group Ac that each training sample belongs to, respec- +tively, where 𝜇𝑐, 𝜎𝑐 ∈ R𝑛, 𝑛 is the number of features. The FallNet +learns the two vectors to parameterize the feature distribution of +𝐴𝑐. A constrain loss is added to minimize the Kullback-Leibler (KL) +divergence between the learned disrtibution of the parametric rep- +resentation and the desired distribution 𝑝 (𝑧|𝑥 ∈ 𝐴𝑐) ∼ N �𝜇𝑐, 𝜎2𝑐 +� +such that: +Lc = −1 +2 +𝑛 +∑︁ +𝑖=1 +� +𝜇2(𝑖) + 𝜎2(𝑖) − log𝜎2(𝑖) − 1 +� +where 𝜇(𝑖) and 𝜎(𝑖) denote the 𝑖-th element of the 𝑛-dimensional +vectors 𝜇 and 𝜎. In such a way, each activity is modeled as a mul- +tivariate Gaussian distribution with 𝑛-dimensional features in the +latent space. Different activities have different mean vectors 𝜇𝑐 and +variance vectors 𝜎𝑐 to represent different Gaussian distributions. +As the number of samples increases, the hidden space gradually +forms a complex Gaussian mixture distribution: +𝑝𝜃 (𝑧) = +𝑐∑︁ 𝑗𝑐 +𝑁 𝑝 +� +𝑥 ∈ 𝐴𝑐 | 𝜇𝑐, 𝜎2 +𝑐 +� +If the latent distribution is valid, correspondingly, any of the latent +space samples from the distribution should be able to reconstruct 𝑥 + +0.1 +0.1 +1.2 +-0.7 +0 +0 +0.5 +0 +0.5 +-0.2-0.5 +x +1.3 +0.8 +1. +目 +0.3 +1.3 +0 +0 +0 +x +X +0.1 +0.4 +0.9-0.1 +-0.2 +0.4 +0 +0.4 +0 +0 +x +indices +values +-0.6 +0.1 +0.5 +0.3 +0 +0 +0.1 +0 +max-pooling +up-samplingSenSys ’22, November 6–9, 2022, Boston, MA, USA +Sijie Ji, Yaxiong Xie, and Mo Li +well. Therefore, the input of the decoder now becomes a 𝑧 that is +stochastically sampled from the corresponding 𝜇 and 𝜎 such that +ˆx = FD +� +𝛿z ∼ N +� +𝜇, 𝜎2� +, ΘD +� +where 𝛿 represents a random sampling operation. On the other +hand, the back-propagation of training neural network requires +deterministic operations at each neural network nodes which iter- +atively pass the gradients and apply the chain rule. The stochas- +tic sampling operation however is not a continuous function and +thus not differentiable to obtain the gradient. To make the neu- +ral network trainable, the FallNet adopts the reparameterization +technique [30]. It generates random 𝜀 from a standard normal dis- +tribution N (0, 1) independent of the neural network nodes. The +latent sample 𝑧 is obtained through scaling and transformation by +𝑧 = 𝜇 + 𝜎 × 𝜀. The reparameterization allows 𝑧 to be sampled from +the corresponding distribution of 𝜇 and 𝜎 at each iteration while +the random sampling itself is not involved in the training process. +As the sampled 𝑧 is deterministic at each iteration its gradient can +be back-propagated to train the entire neural. Consequently, the +objective of the FallNet is revised: +arg min +𝜇,𝜎,ΘD +E𝑥∼X +� +∥𝑥 − FD ((𝜇𝑥 + 𝜎𝑥 × 𝜀, ΘD)∥2� +,𝜀 ∼ N (0, 1) +In addition, the FallNet design also employs data augmentation +scheme to compensate the data scarcity and improve model gener- +ality. The FallNet imposes two specific augmentation schemes: (i) +To simulate a low SNR scenario, before being converted to STFT +spectrums, for each segmented 𝑆(𝑡), we add Gaussian white noises, +which equals to adding noises in the channel domain of the input +tensor; (ii) To alleviate the limitation of time resolution due to the +fixed STFT window length. Each input tensor 𝑥𝑖 goes through three +rounds of random horizontal shift [42], with the shifting length +smaller than the STFT window length. At the end we are able to +fabricate 24× the amount of original data to augment the training +size. +3.4 +Online Detection and Model Updating +After pre-training with a normal activity dataset 𝑋, the FallNet +has established the distribution of the anchor daily activities in the +latent feature space. Let each activity segment 𝑥 go through the +FallNet, we can derive the statistics of reconstruction error of the +dataset including its average 𝛼 and median𝛾. In the online detection +phase, the FallNet takes the real time segmented STFT samples for +inference in a single run, and measures its reconstruction error 𝑒. If +𝑒 > 2𝛼, it is detected as a fall and at the same time 𝛼 and 𝛾 remain +unchanged. If 𝛼 < 𝑒 < 2𝛼, the system takes it as a suspicious daily +activity and saves the segmented samples for feature reference, +but 𝛼 and 𝛾 are recalculated and updated accordingly. Once the +change of𝛾 exceeds a threshold, the system takes it as an indication +of significant change in the environment. If 𝑒 < 𝛼, the system +updates the 𝛼 and 𝛾 and then performs data augmentation where +a mini-batch of augmented data samples are fed to the FallNet for +retraining the model. Therefore, the system keeps evolving with +the feedback of reconstruction error 𝑒 and adaptively updates the +threshold 𝛼 to determine falls. +As the system runs in real-time, the incoming fall-like samples +for inference may bring two types of distribution shift, one being +the semantic shift caused by the individualized movement patterns +across people, the other being the covariance shift due to environ- +ment variation over time. As SiFall eliminates the environment +impact by extracting the dynamics of RF signals, the covariance +shift is well accommodated along with the continuous update of +the FallNet. The saved suspicious daily data samples are utilized +to deal with the semantic shift. Whenever an adequate amount +of suspicious daily data samples (i.e., 50 as set in our current im- +plementation) are collected, SiFall performs principal components +analysis (PCA) to reduce the dimension to 𝑛 and then performs +mean-shift clustering [41] to identify 𝑁 clusters. Two criteria are +applied to handle the cluster points, namely, representativeness +and diversity. We examine the largest cluster as it indicates many +repeatable activities which are unlikely to be human falls. SiFall +retrieves the signal segment of the centroid of the largest cluster, +produces 24× augmented data, and feeds that to the FallNet for +model retraining. SiFall also notices when there is a cluster that +is far away from other clusters. The cluster is taken as a potential +undiscovered user activity group and its signal segments are kept +for later examination when adequate amount of such suspicious +data are collected. The remaining signal segments are discarded and +the counter is updated till next time the number of saved samples +reaches 50. +Based on the above described mechanism of automatic model +update, SiFall does not require explicit human intervention for most +of the time. Only when a "fall" is detected SiFall triggers an alarm +for possible human intervention. The corresponding data samples +are saved with a timestamp regardless whether the detected "fall" +is a true positive or false positive. The human user may examine +the saved "fall" samples at any later time to decide whether they +are true positives in which case the samples are discarded, or false +positives in which case the samples are augmented and fed back to +the FallNet for retraining. +4 +EVALUATION +In this section, we evaluate the performance of SiFall. We first +introduce our experimental settings and then present the results. +4.1 +Experimental Setting +We implement SiFall with two commercial off-the-shelf (COTS) APs +as the Tx and Rx to collect the WiFi CSI, one laptop connected to +the Wi-Fi receiver to serve as the front-end edge server and one +back-end server. We use a camera to capture the ground truth. +Hardware. We use COTS COMPEX WPJ558 equipped with Atheros +SoC QCA9558 in the experiment. We let these two APs transmits +200 packets per second on a 20MHz channel in 2.4GHz frequency +band. We fix the Modulation and Coding Scheme (MCS) to reduce +packet loss and noises. We use Atheros-CSI-Tool [59] to collect raw +CSI data. The receiver forwards the collected CSI to the ThinkPad +T430 laptop with an Intel Core i5-3360M CPU to process the RF sig- +nals and generate STFT segments (as introduced in Section 3.1). We +use a Linux desktop computer equipped with Intel Core i9-9820X +CPU and one Nvidia 2080Ti GPUs to work as the back-end server + +SiFall: Practical Online Fall Detection with RF Sensing +SenSys ’22, November 6–9, 2022, Boston, MA, USA +Figure 8: The environment of the three testbeds and their floor plans. +to maintain the FallNet and perform real-time inference to detect +the falls. +Testbed. We test SiFall based on three testbeds - an emulated "bed- +room" with an enclosed space measured 4.32m × 8.24m for com- +prehensive evaluation (testbed 1), a real apartment room measured +7.85m × 4.47m for system adoption test on a daily basis (testbed +2), as well as a big open area measured 9.54m × 7.05m to test the +effective sensing range of the system (testbed 3). Figure 8 depicts +the three different testbeds. The marked Tx and Rx indicate the +locations of the WiFi Tx and Rx antennas. +Ground Truth. We use a camera to record the detailed human +activities at a frame rate of 30fps, and manually analyze the recorded +video clips to generate the ground truth. We use network time +protocol (NTP) to synchronise the time in the camera recordings +and the collected Wi-Fi CSI data. +FallNet Pretraining. We pre-train the FallNet with the data col- +lected intermittently during 3 months in testbed 1, including 1447 +sets of STFT segments of sitting, jumping, swinging, bowing, run- +ning, and other daily activities, augmented 24 times to produce a +total number of 34,728 samples. Correlation among raw samples is +removed by OpenCV, and the weight parameters are initiated by +kaiming initialization [21]. The model was trained by Adam [29] +optimizer on 4 Nvidia 2080Ti GPU for 2 hours. +Testing Subjects. We recruit 16 volunteers (11 males and five fe- +males) with ages between 21 and 56 to take part in our experimental +evaluation (with IRB approval). Table 1 summarizes the detailed in- +formation of all volunteers. The testing subjects are highly diverse +in their age, weight, and height. Specifically, the body weight of +our volunteers varies from 42kg to 100kg. Their body height varies +from 155cm to 186cm, and their age varies from 21 to 56 years old. +RT-Fall. We compare the performance of SiFall with RT-Fall [53], +which is, to the best of our knowledge, the only RF-based fall de- +tection system which claims being able to achieve real time fall +detection in practice. RT-Fall identifies fall-like activities based on +Table 1: Summary of the testing subjects +#Person +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 12 +13 14 15 +16 +Age +34 25 21 25 28 27 26 +25 +29 +27 25 29 26 22 52 56 +Height(cm) 165 167 177 188 184 171 173 155 186 173 165 175 172 166 173 155 +Weight(kg) 62 52 65 85 73 61 74 42 100 65 52 71 63 53 60 62 +Gender +M +F +M +M +M +M +M +F +M +M +F +M +M +F +M +F +a pre-defined threshold on the measured CSI phase difference be- +tween two Rx antennas and segments the collected CSI stream with +a fixed 3s time window. RT-Fall then feeds the derived statistical +phase and amplitude features of the CSI segment into a pre-trained +SVM model to identify falls. We reproduce the system and train an +SVM classification model of RT-Fall with the data collected from +our testbed, the same as what we use to pretrain our FallNet. +4.2 +End-to-end Evaluation +We first conduct intensive movement experiments with 12 subjects +and report the end-to-end performance. After that, the proposed +system components are evaluated based on the detailed experiment +results. +4.2.1 +Methodology. 12 testing subjects (#P1,#P6-#P16) are involved +to conduct the experiment in a sequential order. Each testing subject +is requested to move freely around one and half an hours inside +the bedroom testbed as depicted in Figure 8. We request each of +them to perform the following actions at their will when they move +around: "jump", "squat", "sit to the floor", "sit to the chair", "knee +down", and "bow" at least three times at different locations and +with different body orientations. Other than the requested type of +movements, they are free to perform any other activities at their +will. We summarize other fall-like movements that are hard to +quantify as "swing". +To mimic unconscious falls as much as possible while meeting +the IRB requirement on risk control, we set up a safety mattress and +experiment with the falls of three categories [4, 32, 46]: (1) for "walk + +A1 +A2 +A3 +RX +dSenSys ’22, November 6–9, 2022, Boston, MA, USA +Sijie Ji, Yaxiong Xie, and Mo Li +Table 2: Types of Falls +Types +Examples +"walk fall" +slip, stumble +scenes: rushing to answer the telephone, slipping in the bathroom, +and tripping over the cable, etc. +"stop fall" +lost balance, lost consciousness +scenes: coming out of bed, epileptic seizure, stroke, +and heart attack, etc. +"slow fall" +dizziness/vertigo, weakness +scenes: arthritis pain, transfer to a dim room, postural hypotension, +and vision disorder, etc. +fall", the subject is asked to walk around the mat and instantly fall on +the mat once a random alarm is triggered by us - the fall is performed +regardless the instant body orientation of the testing subject; (2) +for "stop fall", the subject stands still on the mat and tries to dodge +the tennis balls thrown at her - if she happens to fall the activity is +noted as a valid "stop fall", and as swing activity otherwise; (3) for +"slow fall", the subject keeps standing still until we give a random +alarm when she simulates a slow fall on the mat. Table 2 illustrates +the three categories of falls with corresponding real life scenes and +examples. It is worth noting that regardless of the type of falls, the +falling orientation is random during the experiments based on the +reaction of the subject. During the experiment, SiFall continuously +operates and each of the 12 testing subjects enters the bedroom in +sequence. The total experiment duration for all 12 testing subjects is +about 19.3 hours. The FallNet model is continuously maintained and +updated throughout the experiment. We evaluate the performance +with True Positive Rate (TPR) and False Positive Rate (FPR) metrics, +where TPR is true falls out of SiFall reported falls and FPR is falsely +reported falls out of other activities. The accuracy is calculated by +the percentage of correctly detected falls and non-falls against the +ground truth. +4.2.2 +Overall Performance. During the 19.3 hours experiment, SiFall +captures a total number of 1497 fall-like activities, of which 523 seg- +ments are intentional activities performed by the testing subjects +(including 123 falls and 400 required fall-like activities). Among +the 123 falls, 60 are "walk fall", 33 are "slow fall" and 30 are "stop +fall". We derive the TPR and FPR in about every 20 minutes and +plot the results over time in Figure 9, where TPR is represented +by the black solid line and FPR is represented by the balck dashed +line. Both the TPR and FPR vary over time as the FallNet model +continuously evolves when more training data are collected from +the testing subjects. We see a clear trend of improvement on both +the TPR and FPR. +First, the TPR improves quickly over time. From 83% at the be- +ginning of the experiment, the TPR constantly improves over time +and reaches 100% within 4 hours of operation, which demonstrates +SiFall’s capability in accurately identifying the abnormal falls from +normal daily activities. Second, the FPR of SiFall improves greatly +over time. The falsely reported falls by SiFall are 6.7 per hour in +the first two hours and eventually drops to below 1 per hour in +the last two hours of the experiment. While the TPR shows a clear +trend of improvement over time, the FPR occasionally fluctuates, +especially during the experiment of each individual testing subject. +That is mainly due to the fact that our experiment does not restrict +how each testing subject performs certain activities, and as a result +Figure 9: System end-to-end performance evolution over +time across different test subjects. +some testing subjects may choose to perform more activities simi- +lar to falls, and in different orders. For example, one (#P9) prefers +challenging SiFall system by performing more "sit on the floor" +activity which is more similar to "slow falls" and results fluctuated +FPR during his experiment. If we focus on the FPR statistics by the +end of each testing subject’s experiment (the gray line), we may +see steadily improved performance. At the end of the 19.3 hour +experiment, SiFall is able to achieve 100% TPR and 1.8% FPR. +We simultaneously run RT-Fall for comparison and plot the +achieved TPR and FPR of RT-Fall in red in Figure 9. We find that +during the real time operation RT-Fall achieves a much lower per- +formance, with its TPR of 64.9% and FPR of 49.2%. Since RT-Fall +does not have the ability to self-evolve, it cannot gain performance +over time and it fluctuates across different testing subjects. Overall +the comparative results show huge comparative advantage of SiFall +over the SOTA available real-time RF fall detection approach. +4.2.3 +FallNet Visualization. We visualize the FallNet input and out- +put to demonstrate the rationale when applying FallNet to detect +the falls. Specifically, we impose three checkpoints during the ex- +periment (as indicated in Figure 9 as CKPT1 to CKPT3, after the +test of subject #P6, #P11, and #P15, respectively). At each check- +point, we freeze the FallNet model and memorize it for detailed +investigation. We feed different RF signal segments collected from +normal activities and falls into the restored FallNet models at the +three checkpoints, respectively, to examine the reconstructed out- +put from the FallNet. In Figure 10, we plot the STFT segments of the +input signal and the reconstructed STFT segments by the FallNet +for different types of falls (Figure 10a) and other ordinary activities +(Figure 10b). We clearly see that while most STFT segments of most +ordinary activities can be recovered by the FallNet the STFT of falls +cannot. We also observe that as the model evolves (from CKPT1 +to CKPT3), the reconstruction errors of all the falls increase while +the reconstruction errors of ordinary activities decrease. Note that +the errors is computed as the L2-norm of the difference between +the FallNet input and output. The reconstruction errors of falls are +order of magnitude higher than those of ordinary activities. +The visualization suggests that the FallNet is able to continuously +learn better latent distribution to describe the human daily activities +and based on that make more accurate detection of falls as outliers. +Notably the low-frequency part of the spectrum and the end-of- +motion part remain clear despite the deteriorating quality of the +reconstructed STFT samples of falls, which suggests that the FallNet + +CKPT1 +CKPT2 +CKPT3 +100 +TPR(%) +80 +65 +54 +SiFall +FPR(%) +20 +RTFall +15 +5 +2.5 +#P1 +#P6 +#P7 +#P8 +#P9 +#P10 +#P11 +#P12 +#P13 +#P14 +#P15 +#P16 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +Time(h)SiFall: Practical Online Fall Detection with RF Sensing +SenSys ’22, November 6–9, 2022, Boston, MA, USA +(a) The original and reconstructed STFT segments of Falls. +(b) The original and reconstructed STFT segments of other activities. +Figure 10: Visualization of the FallNet original input and reconstructed output STFT segments. +indeed utilizes the low-frequency and end-of-motion features in +discriminating the samples. +4.2.4 +SiFall Segmentation Performance. The segmentation algo- +rithm of SiFall depends on detecting the status of human movement +and is threshold based. We separately evaluate the accuracy of the +movement detector and the threshold sensitivity. +Movement Detection. We classify the human movement into +three levels. Body level movement refers to the whole body move- +ment involving position change, such as walking and running. Torso +level movement refers to torso movement without position change, +such as bow and squat. Limbs level movement refers to limbs and +hands movements at minor scale such as shaking hands and typing. +Figure 11a reports the percentage of relative error (false negative +rate) of the corresponding movement detection results when testing +subjects move freely in testbed environment 1. The figure plots the +cumulative distribution based on the movements from 12 testing +subjects (#P1, and #P6-16). The experiment logs SiFall movement +detection performance at body level, torso level and limbs level +movement with median error of 0.5%, 1.1%, and 1.8%, and 90th- +percentile error of 1.2%, 1.6% and 3.7%, respectively. +(a) +(b) +Figure 11: SiFall (a) segmentation performance of movement +detection error and (b) the acceleration distribution of differ- +ent types of movements. +Threshold Sensitivity. To quantitatively evaluate the reliability +of the threshold Θ = 2.5 as fixed in the SiFall implementation, we +derive the maximal frequency of each human movement and project +to its acceleration. Figure 11b depicts the cumulative distribution +of the projected acceleration for different types of movements. +We see that all fall activities have their derived acceleration +above the threshold and the majority of other fall-like activities +are also captured with the current threshold setting. On the other +hand, most ordinary walk and run movements are screened out by +the threshold. 12.5% of bow activities are screened out as well. The +result suggests that the threshold setting is effective in screening +falls and fall-like activities. It is also obvious that SiFall is robust to +the threshold setting - its accuracy will not be impaired when the +threshold falls in the range between 2 to 3.7. +Segmentation Length. Following the strategy of "more is better +than less", the SiFall segmentation algorithm aims at capturing the +activities with redundancy in their time durations. Figure 12 com- +pares the lengths of SiFall captured segments and the corresponding +ground truth time durations of the activities. In general the SiFall +segments have an average duration of 5.1s, which is longer than +the actual activity duration averaged at 2.6s. Additional 2.5s signal +data are included in the SiFall segments for redundancy. Overall, +SiFall segmentation algorithm introduces necessary redundancy +in extracting the signal segments while maintaining the signal +processing overhead on the extra signal durations acceptable. +Figure 12: Comparison of the ground truth and SiFall seg- +mented lengths of different activities. + +1 +0.9 +0.5 +c +0.3 +0.2 +-BodyLevelMovement +:TorsoLevelMovement +LimbsLevelMovement +0 +0 +1 +2 +3 +4 +5 +Relative Error(%)0.9 +Walk/Run +Sit +！ +Kneel +0.3 +Bow +Swing +0.2 +Jump +Squat +0 +Fall +0 +2.5 +4 +5 +6 +7 +8 +9 +a-SiFallSegments +6 +GroundTruth +2 +Bow +Squat +Kneel +dwnr +Sit +Swing +FallSenSys ’22, November 6–9, 2022, Boston, MA, USA +Sijie Ji, Yaxiong Xie, and Mo Li +Figure 13: False alarms that occur during the daily life adop- +tion test across three days. +4.3 +Daily Life Adoption Test +In the above experiment, human subjects were asked to contin- +uously perform a large number of falls and fall-like activities in +a short time duration for comprehensive evaluation. To further +challenge our system and understand the long-term performance +in a more realistic setting, we adopt SiFall with pretrained FallNet +from the emulated bedroom (testbed 1) to a real apartment room +(testbed 2). We conduct a continuous three day evaluation with one +testing subject (#P2) working and living inside the testing room +day and night. A total number of 204 fall-like activities (67 on day +1, 63 on day 2, and 74 on day 3, respectively) are captured at the +front end and sent to the FallNet for fall inference. Totally 12 false +alarms are triggered (9 on day 1, 2 on day 2, and 1 on day3, respec- +tively). Figure 13 depicts those false alarms and their occurrence +time. After verifying with the testing subject, the first-day false +alarms mainly come from sitting on the sofa and they are phased +out gradually with the model update. The first false alarm on the +second day is raised when an object falls from the wardrobe and +the testing subject picks it up immediately. The second false alarm +on the same day is raised when the subject jumps and dives into +the sofa from the back of the sofa. The last false alarm on the third +day is triggered when the testing subject does handstand on the +Yoga mat. We see a clear trend that the false alarms dramatically +reduce when the FallNet model is continuously updated over time. +Quantitatively, the false alarm rate decreases from 13.4% on the +first day to 1.4% on the last day. The experiment results suggest +that SiFall has the ability to learn from personalized daily activities, +and build evolved models for more accurate fall detection. However, +some rarely-seen combination of movements may still trigger the +false alarm, which we expect would reduce when SiFall continues +to see more repeated occurrences of such activities over longer time +of deployment. +After the three day continuous monitoring of the daily activities +with SiFall, we perform a purposed experiment to evaluate its de- +tection accuracy of true falls. We freeze the model update of SiFall, +and let the testing subject perform ten emulated "stop falls" and +"slow falls" following the same methodology illustrated in §4.1. The +falls are performed across 5 different locations in the room (shown +as "X" in Figure 8). We then pour a lot of powder on the floor, let the +testing subject wear safety gear, keep jogging in the room till five +"walk falls" are collected. Apart from those, the testing subject also +simulates a fall that rolls from the bed as well as a slipping fall when +trying to sit on the office chair. SiFall can detect all the above falls +except for the rolling fall from the bed with 94.1% detection rate, +demonstrating the high reliability of SiFall in real-life application. +Figure 14: Accuracy of different person at different link dis- +tances +4.4 +Effective Covering Range +As SiFall captures fall-like activities based on sensing the wireless +channel dynamics, we want to evaluate its effective sensing range. +We deploy SiFall with the pretrained FallNet from the "bedroom" +environment (testbed 1) to a bigger open area (testbed 3) without +fine tuning the model. Three testing subjects (#P3,#P4 and #P5) are +requested to perform "jump", "sit to the floor", "swing", and "walk +fall" (each for five times with a random sequence) repeatedly in +three different areas (as depicted in Figure 8) with a distance of 1-3 +meters, 3-5 meters, and 5-7 meters, respectively, to the LOS link +of the Wi-Fi transceivers. Figure 14 reports the TPR and FPR of +the three testing subjects when experimented in the three different +areas. We see that SiFall performance is robust when adopted across +the environment. The accuracy is reasonably good when the testing +subjects move to as far as 5m away from the Wi-Fi link. The average +TPR and FPR in area A1 and area A2 were 73.3% and 20%, 73.3% and +22.2%, respectively. When the distance increases to 7m in area A3, +the average TPR drops significantly to 40% and the FPR drops to +4.4% at the same time due to failures in detecting and segmenting +all fall-like activities. Note that when directly migrating the FallNet +model to a new environment (from the "bedroom" in testbed 1 to the +big open area in testbed 3), SiFall still achieves an average accuracy +of 78.3% which is impressive. We expect the accuracy will further +improve with time when more human activities are captured and +consumed by the FallNet model to evolve. +4.5 +Computation Overhead +We provide a quantitative analysis of the SiFall’s computation over- +head. We utilize the Pytorch-OpCounter tool to measure the compu- +tation cost in flops (floating-point operations per second) of FallNet +and some representative CNN-based models used in other appli- +cations. As Figure 15a depicts, FallNet falls in between the ultra- +lightweight model (e.g., MobileNetV2) and the medium-weight +models (eg., Densenet121 and AlexNet), which indicates FallNet is +a relatively lightweight model. +(a) FallNet model complexity compared with +SOTA CNN-based models. +Inference Time (ms) +23.267 +Update Time (ms) +65.242 +Warning Delay (ms) +14.552 +Alarm Delay (s) +1.670 +(b) Latency of SiFall compo- +nents. +Figure 15: SiFall computation overhead. + +2 +Day 1 +Day 2 +Day 3 +P +0 +10:00 +22:00 +10:00 +22:00 +10:00 +22:00 +10:00 +Time80 +60 +40 +20 +0 +FPR(%) +20 +1#P3 +40 +1#P4 +#P5 +60 +80 +Area1 +Area2 +Area37 +VGG16 +InceptionV3 +6432 +flops(G) +ResNet50 +DenseNet121 +FallNet +1 +AlexNet +MobileNetV2 +10 +55 +100 145 +#Parameters(M)SiFall: Practical Online Fall Detection with RF Sensing +SenSys ’22, November 6–9, 2022, Boston, MA, USA +We also measure the end to end latency of SiFall operation in +our testbed, and summarize the result in Table 15b. The average +inference time and the model parameter update time are measured +on a single NVIDIA 2080Ti GPU. Once SiFall front-end detects a +fall-like activity, it triggers a warning and waits for the FallNet at +the back-end to generate the alarm for confirmed falls. We measure +the warning delay (alarm delay) by averaging the time interval +between the system warning time (alarm time) and the ground +truth ending time of the fall-like activities (fall). The major delay of +the system comes from the signal segmentation, where SiFall keeps +monitoring the channel dynamics for 1s. +5 +RELATED WORK +RF-based fall detection. Existing RF-based fall detection sys- +tems [23, 38, 45, 51, 53, 57] all assume repeatable human fall pat- +terns and follow pre-defined fall templates in the feature space +for detection. Most of them depend on manually segmented signal +clips for inference. Aryokee [51] utilizes CNN to extract features of +human fall as opposed to previous manual feature extraction ap- +proaches [38, 51, 53, 57]. However, the samples are collected offline +with the same length, the Aryokee model is not able to deal with +varying length RF samples and the system cannot run in real-time. +FallDefi [38] improves the performance by using the combined +features of previous approaches and adopting more WiFi links for +gathering RF signals. There are some general RF based human ac- +tivity recognition systems including Witrack [2], CARM [55] and +HAR-SANet [11] which treat fall as one of the ordinary human +activities and can only capture few types of falls. To the best of +our knowledge, RT-Fall is the most practical solution of real time +fall detection, which however as suggested in our experimental +evaluation cannot provide high accuracy with realistic falls occur- +ring in practice. Although Defall [23] claims real-time fall detection +capability, it uses a human-like dummy to do the experiment to +learn the fall template, and thus it can only detect simulated "hard +fall", which is falling from a standstill position at a certain height, +by its nature significantly limits its application in practice. +Other fall detection solutions. Other than RF-based fall detec- +tion, there are CV-based fall detection approaches [13, 17, 64] that +take optical measurements by camera or infrared sensors for analy- +sis. Those solutions are often criticized for compromising human +privacy. Wearable-based fall detection methods either require the +user to carry the device [3, 9] or wear the device [27] which are +intrusive and thus not the most desired way for fall detection [44]. +The acoustic-based [15] method is limited by ambient noise. Sensor +fusion-based fall detection [34, 69] is believed to be more reliable +as various sensors may complement each other in different situa- +tions, but generally leads to higher cost and deployment overhead. +Among them, some works claim they detect falls based on anom- +aly detection [9, 17]. A CV-based approach [17] collects a balanced +dataset with fall and non-fall samples and use a supervised anomaly +detection method. A wearable-based approach [9] learns a fixed +boundary in feature space to separate daily activity and the anomaly +fall, which cannot cope with unseen daily activities and falls. +Deep learning based RF sensing. Deep learning has recently +been widely adopted to various wireless sensing applications, in- +cluding physiological sensing [67], food and liquid sensing [20], +gesture recognition [68], body skeletons reconstructing [36, 65, 66], +localization [5] and etc [8, 18, 25].Those solutions cannot be directly +applied to detecting human falls. Most of them do not support the +neural network update during run time and often require extensive +data collection and annotation to facilitate the model training. +Anomaly detection. While anomaly detection is well-studied in +the literature, anomaly detection for high-dimensional data in real- +time remains challenging [39]. Traditional methods such as One- +Class SVM [47], Kernel Density Estimation [40] and Tree-based +Isolation Forest [35] all fail to operate online due to unsatisfactory +computational scalability and the curse of dimensionality. Thanks +to the rapid development of deep learning technology, a lot of deep +learning based anomaly detection methods have been proposed [54, +62, 70] with similar frameworks that consist of three parts: feature +extraction, feature representation learning, and end-to-end anomaly +score learning. We design FallNet based on this skeleton and make +it capable to run in real-time with unstructured input signal data, to +fill the gap in the literature, as most deep learning based methods +are capable to only structured datasets and lack real-time practices. +Self-supervised learning. Self-supervised learning techniques +support learning representations from a large amount of unlabeled +data and based on that representation to serve downstream classi- +fication tasks with a few labeled instances [26]. As an alternative +solution to establish a representation of daily human activities, self- +supervised learning still faces the challenge in the lack of labels +for unforeseeable human fall types. From a different perspective, +SiFall deals with the domain variations by building an anomaly +detection neural network model and continuously evolving the +model to represent high-level semantics of normal daily activities. +6 +DISCUSSION & CONCLUSION +This paper proposes SiFall, a self-supervised incremental learning +human fall detection system. SiFall leverages Wi-Fi RF signals and +is able to detect daily human falls in real time. Extensive experiment +results demonstrate that SiFall achieves high accuracy in human fall +detection and is resilient to varied human subjects, environment, +and different types of falls. The design of SiFall makes an important +contribution towards building practical and reliable RF-based fall +detection systems. The current study is still limited in its lack of +real fall samples, especially of elderly aged above 60. Since SiFall +relies on wireless channel dynamics to catch human activities, it +is currently limited to working with single room occupancy. We +leave the exploration to the above two limitations to future work +when developing SiFall into higher technology readiness levels. +ACKNOWLEDGMENTS +We sincerely thank the shepherd and reviewers for their insightful +comments and suggestions. We also thank all volunteers for their +participation in our experiments. This research is supported by the +National Research Foundation Singapore under its Industry Align- +ment Fund – Pre-positioning (IAF-PP) Funding Initiative, and Min- +istry of Education Singapore MOE AcRF Tier 2 MOE-T2EP20220- +0004. 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Zero-effort cross-domain gesture recognition with Wi-Fi. +In Proceedings of the 17th Annual International Conference on Mobile Systems, +Applications, and Services. 313–325. +[69] Xu Zhou, Li-Chang Qian, Peng-Jie You, Ze-Gang Ding, and Yu-Qi Han. 2018. Fall +detection using convolutional neural network with multi-sensor fusion. In 2018 +IEEE international conference on Multimedia & Expo Workshops (ICMEW). IEEE. +[70] Bo Zong, Qi Song, Martin Renqiang Min, Wei Cheng, Cristian Lumezanu, Daeki +Cho, and Haifeng Chen. 2018. Deep autoencoding gaussian mixture model for +unsupervised anomaly detection. In International conference on learning represen- +tations. +A +INSTANCE NORM VS BATCH NORM +The equation of Instance Norm is the same as Batch Norm such +that: +BN𝛾,𝛽 (𝑥) ≡ 𝛾 +�𝑥 − 𝜇(𝑥) +𝜎(𝑥) +� ++ 𝛽 +(8) +IN𝛾,𝛽 (𝑥) ≡ 𝛾 +�𝑥 − 𝜇(𝑥) +𝜎(𝑥) +� ++ 𝛽 +(9) +where 𝛾, 𝛽 are affine parameters learned from data; 𝜇(𝑥), 𝜎(𝑥) are +the mean and standard deviation. The difference of the two norm +just the way that how the statistical descriptors 𝜇 and 𝜎 are obtained. +Given an input batch 𝑥 ∈ R𝐵×𝐻×𝑊 ×𝐶, Batch Norm normalizes +the mean and standard deviation for each individual feature channel +to a whole batch: +𝜇𝑐 (𝑥) = +1 +𝐵𝐻𝑊 +𝐵 +∑︁ +𝑛=1 +𝐻 +∑︁ +ℎ=1 +𝑊 +∑︁ +𝑤=1 +𝑥𝑏𝑐ℎ𝑤 +(10) +𝜎𝑐 (𝑥) = +√︂ +1 +𝐵𝐻𝑊 +𝐵 +∑︁ +𝑛=1 +𝐻 +∑︁ +ℎ=1 +𝑊 +∑︁ +𝑤=1 +(𝑥𝑏𝑐ℎ𝑤 − 𝜇𝑐 (𝑥))2 + 𝜖 +(11) +As Batch Norm uses mini-batch statistics during training phase +and replace them with average mean and variance across batches +during inference phase, which implicitly requires consistency distri- +bution of training domain and inference domain. SiFall is an online +detection system and the samples keep generated that might induce +difference across different person and environments so that we use +Instance Norm which normalize as per sample: +𝜇𝑏𝑐 (𝑥) = +1 +𝐻𝑊 +𝐻 +∑︁ +ℎ=1 +𝑊 +∑︁ +𝑤=1 +𝑥𝑏𝑐ℎ𝑤 +(12) +𝜎𝑏𝑐 (𝑥) = +√︂ +1 +𝐻𝑊 +𝐻 +∑︁ +ℎ=1 +𝑊 +∑︁ +𝑤=1 +(𝑥𝑏𝑐ℎ𝑤 − 𝜇𝑏𝑐 (𝑥))2 + 𝜖 +(13) +B +CONVOLUTION OPERATION +INDEPENDENT TO THE INPUT SIZE. +The "convolution" operation in the neural network is different +from the "convolution" in the signal processing domain. Indeed, +the convolution operation is an element-wise multiplication and +summation over a local region of the input tensor. The operation +is repeated in sequential local regions until the whole tensor has +been calculated. +Each learnable filter𝑊 in convolution operation with dimension +𝑊 ∈ R𝑘×𝑘, where 𝑘 denotes the kernel size, i.e., the size of the local +region that calculates the multiplication. Let 𝑋 ∈ R𝐻𝑖𝑛×𝑊𝑖𝑛×𝐶 de- +note the input tensor. The convolution operation calculates output +𝑌 such that: +𝑌𝑝,𝑞 = +𝐶 +∑︁ +𝑛=1 +∑︁ +𝑖,𝑗 ∈N𝑘 +𝑊 ⊤ +𝑖+ 𝑘−1 +2 ,𝑗+ 𝑘−1 +2 +𝑋𝑛 +𝑝+𝑖,𝑞+𝑗 +where (𝑝,𝑞) denotes the location coordinate and +N𝑘 = +� +(𝑖, 𝑗) : 𝑖 = +� +−𝑘−1 +2 , . . . , 𝑘−1 +2 +� +, 𝑗 = +� +−𝑘−1 +2 , . . . , 𝑘−1 +2 +�� +defines +a local neighborhood. A convolution layer specify how the kernel +sliding 𝑖 and 𝑗 through the input tensor by setting stride 𝑠 and how +we want the input tensor be padded by setting 𝑝, as a result the +output 𝑌 ∈ R𝐻𝑜𝑢𝑡 ×𝑊𝑜𝑢𝑡 ×𝑓 can be computed by +(𝐻𝑜𝑢𝑡,𝑊𝑜𝑢𝑡) = +��𝐻𝑖𝑛 + 2 ∗ 𝑝 − 𝑘 +𝑠 +� ++ 1, +�𝑊𝑖𝑛 + 2 ∗ 𝑝 − 𝑘 +𝑠 +� ++ 1 +� +, where 𝑓 is the number of learnable filters. The ’same padding’ +technique help choose proper 𝑠 and 𝑝 so that 𝐻𝑜𝑢𝑡 = 𝐻𝑖𝑛, 𝑊𝑜𝑢𝑡 = +𝑊𝑖𝑛. Consequently, convolution layer are able to adapt to the input +tensor with arbitrary size. + diff --git a/_NE2T4oBgHgl3EQfQwY9/content/tmp_files/load_file.txt b/_NE2T4oBgHgl3EQfQwY9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..64bf56731fb49f93174e0bddd51ae251f9e4c69c --- /dev/null +++ b/_NE2T4oBgHgl3EQfQwY9/content/tmp_files/load_file.txt @@ -0,0 +1,1177 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf,len=1176 +page_content='SiFall: Practical Online Fall Detection with RF Sensing Sijie Ji sijie001@e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='sg Nanyang Technological University Singapore, Singapore Yaxiong Xie yaxiongx@buffalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='edu University at Buffalo Buffalo, New York Mo Li limo@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='sg Nanyang Technological University Singapore, Singapore ABSTRACT Falls are one of the leading causes of death in the elderly people aged 65 and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In order to prevent death by sending prompt fall detection alarms, non-invasive radio-frequency (RF) based fall detection has attracted significant attention, due to its wide cover- age and privacy preserving nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Existing RF-based fall detection systems process fall as an activity classification problem and as- sume that human falls introduce reproducible patterns to the RF signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We, however, argue that the fall is essentially an accident, hence, its impact is uncontrollable and unforeseeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We propose to solve the fall detection problem in a fundamentally different manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Instead of directly identifying the human falls which are difficult to quantify, we recognize the normal repeatable human activities and then identify the fall as abnormal activities out of the normal activity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We implement our idea and build a prototype based on commercial Wi-Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We conduct extensive ex- periments with 16 human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The experiment results show that our system can achieve high fall detection accuracy and adapt to different environments for real-time fall detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' CCS CONCEPTS Human-centered computing → Ubiquitous and mobile com- puting systems and tools;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' • Computer systems organization → Real-time systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' • Applied computing → Health care in- formation systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' • Computing methodologies → Machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' KEYWORDS Self-supervised Learning, Wireless Sensing, Real-time System, Adap- tive Segmentation, Fall Detection, Device-free ACM Reference Format: Sijie Ji, Yaxiong Xie, and Mo Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' SiFall: Practical Online Fall Detection with RF Sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In ACM Conference on Embedded Networked Sensor Systems (SenSys ’22), November 6–9, 2022, Boston, MA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' ACM, Boston, MA, USA, 15 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1145/3560905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3568517 1 INTRODUCTION Fall is an important global public health issue [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Every year there are approximately 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 million fall-related injuries that require Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1145/3560905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3568517 Figure 1: Diversity of falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' medical attention and directly cost $34 billion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Clinical reports show that timely treatment (<1 hour) can prevent deaths from fatal falls [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Therefore, an effective fall detection system is necessary to facilitate timely treatment and benefit the current aging society where more and more elderly people are living alone [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Existing fall detection solutions can be classified into two cate- gories: wearable-based solutions and device-free solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Medical research has reported that wearable-based solutions do not work well in practice due to the burden of carrying and charging those devices from time to time [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In contrast, device-free solutions in- cluding computer vision (CV) based, acoustic-based, and RF-based are more user-friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Among them, the CV-based solutions cannot work under dim light conditions, occlusions and often compromises user privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The acoustic-based solutions are limited by its sensing range (<4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5m) [56] and possibly subject to restriction by ambient loudness (<40dB SPL) [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' However, RF-based solutions are not constrained by the above and also cost-effective as they take the advantage of existing ubiquitous communication infrastructures such as WiFi APs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Existing RF-based fall detection systems [38, 51, 53, 57] consider falls as a type of normal human activity and applies traditional human activity recognition method to identify the falls out of simi- lar activities such as sitting, sleeping and jumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Generally, the solution consists of off-line training and on-line inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' During the off-line training, the system builds up a model based on feature engineering [38, 57] or machine learning [51, 53], to separate the falls from other human activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The RF signals are collected for training purposes when the human being performs a set of pre- defined activities, such as falling, sitting and jumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The system then applies the trained model to identify falls from the received signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' All existing solutions implicitly assume that human falls in- troduce reproducible patterns to the RF signals which can be captured by the trained model and used to differentiate the falls from other activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In this paper, we revisit such a problem and argue that the sig- nal patterns introduced by human falls are full of randomness and consequently hard to be fully captured by trained templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Our key intuition is that the human fall, by its nature, is an accident arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='03773v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='HC] 10 Jan 2023 external factor level-change Slip Stumble Fall on the floor Lying on the bed internalfactor α change Lost consciousness Lost balance Fall on the floor Sit to the chairSenSys ’22, November 6–9, 2022, Boston, MA, USA Sijie Ji, Yaxiong Xie, and Mo Li that is unforeseeable and the human reaction is highly uncon- trollable, introducing highly dynamic disturbance to the wireless signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Specifically, as depicted in Figure 1, there are diverse causes of human falls, such as a stumble, a slip, loss of consciousness, loss of balance, a sudden fright, etc, which may result in randomness, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=', a stumble or a slip may result in displacement of the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In contrast, a person stays at the same place if he loses his consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In addition, the free range of movement in the joints of human body brings in another level of randomness when the human being cannot properly control his behavior during the falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Extracting representative features of the human falls becomes im- practical because of such uncontrolled randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Even collecting adequate data is challenging because one person can hardly repeat real and uncontrollable falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' With the above observation, in this paper we handle the fall detection problem in a fundamentally different manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Instead of seeking features to characterize the unforeseeable and uncontrol- lable human falls, we turn to solving an easier problem: recogniz- ing normal repeatable human activities including but not limited to jumping, sitting, and walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We formulate fall detection as adaptive anomaly detection and identity an abnormal activity that cannot be classified as any of the known activities as a fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Our hypothesis is that after an adequate time period of training, a self- supervised learning process will eventually perfect the model to dif- ferentiate uncontrolled falls from other repeated controlled human activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To prune the search space and speed up the convergence of the model training, we apply analyzable signal processing to early filter out non-fall human activities with distinguishable sig- nal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Our observation suggests that falls change the status of the human body in a short period of time and thus introduce high frequency components to the signal variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We feed the identified suspicious fall-like activities to a deep neural network called FallNet to recognize the true falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Specifically, the FallNet trains an auto-encoder [22] to learn a compressed representation of normal fall-like activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' When used for inference, the auto- decoder is only able to accurately reconstruct the normal fall-like activities but not real human falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The FallNet, therefore, identifies the activities that result in large reconstruction error as falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' After deployment, the FallNet is continuously updated using the freshly collected data in a self-supervised manner, so it evolves to adapt to the local propagation environment and the particular human subjects that the system monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We expect that the FallNet will eventually perfect its detection accuracy and false alarm rate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To realize our idea, we implement a Self-supervised Incremental learning Fall detection system, SiFall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To the best of our knowledge, SiFall is the first RF-based fall detection system that can work in real time for online fall detection on a daily basis across different human, different environments and different types of activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' SiFall possesses the following three advantages: SiFall works with daily human activities in runtime - WiFi CSI samples are dynamically processed, segmented, and dis- criminated to detect ongoing "falls".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' SiFall’s self-learning process can adapt to the variation of human subjects, environment, and types of falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The core anomaly detection model of SiFall evolves during its use;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' SiFall separates the signal processing from its machine learn- ing model, which is designed to be lightweight and may easily be accommodated at the edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The developed SiFall prototype has been comprehensively eval- uated with a total amount of over 92 hours of test data collected from 16 human subjects of different ages and genders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' During our experimental evaluation, SiFall is able to achieve 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3% accuracy in a real-world setting with extensive movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' During a continu- ous three-day adoption in a normal living environment, SiFall is able to detect 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1% falls with only one false alarm in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 2 CHALLENGES AND OPPORTUNITIES This section first discusses the challenges in developing a practical RF-based fall detection system and then presents the key observa- tions and opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 Challenges 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 Fall Ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' There is no uniform quantitative definition of "fall" in medicine, biology, or physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' According to the World Health Organization (WHO), fall is a subjective term, which is measured by the level of discomfort in the human body after a person accidentally lies on the ground or other low level [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As a result, it is hard to identify a quantified signal template to feature the "fall" when performing RF sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Besides, the orientation and the reflection surface of the human body may impact the reflected RF signal which leads to inter-activity similarities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=', falling v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' lying down) [1, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Other factors including deployment layout and individual difference may also contribute to the ambiguity in defining and quantifying the "fall" in the RF signal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2 Data Scarcity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Recent advances in deep learning allow learn- ing powerful discriminative models from a number of represen- tative samples [14], which may bypass the difficulty in defining precise signal templates of "falls".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' However, since the "falls" are high exceptional human activities that often occur uncontrollably, it is extremely difficult to obtain sufficient repeatable real-life data samples containing different types of falls, leading to a data scarcity issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Most existing fall detection studies depend on learning from artificial fall samples collected from the laboratory environment and thus may have gaps in detecting real falls that take place in daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The lack of fall data may also result in class distribution skews where the learned model is biased towards the majority types of falls and may have poor predictive performance for other types of falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As long as the types of falls are not sufficiently emulated, the learned model may be unreliable with poor generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 Unstructured Input Signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Human motions, even of the same type, may last for different durations of time, and as a result, the relevant RF signals are unstructured and of different lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The processing of variable-length input signals is very different from processing fixed-length data samples in many machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Real-time processing of variable-length sequences is par- ticularly difficult because data structurization techniques like se- quence padding or dynamic template mapping can hardly be applied in real time [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In addition, real-time segmentation of the RF sig- nals from consecutive activities is also challenging, the inaccuracy SiFall: Practical Online Fall Detection with RF Sensing SenSys ’22, November 6–9, 2022, Boston, MA, USA (a) STFT segments from the same testing subject #Person 1 (b) STFT segments from different testing subjects #Person 8, 9, and 16 Figure 2: STFT segments across activities and testing subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' of which may lead to inconsistency of features in the machine learn- ing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Most existing fall detection solutions assume pre-defined fixed-length RF signal input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2 Opportunities While the practical challenges suggest extreme difficulties in learn- ing the RF templates of human falls, we observe that there is an opportunity on the other hand to categorize human daily activities as they are usually repeatable and there exist plenty daily data samples for training a model to describe them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To showcase such an observation, Figure 2b visualizes the extracted WiFi signal fea- tures after short time Fourier transformation (STFT) across various human activities (details in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 2a depicts the STFT seg- ments collected from the same testing human subject and Figure 2b depicts those collected from three different human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' It is obvious to see that the daily human activities give very consistent STFT patterns, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=', the kneeling and sitting patterns in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Even across different human subjects the patterns of the same daily activities remain consistent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=', the kneeling and sitting patterns in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The "falls" however appear highly varied and non- repeatable across the types, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=', the "stop fall", "walk fall", and "slow fall" (details in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2), as well as the testing human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The above observations suggest that it is reliable to train a model to accurately describe the normal daily activities and as an oppor- tunity to identify "falls" as abnormal outlier output from such a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As there are plenty of daily activities to see when the sys- tem is deployed in reality, a self-supervised learning scheme may continuously perfect the trained model with improved accuracy in distinguishing the falls from normal daily activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 3 SYSTEM DESIGN A desired fall detection system should have the following charac- teristics: (i) it must work in real-time and detect falls with run-time data input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' (ii) it must be able to evolve itself without involving human efforts to label the data samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' (iii) it must adapt to envi- ronment and different users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In this section, we present the design of SiFall, a system that accommodates the above design consider- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We begin with the system overview followed by fall-like activity segmentation and the design of FallNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 Overview SiFall consists of a front-end to process RF signals and a back-end server to train the neural network model and detect the fall, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' SiFall’s front-end collects WiFi channel state information (CSI) measurements, denoises the CSI and extracts the dynamic compo- nent of the CSI to obtain an approximate RF-signal description of human movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Finally, a lightweight algorithm is used to quan- tify the motion intensity and segment the RF signals accordingly (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In the end, SiFall applies short-time Fourier transformation (STFT) to derive the time-frequency spectrum of each piece of segmented RF signal clip and supplies the STFT spectrum to the back-end server for fall detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The purpose of the front-end signal processing is two folded: to early rule out normal activities that possess clear daily activity features, and to present segmented RF signals with data cleansing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Typical daily human movements without high-frequency components are expected to be filtered out to narrow down the learning space of the back-end neural network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In the back-end server, a self-evolving deep neural network called FallNet takes the segmented RF signal as input and identify the falls from the normal fall-like activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The FallNet is designed based on the auto-encoder framework to do the self-supervised learning where the encoder learns a nonlinear mapping from the unstructured RF-signal space to uniformed compact latent feature space and thus addresses challenge from the unstructured input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The decoder learns the mapping from the latent space back ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='to the RF-signal space with the goal of reconstructing the original ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='SlowFall ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='WalkFall ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='StopFall ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='Kneel ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='、 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='Time (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='Time (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='Time (s)SenSys ’22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' November 6–9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' USA Sijie Ji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Yaxiong Xie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' and Mo Li Figure 3: Overview of SiFall RF-signal as closely as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' After training with a large num- ber of repeated regular human activities, the FallNet establishes a Gaussian mixture distribution of normal human activities in the latent space (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3) and thus is capable of accurately recognizing and recovering the RF-signal clips of normal activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' When de- ployed, the FallNet classifies the fall-like RF-signal clips that can be well reconstructed as normal daily activities and those RF-signal clips that cannot be reconstructed as falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The FallNet is continu- ously updated with RF-signal clips of repeatedly-appearing normal human activities fed from the front-end (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2 RF signal Segmentation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 CSI Extraction and Denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The received WiFi signal can be modeled as: 𝑌 (𝑓 ,𝑡) = 𝐻 (𝑓 ,𝑡) × 𝑋 (𝑓 ,𝑡) (1) where 𝑋 (𝑓 ,𝑡) represents the signals carried at subcarrier fre- quency 𝑓 and time point 𝑡 and 𝐻 (𝑓 ,𝑡) denotes the CSI value at 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The CSI describes how the RF signals are transformed by the current wireless channel - the amplitude attenuation and phase rotation of different frequency components due to multipath re- flection, diffraction, and scattering by objects in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' On top of that, RF chipset processing at WiFi transceivers may introduce additional distortion and noises [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Therefore, we perform necessary data cleaning to eliminate the impact of the hardware imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The CSI 𝐻 consists of a static part induced by ambient environ- ment 𝐻𝑠 and a dynamic part related to human movement 𝐻𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' CSI is also subject to WiFi hardware distortion 𝐻ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' we model the overall CSI as: 𝐻 (𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='𝑡) = (𝐻𝑠 (𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='𝑡) + 𝐻𝑑 (𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='𝑡)) · 𝐻ℎ (𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='𝑡) = (𝐻𝑠 (𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='𝑡) + 𝐻𝑑 (𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='𝑡)) · 𝜀1 (𝑡) 𝑒𝜀2(𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='𝑡)+𝜀3(𝑡)+𝜀4 (2) where 𝜀1(𝑡) is the amplitude scaling caused by automatic gain control (AGC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 𝜀2(𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='𝑡) represents the phase offset introduced by the combination of packet detection delay (PDD),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' sampling frequency offset (SFO) and sampling time offset (STO),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 𝜀3(𝑡) is the phase offset caused by the carrier frequency offset (CFO),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' and 𝜀4 is the initial Figure 4: Static CSI Amplitude (upper left),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Dynamic CSI Am- plitude (upper right) and the Extract Channel Dynamic 𝑆(𝑡),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' gray is the ground truth static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' phase offset of the radio chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We utilize relatively clean CSI amplitude and mitigate the impact of noisy CSI phase by calculating the conjugate multiplication of CSI as ˆ𝐻 (𝑓 ,𝑡) for each subcarrier: ˆ𝐻 (𝑓 ,𝑡) ≡ 𝐻 (𝑓 ,𝑡) 𝐻 (𝑓 ,𝑡) = 𝜀2 1 (𝑡) |𝐻𝑠 (𝑓 ,𝑡) + 𝐻𝑑 (𝑓 ,𝑡)|2 , (3) The resulting ˆ𝐻 (𝑓 ,𝑡) is still affected by the amplitude scaling 𝜀1(𝑡) that AGC introduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To visualize the impact of 𝜀1(𝑡), we collect CSI measurements from a static environment and calculate CSI amplitude across subcarriers in Figure 4 (upper left), from which we see that the CSI amplitude curves across subcarriers are similar but not identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The reason is that the amplitude scaling factor 𝜀1(𝑡) is time-varying but consistent across subcarriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We note that, because of the amplitude scaling factor, the CSI amplitude of a single subcarrier ˆ𝐻 (𝑓 ,𝑡) is time-varying even when the environment is static and thus cannot capture the dynamics introduced by the human motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2 Capturing Channel Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We use the variations of the CSI amplitude curve to capture the channel dynamics introduced by human motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To illustrate the intuition, we plot the CSI amplitude when the human is moving in Figure 4 (upper right), from which we see that the shape of amplitude curve varies significantly in non-static environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We use cosine similarity to quantify the similarity between consecutive CSI measurements: 𝑆(𝑡𝑛) = ⟨ ˆ𝐻 (𝑡𝑛), ˆ𝐻 (𝑡𝑛−1)⟩ | ˆ𝐻 (𝑡𝑛)|| ˆ𝐻 (𝑡𝑛−1)| (4) where ˆ𝐻 (𝑡𝑛) = [ ˆ𝐻 (𝑓1,𝑡𝑛), · · · ˆ𝐻 (𝑓𝑀,𝑡𝑛)] represents the CSI ampli- tude vector of all 𝑀 subcarriers sampled at 𝑛-th time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We plot the calculated S(t) for CSI collected from both static and non-static environment in Figure 4 (bottom), from which we see the variation of the 𝑆(𝑡) accurately captures the dynamics of the wireless chan- nels, because the normalization operation to compute similarity essentially removes the effect of AGC and thus 𝜀1(𝑡) is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We note that, the similarity 𝑆(𝑡) is affected by CSI sampled at two time point, so its value may also vary when the sampling interval Detecting&Segmenting Self-learning Signal Preprocessing FallNet Model Updating MovementDetection pe(x) Fall-like Segmentation FALLALARM70 Amplitude(dB) 65 static dynamic 60 10 20 30 40 50 10 20 30 40 50 Subcarrier Subcarrier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='9 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='8 0 1 2 3 4 Time(s)SiFall: Practical Online Fall Detection with RF Sensing SenSys ’22, November 6–9, 2022, Boston, MA, USA (a) Spectrum (b) 𝑎(𝑡) Figure 5: The STFT spectrum and the corresponding acceler- ation of channel dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' varies, adding another unpredictable factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In our implementation, we introduce a reference vector �𝑟 = [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' , 1] and derive 𝑆(𝑡) as the similarity between the ˆ𝐻 (𝑡𝑛) and the reference vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We use the variance of S(t) across 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1s and above a threshold Γ to detect the human movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 Segmenting Fall-like Activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To forge efficient online de- tection and relatively consistent feature extraction, we propose a heuristic algorithm to segment fall-like activities from continuous monitored RF signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The key observation is that a fall and fall-like activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Sit, Jump and Squat) usually comes to a full pause at the end of the motion before transitioning to next movement, which may be due to the direction change of the movement (vertical to horizontal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Similar observation has been reported in previous studies with WiFi [53] and RFID [10] signals as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Therefore, we segment 𝑆(𝑡) in a backtracking manner from an observation of motion pause, which is easier to capture than the actual start of an activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Meanwhile, as the channel dynamic is caused by the human movements, we derive an approximate acceleration descrip- tor 𝑎 to help further filter out daily movements accompanied by a pause with low-intensity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' walk and stop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In addition, we assume the RF signals collected after a fall are also useful and thus a greedy algorithm is used to keep monitoring the 𝑆(𝑡) to window the entire fall-like activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The approximate 𝑎 is computed by using the relationship [43]: 𝑎(𝑡) = d2 dt2 𝑆(𝑡) = 𝜆 d dt 𝑓𝐷 (𝑡) (5) where 𝜆 is wave-length of the subcarrier wave, 𝑓𝐷 (𝑡) is the Doppler frequency shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We approximate d d𝑡 𝑓𝐷 (𝑡) by computing STFT of 𝑆(𝑡) as STFT is used to capture the frequency component in a small time duration and the frequency component change is caused by the relative movement between transceivers and the reflecting human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Denote the STFT spectrum as S ∈ R𝐹×𝑇 , where 𝐹 is the fix frequency bins and 𝑇 is the number of time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' For each time bins, we have a vector of approximate d d𝑡 𝑓𝐷 (𝑡) denote as �𝑣, �𝑣 ∈ R𝐹 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We search 𝑚𝑎𝑥(𝑣) as the function of indices of frequency bins that exceed the noise floor via dynamic programming such that: max(v) = argmax 𝑓1,···,𝑓𝑇 𝑇 ∑︁ 𝑖=1 S𝑖,𝑓𝑖, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' |𝑓𝑖 − 𝑓𝑖−1| <= 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='𝑖 = 2, · · · ,𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' (6) 𝑚𝑎𝑥(𝑣) ≜ d d𝑡 𝑓𝐷 (𝑡) so 𝑎(𝑡) may be obtained by calculating 𝜆𝑚𝑎𝑥(𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We consider 𝑎(𝑡) > Θ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 indicates a potential fall-like activity as the human normal acceleration in walking is less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5𝑚/𝑠2 [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 5a is the STFT spectrum derived from 𝑆(𝑡) contained in Fig- ure 4 and Figure 5b is the 𝑚𝑎𝑥(𝑣) derived from the STFT contained in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' When applied in real time, once the variance of 𝑆(𝑡) is estimated below Γ, suggesting a pause after a move, SiFall records the time as 𝑡𝑒𝑛𝑑 and then searches if there exists 𝑎(𝑡) > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 in the past five seconds (𝑡 ∈ [𝑡𝑒𝑛𝑑 − 5,𝑡𝑒𝑛𝑑]) and records the 𝑚𝑎𝑥(𝑎) and its corresponding time as 𝑡𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' An online greedy change point detection algorithm [28] is ap- plied to continuously update 𝑡𝑒𝑛𝑑 for one second afterwards to obtain 𝑡∗ 𝑒𝑛𝑑: C � 𝑆(𝑡𝑒𝑛𝑑 : 𝑡∗ 𝑒𝑛𝑑) � + 𝛽 < C � 𝑆(𝑡𝑒𝑛𝑑 : 𝑡∗ 𝑒𝑛𝑑+1) � (7) where C stands for the error of the linear regression and 𝛽 is a penalty value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The rationale behind is that the RF signals collected after the fall-like activity may also contain useful information for identifying the fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In the end, SiFall extracts 𝑆(𝑡) between � 𝑡𝑚𝑎𝑥 − 3𝑠,𝑡∗ 𝑒𝑛𝑑 � and per- forms STFT on 𝑆(𝑡) to obtain the fall-like segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The additional three-second time before 𝑡𝑚𝑎𝑥 is used to include as complete fall- like activity as possible because we would rather contain redundant signal data as compared to missing any possible important data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Note that the lengths of STFT segments and their corresponding spectrums are variable because the time between 𝑡𝑚𝑎𝑥 and 𝑡∗ 𝑒𝑛𝑑 depends on the duration of the captured activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Following that, the STFT segment of the fall-like activity is supplied to the neural network in the back-end for affirmative fall detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Algorithm 1 defines the whole backtracking segmentation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Algorithm 1: Fall-like Segmentation Algorithm Input: 𝑆(𝑡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Threshold: Θ, Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Penalty: 𝛽;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 𝑓 𝑠 if movstd(S(t),fs/10) < Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' then record 𝑡 as 𝑡𝑒𝑛𝑑, S=STFT([S(t-5fs),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=',S(t)]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' if 𝑚𝑎𝑥(𝑣) > Θ then record 𝑡𝑚𝑎𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' while 𝑡 < 𝑡𝑒𝑛𝑑+𝑓 𝑠 do err(𝑡)=C (𝑡𝑒𝑛𝑑 : 𝑡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' if err(t) > err(t-1) + 𝛽 then 𝑡∗ 𝑒𝑛𝑑=𝑡 − 1 continue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' temp = [𝑆(𝑡𝑚𝑎𝑥 − 3𝑓 𝑠),𝑆(𝑡∗ 𝑒𝑛𝑑)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' seg = STFT(temp);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 FallNet Design The difficulty now lies in identifying ongoing falls from those RF clips of fall-like activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' This section elaborates on the design of FallNet, which is able to further identify falls from the RF clips of fall-like activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Specifically, we learn the complicated distri- bution of normal fall-like activities by a variational auto-encoder 80 Fall 60 Freguency Jump 40 Squat 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 Time (s)10 max(v) Acceleration(m/s smooth 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 Time (s)SenSys ’22, November 6–9, 2022, Boston, MA, USA Sijie Ji, Yaxiong Xie, and Mo Li ������ �� � � ������� � ��� �� � � �� ��� σ ���� μ ɛ iteration constrain � ��� ��� ��� � � �� � Conv + IN + LeakyReLU Pooling Uppooling Pooling Indices encoder decoder Figure 6: The FallNet Architecture: the encoder, decoder and bottleneck layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' based FallNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' When used for inference, the FallNet is only able to accurately reconstruct the normal fall-like activities but not real human falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We, therefore, identify the activities that result in large reconstruction error as falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We first construct the core encoder-decoder architecture of FallNet, which does not rely on data annotation and is able to accept unstructured input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Then, we elaborate on some special designs of FallNet to cope with partic- ular issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Finally, we import the variational inference technique to FallNet to make it more generalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 FallNet Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We design FallNet based on autoen- coder architecture which is a well-known deep learning framework to compress data without labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The ability to compress data shows its high ability to understand the intrinsic relationship between the compressed data and the original data, hence, a trained encoder is also widely used as a feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We train the encoder- decoder only based on the fall-like STFT segments collected from daily activities so the FallNet learns the representative features of daily activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' When used for inference, the encoder-decoder is able to fully reconstruct the signals of those repeatedly seen normal activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The input to the network is the signal clip of the 𝑖th activity 𝑥𝑖 ∈ R𝐹×𝑇 (𝑖)×𝐶 from a total number of 𝑁 activities, where the 𝐹 is a chosen frequency resolution of the STFT image,𝑇 (𝑖) is the time duration of the activity, which might vary across activities, and 𝐶 is the number of spatial streams(between Tx and Rx antennas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Therefore, the complete information of the three domains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=', time, frequency and spatial, are fed into the FallNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The encoder of the FallNet learns a nonlinear transformation FE : X → Z that maps the original data space X ⊆ R𝑚(𝑖) with variable dimensions and inconsistency to a compact latent feature space Z ⊆ R𝑛 with uniform dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 𝑚(𝑖) denotes the flattened dimension of 𝑥𝑖 and 𝑛 represents the dimension of the latent space of features that are most representative to describe the activities such that: z = FE (x, ΘE) where ΘE is a set of parameters of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As the encoder learns the most representative features and automatically filters out the redundancy, it works well with the STFT segments, which may be longer than the actual activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The ability of the encoder to project variable-length data space into a uniform latent feature space is owing to our fully convolu- tional network structure design of the building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The convo- lution operation itself intrinsically can cope with input of varying lengths, although many people don’t notice this because the con- volution operation is usually used to process images that are of same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The convolution operator in fact works on local tensor regions and depends only on relative spatial coordinates determi- nated by the convolution kernel size [19] (refer to Appendix B for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As a result, when using the "same padding" [19] in a convolution layer, for an input with dimension 𝐹 × 𝑇 (𝑖) × 𝐶, the output will be with the dimension of 𝐹 ×𝑇 (𝑖) × 𝐶′, where the only change is the channel dimension 𝐶′, depending on the number of convolution filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In particular, the encoder of FallNet consists of five building blocks of decreased size that are stacked together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Each building block consists of two convolution layers with instance nor- malization (IN) [52], an activation function of LeakyReLU [61], and a max-pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' All convolution layers fix the convolution filter size to 3 which simulates a larger filter while keeping the benefits of smaller filter sizes in order to reduce the computational overhead [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' IN is used to cope with the antenna imbalance is- sue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' LeakyReLU is the activation function to bring in non-linearity ability of the network and it can avoid the dying ReLU problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Max-pooling is used to achieve translation in-variance over small spatial shifts in the input tensor [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The max-pooling layer will decrease the size of the input to half so that the final output size of each building block is 𝐹/2 ×𝑇 (𝑖)/2 × 𝐶′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' At the end of the five building blocks, we first average pooling the feature values along the time dimension with an index to record its dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Note that 𝐶′ is determinated by the number of convolution filters which is controlled by us and the 𝐹 is fixed, a fully connected layer hence can be used to conduct channel-wise linear transformation to map the tensor to a fixed-length vector 𝑧 with 𝑛 dimension that represents the extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The decoder learns to reconstruct the input signal 𝑥𝑖 from the output 𝑧 of the encoder, such that ˆx = FD (z, ΘD) where ˆx is the reconstructed signal, and ΘD is a set of parameters of the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To reconstruct ˆ𝑥, the decoder needs up-sampling EZSiFall: Practical Online Fall Detection with RF Sensing SenSys ’22, November 6–9, 2022, Boston, MA, USA Figure 7: Up-pooling diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' oprations to map 𝑧 back to the size𝑚(𝑖) of the original input smaple 𝑥(𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In consequence, the decoder and the encoder are symmetric with the same number of building blocks, except that the max- pooling layers at the encoder is replaced by up-pooling layers at the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As up-pooling [6] utilizes the 2-bit indices stored during max-pooling operation in the encoding phase and up-samples the feature map by filling the values directly to the index position and zero-padding the remaining positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' It avoids parameter learning to reduce the computation overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 7 illustrates the up- pooling operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='Another small detail is that the decoder first uses the record index from the previous average pooling operation to zero-pad the 𝑧 back to the dimension before the fully connected layer, then goes through the five identical building blocks of the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Consequently, the goal of the FallNet is to learn the parameter sets of encoder and decoder satisfying: � ˆΘE, ˆΘD � = arg min ΘE,ΘD E𝑥∼X � ∥x − FD (FE (x, ΘE) , ΘD) ∥2� It is worth noting that this learning process only needs the input sample 𝑥 and does not require any labelled data, therefore, it can benefit from substantial and easily accessible RF samples of daily activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2 Coping with Antenna Imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' CSI collcted from different antennas may have different amplitudes, which lead to the imbal- ance of the power of STFT spectrums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The removal of AGC impact in the CSI denoising phase further amplifies this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As the 𝐶 channels of input tensor corresponds to different Tx-Rx antenna streams, the FallNet adopts IN that normalizes the antenna streams with learnable affine parameters 𝛾 , 𝛽 to cope with the antenna imbalance: IN𝛾,𝛽 (𝑋) ≡ 𝛾 � 𝑋 + 𝛽, where � 𝑋 = 𝑋 − 𝜇 √ 𝜎2 + 𝜖 where 𝜇 and 𝜎2 are computed across spatial dimensions indepen- dently for each channel so that every spectrum has the same range of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 𝜖 is a small constant added for numerical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Noted that the FallNet removes the commonly adopted Batch Normaliza- tion (BN), as the data samples in our case are generated online and may follow different distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' IN has the same characteristics as BN does, which helps the entire neural network to alleviate gra- dient saturation and accelerate convergence [24] (refer to Appendix A for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 Coping with RF Data Scarcity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Although the training of the FallNet is free from data annotation, making it possible to con- tinuously learn from daily fall-like activities, it is not realistic to enumerate all possible fall-like activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Besides, some types of activities may be relatively dominant owing to specific user activity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As a result, FallNet may be prone to be overfitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To make the FallNet resistant to such overfitting and be generalized to function properly, instead of using a vector 𝑧 with 𝑛 dimension to represent the learned fall-like activity features, we adopt a bottle- neck layer with stochastic sampling operation to make the FallNet become probabilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The reason for doing this is based on our observation (Figure 2) that fall-like activities of the same type are similar, though not identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' By introducing this prior knowledge, we can construct the obtained samples with certain distributions and assume that the same type of activities come from the corresponding distribu- tion to obtain more general sample characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We, therefore, import such prior knowledge into the network, allowing the neu- ral network to learn more generalizable features from the limited data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In particular, we assume that each of the 𝑛 features of the RF signals follows a normal distribution due to different body shapes or orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Refer to Figure 2b to see that the same actions performed by a single person or multiple persons have similarity due to the kinematic consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Thus, in the feature space, sam- ples from each normal activity group 𝐴𝑐 are supposed to follow an 𝑛-dimensional Gaussian distribution as the activities from the same group (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=', sit, bow, or jump) are repeated and controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We denote a certain activity group as 𝐴𝑐 with number of 𝑗 (𝑐) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Ideally, all normal samples from different daily activities together form a mixture distribution of Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' With such prior knowledge, we therefore impose the constraint to FallNet’s learning process and force it to learn a mixture Gaussian distribution over the latent feature space, rather than learning a vec- tor of feature representations 𝑧 that may be over-fitted with limited data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To this end, we modify the output of the encoder from 𝑧 to two vectors 𝜇𝑐 and 𝜎𝑐 that represents mean and variance of the activity group Ac that each training sample belongs to, respec- tively, where 𝜇𝑐, 𝜎𝑐 ∈ R𝑛, 𝑛 is the number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The FallNet learns the two vectors to parameterize the feature distribution of 𝐴𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' A constrain loss is added to minimize the Kullback-Leibler (KL) divergence between the learned disrtibution of the parametric rep- resentation and the desired distribution 𝑝 (𝑧|𝑥 ∈ 𝐴𝑐) ∼ N �𝜇𝑐, 𝜎2𝑐 � such that: Lc = −1 2 𝑛 ∑︁ 𝑖=1 � 𝜇2(𝑖) + 𝜎2(𝑖) − log𝜎2(𝑖) − 1 � where 𝜇(𝑖) and 𝜎(𝑖) denote the 𝑖-th element of the 𝑛-dimensional vectors 𝜇 and 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In such a way, each activity is modeled as a mul- tivariate Gaussian distribution with 𝑛-dimensional features in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Different activities have different mean vectors 𝜇𝑐 and variance vectors 𝜎𝑐 to represent different Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As the number of samples increases, the hidden space gradually forms a complex Gaussian mixture distribution: 𝑝𝜃 (𝑧) = 𝑐∑︁ 𝑗𝑐 𝑁 𝑝 � 𝑥 ∈ 𝐴𝑐 | 𝜇𝑐, 𝜎2 𝑐 � If the latent distribution is valid, correspondingly, any of the latent space samples from the distribution should be able to reconstruct 𝑥 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='7 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 目 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 0 0 0 x X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='9-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='4 0 0 x indices values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 0 max-pooling up-samplingSenSys ’22, November 6–9, 2022, Boston, MA, USA Sijie Ji, Yaxiong Xie, and Mo Li well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Therefore, the input of the decoder now becomes a 𝑧 that is stochastically sampled from the corresponding 𝜇 and 𝜎 such that ˆx = FD � 𝛿z ∼ N � 𝜇, 𝜎2� , ΘD � where 𝛿 represents a random sampling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' On the other hand, the back-propagation of training neural network requires deterministic operations at each neural network nodes which iter- atively pass the gradients and apply the chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The stochas- tic sampling operation however is not a continuous function and thus not differentiable to obtain the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To make the neu- ral network trainable, the FallNet adopts the reparameterization technique [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' It generates random 𝜀 from a standard normal dis- tribution N (0, 1) independent of the neural network nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The latent sample 𝑧 is obtained through scaling and transformation by 𝑧 = 𝜇 + 𝜎 × 𝜀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The reparameterization allows 𝑧 to be sampled from the corresponding distribution of 𝜇 and 𝜎 at each iteration while the random sampling itself is not involved in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As the sampled 𝑧 is deterministic at each iteration its gradient can be back-propagated to train the entire neural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Consequently, the objective of the FallNet is revised: arg min 𝜇,𝜎,ΘD E𝑥∼X � ∥𝑥 − FD ((𝜇𝑥 + 𝜎𝑥 × 𝜀, ΘD)∥2� ,𝜀 ∼ N (0, 1) In addition, the FallNet design also employs data augmentation scheme to compensate the data scarcity and improve model gener- ality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The FallNet imposes two specific augmentation schemes: (i) To simulate a low SNR scenario, before being converted to STFT spectrums, for each segmented 𝑆(𝑡), we add Gaussian white noises, which equals to adding noises in the channel domain of the input tensor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' (ii) To alleviate the limitation of time resolution due to the fixed STFT window length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Each input tensor 𝑥𝑖 goes through three rounds of random horizontal shift [42], with the shifting length smaller than the STFT window length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' At the end we are able to fabricate 24× the amount of original data to augment the training size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='4 Online Detection and Model Updating After pre-training with a normal activity dataset 𝑋, the FallNet has established the distribution of the anchor daily activities in the latent feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Let each activity segment 𝑥 go through the FallNet, we can derive the statistics of reconstruction error of the dataset including its average 𝛼 and median𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In the online detection phase, the FallNet takes the real time segmented STFT samples for inference in a single run, and measures its reconstruction error 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' If 𝑒 > 2𝛼, it is detected as a fall and at the same time 𝛼 and 𝛾 remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' If 𝛼 < 𝑒 < 2𝛼, the system takes it as a suspicious daily activity and saves the segmented samples for feature reference, but 𝛼 and 𝛾 are recalculated and updated accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Once the change of𝛾 exceeds a threshold, the system takes it as an indication of significant change in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' If 𝑒 < 𝛼, the system updates the 𝛼 and 𝛾 and then performs data augmentation where a mini-batch of augmented data samples are fed to the FallNet for retraining the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Therefore, the system keeps evolving with the feedback of reconstruction error 𝑒 and adaptively updates the threshold 𝛼 to determine falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As the system runs in real-time, the incoming fall-like samples for inference may bring two types of distribution shift, one being the semantic shift caused by the individualized movement patterns across people, the other being the covariance shift due to environ- ment variation over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As SiFall eliminates the environment impact by extracting the dynamics of RF signals, the covariance shift is well accommodated along with the continuous update of the FallNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The saved suspicious daily data samples are utilized to deal with the semantic shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Whenever an adequate amount of suspicious daily data samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=', 50 as set in our current im- plementation) are collected, SiFall performs principal components analysis (PCA) to reduce the dimension to 𝑛 and then performs mean-shift clustering [41] to identify 𝑁 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Two criteria are applied to handle the cluster points, namely, representativeness and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We examine the largest cluster as it indicates many repeatable activities which are unlikely to be human falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' SiFall retrieves the signal segment of the centroid of the largest cluster, produces 24× augmented data, and feeds that to the FallNet for model retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' SiFall also notices when there is a cluster that is far away from other clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The cluster is taken as a potential undiscovered user activity group and its signal segments are kept for later examination when adequate amount of such suspicious data are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The remaining signal segments are discarded and the counter is updated till next time the number of saved samples reaches 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Based on the above described mechanism of automatic model update, SiFall does not require explicit human intervention for most of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Only when a "fall" is detected SiFall triggers an alarm for possible human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The corresponding data samples are saved with a timestamp regardless whether the detected "fall" is a true positive or false positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The human user may examine the saved "fall" samples at any later time to decide whether they are true positives in which case the samples are discarded, or false positives in which case the samples are augmented and fed back to the FallNet for retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 4 EVALUATION In this section, we evaluate the performance of SiFall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We first introduce our experimental settings and then present the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 Experimental Setting We implement SiFall with two commercial off-the-shelf (COTS) APs as the Tx and Rx to collect the WiFi CSI, one laptop connected to the Wi-Fi receiver to serve as the front-end edge server and one back-end server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We use a camera to capture the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We use COTS COMPEX WPJ558 equipped with Atheros SoC QCA9558 in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We let these two APs transmits 200 packets per second on a 20MHz channel in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='4GHz frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We fix the Modulation and Coding Scheme (MCS) to reduce packet loss and noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We use Atheros-CSI-Tool [59] to collect raw CSI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The receiver forwards the collected CSI to the ThinkPad T430 laptop with an Intel Core i5-3360M CPU to process the RF sig- nals and generate STFT segments (as introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We use a Linux desktop computer equipped with Intel Core i9-9820X CPU and one Nvidia 2080Ti GPUs to work as the back-end server SiFall: Practical Online Fall Detection with RF Sensing SenSys ’22, November 6–9, 2022, Boston, MA, USA Figure 8: The environment of the three testbeds and their floor plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' to maintain the FallNet and perform real-time inference to detect the falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We test SiFall based on three testbeds - an emulated "bed- room" with an enclosed space measured 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='32m × 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='24m for com- prehensive evaluation (testbed 1), a real apartment room measured 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='85m × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='47m for system adoption test on a daily basis (testbed 2), as well as a big open area measured 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='54m × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='05m to test the effective sensing range of the system (testbed 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 8 depicts the three different testbeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The marked Tx and Rx indicate the locations of the WiFi Tx and Rx antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Ground Truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We use a camera to record the detailed human activities at a frame rate of 30fps, and manually analyze the recorded video clips to generate the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We use network time protocol (NTP) to synchronise the time in the camera recordings and the collected Wi-Fi CSI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' FallNet Pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We pre-train the FallNet with the data col- lected intermittently during 3 months in testbed 1, including 1447 sets of STFT segments of sitting, jumping, swinging, bowing, run- ning, and other daily activities, augmented 24 times to produce a total number of 34,728 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Correlation among raw samples is removed by OpenCV, and the weight parameters are initiated by kaiming initialization [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The model was trained by Adam [29] optimizer on 4 Nvidia 2080Ti GPU for 2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Testing Subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We recruit 16 volunteers (11 males and five fe- males) with ages between 21 and 56 to take part in our experimental evaluation (with IRB approval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Table 1 summarizes the detailed in- formation of all volunteers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The testing subjects are highly diverse in their age, weight, and height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Specifically, the body weight of our volunteers varies from 42kg to 100kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Their body height varies from 155cm to 186cm, and their age varies from 21 to 56 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' RT-Fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We compare the performance of SiFall with RT-Fall [53], which is, to the best of our knowledge, the only RF-based fall de- tection system which claims being able to achieve real time fall detection in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' RT-Fall identifies fall-like activities based on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='Table 1: Summary of the testing subjects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='#Person ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='a pre-defined threshold on the measured CSI phase difference be- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='tween two Rx antennas and segments the collected CSI stream with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='a fixed 3s time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' RT-Fall then feeds the derived statistical phase and amplitude features of the CSI segment into a pre-trained SVM model to identify falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We reproduce the system and train an SVM classification model of RT-Fall with the data collected from our testbed, the same as what we use to pretrain our FallNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2 End-to-end Evaluation We first conduct intensive movement experiments with 12 subjects and report the end-to-end performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' After that, the proposed system components are evaluated based on the detailed experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1 Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 12 testing subjects (#P1,#P6-#P16) are involved to conduct the experiment in a sequential order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Each testing subject is requested to move freely around one and half an hours inside the bedroom testbed as depicted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We request each of them to perform the following actions at their will when they move around: "jump", "squat", "sit to the floor", "sit to the chair", "knee down", and "bow" at least three times at different locations and with different body orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Other than the requested type of movements, they are free to perform any other activities at their will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We summarize other fall-like movements that are hard to quantify as "swing".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To mimic unconscious falls as much as possible while meeting the IRB requirement on risk control, we set up a safety mattress and experiment with the falls of three categories [4, 32, 46]: (1) for "walk A1 A2 A3 RX dSenSys ’22, November 6–9, 2022, Boston, MA, USA Sijie Ji, Yaxiong Xie, and Mo Li Table 2: Types of Falls Types Examples "walk fall" slip, stumble scenes: rushing to answer the telephone, slipping in the bathroom, and tripping over the cable, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' "stop fall" lost balance, lost consciousness scenes: coming out of bed, epileptic seizure, stroke, and heart attack, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' "slow fall" dizziness/vertigo, weakness scenes: arthritis pain, transfer to a dim room, postural hypotension, and vision disorder, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' fall", the subject is asked to walk around the mat and instantly fall on the mat once a random alarm is triggered by us - the fall is performed regardless the instant body orientation of the testing subject;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' (2) for "stop fall", the subject stands still on the mat and tries to dodge the tennis balls thrown at her - if she happens to fall the activity is noted as a valid "stop fall", and as swing activity otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' (3) for "slow fall", the subject keeps standing still until we give a random alarm when she simulates a slow fall on the mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Table 2 illustrates the three categories of falls with corresponding real life scenes and examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' It is worth noting that regardless of the type of falls, the falling orientation is random during the experiments based on the reaction of the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' During the experiment, SiFall continuously operates and each of the 12 testing subjects enters the bedroom in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The total experiment duration for all 12 testing subjects is about 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The FallNet model is continuously maintained and updated throughout the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We evaluate the performance with True Positive Rate (TPR) and False Positive Rate (FPR) metrics, where TPR is true falls out of SiFall reported falls and FPR is falsely reported falls out of other activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The accuracy is calculated by the percentage of correctly detected falls and non-falls against the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2 Overall Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' During the 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 hours experiment, SiFall captures a total number of 1497 fall-like activities, of which 523 seg- ments are intentional activities performed by the testing subjects (including 123 falls and 400 required fall-like activities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Among the 123 falls, 60 are "walk fall", 33 are "slow fall" and 30 are "stop fall".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We derive the TPR and FPR in about every 20 minutes and plot the results over time in Figure 9, where TPR is represented by the black solid line and FPR is represented by the balck dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Both the TPR and FPR vary over time as the FallNet model continuously evolves when more training data are collected from the testing subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We see a clear trend of improvement on both the TPR and FPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' First, the TPR improves quickly over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' From 83% at the be- ginning of the experiment, the TPR constantly improves over time and reaches 100% within 4 hours of operation, which demonstrates SiFall’s capability in accurately identifying the abnormal falls from normal daily activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Second, the FPR of SiFall improves greatly over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The falsely reported falls by SiFall are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='7 per hour in the first two hours and eventually drops to below 1 per hour in the last two hours of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' While the TPR shows a clear trend of improvement over time, the FPR occasionally fluctuates, especially during the experiment of each individual testing subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' That is mainly due to the fact that our experiment does not restrict how each testing subject performs certain activities, and as a result Figure 9: System end-to-end performance evolution over time across different test subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' some testing subjects may choose to perform more activities simi- lar to falls, and in different orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' For example, one (#P9) prefers challenging SiFall system by performing more "sit on the floor" activity which is more similar to "slow falls" and results fluctuated FPR during his experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' If we focus on the FPR statistics by the end of each testing subject’s experiment (the gray line), we may see steadily improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' At the end of the 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 hour experiment, SiFall is able to achieve 100% TPR and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='8% FPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We simultaneously run RT-Fall for comparison and plot the achieved TPR and FPR of RT-Fall in red in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We find that during the real time operation RT-Fall achieves a much lower per- formance, with its TPR of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='9% and FPR of 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Since RT-Fall does not have the ability to self-evolve, it cannot gain performance over time and it fluctuates across different testing subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Overall the comparative results show huge comparative advantage of SiFall over the SOTA available real-time RF fall detection approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 FallNet Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We visualize the FallNet input and out- put to demonstrate the rationale when applying FallNet to detect the falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Specifically, we impose three checkpoints during the ex- periment (as indicated in Figure 9 as CKPT1 to CKPT3, after the test of subject #P6, #P11, and #P15, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' At each check- point, we freeze the FallNet model and memorize it for detailed investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We feed different RF signal segments collected from normal activities and falls into the restored FallNet models at the three checkpoints, respectively, to examine the reconstructed out- put from the FallNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In Figure 10, we plot the STFT segments of the input signal and the reconstructed STFT segments by the FallNet for different types of falls (Figure 10a) and other ordinary activities (Figure 10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We clearly see that while most STFT segments of most ordinary activities can be recovered by the FallNet the STFT of falls cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We also observe that as the model evolves (from CKPT1 to CKPT3), the reconstruction errors of all the falls increase while the reconstruction errors of ordinary activities decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Note that the errors is computed as the L2-norm of the difference between the FallNet input and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The reconstruction errors of falls are order of magnitude higher than those of ordinary activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The visualization suggests that the FallNet is able to continuously learn better latent distribution to describe the human daily activities and based on that make more accurate detection of falls as outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Notably the low-frequency part of the spectrum and the end-of- motion part remain clear despite the deteriorating quality of the reconstructed STFT samples of falls, which suggests that the FallNet CKPT1 CKPT2 CKPT3 100 TPR(%) 80 65 54 SiFall FPR(%) 20 RTFall 15 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 #P1 #P6 #P7 #P8 #P9 #P10 #P11 #P12 #P13 #P14 #P15 #P16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Time(h)SiFall: Practical Online Fall Detection with RF Sensing SenSys ’22, November 6–9, 2022, Boston, MA, USA (a) The original and reconstructed STFT segments of Falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' (b) The original and reconstructed STFT segments of other activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 10: Visualization of the FallNet original input and reconstructed output STFT segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' indeed utilizes the low-frequency and end-of-motion features in discriminating the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='4 SiFall Segmentation Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The segmentation algo- rithm of SiFall depends on detecting the status of human movement and is threshold based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We separately evaluate the accuracy of the movement detector and the threshold sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Movement Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We classify the human movement into three levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Body level movement refers to the whole body move- ment involving position change, such as walking and running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Torso level movement refers to torso movement without position change, such as bow and squat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Limbs level movement refers to limbs and hands movements at minor scale such as shaking hands and typing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 11a reports the percentage of relative error (false negative rate) of the corresponding movement detection results when testing subjects move freely in testbed environment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The figure plots the cumulative distribution based on the movements from 12 testing subjects (#P1, and #P6-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The experiment logs SiFall movement detection performance at body level, torso level and limbs level movement with median error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1%, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='8%, and 90th- percentile error of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='6% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='7%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' (a) (b) Figure 11: SiFall (a) segmentation performance of movement detection error and (b) the acceleration distribution of differ- ent types of movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Threshold Sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To quantitatively evaluate the reliability of the threshold Θ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 as fixed in the SiFall implementation, we derive the maximal frequency of each human movement and project to its acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 11b depicts the cumulative distribution of the projected acceleration for different types of movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We see that all fall activities have their derived acceleration above the threshold and the majority of other fall-like activities are also captured with the current threshold setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' On the other hand, most ordinary walk and run movements are screened out by the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5% of bow activities are screened out as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The result suggests that the threshold setting is effective in screening falls and fall-like activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' It is also obvious that SiFall is robust to the threshold setting - its accuracy will not be impaired when the threshold falls in the range between 2 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Segmentation Length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Following the strategy of "more is better than less", the SiFall segmentation algorithm aims at capturing the activities with redundancy in their time durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 12 com- pares the lengths of SiFall captured segments and the corresponding ground truth time durations of the activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' In general the SiFall segments have an average duration of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1s, which is longer than the actual activity duration averaged at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Additional 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5s signal data are included in the SiFall segments for redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Overall, SiFall segmentation algorithm introduces necessary redundancy in extracting the signal segments while maintaining the signal processing overhead on the extra signal durations acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 12: Comparison of the ground truth and SiFall seg- mented lengths of different activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2 BodyLevelMovement :TorsoLevelMovement LimbsLevelMovement 0 0 1 2 3 4 5 Relative Error(%)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='9 Walk/Run Sit ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Kneel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 Bow Swing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2 Jump Squat 0 Fall 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 4 5 6 7 8 9 a-SiFallSegments 6 GroundTruth 2 Bow Squat Kneel dwnr Sit Swing FallSenSys ’22, November 6–9, 2022, Boston, MA, USA Sijie Ji, Yaxiong Xie, and Mo Li Figure 13: False alarms that occur during the daily life adop- tion test across three days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3 Daily Life Adoption Test In the above experiment, human subjects were asked to contin- uously perform a large number of falls and fall-like activities in a short time duration for comprehensive evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To further challenge our system and understand the long-term performance in a more realistic setting, we adopt SiFall with pretrained FallNet from the emulated bedroom (testbed 1) to a real apartment room (testbed 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We conduct a continuous three day evaluation with one testing subject (#P2) working and living inside the testing room day and night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' A total number of 204 fall-like activities (67 on day 1, 63 on day 2, and 74 on day 3, respectively) are captured at the front end and sent to the FallNet for fall inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Totally 12 false alarms are triggered (9 on day 1, 2 on day 2, and 1 on day3, respec- tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 13 depicts those false alarms and their occurrence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' After verifying with the testing subject, the first-day false alarms mainly come from sitting on the sofa and they are phased out gradually with the model update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The first false alarm on the second day is raised when an object falls from the wardrobe and the testing subject picks it up immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The second false alarm on the same day is raised when the subject jumps and dives into the sofa from the back of the sofa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The last false alarm on the third day is triggered when the testing subject does handstand on the Yoga mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We see a clear trend that the false alarms dramatically reduce when the FallNet model is continuously updated over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Quantitatively, the false alarm rate decreases from 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='4% on the first day to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='4% on the last day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The experiment results suggest that SiFall has the ability to learn from personalized daily activities, and build evolved models for more accurate fall detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' However, some rarely-seen combination of movements may still trigger the false alarm, which we expect would reduce when SiFall continues to see more repeated occurrences of such activities over longer time of deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' After the three day continuous monitoring of the daily activities with SiFall, we perform a purposed experiment to evaluate its de- tection accuracy of true falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We freeze the model update of SiFall, and let the testing subject perform ten emulated "stop falls" and "slow falls" following the same methodology illustrated in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The falls are performed across 5 different locations in the room (shown as "X" in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We then pour a lot of powder on the floor, let the testing subject wear safety gear, keep jogging in the room till five "walk falls" are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Apart from those, the testing subject also simulates a fall that rolls from the bed as well as a slipping fall when trying to sit on the office chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' SiFall can detect all the above falls except for the rolling fall from the bed with 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='1% detection rate, demonstrating the high reliability of SiFall in real-life application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 14: Accuracy of different person at different link dis- tances 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='4 Effective Covering Range As SiFall captures fall-like activities based on sensing the wireless channel dynamics, we want to evaluate its effective sensing range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We deploy SiFall with the pretrained FallNet from the "bedroom" environment (testbed 1) to a bigger open area (testbed 3) without fine tuning the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Three testing subjects (#P3,#P4 and #P5) are requested to perform "jump", "sit to the floor", "swing", and "walk fall" (each for five times with a random sequence) repeatedly in three different areas (as depicted in Figure 8) with a distance of 1-3 meters, 3-5 meters, and 5-7 meters, respectively, to the LOS link of the Wi-Fi transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 14 reports the TPR and FPR of the three testing subjects when experimented in the three different areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We see that SiFall performance is robust when adopted across the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The accuracy is reasonably good when the testing subjects move to as far as 5m away from the Wi-Fi link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The average TPR and FPR in area A1 and area A2 were 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3% and 20%, 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3% and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='2%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' When the distance increases to 7m in area A3, the average TPR drops significantly to 40% and the FPR drops to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='4% at the same time due to failures in detecting and segmenting all fall-like activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Note that when directly migrating the FallNet model to a new environment (from the "bedroom" in testbed 1 to the big open area in testbed 3), SiFall still achieves an average accuracy of 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='3% which is impressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We expect the accuracy will further improve with time when more human activities are captured and consumed by the FallNet model to evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='5 Computation Overhead We provide a quantitative analysis of the SiFall’s computation over- head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We utilize the Pytorch-OpCounter tool to measure the compu- tation cost in flops (floating-point operations per second) of FallNet and some representative CNN-based models used in other appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As Figure 15a depicts, FallNet falls in between the ultra- lightweight model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=', MobileNetV2) and the medium-weight models (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=', Densenet121 and AlexNet), which indicates FallNet is a relatively lightweight model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' (a) FallNet model complexity compared with SOTA CNN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Inference Time (ms) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='267 Update Time (ms) 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='242 Warning Delay (ms) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='552 Alarm Delay (s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='670 (b) Latency of SiFall compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Figure 15: SiFall computation overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 2 Day 1 Day 2 Day 3 P 0 10:00 22:00 10:00 22:00 10:00 22:00 10:00 Time80 60 40 20 0 FPR(%) 20 1#P3 40 1#P4 #P5 60 80 Area1 Area2 Area37 VGG16 InceptionV3 6432 flops(G) ResNet50 DenseNet121 FallNet 1 AlexNet MobileNetV2 10 55 100 145 #Parameters(M)SiFall: Practical Online Fall Detection with RF Sensing SenSys ’22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' November 6–9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' USA We also measure the end to end latency of SiFall operation in our testbed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' and summarize the result in Table 15b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The average inference time and the model parameter update time are measured on a single NVIDIA 2080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Once SiFall front-end detects a fall-like activity, it triggers a warning and waits for the FallNet at the back-end to generate the alarm for confirmed falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We measure the warning delay (alarm delay) by averaging the time interval between the system warning time (alarm time) and the ground truth ending time of the fall-like activities (fall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The major delay of the system comes from the signal segmentation, where SiFall keeps monitoring the channel dynamics for 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 5 RELATED WORK RF-based fall detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Existing RF-based fall detection sys- tems [23, 38, 45, 51, 53, 57] all assume repeatable human fall pat- terns and follow pre-defined fall templates in the feature space for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Most of them depend on manually segmented signal clips for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Aryokee [51] utilizes CNN to extract features of human fall as opposed to previous manual feature extraction ap- proaches [38, 51, 53, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' However, the samples are collected offline with the same length, the Aryokee model is not able to deal with varying length RF samples and the system cannot run in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' FallDefi [38] improves the performance by using the combined features of previous approaches and adopting more WiFi links for gathering RF signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' There are some general RF based human ac- tivity recognition systems including Witrack [2], CARM [55] and HAR-SANet [11] which treat fall as one of the ordinary human activities and can only capture few types of falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' To the best of our knowledge, RT-Fall is the most practical solution of real time fall detection, which however as suggested in our experimental evaluation cannot provide high accuracy with realistic falls occur- ring in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Although Defall [23] claims real-time fall detection capability, it uses a human-like dummy to do the experiment to learn the fall template, and thus it can only detect simulated "hard fall", which is falling from a standstill position at a certain height, by its nature significantly limits its application in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Other fall detection solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Other than RF-based fall detec- tion, there are CV-based fall detection approaches [13, 17, 64] that take optical measurements by camera or infrared sensors for analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Those solutions are often criticized for compromising human privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Wearable-based fall detection methods either require the user to carry the device [3, 9] or wear the device [27] which are intrusive and thus not the most desired way for fall detection [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The acoustic-based [15] method is limited by ambient noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Sensor fusion-based fall detection [34, 69] is believed to be more reliable as various sensors may complement each other in different situa- tions, but generally leads to higher cost and deployment overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Among them, some works claim they detect falls based on anom- aly detection [9, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' A CV-based approach [17] collects a balanced dataset with fall and non-fall samples and use a supervised anomaly detection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' A wearable-based approach [9] learns a fixed boundary in feature space to separate daily activity and the anomaly fall, which cannot cope with unseen daily activities and falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Deep learning based RF sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Deep learning has recently been widely adopted to various wireless sensing applications, in- cluding physiological sensing [67], food and liquid sensing [20], gesture recognition [68], body skeletons reconstructing [36, 65, 66], localization [5] and etc [8, 18, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='Those solutions cannot be directly applied to detecting human falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Most of them do not support the neural network update during run time and often require extensive data collection and annotation to facilitate the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' While anomaly detection is well-studied in the literature, anomaly detection for high-dimensional data in real- time remains challenging [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Traditional methods such as One- Class SVM [47], Kernel Density Estimation [40] and Tree-based Isolation Forest [35] all fail to operate online due to unsatisfactory computational scalability and the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Thanks to the rapid development of deep learning technology, a lot of deep learning based anomaly detection methods have been proposed [54, 62, 70] with similar frameworks that consist of three parts: feature extraction, feature representation learning, and end-to-end anomaly score learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We design FallNet based on this skeleton and make it capable to run in real-time with unstructured input signal data, to fill the gap in the literature, as most deep learning based methods are capable to only structured datasets and lack real-time practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Self-supervised learning techniques support learning representations from a large amount of unlabeled data and based on that representation to serve downstream classi- fication tasks with a few labeled instances [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' As an alternative solution to establish a representation of daily human activities, self- supervised learning still faces the challenge in the lack of labels for unforeseeable human fall types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' From a different perspective, SiFall deals with the domain variations by building an anomaly detection neural network model and continuously evolving the model to represent high-level semantics of normal daily activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 6 DISCUSSION & CONCLUSION This paper proposes SiFall, a self-supervised incremental learning human fall detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' SiFall leverages Wi-Fi RF signals and is able to detect daily human falls in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Extensive experiment results demonstrate that SiFall achieves high accuracy in human fall detection and is resilient to varied human subjects, environment, and different types of falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The design of SiFall makes an important contribution towards building practical and reliable RF-based fall detection systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The current study is still limited in its lack of real fall samples, especially of elderly aged above 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Since SiFall relies on wireless channel dynamics to catch human activities, it is currently limited to working with single room occupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We leave the exploration to the above two limitations to future work when developing SiFall into higher technology readiness levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' ACKNOWLEDGMENTS We sincerely thank the shepherd and reviewers for their insightful comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' We also thank all volunteers for their participation in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' This research is supported by the National Research Foundation Singapore under its Industry Align- ment Fund – Pre-positioning (IAF-PP) Funding Initiative, and Min- istry of Education Singapore MOE AcRF Tier 2 MOE-T2EP20220- 0004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Any opinions, findings and 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' 𝜇(𝑥), 𝜎(𝑥) are the mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The difference of the two norm just the way that how the statistical descriptors 𝜇 and 𝜎 are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Given an input batch 𝑥 ∈ R𝐵×𝐻×𝑊 ×𝐶,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Batch Norm normalizes the mean and standard deviation for each individual feature channel to a whole batch: 𝜇𝑐 (𝑥) = 1 𝐵𝐻𝑊 𝐵 ∑︁ 𝑛=1 𝐻 ∑︁ ℎ=1 𝑊 ∑︁ 𝑤=1 𝑥𝑏𝑐ℎ𝑤 (10) 𝜎𝑐 (𝑥) = √︂ 1 𝐵𝐻𝑊 𝐵 ∑︁ 𝑛=1 𝐻 ∑︁ ℎ=1 𝑊 ∑︁ 𝑤=1 (𝑥𝑏𝑐ℎ𝑤 − 𝜇𝑐 (𝑥))2 + 𝜖 (11) As Batch Norm uses mini-batch statistics during training phase and replace them with average mean and variance across batches during inference phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' which implicitly requires consistency distri- bution of training domain and inference domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' SiFall is an online detection system and the samples keep generated that might induce difference across different person and environments so that we use Instance Norm which normalize as per sample: 𝜇𝑏𝑐 (𝑥) = 1 𝐻𝑊 𝐻 ∑︁ ℎ=1 𝑊 ∑︁ 𝑤=1 𝑥𝑏𝑐ℎ𝑤 (12) 𝜎𝑏𝑐 (𝑥) = √︂ 1 𝐻𝑊 𝐻 ∑︁ ℎ=1 𝑊 ∑︁ 𝑤=1 (𝑥𝑏𝑐ℎ𝑤 − 𝜇𝑏𝑐 (𝑥))2 + 𝜖 (13) B CONVOLUTION OPERATION INDEPENDENT TO THE INPUT SIZE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The "convolution" operation in the neural network is different from the "convolution" in the signal processing domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Indeed, the convolution operation is an element-wise multiplication and summation over a local region of the input tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The operation is repeated in sequential local regions until the whole tensor has been calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Each learnable filter𝑊 in convolution operation with dimension 𝑊 ∈ R𝑘×𝑘, where 𝑘 denotes the kernel size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=', the size of the local region that calculates the multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Let 𝑋 ∈ R𝐻𝑖𝑛×𝑊𝑖𝑛×𝐶 de- note the input tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The convolution operation calculates output 𝑌 such that: 𝑌𝑝,𝑞 = 𝐶 ∑︁ 𝑛=1 ∑︁ 𝑖,𝑗 ∈N𝑘 𝑊 ⊤ 𝑖+ 𝑘−1 2 ,𝑗+ 𝑘−1 2 𝑋𝑛 𝑝+𝑖,𝑞+𝑗 where (𝑝,𝑞) denotes the location coordinate and N𝑘 = � (𝑖, 𝑗) : 𝑖 = � −𝑘−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' , 𝑘−1 2 � , 𝑗 = � −𝑘−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' , 𝑘−1 2 �� defines a local neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' A convolution layer specify how the kernel sliding 𝑖 and 𝑗 through the input tensor by setting stride 𝑠 and how we want the input tensor be padded by setting 𝑝, as a result the output 𝑌 ∈ R𝐻𝑜𝑢𝑡 ×𝑊𝑜𝑢𝑡 ×𝑓 can be computed by (𝐻𝑜𝑢𝑡,𝑊𝑜𝑢𝑡) = ��𝐻𝑖𝑛 + 2 ∗ 𝑝 − 𝑘 𝑠 � + 1, �𝑊𝑖𝑛 + 2 ∗ 𝑝 − 𝑘 𝑠 � + 1 � , where 𝑓 is the number of learnable filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' The ’same padding’ technique help choose proper 𝑠 and 𝑝 so that 𝐻𝑜𝑢𝑡 = 𝐻𝑖𝑛, 𝑊𝑜𝑢𝑡 = 𝑊𝑖𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} +page_content=' Consequently, convolution layer are able to adapt to the input tensor with arbitrary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE2T4oBgHgl3EQfQwY9/content/2301.03773v1.pdf'} diff --git a/a9FST4oBgHgl3EQfCDgc/content/tmp_files/2301.13705v1.pdf.txt b/a9FST4oBgHgl3EQfCDgc/content/tmp_files/2301.13705v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4aaa5a804a788c1e6b52be691a92ab2f3e21c2cc --- /dev/null +++ b/a9FST4oBgHgl3EQfCDgc/content/tmp_files/2301.13705v1.pdf.txt @@ -0,0 +1,732 @@ +Clever Design, Unexpected Obstacles: Insights on +Implementing a Quantum Boltzmann Machine +Felix Paul +StoneOne AG +Berlin, Germany +felix.paul@stoneone.de +Michael Falkenthal +StoneOne AG +Berlin, Germany +michael.falkenthal@stoneone.de +Sebastian Feld +Delft University of Technology +Delft, The Netherlands +s.feld@tudelft.nl +Abstract—We have implemented a gated-based quantum ver- +sion of a restricted Boltzmann machine for approximating the +ground state of a Pauli-decomposed qubit Hamiltonian. During +the implementation and evaluation, we have noticed a variety of +unexpected topics. It starts from limitations due to the structure +of the algorithm itself and continues with constraints induced by +specific quantum software development kits, which did not (yet) +support necessary features for an efficient implementation. In this +paper we systematically summarize our findings and categorize +them according to their relevance for the implementation of +similar quantum algorithms. We also discuss the feasibility of +executing such implementations on current NISQ devices. +Index Terms—quantum software engineering, NISQ devices, +quantum machine learning, limitations and constraints +I. INTRODUCTION +The advent of quantum computers accessible to the public +has given an immense boost to the development of new quan- +tum algorithms in the last decade. Many algorithms promise +advantages over previously known classical ones, e.g., with +regard to a speedup [1] or enhanced solution quality [2]. +However, to raise these advantages the conceptual algorithms +first have to be applied to actual use cases. Second, they +have to be implemented to run on specific quantum computers +which are still NISQ devices providing limitations with respect +to decoherence time, gate and measurement failure rates [3]. +In practice, this leads to major difficulties when transferring +conceptual algorithms into actual implementations. In order +for them to be executable on NISQ-devices, the limitations +of the specific quantum hardware must be taken into account +accordingly. For example, the available number of qubits and +the fidelity of their states must be considered such that an +implemented algorithm can (i) be transferred to the available +qubits at all and (ii) be executed by the quantum computer +in an amount of time that still guarantees tolerable error rates +and, thus, allows to read out meaningful results [4]. +However, these limitations have a strong influence on +whether the theoretically described advantages of an algorithm +can be raised at all [5]. Therefore, it is valuable to examine +algorithms for their practicability on NISQ computers and to +recognize limitations early on. On this basis, it is possible to +investigate and elaborate on appropriate mitigation. +This work was partially funded by the project PlanQK (01MK20005N) +supported by the German Federal Ministry for Economic Affairs and Climate +Action. +In this work, we present details and findings regarding +the implementation of a quantum version of a restricted +Boltzmann machine (QBM) based on the algorithm introduced +by Xia and Kais [6]. We provide insights into what kind +of obstacles we encountered when translating a theoretically +proposed quantum algorithm into an implementation that can +be executed on one of the quantum backends currently avail- +able. Some of these issues are caused by implementing the +algorithm within a given software stack or quantum software +development kit (quantum SDK). Others only arise when +trying to execute the code on quantum backends, since their +properties play a crucial role for gaining meaningful results. +We also discuss aspects regarding the possible solution +quality of the chosen model: There are limitations induced +by transferring quantum data, stored within the hardware, into +classical data and also some limitations that are induced by the +variational model itself and the conditions it is built on. For +some of the mentioned issues we suggest possible approaches, +as to how the impact of these may be reduced. Those rec- +ommendations provide guidance for implementing other algo- +rithms which make use of similar concepts. With this in mind, +we classify the encountered problems in terms of how the +same or similar ones may arise during the implementation of +other algorithms. Since numerous algorithms proposed make +use of similar subroutines or concepts (e.g., using parametrized +rotational gates for variational models), we believe that by +systematically classifying common issues, some of these can +be managed better within future implementations. Thus, we +aim on giving an initial list of indicators which need to +be considered in order to access the practical feasibility of +algorithms proposed. Since our findings will be explained in +the context of a QBM algorithm, we also summarize the key +aspects of it which we supplement with exemplary calculations +to illustrate relevant procedures in more detail. +The remainder of this paper is structured as follows: We +introduce the essential concepts of the quantum Boltzmann +machine algorithm by Xia and Kais [6] in Sec. II. We identify +and discuss findings and restrictions of the QBM algorithm +with respect to currently available quantum computers and +software development stacks and development kits in Sec. III. +Finally, we conclude the paper with an outlook on future work +in Sec. IV. +arXiv:2301.13705v1 [quant-ph] 31 Jan 2023 + +II. ALGORITHM DETAILS +The goal of the paper is to highlight practical problems that +arise during the implementation of quantum algorithms, which +is done on the example of a QBM. For this purpose, we will +present the algorithmic details in the following, which will be +referred to in Section III. +The goal of the approach described by Xia and Kais [6] is +to approximate the ground state of a given Hamiltonian using +a hybrid quantum-classical algorithm. The utilized QBM is +made up of three layers: a visible layer with n qubits, a hidden +layer with m qubits, and a classical sign layer realizing relative +signs between different states of the visible layer (see Fig. 1). +For explicit values of n and m, we will refer to this model +as a (n, m)-QBM. The wave function to be generated by the +QBM and which should approximate the ground state is +|ψ⟩ = +� +{v} s(v)φ(v) |v⟩ . +(1) +Here, v denotes a single binary state consisting of n bits and +{v} represents the set of all possible binary states of the +visible layer (e.g., n = 2 leads to {v} = {00, 01, 10, 11}). +Furthermore, φ(v) is the real-valued amplitude associated to +state |v⟩, while s(v) is the corresponding value of the sign +node. The probability distribution p(v) of the qubit states +of the visible layer is generated by using the QBM, and it +determines the amplitudes of the target wave function: +p(v) = φ2(v) = 1 +Z +� +{h} exp (E(v, h)) , +Z = +� +{v,h} exp (E(v, h)) , +(2) +where � +{h} indicates the summation over all configurations +of the hidden layer for a fixed configuration of the visible +layer. The ”energy” E(v, h) of a given state of the visible and +hidden layer depends on real biases {ai}, {bj} and weights +{wij}, which are all subject to the optimization procedure. +The energy is given by +E(v, h) = +� +i aivi + +� +j bjhj + +� +ij wijvihj . +(3) +In (3), vi, hj ∈ {±1} are the σz-eigenvalues of the computa- +tional basis states (σz |0⟩ = |0⟩ , σz |1⟩ = − |1⟩) for the i-th +and j-th qubit in the visible and hidden layer, respectively. +The sign node is a smooth function that depends on the state +of the qubits from the visible layer as well as on parameters +{ci} and d, which are to be optimized, and it is given by +s(v) = tanh +�� +i civi + d +� +(4) +In fact, the algorithm proposed in [6] does not actually +generate the probability distribution described in (2), but a +modified one with an additional regulator k = O(� +ij |wij|). +This regulator normalizes all parameters p ∈ {ai, bj, wij} +according to p → p/k in order to increase the probability for +successfully generating the distribution initially wanted (see +Section II-B). For simplicity, including the regulator will be +omitted in the upcoming sections, but when implementing the +algorithm it needs to be considered. +Fig. 1: Exemplary network architecture of a (2, 3)-QBM. +Vertices v1 and v2 represent qubits from the visible layer, +while h1, h2, and h3 represent qubits from the hidden layer. +s corresponds to a sign node realizing relative signs between +states of the visible layer. As part of the algorithm, qubits from +the visible and hidden layer are entangled with each other via +3-qubit gates. +In the following sections we will describe the necessary +steps for generating the probability distribution given in (2) in +two steps – namely generating the linear and quadratic terms +of (3). After that, the basic optimization procedure and the +analytical gradients used in it are introduced. +A. Linear terms +The linear terms in (3) are generated by performing Ry- +rotations on all qubit states from the visible and hidden layer +with angle +θℓ = 2 arcsin +�� +e−pℓ +epℓ + e−pℓ +� +, +(5) +where pℓ ∈ {ai, bj} depends on whether the gate acts on a +qubit state from the visible or hidden layer and ℓ to be taken +from the corresponding index set. To give an example, (6) +shows the action on a single |0⟩-state which generates the +correct sign within the amplitude of each state: +Ry(θℓ) |0⟩ = +1 +√ +epℓ + e−pℓ +� +epℓ/2 |0⟩ + e−pℓ/2 |1⟩ +� +(6) +Note, that by rotating the qubit states in the described +manner, the probability distribution of (2) is generated for +wij = 0, ∀ i, j. +B. Quadratic terms +In order to include the interaction term of the i-th visible +and j-th hidden qubit in (3), a series of four doubly-controlled +Ry-rotation gates has to act on an ancillary qubit. The schema +of one such entangling layer is depicted in Fig. 2 and shows +four 3-qubit gates which all get controlled by one of the four +2-qubit basis states {|00⟩ , |01⟩ , |10⟩ , |11⟩}. + +h1 +U1 +h2 +S +~2 +h3vi : +... +hj : +... +a1 : +θ+ +ij +θ− +ij +θ− +ij +θ+ +ij +... +Fig. 2: Circuit diagram representation of an entangling layer +generating the interaction term between the i-th qubit from the +visible and the j-th qubit from the hidden layer. The gates are +Ry-gates with the arguments denoted in the box and defined +in (7). +For each entangling layer two angles are necessary, which +both depend on the weight wij between qubit i and j in the +following way: +θ± +ij = 2 arcsin +�� +e±wij +e|wij| +� +, +(7) +where θ+ +ij is used when the Ry-gate is controlled by states +with even parity (|00⟩ and |11⟩), while θ− +ij is accordingly used +for control states with odd parity (|01⟩ and |10⟩). This can +better be understood when looking at the energy in (3): If +visible qubit i and hidden qubit j are in the same state (either +both |0⟩ or both |1⟩), the product of their σz-eigenvalues is ++1 (since (+1)2 = (−1)2 = +1), whereas them being in +different states results in an eigenvalue product of −1 (since +(−1)(+1) = (+1)(−1) = −1). Now, in order to understand +the action of the doubly-controlled Ry-gates in more detail, +we will look at the action of a single-qubit Ry-gate on an +ancillary qubit with the angle defined in (7) (omitting the i, j +indices for simplicity): +Ry(θ±) |0⟩ = +1 +e|w/2| +�� +e|w| − e±w |0⟩ + e±w/2 |1⟩ +� +. (8) +Eq. (8) shows that with probability ∼ e±w the ancillary qubit +will be in state |1⟩, thus giving the correct contribution to the +energy defined in (3) when measured to be in that particular +state. +In order to successfully generate the desired distribution, +one ancillary qubit for every combination of qubits from the +visible and hidden layer has to be prepared according to the +doubly-controlled rotation layer depicted in Fig. 2, and it has +to be measured to be in state |1⟩ directly after the action of +the layer. This results in an additional amount of nm qubits +required besides the n + m qubits making up the visible and +hidden layer, respectively. However, the number of required +ancillaries could be reduced to 1 if it is possible to reliably +reinitialize it back to the computational ground state |0⟩ after +each measurement, resulting in n + m + 1 required qubits in +total. +Furthermore, when looking at the modulus in the normal- +ization factor in (8) it might seem out of place compared +to the partition function-like normalization from (2). But +after measuring the ancillary qubit to be in state |1⟩, the +normalization of the wave function adapts accordingly. +After following the steps described above, the probability +distribution of the visible qubit states follows the one given +in (2), which allows for the sampling of the target wave +function defined in (1). In order to optimize the involved +parameters, the sampled wave function then has to be used +for calculating expectation values. These are necessary for +calculating analytic gradients w.r.t. the parameters which will +be described in the following. +C. Optimization & Analytic gradients +It is common to present a problem Hamiltonian in its Pauli- +decomposed form according to +H = +N +� +k=1 +ck +n +� +i=1 +P (k) +i +, P (k) +i +∈ {I, σx, σy, σz} . +(9) +The Hamiltonian in (9) consists of N terms contributing to +the sum, each being a so-called Pauli-word or Pauli-string, a +tensor product of operators taken from the set of Pauli matrices +including the identity, multiplied with a real coefficient ck. As +an example, a Pauli-decomposed Hamiltonian for n = 3 and +N = 2 might read as +H = 2 σx ⊗ I ⊗ σz − 3 I ⊗ σy ⊗ σy . +(10) +The optimization procedure is straight-forward: For a given +set of parameters p ∈ {ai, bj, wij, ci, d}, the expectation value +⟨H⟩ = ⟨ψ|H|ψ⟩ of the Hamiltonian w.r.t. the sampled wave +function is the objective function used for the gradient-based +optimization of the parameters. For plain gradient descent with +a constant learning rate η, the parameters in the k-th iteration +step are adjusted according to +pk+1 = pk − η∂p ⟨H⟩ . +(11) +Given the explicitly known dependence of the amplitude +a(v) := s(v)φ(v) on the parameters p and exploiting the +fact that the amplitudes are real, the following covariance-like +structure can be derived for the derivative of the expectation +value (we refer to the supplementary notes of [6] for a detailed +derivation): +∂p ⟨H⟩ = 2 ⟨ElocDp⟩ − 2 ⟨Eloc⟩ ⟨Dp⟩ , +(12) +with the local energy Eloc(v) and the logarithmic amplitude +derivative Dp(v) defined as follows +Eloc(v) = ⟨v|H|ψ⟩ +a(v) +, +(13) +Dp(v) = ∂p log (a(v)) = ∂pa(v) +a(v) +. +(14) + +For each parameter, the explicit expression of Dp can be +worked out analytically to give +Dai(v) = 1 +2vi − 1 +2 ⟨vi⟩QBM , +(15) +Dbj(v) = 1 +2 tanh (gj) − 1 +2 ⟨hj⟩QBM , +(16) +Dwij(v) = 1 +2 tanh (gj) vi − 1 +2 ⟨vihj⟩QBM , +(17) +Ddi(v) = vi +� 1 +s(v) − s(v) +� +, +(18) +Dc(v) = +1 +s(v) − s(v) , +(19) +with gj = ∂E/∂bj = hj + � +i wijvi, vi and hj again being +the corresponding σz-eigenvalues for the i-th and j-th qubit +state in the visible and hidden layer, respectively, and s(v) +being the sign node from (4). Due to the covariant structure +of (12), the constant shifts ⟨...⟩QBM in (15)–(17) cancel out +and do not have to be calculated. +With the QBM approach just explained, the underlying +exponentially growing, complex distribution of 2n states can +be generated using quadratically growing resources, namely +the circuit width (assuming m ∼ O(n)), circuit depth (ac- +cording to nm entangling layers necessary to generate the +quadratic terms), and required parameters. However, during +the investigation and implementation of the approach, several +(unexpected) observations have been made that, when applied, +diminish the practicability of the otherwise cleverly designed +theoretical algorithm. These points will be discussed in the +following section. +III. IMPLEMENTATION INSIGHTS +AND HURDLES IN THE NISQ-ERA +Besides problems that can occur when executing circuits +on today’s NISQ devices, we have used the presented QBM +algorithm as an example to work out points that can lead to is- +sues when transferring a quantum algorithm into an executable +implementation. In addition to that, in the rapidly growing +domain of quantum software stacks, not all available SDKs +support necessary features for an efficient implementation. A +combination of these points will be addressed in this section, +combined with the perspective for what types of algorithms +these points might be an obstacle and, if possible, how to +mitigate some of these issues. +To ensure the verifiability of our results, we provide the +source code of our implementation here [7]. The code was +written within the open-source quantum computing framework +Qiskit [8] at version 0.31.0. The implementation was devel- +oped in the context of a quantum chemistry use case with the +goal of approximating electronic ground states of molecules. +A. Scaling of sampling quality +In order to sample the wave function from the distribution, +the circuit must be executed multiple times. However, the +sample size (i.e., the number of shots) should be the same +order of magnitude as there are states to be represented. +Following this argument and assuming that for a given n- +qubit Hamiltonian a large portion of the possible basis states +contribute to the ground state, the number of required shots +scales exponentially as O(2n). Thus, in the regime where +a complex 2n-dimensional distribution might be difficult to +access classically, an exponentially growing number of shots +has to be performed. Even for usual single-shot, small-depth +circuit execution times of ∼ O(µs) an exponentially growing +number of repetitions might be a limiting factor. As a refer- +ence, currently available NISQ-devices support a maximum +of around 20,000 - 100,000 shots. Even if technically possible +to allow for a much larger number of shots, stability of the +calibration of the underlying hardware must be ensured in +order to get meaningful results. If this is the case, an efficient +generation of the probability distribution on the quantum +register is possible at the expense of many circuit executions in +order to sample it. This, however, is not just an limiting aspect +of the discussed QBM algorithm but is rather an issue for any +quantum algorithm that relies on sampling for representing a +distribution of states encoded in a quantum register. +B. Classical Post Processing +Conventional variational approaches like, e.g., the Varia- +tional Quantum Eigensolver (VQE) [9], [10], encode the target +wave function and the problem Hamiltonian onto the quantum +hardware in the form of quantum logic gates and allow, e.g., +for the efficient evaluation of expectation values. In contrast, +the QBM approach generates a probability distribution as a +basis for classically calculating the target wave function by +summing over hidden layer configurations for a given visible +layer configuration. In order to evaluate expectation values +necessary for the parameter optimization, the action of the +problem Hamiltonian on the wave function must be calculated +classically as well. This essentially results in calculating the +action of exponentially large (yet sparse) matrices on state vec- +tors which ,especially in the context of computing expectation +values, can be efficiently performed by quantum computers. +However, this could be a common problem for algorithms, +which need to further process information about quantum +states after it has been transferred to classical data. Thus, it +is important to be aware of additional classical calculations +after the actual quantum computation in order to not lose the +gained advantage by introducing a quantum step. +C. Mid-Circuit measurements +As described in Section II-B, besides the data qubits from +the visible and hidden registers, additional ancillary qubits are +necessary in order to ensure a successful sampling. Naively, +for n qubits in the visible and m qubits in the hidden layer, +it requires additional nm ancillary qubits to generate the +probability distribution. In this scenario, all measurements of +the data and ancillary qubits can be performed at the end of +the circuit, as it is usual for most circuits. But by reusing a +single ancillary qubit for all connections between visible and +hidden layers, the required qubit resources reduce from O(n2) +to O(n). This, however, requires both, the measurement and + +fast, reliable relaxation of the ancillary during the execution of +the circuit. Algorithmically, neither the relaxation nor the mid- +circuit measurement pose a problem. Including these features +from a hardware and software perspective, however, is more +challenging since they inherently affect the way quantum cir- +cuits and their execution results must be represented. However, +in order to efficiently implement the QBM, these features +are necessary requirements for the hardware and software +stack and are not yet supported by some, which limited the +available options for the framework of choice. As a step +further, allowing for mid-circuit measurements would also +open up the possibility of including conditional operations or +operation layers on the register, based on the measurement +results as, in principle, is intended in the discussed QBM +algorithm as well. +D. Ansatz universality +The ansatz for the target wave function given in (1) allows +for the generation of real amplitudes with different signs for +the basis states in order to approximate the ground state of the +problem Hamiltonian. Since the amplitudes of the ground state +of an arbitrary Hamiltonian can be complex-valued, the ansatz +itself does not allow for an arbitrary good approximation of +the actual ground state. As with many optimization problems, +it is in fact difficult to compare the quality of the best known +solution in the context of the method, with the globally optimal +solution. In order to address this issue, the originally proposed +algorithm has been extended to allow for relative phases +between basis states (see [11], [12]). This can be done by +including an imaginary part in the sign-node in (4) according +to +s(v) = tanh +�� +k(ck + iγk)vi + d + iδ +� +, +(20) +with {γk}, δ being n + 1 additional parameters. But by using +a phase-node, the covariant structure of the analytic gradient +in (12) is lost by then including real and imaginary parts of the +involved quantities, thus not automatically cancelling out the +shifts in (15)–(17). Besides that, the sign- and phase-node pose +another not directly apparent issue, which various Quantum +Machine Learning models might struggle with. This will be +discussed next. +E. Ansatz expressivity +Now, even when assuming that the ground state of a +given problem Hamiltonian has real-valued amplitudes, the +algorithm proposed by Xia and Kais [6] would still not be +able to approximate any ground state arbitrarily well. This +is due to the fact, that on the one hand, the most general +real-valued n-qubit wave function has 2n − 1 degrees of +freedom – one amplitude for each of the 2n states minus +one fixed amplitude due to normalization. On the other hand, +the number of parameters making up the model scales as +O(n2). This information gap becomes especially apparent for +the expressivity of the sign-node: Ideally, the node is supposed +to realize relative signs between contributing states. But since +it is built from only n + 1 parameters, it is not able to realize +θ += +θ/2 +−θ/2 +θ/2 +(a) +θ += +θ/2 +−θ/2 +(b) +Fig. 3: Decomposition of (a) a double-controlled rotation gate +into controlled-rotation gates and CNOTs and (b) a controlled- +rotation gate into one-qubit rotations and CNOTs. The gates +can either be Rz- or Ry rotations. All gates are Ry-rotation +gates with the argument denoted in the box, although these +decompositions hold for Rx- and Rz-rotations as well (for +Rx-rotations replace the CNOTs in (b) with CZ-gates). +relative signs for all of the 2n possible states. For certain +Hamiltonians this already posed a major issue for the solution +quality obtained with as few as 2 qubits in the visible layer. +Note, that even by replacing the sign- with a phase-node, the +limited expressivity is not resolved since it merely doubles the +number of available parameters which can only contribute to +the imaginary part of the amplitude. In general, when building +variational models, it is difficult to find a good compromise +between expressivity and the number of parameters involved +in building the model. It is reasonable to assume that in +the context of the QBM algorithm for a Hamiltonian, good +approximations can only be found for the ground state if it +is composed of only a few basis states and if there is only a +small number of relative signs between basis states. +F. Width & Depth +As stated at the end of Section II-C, the width and depth +requirements necessary for implementing the QBM algorithm +scale as O(n2) when assuming equally large visible and +hidden layers (i.e., m ∼ O(n)). At a first glance, this scaling +seems fairly tolerable. However, when implementing said +algorithm and trying to execute it on current quantum devices, +one might stumble over some points, which are not necessarily +obvious: For example, in the NISQ era, the prefactor of the +depth scaling plays a non-negligible role. Of course, in the +context of complexity theory any prefactors and non-dominant +terms can be neglected, but for implementing algorithms on +today’s NISQ-devices, e.g. just doubling the depth of an +algorithm has a huge impact on the quality of the results. +This prefactor actually becomes quite large for the necessary +double-controlled rotation gates when they are decomposed +into gates, which can be executed on quantum backends. This +is necessary, because current backends support only a limited + +number of one- and two-qubit gates, into which any other gate +described in an algorithm must be decomposed. +As a small example, assume that a backend supports +Ry-gates and CNOTs. Following the decomposition rules +of Fig. 3a and Fig. 3b, a single doubly-controlled rotation +gate requires 8 CNOT- and 6 Ry-gates. Thus, in order to +implement all n2 entangling layers necessary for generating +the quadratic terms in the probability distribution, it actually +requires 4n2(2+3·2) = 32n2 two-qubit gates and 4n2(3·2) = +24n2 one-qubit gates. +In addition to the gate decomposition, even more two-qubit +SWAP-gates are necessary if the interacting qubits cannot be +directly entangled due to the quantum processor’s topology. +Referring to the work of Leymann and Barzen [4], this exam- +ple was intended to point out that the choice of a particular +backend has a major impact on the circuit depth and, thus, +the quality of the results. By analyzing the structure of an +algorithm and the features of available backends, it is possible +to improve on the solution quality. +IV. CONCLUSION & OUTLOOK +To gain insight into the practicability of theoretically for- +malized quantum algorithms on current NISQ devices, we have +implemented an algorithm for a quantum Boltzmann machine +(QBM) proposed by Xia and Kais [6] and systematically +summarized obstacles and limitations we have faced in the +process. Thereby, we have identified discrepancies between +the domain of theoretical algorithm design and the practi- +cal application of quantum algorithms for actual use cases +on current quantum hardware. The systematically presented +obstacles, limitations, and initially identified mitigation ideas +can guide algorithm developers and practitioners to apply +and implement a QBM, as well as similar algorithms. One +of the key findings is the following: Besides issues when +transferring quantum data into classical data, e.g., when sam- +pling from a distribution stored on the quantum register, to +further process this data, we pointed out the desirable support +of quantum hardware and quantum software for mid-circuit +measurements. In our opinion, this feature holds potential for +novel quantum algorithms, especially when considering the +interchangeability of whole circuit operations conditioned on +(multiple) measurement results. Based on the decomposition +of gates necessary for the algorithm at hand, we argue that +the choice of an appropriate backend supporting a favorable +set of gates is crucial for reducing the circuit depth and +improving on the quality of results. In this context, we would +like to draw attention to promising automated backend and +implementation selection approaches based on algorithm and +hardware properties, as described in the work by Salm et +al. [13]. Whilst experimenting with the implementation of +different Hamiltonians (as described in Section III-E), the +question has come up, in which way it might be possible +to estimate the solution quality based on properties of the +Hamiltonian and the ansatz of the variational circuit. +Furthermore, we are going to share our implementation of +the QBM via the collaborative quantum software platform +PlanQK [14], [15] to present and discuss our findings with +further developers, quantum algorithm experts, and scientists +in the quantum algorithm and quantum software development +community. We are eager to abstract and generalize the +essence of our findings along with proven mitigation ideas into +design patterns for quantum algorithms and contribute them to +the body of knowledge on quantum pattern languages started +by Weigold et al. [16]. +ACKNOWLEDGEMENTS +This work was partially funded by the project PlanQK +(01MK20005N) supported by the German Federal Ministry +for Economic Affairs and Climate Action. +REFERENCES +[1] A. W. Harrow, A. Hassidim, and S. Lloyd, “Quantum algorithm for +linear systems of equations,” Physical Review Letters, vol. 103, no. 15, +p. 150502, 2009. +[2] V. Havlicek, A. D. C´orcoles, K. Temme, A. W. Harrow, A. Kandala, +J. M. Chow, and J. M. Gambetta, “Supervised learning with quantum +enhanced feature spaces,” Nature, vol. 567, pp. 209—-212, 2019. +[3] J. Preskill, “Quantum Computing in the NISQ era and beyond,” Quan- +tum, vol. 2, p. 79, 2018. +[4] F. Leymann and J. Barzen, “The bitter truth about gate-based quantum +algorithms in the NISQ era,” Quantum Science and Technology, vol. 5, +no. 4, 2020. +[5] S. Aaronson, “Read the Fine Print,” Nature Physics, vol. 11, pp. 291– +293, 2015. +[6] R. Xia and S. Kais, “Quantum machine learning for electronic structure +calculations,” Nature Communications, vol. 9, no. 4195, 2018. +[7] Paul, Felix, “QBM Implementation,” 2022, last accessed 31.03.2022. +[Online]. Available: https://github.com/PlanQK/QBM +[8] MD +SAJID +ANIS +et +al., +“Qiskit: +An +open-source +framework +for quantum computing,” 2021, last accessed 31.03.2022. [Online]. +Available: https://github.com/Qiskit/qiskit +[9] A. Peruzzo, J. McClean, P. Shadbolt, M. H. Yung, X. Q. Zhou, P. J. Love, +A. Aspuru-Guzik, and J. L. O’Brien, “A variational eigenvalue solver on +a photonic quantum processor,” Nature Communications, vol. 5, no. 2, +pp. 1–10, 2014. +[10] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, “The +theory of variational hybrid quantum-classical algorithms,” New Journal +of Physics, vol. 18, no. 2, pp. 1–20, 2016. +[11] S. +Kanno +and +T. +Tada, +“Many-body +calculations +for +periodic +materials +via +quantum +machine +learning,” +arXiv: +Computational +Physics, 2019, last accessed 31.03.2022. [Online]. Available: http: +//arxiv.org/abs/1911.10330 +[12] S. H. Sureshbabu, M. Sajjan, S. Oh, and S. Kais, “Implementation +of Quantum Machine Learning for Electronic Structure Calculations +of Periodic Systems on Quantum Computing Devices,” Journal of +Chemical Information and Modeling, 2021. +[13] M. Salm, “The NISQ Analyzer: Automating the Selection of Quantum +Computers for Quantum Algorithms,” in Proceedings of the 14th Sympo- +sium and Summer School on Service-Oriented Computing (SummerSOC +2020). +Springer International Publishing, Dec. 2020, pp. 66–85. +[14] F. Leymann, J. Barzen, M. Falkenthal, D. Vietz, B. Weder, and K. Wild, +“Quantum in the Cloud: Application Potentials and Research Opportu- +nities,” in Proceedings of the 10th International Conference on Cloud +Computing and Services Science (CLOSER 2020). +SciTePress, 2020, +pp. 9—-24. +[15] PlanQK Community, “Planqk platform,” 2022, last accessed 31.03.2022. +[Online]. Available: https://platform.planqk.de +[16] M. Weigold, J. Barzen, F. Leymann, and D. Vietz, “Patterns For Hybrid +Quantum Algorithms,” in Proceedings of the 15th Symposium and +Summer School on Service-Oriented Computing (SummerSOC 2021). +Springer, 2021, pp. 34—-51. + diff --git a/a9FST4oBgHgl3EQfCDgc/content/tmp_files/load_file.txt b/a9FST4oBgHgl3EQfCDgc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c2bdbea503417b9d081bc37fd70ab24da4f97858 --- /dev/null +++ b/a9FST4oBgHgl3EQfCDgc/content/tmp_files/load_file.txt @@ -0,0 +1,351 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf,len=350 +page_content='Clever Design, Unexpected Obstacles: Insights on Implementing a Quantum Boltzmann Machine Felix Paul StoneOne AG Berlin, Germany felix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='paul@stoneone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='de Michael Falkenthal StoneOne AG Berlin, Germany michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='falkenthal@stoneone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='de Sebastian Feld Delft University of Technology Delft, The Netherlands s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='feld@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='nl Abstract—We have implemented a gated-based quantum ver- sion of a restricted Boltzmann machine for approximating the ground state of a Pauli-decomposed qubit Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' During the implementation and evaluation, we have noticed a variety of unexpected topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' It starts from limitations due to the structure of the algorithm itself and continues with constraints induced by specific quantum software development kits, which did not (yet) support necessary features for an efficient implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In this paper we systematically summarize our findings and categorize them according to their relevance for the implementation of similar quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' We also discuss the feasibility of executing such implementations on current NISQ devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Index Terms—quantum software engineering, NISQ devices, quantum machine learning, limitations and constraints I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' INTRODUCTION The advent of quantum computers accessible to the public has given an immense boost to the development of new quan- tum algorithms in the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Many algorithms promise advantages over previously known classical ones, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=', with regard to a speedup [1] or enhanced solution quality [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' However, to raise these advantages the conceptual algorithms first have to be applied to actual use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Second, they have to be implemented to run on specific quantum computers which are still NISQ devices providing limitations with respect to decoherence time, gate and measurement failure rates [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In practice, this leads to major difficulties when transferring conceptual algorithms into actual implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In order for them to be executable on NISQ-devices, the limitations of the specific quantum hardware must be taken into account accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' For example, the available number of qubits and the fidelity of their states must be considered such that an implemented algorithm can (i) be transferred to the available qubits at all and (ii) be executed by the quantum computer in an amount of time that still guarantees tolerable error rates and, thus, allows to read out meaningful results [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' However, these limitations have a strong influence on whether the theoretically described advantages of an algorithm can be raised at all [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Therefore, it is valuable to examine algorithms for their practicability on NISQ computers and to recognize limitations early on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' On this basis, it is possible to investigate and elaborate on appropriate mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This work was partially funded by the project PlanQK (01MK20005N) supported by the German Federal Ministry for Economic Affairs and Climate Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In this work, we present details and findings regarding the implementation of a quantum version of a restricted Boltzmann machine (QBM) based on the algorithm introduced by Xia and Kais [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' We provide insights into what kind of obstacles we encountered when translating a theoretically proposed quantum algorithm into an implementation that can be executed on one of the quantum backends currently avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Some of these issues are caused by implementing the algorithm within a given software stack or quantum software development kit (quantum SDK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Others only arise when trying to execute the code on quantum backends, since their properties play a crucial role for gaining meaningful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' We also discuss aspects regarding the possible solution quality of the chosen model: There are limitations induced by transferring quantum data, stored within the hardware, into classical data and also some limitations that are induced by the variational model itself and the conditions it is built on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' For some of the mentioned issues we suggest possible approaches, as to how the impact of these may be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Those rec- ommendations provide guidance for implementing other algo- rithms which make use of similar concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' With this in mind, we classify the encountered problems in terms of how the same or similar ones may arise during the implementation of other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Since numerous algorithms proposed make use of similar subroutines or concepts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=', using parametrized rotational gates for variational models), we believe that by systematically classifying common issues, some of these can be managed better within future implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Thus, we aim on giving an initial list of indicators which need to be considered in order to access the practical feasibility of algorithms proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Since our findings will be explained in the context of a QBM algorithm, we also summarize the key aspects of it which we supplement with exemplary calculations to illustrate relevant procedures in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The remainder of this paper is structured as follows: We introduce the essential concepts of the quantum Boltzmann machine algorithm by Xia and Kais [6] in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' We identify and discuss findings and restrictions of the QBM algorithm with respect to currently available quantum computers and software development stacks and development kits in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Finally, we conclude the paper with an outlook on future work in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='13705v1 [quant-ph] 31 Jan 2023 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' ALGORITHM DETAILS The goal of the paper is to highlight practical problems that arise during the implementation of quantum algorithms, which is done on the example of a QBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' For this purpose, we will present the algorithmic details in the following, which will be referred to in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The goal of the approach described by Xia and Kais [6] is to approximate the ground state of a given Hamiltonian using a hybrid quantum-classical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The utilized QBM is made up of three layers: a visible layer with n qubits, a hidden layer with m qubits, and a classical sign layer realizing relative signs between different states of the visible layer (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' For explicit values of n and m, we will refer to this model as a (n, m)-QBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The wave function to be generated by the QBM and which should approximate the ground state is |ψ⟩ = � {v} s(v)φ(v) |v⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (1) Here, v denotes a single binary state consisting of n bits and {v} represents the set of all possible binary states of the visible layer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=', n = 2 leads to {v} = {00, 01, 10, 11}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Furthermore, φ(v) is the real-valued amplitude associated to state |v⟩, while s(v) is the corresponding value of the sign node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The probability distribution p(v) of the qubit states of the visible layer is generated by using the QBM, and it determines the amplitudes of the target wave function: p(v) = φ2(v) = 1 Z � {h} exp (E(v, h)) , Z = � {v,h} exp (E(v, h)) , (2) where � {h} indicates the summation over all configurations of the hidden layer for a fixed configuration of the visible layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The ”energy” E(v, h) of a given state of the visible and hidden layer depends on real biases {ai}, {bj} and weights {wij}, which are all subject to the optimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The energy is given by E(v, h) = � i aivi + � j bjhj + � ij wijvihj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (3) In (3), vi, hj ∈ {±1} are the σz-eigenvalues of the computa- tional basis states (σz |0⟩ = |0⟩ , σz |1⟩ = − |1⟩) for the i-th and j-th qubit in the visible and hidden layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The sign node is a smooth function that depends on the state of the qubits from the visible layer as well as on parameters {ci} and d, which are to be optimized, and it is given by s(v) = tanh �� i civi + d � (4) In fact, the algorithm proposed in [6] does not actually generate the probability distribution described in (2), but a modified one with an additional regulator k = O(� ij |wij|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This regulator normalizes all parameters p ∈ {ai, bj, wij} according to p → p/k in order to increase the probability for successfully generating the distribution initially wanted (see Section II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' For simplicity, including the regulator will be omitted in the upcoming sections, but when implementing the algorithm it needs to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' 1: Exemplary network architecture of a (2, 3)-QBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Vertices v1 and v2 represent qubits from the visible layer, while h1, h2, and h3 represent qubits from the hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' s corresponds to a sign node realizing relative signs between states of the visible layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' As part of the algorithm, qubits from the visible and hidden layer are entangled with each other via 3-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In the following sections we will describe the necessary steps for generating the probability distribution given in (2) in two steps – namely generating the linear and quadratic terms of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' After that, the basic optimization procedure and the analytical gradients used in it are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Linear terms The linear terms in (3) are generated by performing Ry- rotations on all qubit states from the visible and hidden layer with angle θℓ = 2 arcsin �� e−pℓ epℓ + e−pℓ � , (5) where pℓ ∈ {ai, bj} depends on whether the gate acts on a qubit state from the visible or hidden layer and ℓ to be taken from the corresponding index set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' To give an example, (6) shows the action on a single |0⟩-state which generates the correct sign within the amplitude of each state: Ry(θℓ) |0⟩ = 1 √ epℓ + e−pℓ � epℓ/2 |0⟩ + e−pℓ/2 |1⟩ � (6) Note, that by rotating the qubit states in the described manner, the probability distribution of (2) is generated for wij = 0, ∀ i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Quadratic terms In order to include the interaction term of the i-th visible and j-th hidden qubit in (3), a series of four doubly-controlled Ry-rotation gates has to act on an ancillary qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The schema of one such entangling layer is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' 2 and shows four 3-qubit gates which all get controlled by one of the four 2-qubit basis states {|00⟩ , |01⟩ , |10⟩ , |11⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' h1 U1 h2 S ~2 h3vi : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' hj : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' a1 : θ+ ij θ− ij θ− ij θ+ ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' 2: Circuit diagram representation of an entangling layer generating the interaction term between the i-th qubit from the visible and the j-th qubit from the hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The gates are Ry-gates with the arguments denoted in the box and defined in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' For each entangling layer two angles are necessary, which both depend on the weight wij between qubit i and j in the following way: θ± ij = 2 arcsin �� e±wij e|wij| � , (7) where θ+ ij is used when the Ry-gate is controlled by states with even parity (|00⟩ and |11⟩), while θ− ij is accordingly used for control states with odd parity (|01⟩ and |10⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This can better be understood when looking at the energy in (3): If visible qubit i and hidden qubit j are in the same state (either both |0⟩ or both |1⟩), the product of their σz-eigenvalues is +1 (since (+1)2 = (−1)2 = +1), whereas them being in different states results in an eigenvalue product of −1 (since (−1)(+1) = (+1)(−1) = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Now, in order to understand the action of the doubly-controlled Ry-gates in more detail, we will look at the action of a single-qubit Ry-gate on an ancillary qubit with the angle defined in (7) (omitting the i, j indices for simplicity): Ry(θ±) |0⟩ = 1 e|w/2| �� e|w| − e±w |0⟩ + e±w/2 |1⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (8) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (8) shows that with probability ∼ e±w the ancillary qubit will be in state |1⟩, thus giving the correct contribution to the energy defined in (3) when measured to be in that particular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In order to successfully generate the desired distribution, one ancillary qubit for every combination of qubits from the visible and hidden layer has to be prepared according to the doubly-controlled rotation layer depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' 2, and it has to be measured to be in state |1⟩ directly after the action of the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This results in an additional amount of nm qubits required besides the n + m qubits making up the visible and hidden layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' However, the number of required ancillaries could be reduced to 1 if it is possible to reliably reinitialize it back to the computational ground state |0⟩ after each measurement, resulting in n + m + 1 required qubits in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Furthermore, when looking at the modulus in the normal- ization factor in (8) it might seem out of place compared to the partition function-like normalization from (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' But after measuring the ancillary qubit to be in state |1⟩, the normalization of the wave function adapts accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' After following the steps described above, the probability distribution of the visible qubit states follows the one given in (2), which allows for the sampling of the target wave function defined in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In order to optimize the involved parameters, the sampled wave function then has to be used for calculating expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' These are necessary for calculating analytic gradients w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' the parameters which will be described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Optimization & Analytic gradients It is common to present a problem Hamiltonian in its Pauli- decomposed form according to H = N � k=1 ck n � i=1 P (k) i , P (k) i ∈ {I, σx, σy, σz} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (9) The Hamiltonian in (9) consists of N terms contributing to the sum, each being a so-called Pauli-word or Pauli-string, a tensor product of operators taken from the set of Pauli matrices including the identity, multiplied with a real coefficient ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' As an example, a Pauli-decomposed Hamiltonian for n = 3 and N = 2 might read as H = 2 σx ⊗ I ⊗ σz − 3 I ⊗ σy ⊗ σy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (10) The optimization procedure is straight-forward: For a given set of parameters p ∈ {ai, bj, wij, ci, d}, the expectation value ⟨H⟩ = ⟨ψ|H|ψ⟩ of the Hamiltonian w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' the sampled wave function is the objective function used for the gradient-based optimization of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' For plain gradient descent with a constant learning rate η, the parameters in the k-th iteration step are adjusted according to pk+1 = pk − η∂p ⟨H⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (11) Given the explicitly known dependence of the amplitude a(v) := s(v)φ(v) on the parameters p and exploiting the fact that the amplitudes are real,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' the following covariance-like structure can be derived for the derivative of the expectation value (we refer to the supplementary notes of [6] for a detailed derivation): ∂p ⟨H⟩ = 2 ⟨ElocDp⟩ − 2 ⟨Eloc⟩ ⟨Dp⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (12) with the local energy Eloc(v) and the logarithmic amplitude derivative Dp(v) defined as follows Eloc(v) = ⟨v|H|ψ⟩ a(v) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (13) Dp(v) = ∂p log (a(v)) = ∂pa(v) a(v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (14) For each parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' the explicit expression of Dp can be worked out analytically to give Dai(v) = 1 2vi − 1 2 ⟨vi⟩QBM ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (15) Dbj(v) = 1 2 tanh (gj) − 1 2 ⟨hj⟩QBM ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (16) Dwij(v) = 1 2 tanh (gj) vi − 1 2 ⟨vihj⟩QBM ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (17) Ddi(v) = vi � 1 s(v) − s(v) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (18) Dc(v) = 1 s(v) − s(v) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' (19) with gj = ∂E/∂bj = hj + � i wijvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' vi and hj again being the corresponding σz-eigenvalues for the i-th and j-th qubit state in the visible and hidden layer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' and s(v) being the sign node from (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Due to the covariant structure of (12), the constant shifts ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='⟩QBM in (15)–(17) cancel out and do not have to be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' With the QBM approach just explained, the underlying exponentially growing, complex distribution of 2n states can be generated using quadratically growing resources, namely the circuit width (assuming m ∼ O(n)), circuit depth (ac- cording to nm entangling layers necessary to generate the quadratic terms), and required parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' However, during the investigation and implementation of the approach, several (unexpected) observations have been made that, when applied, diminish the practicability of the otherwise cleverly designed theoretical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' These points will be discussed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' IMPLEMENTATION INSIGHTS AND HURDLES IN THE NISQ-ERA Besides problems that can occur when executing circuits on today’s NISQ devices, we have used the presented QBM algorithm as an example to work out points that can lead to is- sues when transferring a quantum algorithm into an executable implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In addition to that, in the rapidly growing domain of quantum software stacks, not all available SDKs support necessary features for an efficient implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' A combination of these points will be addressed in this section, combined with the perspective for what types of algorithms these points might be an obstacle and, if possible, how to mitigate some of these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' To ensure the verifiability of our results, we provide the source code of our implementation here [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The code was written within the open-source quantum computing framework Qiskit [8] at version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The implementation was devel- oped in the context of a quantum chemistry use case with the goal of approximating electronic ground states of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Scaling of sampling quality In order to sample the wave function from the distribution, the circuit must be executed multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' However, the sample size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=', the number of shots) should be the same order of magnitude as there are states to be represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Following this argument and assuming that for a given n- qubit Hamiltonian a large portion of the possible basis states contribute to the ground state, the number of required shots scales exponentially as O(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Thus, in the regime where a complex 2n-dimensional distribution might be difficult to access classically, an exponentially growing number of shots has to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Even for usual single-shot, small-depth circuit execution times of ∼ O(µs) an exponentially growing number of repetitions might be a limiting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' As a refer- ence, currently available NISQ-devices support a maximum of around 20,000 - 100,000 shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Even if technically possible to allow for a much larger number of shots, stability of the calibration of the underlying hardware must be ensured in order to get meaningful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' If this is the case, an efficient generation of the probability distribution on the quantum register is possible at the expense of many circuit executions in order to sample it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This, however, is not just an limiting aspect of the discussed QBM algorithm but is rather an issue for any quantum algorithm that relies on sampling for representing a distribution of states encoded in a quantum register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Classical Post Processing Conventional variational approaches like, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=', the Varia- tional Quantum Eigensolver (VQE) [9], [10], encode the target wave function and the problem Hamiltonian onto the quantum hardware in the form of quantum logic gates and allow, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=', for the efficient evaluation of expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In contrast, the QBM approach generates a probability distribution as a basis for classically calculating the target wave function by summing over hidden layer configurations for a given visible layer configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In order to evaluate expectation values necessary for the parameter optimization, the action of the problem Hamiltonian on the wave function must be calculated classically as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This essentially results in calculating the action of exponentially large (yet sparse) matrices on state vec- tors which ,especially in the context of computing expectation values, can be efficiently performed by quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' However, this could be a common problem for algorithms, which need to further process information about quantum states after it has been transferred to classical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Thus, it is important to be aware of additional classical calculations after the actual quantum computation in order to not lose the gained advantage by introducing a quantum step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Mid-Circuit measurements As described in Section II-B, besides the data qubits from the visible and hidden registers, additional ancillary qubits are necessary in order to ensure a successful sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Naively, for n qubits in the visible and m qubits in the hidden layer, it requires additional nm ancillary qubits to generate the probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In this scenario, all measurements of the data and ancillary qubits can be performed at the end of the circuit, as it is usual for most circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' But by reusing a single ancillary qubit for all connections between visible and hidden layers, the required qubit resources reduce from O(n2) to O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This, however, requires both, the measurement and fast, reliable relaxation of the ancillary during the execution of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Algorithmically, neither the relaxation nor the mid- circuit measurement pose a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Including these features from a hardware and software perspective, however, is more challenging since they inherently affect the way quantum cir- cuits and their execution results must be represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' However, in order to efficiently implement the QBM, these features are necessary requirements for the hardware and software stack and are not yet supported by some, which limited the available options for the framework of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' As a step further, allowing for mid-circuit measurements would also open up the possibility of including conditional operations or operation layers on the register, based on the measurement results as, in principle, is intended in the discussed QBM algorithm as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Ansatz universality The ansatz for the target wave function given in (1) allows for the generation of real amplitudes with different signs for the basis states in order to approximate the ground state of the problem Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Since the amplitudes of the ground state of an arbitrary Hamiltonian can be complex-valued, the ansatz itself does not allow for an arbitrary good approximation of the actual ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' As with many optimization problems, it is in fact difficult to compare the quality of the best known solution in the context of the method, with the globally optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In order to address this issue, the originally proposed algorithm has been extended to allow for relative phases between basis states (see [11], [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This can be done by including an imaginary part in the sign-node in (4) according to s(v) = tanh �� k(ck + iγk)vi + d + iδ � , (20) with {γk}, δ being n + 1 additional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' But by using a phase-node, the covariant structure of the analytic gradient in (12) is lost by then including real and imaginary parts of the involved quantities, thus not automatically cancelling out the shifts in (15)–(17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Besides that, the sign- and phase-node pose another not directly apparent issue, which various Quantum Machine Learning models might struggle with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This will be discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Ansatz expressivity Now, even when assuming that the ground state of a given problem Hamiltonian has real-valued amplitudes, the algorithm proposed by Xia and Kais [6] would still not be able to approximate any ground state arbitrarily well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This is due to the fact, that on the one hand, the most general real-valued n-qubit wave function has 2n − 1 degrees of freedom – one amplitude for each of the 2n states minus one fixed amplitude due to normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' On the other hand, the number of parameters making up the model scales as O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This information gap becomes especially apparent for the expressivity of the sign-node: Ideally, the node is supposed to realize relative signs between contributing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' But since it is built from only n + 1 parameters, it is not able to realize θ = θ/2 −θ/2 θ/2 (a) θ = θ/2 −θ/2 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' 3: Decomposition of (a) a double-controlled rotation gate into controlled-rotation gates and CNOTs and (b) a controlled- rotation gate into one-qubit rotations and CNOTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The gates can either be Rz- or Ry rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' All gates are Ry-rotation gates with the argument denoted in the box, although these decompositions hold for Rx- and Rz-rotations as well (for Rx-rotations replace the CNOTs in (b) with CZ-gates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' relative signs for all of the 2n possible states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' For certain Hamiltonians this already posed a major issue for the solution quality obtained with as few as 2 qubits in the visible layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Note, that even by replacing the sign- with a phase-node, the limited expressivity is not resolved since it merely doubles the number of available parameters which can only contribute to the imaginary part of the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In general, when building variational models, it is difficult to find a good compromise between expressivity and the number of parameters involved in building the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' It is reasonable to assume that in the context of the QBM algorithm for a Hamiltonian, good approximations can only be found for the ground state if it is composed of only a few basis states and if there is only a small number of relative signs between basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Width & Depth As stated at the end of Section II-C, the width and depth requirements necessary for implementing the QBM algorithm scale as O(n2) when assuming equally large visible and hidden layers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=', m ∼ O(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' At a first glance, this scaling seems fairly tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' However, when implementing said algorithm and trying to execute it on current quantum devices, one might stumble over some points, which are not necessarily obvious: For example, in the NISQ era, the prefactor of the depth scaling plays a non-negligible role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Of course, in the context of complexity theory any prefactors and non-dominant terms can be neglected, but for implementing algorithms on today’s NISQ-devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' just doubling the depth of an algorithm has a huge impact on the quality of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This prefactor actually becomes quite large for the necessary double-controlled rotation gates when they are decomposed into gates, which can be executed on quantum backends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' This is necessary, because current backends support only a limited number of one- and two-qubit gates, into which any other gate described in an algorithm must be decomposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' As a small example, assume that a backend supports Ry-gates and CNOTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Following the decomposition rules of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' 3a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' 3b, a single doubly-controlled rotation gate requires 8 CNOT- and 6 Ry-gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Thus, in order to implement all n2 entangling layers necessary for generating the quadratic terms in the probability distribution, it actually requires 4n2(2+3·2) = 32n2 two-qubit gates and 4n2(3·2) = 24n2 one-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In addition to the gate decomposition, even more two-qubit SWAP-gates are necessary if the interacting qubits cannot be directly entangled due to the quantum processor’s topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Referring to the work of Leymann and Barzen [4], this exam- ple was intended to point out that the choice of a particular backend has a major impact on the circuit depth and, thus, the quality of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' By analyzing the structure of an algorithm and the features of available backends, it is possible to improve on the solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' CONCLUSION & OUTLOOK To gain insight into the practicability of theoretically for- malized quantum algorithms on current NISQ devices, we have implemented an algorithm for a quantum Boltzmann machine (QBM) proposed by Xia and Kais [6] and systematically summarized obstacles and limitations we have faced in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Thereby, we have identified discrepancies between the domain of theoretical algorithm design and the practi- cal application of quantum algorithms for actual use cases on current quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' The systematically presented obstacles, limitations, and initially identified mitigation ideas can guide algorithm developers and practitioners to apply and implement a QBM, as well as similar algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' One of the key findings is the following: Besides issues when transferring quantum data into classical data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=', when sam- pling from a distribution stored on the quantum register, to further process this data, we pointed out the desirable support of quantum hardware and quantum software for mid-circuit measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In our opinion, this feature holds potential for novel quantum algorithms, especially when considering the interchangeability of whole circuit operations conditioned on (multiple) measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Based on the decomposition of gates necessary for the algorithm at hand, we argue that the choice of an appropriate backend supporting a favorable set of gates is crucial for reducing the circuit depth and improving on the quality of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' In this context, we would like to draw attention to promising automated backend and implementation selection approaches based on algorithm and hardware properties, as described in the work by Salm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Whilst experimenting with the implementation of different Hamiltonians (as described in Section III-E), the question has come up, in which way it might be possible to estimate the solution quality based on properties of the Hamiltonian and the ansatz of the variational circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' Furthermore, we are going to share our implementation of the QBM via the collaborative quantum software platform PlanQK [14], [15] to present and discuss our findings with further developers, quantum algorithm experts, and scientists in the quantum algorithm and quantum software development community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' We are eager to abstract and generalize the essence of our findings along with proven mitigation ideas into design patterns for quantum algorithms and contribute them to the body of knowledge on quantum pattern languages started by Weigold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FST4oBgHgl3EQfCDgc/content/2301.13705v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work was partially funded by the project PlanQK (01MK20005N) supported by the German Federal Ministry for Economic Affairs and 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--git a/cNE4T4oBgHgl3EQfow2y/vector_store/index.pkl b/cNE4T4oBgHgl3EQfow2y/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..0eb7aacac3a962df9473259ab6dda84612a92858 --- /dev/null +++ b/cNE4T4oBgHgl3EQfow2y/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c74f7c220f2343b72f3a3c20199c84214ce591cdb42dd7594443733797053f88 +size 149581 diff --git a/ctFJT4oBgHgl3EQfRSzM/content/tmp_files/2301.11495v1.pdf.txt b/ctFJT4oBgHgl3EQfRSzM/content/tmp_files/2301.11495v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6fc14bdb81080bf6896f6435e22cafea0645e43c --- /dev/null +++ b/ctFJT4oBgHgl3EQfRSzM/content/tmp_files/2301.11495v1.pdf.txt @@ -0,0 +1,1802 @@ +1 +Skeleton-based Action Recognition through +Contrasting Two-Stream Spatial-Temporal Networks +Chen Pang, Xuequan Lu, Lei Lyu∗ +Abstract—For pursuing accurate skeleton-based action recog- +nition, most prior methods use the strategy of combining Graph +Convolution Networks (GCNs) with attention-based methods in +a serial way. However, they regard the human skeleton as a +complete graph, resulting in less variations between different +actions (e.g., the connection between the elbow and head in action +“clapping hands”). For this, we propose a novel Contrastive +GCN-Transformer Network (ConGT) which fuses the spatial and +temporal modules in a parallel way. The ConGT involves two +parallel streams: Spatial-Temporal Graph Convolution stream +(STG) and Spatial-Temporal Transformer stream (STT). The +STG is designed to obtain action representations maintaining +the natural topology structure of the human skeleton. The STT +is devised to acquire action representations containing the global +relationships among joints. Since the action representations pro- +duced from these two streams contain different characteristics, +and each of them knows little information of the other, we +introduce the contrastive learning paradigm to guide their output +representations of the same sample to be as close as possible +in a self-supervised manner. Through the contrastive learning, +they can learn information from each other to enrich the action +features by maximizing the mutual information between the +two types of action representations. To further improve action +recognition accuracy, we introduce the Cyclical Focal Loss (CFL) +which can focus on confident training samples in early training +epochs, with an increasing focus on hard samples during the +middle epochs. We conduct experiments on three benchmark +datasets, which demonstrate that our model achieves state-of- +the-art performance in action recognition. +Index Terms—Skeleton-based action recognition, Graph con- +volutional network, Transformer, Contrastive learning +I. INTRODUCTION +H +UMAN action recognition has become a fundamental +task in computer vision which is extensively applied +in many real-world applications, such as intelligent security +[1], virtual reality [2], and human–machine interaction [3]. +Skeleton-based action recognition task has received significant +attention due to its computation efficiency and robustness +against viewpoints or appearance. +The core of skeleton-based action recognition is to learn +the discriminative representations of skeleton sequences. At +present, many deep learning based methods have achieved +excellent performance by using Convolutional Neural Net- +works (CNNs) [4], [5], Recurrent Neural Networks (RNNs) +[6], [7] to learn the action representations based on the specific +recognition task. However, these methods rarely consider the +Chen Pang and Lei Lyu are with School of Information Science and +Engineering, Shandong Normal University, Jinan 250358, China. +Xuequan Lu is with School of Information Technology, Deakin University, +Geelong, Australia. +∗Corresponding author: Lei Lyu (e-mail: lvlei@sdnu.edu.cn) +co-dependency contained in body joints and ignore some +important motion information. To better capture joint depen- +dencies, Graph Convolutional Networks (GCNs) are exploited +to aggregate information based on body structures. Spatio- +Temporal Graph Convolutional Network (ST-GCN) [8] is a +pioneering work to model the skeleton data as a spatio- +temporal graph with the joints as graph nodes and natural +connections in both human body structures and time as +graph edges. Later, many variants [9]–[11] are extended based +on ST-GCN, following the same strategy. Although GCNs +have been proved to perform well on skeleton data, they +still have some limitations. First, in GCN-based methods, +the human body is represented as a predefined graph fixed +over all actions, ignoring certain implicit relations between +nonadjacent joints, such as the connection between hand and +head during touching head. Second, with the deepening of +graph convolutional layers, the probability of over-smoothing +problem will increase that the representations of neighbor +nodes tend to converge to the same value, which causes +confusion between joints. Third, in most existing GCN-based +methods, the temporal connections between remote frames are +underestimated since the temporal convolution operations are +limited in a local neighborhood. To cope with these defects, +researchers introduced attention modules behind the GCN +layers to effectively capture the long-distance relations in +the supervised manner. Shi et al. [12] designed a decoupled +spatial-temporal attention network to calculate the connections +between each pair of joints without knowing their positions +or mutual connections. In [13], ST-TR used transformer to +capture the relations of each pair of nodes, ignoring the inher- +ent topology of the human skeleton. Although the recognition +accuracy can be improved by combining the GCN layers with +attention modules in a serial manner, each node is treated in +isolation and the human skeleton is regarded as a complete +graph with connections built between each joint and the rest +joints, resulting in less variations between different actions. +Taking the two actions of “clapping hands” and “touching +nose” for example, the connections among the most joints +are considered to be same, except for the stronger connection +between the hands in “clapping hands” and the stronger +dependence between the hands and nose in “touching nose”. +While the connection between the elbow and head should not +be considered in “clapping hands”, it is helpful in “touching +nose”. Therefore, it is essential to capture the long-distance +relations for better action recognition while preserving the +primitive human skeleton structure. +To this end, we propose a novel Contrastive GCN- +Transformer Network (ConGT) that considers GCN and at- +arXiv:2301.11495v1 [cs.CV] 27 Jan 2023 + +2 +tention model in a parallel manner. Specifically, the network +contains two parallel streams, i. e. Spatial-Temporal Graph +Convolution stream (STG) and Spatial-Temporal Transformer +stream (STT). The STG is used to extract the joint relation- +ships based on the topology of the human skeleton graph, +consisting of adaptive GCN module (AGCN) and temporal +convolutional network module (TCN). In particular, the AGCN +is designed to enforce the generated graph to reflect the +relationships of joints flexibly. Different from the STG, the +STT is primarily responsible for accurately capturing the +relationships among arbitrary joints in the intra- and inter- +frames, which is comprised of spatial transformer module and +temporal transformer module. Since the action representations +produced from these two streams contain different character- +istics and each of them knows little information of the other, +we introduce the contrastive learning paradigm to guide their +output representations of the same sample to be as close as +possible in the embedding space in a self-supervised manner. +Specifically, the action representations learned by STG involve +the natural topology of the human skeleton and the action +representations learned by STT involve long-distance relations. +Through the contrastive learning paradigm, they can learn +information from each other to enrich the action features by +maximizing the mutual information between the two types +of action representations. Moreover, to further improve the +performance of our model, we introduce Cyclical Focal Loss +(CFL) instead of Cross-Entropy loss as our learning objective. +In contrast to the Cross-Entropy loss, the CFL can focus on +confident training samples in early training epochs of our +model, with an increasing focus on hard samples during the +middle epochs. +In summary, the main contributions of our work can be +concluded as follows: +• We propose a novel Contrastive GCN-Transformer Net- +work (ConGT), which can capture the relationships be- +tween arbitrary joints in intra- and inter- frames more +accurately while maintaining the topology structure of +human skeleton graph. +• We propose a Spatial-Temporal Graph Convolution +stream (STG) with an adaptive graph strategy and a +Spatial-Temporal Transformer stream (STT) to learn the +action representations containing local and global joints +relations. +• We introduce the contrastive learning paradigm to inte- +grate the information of two types of action representa- +tions by maximizing mutual information between them. +• We introduce the Cyclical Focal Loss (CFL) as the +learning objective of our network to improve the action +accuracy. +II. RELATED WORK +A. Action Recognition +In this section, we will briefly review the related works +about the three fields on skeleton-based action recognition: +Convolutional Neural Networks (CNNs) based methods, Re- +current Neural Networks (RNNs) based methods and Graph +Convolutional Networks (GCNs) based methods. +1) CNN-based Methods: In the early work, the mainstream +network is based on CNN and RNN. CNN is mainly used +to process 2D images, which can easily learn the high- +level semantic features of images. Thus, most CNN-based +methods generally encode skeleton features to 2D pseudo +images. In [4], each skeleton sequence was transformed into +three clips which were correspond to every channel of the +cylindrical coordinates of the skeleton sequence. Then the +frames of the clips are jointly processed by a multi-task +learning network. Wang et al. [5] encoded the spatio-temporal +information carried in skeleton sequence as joint trajectory +maps. Li et al. [14] proposed an end-to-end hierarchical co- +occurrence network to learn the con-occurrence feature with a +hierarchical methodology, where different levels of contextual +information is aggregated gradually. They used the 3D coor- +dinates of joints to generate several 2D images that are sent +into the pretrained VGG-19 for action recognition. Although +the spatial features can be reserved in these pseudo images, +the motion information contained in actions is ignored. To +solve this problem, Rong et al. [15] used geometric algebra to +learn the shape-motion representations and applied the multi- +stream CNN models to fuse the complementary shape-motion +representations. But 2D CNNs have a weak ability to capture +the temporal and spatial features in the human skeleton. Later, +Duan et al. [16] proposed C3D network instead of 2D CNNs, +which uses a heat map to denote the spatial correlations of +joints and uses a stack to present the temporal sequence. +Although this approach has yielded good results, it still ignores +the motion correlation of the skeleton data which is a common +challenge of CNN-based methods. +2) RNN-based +Methods: +Recurrent +neural +networks +(RNNs) [17] can process sequence data with variable lengths +because its cellular states can determine which temporal states +should be left and which should be forgotten. Therefore, it +has more advantages in processing temporal sequences. In +the field of action recognition, RNN-based methods represent +the skeleton data as a vector sequence that contains the +location information of all joints in one frame. Du et al. [6] +divided the human skeleton into five parts and fed them to +five hierarchical RNN networks separately. In [7], Liu et al. +introduced a new gating mechanism into Long Short Term +Memory(LSTM) network to handle the noise and occlusion in +3D skeleton data. Lee et al. [18] proposed ensemble Temporal +Sliding LSTM (TS-LSTM) networks containing short-term, +medium-term and long-term TS-LSTM. They focused on +the temporal correlation of various human body parts but +ignored the spatial structure of the skeleton. In most of the +actions, the variation range of action in the space domain is +larger than that in the time domain. So, researchers have also +been trying to design some RNN-based networks to process +spatial information. For example, in [19], Liu et al. proposed +a global-aware attention LSTM to make use of the global +contextual information, which can selectively focus on the +informative joints in each frame of the skeleton sequence and +further enhance attention to spatial information. However, +how to perceive the spatial correlations of the human skeleton +remains a burning challenge for RNNs. + +3 +3) GCN-based Methods: Because the topology of skeleton +data is encoded in the form of graphs rather than two- +dimensional grids or vector sequences, CNN-based methods or +RNN-based methods may not be the optimal choice. Recently, +Graph Convolution Networks (GCNs) have achieved remark- +able results in many works based on graph structure data, +which can be divided into two types: spatial GCNs [7], [20], +[21] and spectral GCNs [22]–[24]. For spectral GCNs, the +input graphs are first transformed into the spectral domains and +then operated by means of Fourier transform. Spatial GCNs are +applied directly to the nodes of the graph and their neighbors, +which are more similar to the traditional convolution neural +networks. Our work follows the spatial GCN methods. +Spatial-Temporal Graph Convolutional Networks (ST-GCN) +[8] is the pioneering method to model the skeleton data, +which breaks the limitations that the previous method can- +not effectively extract spatial and temporal features at the +same time. ST-GCN models the joint connection and extracts +correlated features as a spatio-temporal graph, where the +graph convolution operates on the spatial features, and the +2D convolution operates on the temporal motion correlations. +Recently, many works also adopt the same strategy. Li et al. +[10] combined the actional links and structural links into a +generalized skeleton graph and used the actional-structural +graph convolution and temporal convolution to learn the +spatial-temporal features. In [11], Li et al. proposed a spatio- +temporal graph routing scheme to adaptively learn the high- +order connectivity relationships for physically-apart skeleton +joints. Specifically, spatial graph routing aims at spatial rela- +tionships based on sub-group clustering, while the temporal +graph routing explores the temporal correlations. Shi et al. [9] +proposed a two-stream adaptive graph convolutional network +to make the value of the adjacency matrix be variable. With the +adaptive strategy and the two-stream pattern, this method can +model both human joints features and human bones features +simultaneously. DGNN [25] leveraged an alternating spatial +aggregation scheme to update the joint and bone features. +Liu et al. [26] proposed a disentangling and unifying graph +convolutional network, including a simple disentangled multi- +scale graph convolution and G3D module. The former is +used to disentangle the importance of nodes in different +neighborhoods, which can model the long-range relationships. +The latter is presented to directly propagate the information +across the spatial-temporal graph by leveraging the cross- +spacetime edges as skip connections. +B. Transformer in Computer Vision +Transformer [27] was proposed for the Natural Language +Processing tasks to make up for the shortcoming of the +RNN methods. The great contribution of transformer is the +self-attention, which can dynamically focus on the global +context information. Alexey et al. [28] applied a pure vision +transformer to sequences of image patches, which has achieved +excellent performance on the image classification task. In the +object detection field, Carion et al. [29] proposed detection +transformer reasoning about the relations of the objects and +the global image context. Wang et al. [30] proposed Max- +DeepLab for semantic segmentation, which directly predicts +class-labelled mask with a mask transformer. Zhou et al. [31] +proposed a masked transformer for video understanding tasks. +For skeleton-based action recognition, Shi et al. [12] proposed +a pure transformer network to model the correlation between +joints without using the traditional skeleton graph represen- +tation. They designed a decoupled spatial-temporal attention +network to calculate the attention score between each pair of +joints without knowing their positions or mutual connections. +Similarly, Plizzari et al. [13] proposed a spatial and temporal +transformer network. The spatial self-attention module is used +to capture the intra-frame correlations between human joints, +while the temporal self-attention module is used to model +the inter-frame relationships. However, these methods have a +common problem that they ignore the inherent topology of the +human skeleton and overestimate the correlation between some +joints. In this way, the relationship that does not exist between +some joints in some actions will also be forced to be imposed +through the calculation of attention score. For example, in the +action of “sitting down”, it is not necessary to capture the +relationship between the left hand and the right hand that will +bring trouble for the model to recognize actions. In our work, +we use the transformer structure to enhance the ability of the +GCN to capture relationships between joints. Different from +the original transformer and its variants, the position encoding +is not used in our work because of the topological invariance +of graphs. +C. Self-Supervised Learning +The intention of self-supervised learning (SSL) is to learn +the internal structures of data and the feature representations +from the unlabeled data. In [32], it has been verified that +self-supervised pre-training also can bring some assistance for +supervised learning. SSL was first used in visual representation +[33], and now it has a number of applications in computer +vision fields [34], [35], [36]. It is usually achieved by pretext +task which is a hot topic of research. In [37], they predicted the +arrangement of multiple shuffled image patches by using SSL +to learn the spatial relationships. Chen et al. [38] proposed a +simple framework for contrastive learning of visual represen- +tations. They force the feature representation between positive +samples to be more similar than those between negative ones. +For the sequential data, some methods [39], [40] learn the +temporal features by predicting the sequential order of sampled +frames or clips. Cho et al. [41] proposed a video representation +method via a prediction task. For skeleton-based tasks, Lin +et al. [42] integrated prediction task, recognition task, and +contrastive learning paradigm to learn skeleton features from +different aspects. Zheng et al. [43] explored an unsupervised +representation learning approach to compactly encode long- +term global motion dynamics. Su et al. [44] proposed an un- +supervised encoder-decoder recurrent neural network to cluster +similar movements. Xu et al. [45] proposed an unsupervised +framework based on encoder-decoder structure to extract more +discriminative temporal features and explored the inherent +action similarity within the action encoding by clustering. Rao +et al. [46] utilized a variety of data enhancement strategies on +unlabeled data to obtain the action representations with the + +4 +Class +Score +ST&TT +AGCN&TCN +0 +Contrastive +learning +Prediction +Softmax +Action +representation +Action +representation +Temporal Convolutional +Module +Spatial Transformer +Module +Temporal Transformer +Module +ReLU +Linear +Linear +Linear +Q +V +K +Multi-Head Self- +Attention +Norm +MLP + +Adaptive GCN +Module +Convolution +Layer +BatchNorm +Layer +Skeleton sequence +STG +STT +Fig. 1: The overall architecture of the proposed ConGT. The skeleton sequence is first fed into two streams, where the spatial- +temporal GCN stream (STG) processes the input graph through the adaptive GCN module (AGCN) and temporal convolutional +module (TCN). The spatial-temporal transformer stream (STT) operates on the input graph with spatial transformer (ST) and +temporal transformer (TT) modules. The ST and TT modules have the same structure. Then the contrastive learning maximizes +the mutual information across the two streams. Finally, the classifier is added for action classification. And in the test phase, +we choose the prediction result of the output feature of STG as the final result of the network. +contrastive learning paradigm. Li et al. [47] proposed a cross- +view contrastive learning model by leveraging multi-view +complementary supervision signal. Wang et al. [48] proposed +the contrast-reconstruction representation learning network to +capture postures and motion dynamics simultaneously. In [49], +Guo et al. utilized the abundant information mining strategy to +make better use of the movement patterns. In [50], [51], it is +suggested that contrasting congruent and incongruent views of +graphs with mutual information maximization can help encode +rich representations. Inspired by them, we also integrate con- +trastive learning into the training of our network to enhance +graph modelling by maximizing mutual information between +two types of action representations and improve the accuracy +of action recognition. +III. METHOD +In this section, we first present an overview of Contrastive +GCN-Transformer Network (ConGT) and describe the details +of each component of the ConGT. Then, we show the process +of enhancing ConGT with contrastive learning. Finally, we +depict the details of Cyclical Focal Loss (CFL). +A. ConGT +Our proposed ConGT includes two streams: Spatial- +Temporal Graph Convolution stream (STG) and Spatial- +Temporal Transformer stream (STT), as illustrated in Fig. 1. +Specifically, STG is used to obtain the action features based +on the topology of the human skeleton graph, consisting of +adaptive graph convolutional network module (AGCN) and +temporal convolutional network module (TCN). The AGCN +learns the topology of the graph for different layers and +skeleton samples while the TCN models the temporal connec- +tions between adjacent frames in time dimension. Likewise, +the STT contains spatial transformer module and temporal +transformer module, which are primarily responsible for ac- +curately capturing the relationships between arbitrary joints in +the intra- and inter- frames. Both streams can output action +representations, but the generated action representations have +different characteristics and each knows little information +of the other. For this reason, we introduce the contrastive +learning paradigm into ConGT to enforce the two streams +learn more distinctive information. Through the contrastive +learning, we can maximize the interactive information from +the representations learned by these two streams. In this end, +we can obtain the predicted class for the input skeleton graph +with the classifier. +B. Spatial-Temporal Graph Convolution Stream +Notation. For skeleton-based action recognition, the human +action is represented as a sequence of skeleton frames. In a +frame, each skeleton is expressed as a graph G = (V, E), +in which V = {vi|(i = 1, . . . N)} denotes the collection of +vertices representing N human joints and E = {ei,j|(vi, vj)} +denotes the collection of edges representing the human bones. +Formally, the adjacent matrix A ∈ RN×N is used to describe +the structure of skeleton graph, where the value of Aij is 0 or +1 indicating whether an edge exists between joints vi and vj. +And the feature tensor X ∈ RC×T ×N denotes the sequence of +skeleton frames, where C represents the coordinate dimension, +T denotes the total number of skeleton frames contained in +the sequence, and N is the total number of human joints. +Graph Convolutional Network. The process of updating +a graph by GCN is to aggregate nodes information with + +5 + + + +Spatial Adaptive GCN +Adjacency graph ෩𝑨 +(CT,N) +(N,CT) +(N,N) +(N,N) +Learnable graph L +Embedded +graph E +Skeleton graph +Adaptive skeleton +tensor +Embedding +function +SoftMax +layer +(C,T,N) +1×1 conv +Parameterized + +Elements +multiplication + +Elements +summation +(C,T,N) +(N,N) +Skeleton +tensor +(N,N) +Fig. 2: The schematic diagram of the spatial adaptive graph convolution. The input consists of an adjacency matrix of the +skeleton graph and a skeleton tensor. The learnable matrix is a parameterized matrix and is trained with the whole model, and +the skeleton tensor passes through the embedding function to obtain the embedded matrix. Then the three types of matrices +are added together and multiplied with the skeleton tensor to get the output adaptive skeleton tensor. +edge information to generate a new graph representation. For +skeleton-based action recognition, the action representations +can be obtained through GCN operating on the adjacent matrix +A. Let H denote the action representation which is initialized +to H(0) = Xin = +� +X1 +in, X2 +in, . . . , Xn +in +� +∈ RC×T ×N, where +Xin is the graph representation of input skeleton frame, T +and N are the total number of the frames and human joints, +respectively. Then the graph convolution operation can be +expressed as: +H(l+1) +t += D− 1 +2 ¯AD +1 +2 Hl +tωl +(1) +where ¯A = A + I, I is the identity matrix, D is the degree +matrix of ¯A, Hl +t denotes the skeleton tensor in the t-th frame +at the l-th layer, and ωl ∈ RCl×C(l+1) is a learnable weight +matrix. +In the implementation of GCN, the higher-order polynomial +of the adjacency matrix A is applied to aggregate the skeleton +tensor H to get the high-order neighbor information. Thus, +Eq. 1 can be rewritten as: +H(l+1) +t += σ +� Kv +� +k=0 +� +˜AkHl +t +�� +⊙ ωl +k +(2) +where Kv denotes kernel size in the spatial dimension. ˜A +represents the normalized adjacency matrix, while �Ak repre- +sents k-power of the normalized adjacency matrix which can +represent the relationship between adjacent nodes of order k. +ωl +k ∈ RCl×C(l+1) is a learnable weight matrix, ⊙ denotes the +dot product, and σ(·) denotes the activation function. +Adaptive graph strategy. In the traditional graph convo- +lution, a common way to calculate graph topological rela- +tionships is using the adjacency matrix. Each element of the +adjacency matrix A has a value of 0 or 1. When A(i, j) = 0, +it means there is no connection between the joints vi and vj, +otherwise adjacent. Note that even if after several multipli- +cation operations between the adjacency matrix and skeleton +tensor, A(i, j) = 0 will still exist, indicating that there is +no relationship between joints vi and vj during the model +training. This will ignore the connection between some joints +and further affect the recognition accuracy. For instance, the +two hands in action “clapping hands” are not directly linked, +but the interaction between them is actually very useful for +recognizing this action. To this end, we employ the adaptive +graph strategy [9] to adaptively reflect the relationships of +joints. +As shown in Fig. 2, the adaptive graph strategy is imple- +mented by adding the adjacency matrix, the learnable matrix, +and the embedded matrix together. Then the graph convolution +integrated with the adaptive graph strategy can be formulated +as: +H(l+1) +t += σ +� Kv +� +k=0 +�� +�A + L + E +�k +Hl +t +�� +⊙ ωl +k +(3) +where the normalized adjacency matrix �A ∈ RN×N denotes +the original topology of the skeleton graph. L ∈ RN×N is a +learnable matrix which is initialized with the adjacent matrix +A to accelerate the convergence of the network. As we perform +actions, our joints move in groups, but the importance of each +joint is different in different groups. Therefore, we need to +define an importance weight to scale the contribution of node +features to neighboring nodes. The learned matrix L is an +attention map that indicates the importance of each node. In +this way, the data-driven dependencies between nonadjacent +joints vi and vj can be generated as the depth of the network +increases. The embedded matrix E +∈ RN×N learns an +individual graph for each sample. In these individual graphs, +the weights of edges are calculated by measuring the similarity +of graph nodes which can be obtained by the normalized +embedded Gaussian function. The elements of the embedded + +6 +matrix can represent the strength of the dependency between +two joints, and the value of them ranges from 0 to 1. The +embedded matrix E is described mathematically as follows: +E = softmax +�� +θ1 +� +f (l)� +· δ(l) +1 +�T +· +� +θ2 +� +f (l)� +· δ(l) +2 +�� +(4) +where θ represents the embedding function that can map +any two joint vectors to the same vector space. δ denotes +the parameters of the embedding function. f (l) ∈ RC×T ×N +denotes the feature tensor of the l-th layer. And it will be +converted into two intermediate embeddings by embedding +functions θ1 and θ2. Then the two embeddings are multiplied +to get the embedded graph E. +Temporal convolution Network. Unlike the spatial topol- +ogy, the temporal topology of joints is linear. Thus, temporal +relationships are usually captured by using ordinary convolu- +tion operation rather than graph convolution. +In general, the spatial-temporal convolution on the skeleton +graph can be formulated as: +H(l+1) = TCN +� +AGCN +� +H(l)�� +(5) +where TCN(·) is a temporal convolution with a kernel size +KT × 1, and KT = 9 in this work. AGCN(·) denotes the +adaptive graph convolution network, and H(l) denotes the +action representation in the l-th layer. +C. Spatial-Temporal Transformer Stream +Although the adaptive graph strategy can make up for the +defect that A(i, j) = 0 cannot be replaced in multiplication, +the long-distance connections are still easily underestimated. +Moreover, with the stacking of layers, the risk of over- +smoothing of graph convolution will increase, leading to poor +ability to accurately capture the relationships between joints +which are depicted by the action representations. For these +reasons, we propose the spatial-temporal transformer stream +(STT) to enhance the dependence of local neighboring joints +and further effectively capture long-distance relationships in +both spatial and temporal dimensions. +The work flow of the spatial-temporal transformer is de- +picted in Fig. 1. In space, for each node nt +i of the skeleton +graph in the frame t, a query vector qt +i, a key vector kt +i, and a +value vector vt +i can be calculated by the trainable linear trans- +formations with three parameters matrices Wq ∈ RCin×dq, +Wk ∈ RCin×dk, and Wv ∈ RCin×dv, which are shared by +all nodes. With these vectors, we use the multi-head attention +to calculate the attention score to weight the value vector vt +i +that corresponds to the query vectors. In time dimension, a +node in one frame will pay attention to nodes representing the +same joint in other frames. Similar to the spatial transformer, +in temporal transformer, the first step is also to calculate a +query vector qfj, a key vector kfj, and a value vector vfj +for each node nfj in different frame f. Then, we compute +the attention score with multi-head attention, which is used to +determine how much attention one node is paid to other nodes +that represent the same joints along the temporal dimension. +As the core component of transformer, the multi-head +attention is detailed in Fig. 3. The input consists of the query +vector q, the key vector k, and the value vector v. Then, the +dot products of the query with all keys are calculated to get +the attention scores between all nodes. In order to prevent the +vanishing gradient in the process of backpropagation, we scale +the attention scores by +1 +√dk , where dk denotes the dimension +of query vector and key vector. Subsequently, we apply the +softmax function on the attention scores to weight the value +vectors. Finally, the attention-enhanced value vectors of nodes +are aggregated in a summation manner. The above process is +repeated Nh times with different queries, keys and values, +where Nh = 8 in our work. Then the results of the Nh +attention heads are concatenated together to constitute the +output representation. Formally, this process is expressed as: +O(l) +i += +Nh +� +1 +� +� � +j∈Ni +softmax +� +S(l) +i,j +√dk +� +v(l) +j +� +� +(6) +S(l) +i,j = q(l) +i +· k(l) +j +(7) +where O(l) +i +denotes the output of the multi-head attention. +�Nh +1 (·) denotes the concatenation of Nh heads. q(l) +i , k(l) +i , and +v(l) +i +denote the query vector, the key vector, and the value +vector of node i in the frame t at the l-th layer, respectively. +S(l) +i,j is the attention score between nodes i and j, which is +treated as a weight when aggregating the values of different +nodes. +Next, the output of the multi-head attention passes through a +batch normalization layer and a Multi-Layer Perceptron (MLP) +block to obtain the input tensor of the next layer or the final +output of STT in the last layer. The STT can be formulated +as: +H(l+1) = TT +� +ST +� +H(l)�� +(8) +where TT(·) and ST(·) denote the temporal transformer +module and the spatial transformer module, respectively. H(l) +denotes the generated action representation in the l-th trans- +former layer. +D. Enhance ConGT with Contrastive Learning +The STG can learn the topology of the graph for different +GCN layers and skeleton samples in an end-to-end manner, +obtaining the action representations based on the topology +of the human skeleton graph. However, the implicit relations +between nonadjacent joints are ignored, such as the connection +between hand and head during touching head. Meanwhile, the +temporal connection between remote frames is underestimated +due to the limitation of temporal convolution kernel size. +To solve them, we bridge GCN and attention module in a +parallel way with contrastive learning to enrich the action +features. We design the STT based on transformer to capture +the long-distance correlations between each pair of joints in +both spatial and temporal dimensions without knowing their +positions or mutual connections. Since each stream encodes a +representation that only depicts either the topology of human +skeleton graph or the long-distance correlations between each +pair of joints, the two types of action representations know +little about each other but may mutually complement each + +7 +Attention +Score +𝑞𝑖 +𝑡 +𝑘𝑖 +𝑡 +𝑣𝑖 +𝑡 +Scaling +Dot +Products +𝑂𝑡,𝑖 +𝑙 +𝑁ℎ heads +S +Summation +Dot +Products +V +C +S +Softmax +C +Concatenation +Fig. 3: The detail of multi-head attention. +other. Specifically, they can be the ground-truth of each other +for contrastive learning. Then, by maximizing the mutual +information between the action representations learned via +the two streams through contrastive learning, our network +can learn more distinctive information to enhance recognition +accuracy. +We adopt InfoNCE as the contrastive learning objective, +which is defined as: +Lcon = − log σ +� +fd +� +ag +i , at +i +�� +− log σ +� +1 − fd +� +˜ag +i , at +i +�� +(9) +where ag +i and at +i denote the action representations obtained +through the STG and STT, respectively. �ag +i represents the neg- +ative samples obtained by corrupting ag +i with both row-wise +and column-wise shuffling. fd(·) is a discriminant function: +Rd×Rd → R, which scores the agreement between the two in- +put vectors. The contrastive learning objective is to maximize +the agreement of two different types of representations, while +minimizing the agreement with other negative representations. +In this way, both STG and STT can acquire information from +each other and further enrich the action features. +In the end, we combine the contrastive learning task with +the action recognition task to form multi-task learning, where +contrastive learning is the auxiliary task. We optimize our +model by means of joint learning that is defined as: +L = LCF L + βLcon +(10) +where LCF L is cyclical focal loss that is the learning objective +of action recognition task and β is a hyper-parameter used to +control the magnitude of the contrastive task. +E. Cyclical Focal Loss +We define the learning objective of action recognition task +as the cyclical focal loss which is formed of the focal loss +and the general cyclical training principle. Cyclical focal loss +can focus on confident predictions in the early epochs of the +network training. As the number of training epochs increases, +it will focus more on misclassified samples. In the following, +we will describe the details of the cyclical focal loss. +Focal Loss is mostly used for binary classification problems. +It modifies cross-entropy softmax loss to reduce the weight of +easily classified samples so that the model can focus more +on samples that are difficult to classify during training. It is +defined as: +Llc = −(1 − pt)γlog(pt) +(11) +where pt denotes the softmax probabilities, and γ ≥ 0 is a +tunable hyper-parameter. For the confident training samples, +the value of pt tends to be 1, and the weight (1 − pt)γ for +the loss will drive the loss to zero faster than that for cross- +entropy. Although the focus loss is beneficial in tasks with +unbalanced class data, it often affects performance when the +dataset is more balanced. Therefore, in most applications, the +focus loss is not the best choice. +Combining the focus loss with the general cyclical training +principle, the cyclical focal loss [52] is proposed. In [53], the +general cyclical training of a neural network is considered +as a combination of curriculum learning in the early epochs +with fine-tuning at the end of training. Specifically, the easy +and confident training samples are used at the start and end +stages of network training, and the hard training samples are +processed at the middle stage of training. For this purpose, +Smith et al. [52] proposed a new loss: +Lhc = − (1 + pt)γhc log (pt) +(12) +where pt denotes the softmax probabilities, and γhc ≥ 0 +is a tunable hyper-parameter. In this manner, the loss can +pay more attention to the confident training samples. And +the hard training samples can be heavily weighted via Eq. +11. Therefore, the cyclical focal loss can be accomplished +by combining Eq. 11 and Eq. 12 in a reasonable manner. In +[52], they used a linear schedule and defined a parameter ξ to + +8 +combine them that varies with the training epoch as: +ξ = +� +1 − fc +epi +epn +if fc × epi ≤ epn +� +fc +epi +epn − 1 +� +/ (fc − 1) +otherwise +(13) +where epi denotes the number of current training epochs +and epn is the total number of training epochs. fc denotes +the cyclical factor that provides adaptability for the cyclical +schedule. +Integrating Eq. 11, Eq. 12 and Eq. 13, the cyclical focal +loss can be defined as: +CFL(p, y) = ξLhc + (1 − ξ)Llc +(14) +In our experiments, we keep the values of the hyper- +parameters consistent with those in the original work, where +γlc = 2, γhc = 2, fc = 4. +IV. EXPERIMENTS +To verify the effect of the proposed method, we conduct +extensive experiments on three widely used datasets: NTU- +RGBD 60 [54], NTU-RGBD 120 [55], and Northwestern- +UCLA [56]. In this section, we first give the description of the +three datasets in detail. Next, we will describe the experiment +settings. Then, the comparisons between the proposed method +and several state-of-the-art methods are introduced. Finally, +we investigate the contributions of each component in the +proposed method. +A. Datasets +NTU-RGBD 60 (NTU-60): NTU-60 is one of the widely +used datasets for skeleton-based action recognition tasks, +which contains 56,880 samples of 60 different classes. The +dataset is captured by a Microsoft Kinect V2 camera. The +skeleton data is formed of the 3D joint locations (X, Y, Z) of 25 +joints. In each video, there are no more than two persons. The +dataset is divided into Cross-View (X-View) Setting and Cross- +Subject (X-Sub) Setting. In X-View, the actions are captured +by three cameras with the same height in the vertical direction +and different angles −45◦, 0◦, 45◦ in the horizontal direction, +where the training set contains 37,920 samples and the testing +set includes 18,960 samples. In X-Sub, the subjects of the +training set and the testing set are different. The training set +contains 40,320 videos, and the testing set contains 16,560 +videos. +NTU-RGBD 120 (NTU-120): NTU-120 is an extension +of NTU-60 with additional 57,367 skeleton sequences, which +totally contains 114,480 samples. The action categories in +this dataset can be divided into three major groups: 82 daily +actions (eating, sitting down, standing up, etc), 12 health- +related actions (falling down, blowing nose, etc), and 26 +mutual actions (hugging, shaking hands, etc). Similar to NTU- +60, NTU-120 also has two benchmarks: 1) cross-subject (X- +Sub), 2) cross-setup (X-Set). In cross-subject, the 106 subjects +are equally split into the training set and the testing set. In +cross-setup, the samples whose ID is even belongs to the +training set, while the samples whose ID is odd are treated +as the testing set. +Northwestern-UCLA (NW-UCLA): NW-UCLA [56] con- +tains 1,494 video clips covering 10 categories and is captured +by three Kinect cameras simultaneously from a variety of +viewpoints. Each action sample is performed by 10 different +actors. Following [56], we adopt the same protocol to divide +the dataset that the training set is composed of the samples of +the first and second viewpoints, while the testing set is made +up of samples of the third viewpoint. +B. Implementation Details +The implementation of our model is conducted on the +PyTorch framework. We use the stochastic gradient descent +(SGD) as the optimizer to train our network, where the +momentum is set to 0.9 and the weight decay is set to 0.001. +For NTU-60 and NTU-120, we set the number of training +epochs to 75 and the initial learning rate to 0.001. The learning +rate decays at the 45th and the 55th epoch. For the NW-UCLA +dataset, the number of training epochs is set to 65, and the +learning rate decays at the 50th epoch, which is initially set to +0.01. In contrastive learning, the hyper-parameter β is set to +0.01 for NTU-60, 0.05 for NTU-120, and 0.1 for NW-UCLA, +which is used to control the magnitude of the contrastive +learning task. For the cyclical focal loss, we set the cyclical +factor fc to 4 and both hyper-parameters γlc and γhc to 2. +C. Comparison Against the State of the Art +To verify the effectiveness of our network, we compare our +prediction accuracy with the current state-of-the-art methods +on NTU-60, NTU-120 and NW-UCLA datasets under two +evaluation protocols, including linear evaluation protocol and +fine-tuning protocol. +Linear Evaluation Protocol. With this protocol, we eval- +uate the quality of the representations learned by our method +with training a linear classifier (including a fully-connected +layer and a softmax layer) and freezing the parameters of the +other part. We report the comparison results in Table I, Table +II, and Table III. +TABLE I: Performance comparison on the NTU-60 dataset +with linear evaluation protocol. +Methods +X-View(%) +X-Sub(%) +LongT GAN [43] +39.1 +48.1 +MS2L [42] +- +52.6 +PCRP [45] +63.5 +53.9 +AS-CAL [46] +64.8 +58.5 +CRRL [48] +73.8 +67.6 +3s-CrossSCLR [47] +83.4 +77.8 +3s-AimCLR [49] +83.8 +78.9 +BRL [57] +91.2 +86.8 +ConGT +92.0 +86.2 +Table I shows the comparisons with previous related meth- +ods on NTU-60. Our method achieves the accuracy of 92.0% +on X-View and 86.2% on X-Sub. Compared with LongT +GAN [43], MS2L [42], PCRP [45], and AS-CAL [46], our + +9 +method achieves an overwhelming performance. CRRL [48] is +a contrast-reconstruction representation learning network that +can simultaneously capture postures and motion dynamics for +unsupervised skeleton-based action recognition. By contrast, +our method performs 18.2% better on X-View and 18.6% on +X-Sub. 3s-CrosSCLR [47] also attains superior performance +due to its multi-view strategy. Our ConGT achieves excellent +performance that outperforms it 8.6% on X-View and 8.4% on +X-Sub. 3s-AimCLR [49] is a contrastive learning framework +with utilizing abundant information mining for self-supervised +action representation. Compared with it, the performance of +our method is 8.2% and 7.3% higher on X-View and X- +Sub under Top-1 recognition accuracy, respectively. Compared +to other unsupervised methods, BRL [57] gains remarkable +results by using the data augmentation and multi-viewpoint +sampling strategies, which achieves the accuracy of 91.2% on +X-View and 86.8% on X-Sub. Our method still works better +than it on X-View. +TABLE II: Performance comparison on the NTU-120 dataset +with linear evaluation protocol. +Methods +X-Set(%) +X-Sub(%) +LongT GAN [43] +39.7 +35.6 +PCRP [45] +45.1 +41.7 +AS-CAL [46] +49.2 +48.6 +CRRL [48] +57.0 +56.2 +3s-CrossSCLR [47] +66.7 +67.9 +3s-AimCLR [49] +68.8 +68.2 +ISC [58] +67.1 +67.9 +BRL [57] +79.2 +77.1 +ConGT +80.5 +78.6 +In Table II, we conduct the comparative experiment on +NTU-120 dataset on both X-Sub and X-Set benchmarks. We +also follow the standard practice in the literature, reporting +the top-1 classification accuracies on both benchmarks. The +competitive results in Table II verify the superiority of our +proposed method over all methods. +TABLE III: Performance comparison on the NW-UCLA +dataset with linear evaluation protocol. +Methods +Accuracy(%) +LongT GAN [43] +74.3 +MS2L [42] +76.8 +CRRL [48] +83.8 +ConGT +85.3 +As shown in Table III, the proposed ConGT achieves the +best accuracy of 85.3% on the NW-UCLA dataset, surpassing +the previous state-of-the-art methods. The NW-UCLA contains +ten categories of actions: pick up with on hand, pick up +with two hands, drop trash, walk around, sit down, stand up, +donning, doffing, throw, and carry. For each specific action, +we use the boxplots to show the training accuracy of every +5 epochs in Fig. 4, where these ten classes are denoted by +numbers 1-10. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Class +20 +40 +60 +80 +100 +Accuracy(%) +Fig. 4: Different color boxes indicate the accuracy range of +several categories, the black line inside each box represents +the median value, boxes limits include interquartile ranges +from 25% to 75% of samples, upper and lower whiskers are +computed as 1.5 times the distance of upper and lower limits +of the box, and all values outside the whiskers are considered +as outliers. +TABLE IV: Performance comparison on the NTU-60 and +NTU-120 dataset with fine-tuning protocol. +Methods +NTU-60 +NTU-120 +X-View +X-Sub +X-Set +X-Sub +SkeletonCLR [47] +88.9 +82.2 +75.3 +73.6 +AimCLR [49] +89.2 +83.0 +76.7 +76.4 +ConGT +91.6 +84.6 +80.5 +79.4 +Fine-tuning Protocol. Following [47], we first pre-train +STG, STT and contrastive learning and then append a linear +classifier to retrain the whole model, where the parameters of +each layer in our network are updated with the backpropaga- +tion. We compare our model with the state-of-the-art methods +in Table IV. +To make a fair comparison with SkeletonCLR [47] and +AimCLR [49], we only use the bone data to compare the +finetuned results. As shown in Table IV, our ConGT defeats +them both on NTU-60 and NTU-120. Specifically, on X-View +and X-Sub of NTU-60, our method surpasses SkeletonCLR +by 2.7% and 2.4%, respectively, and outperforms AimCLR +by 2.4% and 1.6%, respectively. On NTU-120, compared to +SkeletonCLR, the improvements reach 5.2% and 5.8% on +X-Set and X-Sub, respectively. Compared to AimCLR, our +method surpasses it by 3.8% and 3.0% on X-Set and X-Sub, +respectively. The results demonstrate that our model with the +contrastive learning paradigm can effectively learn rich action +representations of human actions. +D. Ablation Study +In this section, we design ablation experiments to investigate +the effectiveness of the proposed approach. We first validate +the effectiveness of each component of our model. And we + +10 +demonstrate that the existence of over-smoothing problem +during the accumulation of GCN layers. Then we test the +influence of the hyper-parameter β that controls the magnitude +of contrastive learning. Finally, we verify the effectiveness of +the cyclical focal loss. +TABLE V: The comparison performance of ConGT with +different parts on X-View of NTU-60. +Subnet +STG +STT +CL +Accuracy(%) +N-STG +✓ +89.6 +N-STT +✓ +32.5 +ST-GT +✓ +✓ +73.3 +ST-MGT +✓ +✓ +90.2 +ConGT +✓ +✓ +✓ +92.4 +1) Impact of Each Component in ConGT: As three primary +components of our proposed ConGT, the Spatial-Temporal +Graph Convolution stream (STG), Spatial-Temporal Trans- +former stream (STT), and Contrastive Learning (CL) are +also the main contributions in this work. To evaluate the +effectiveness of STG, STT and CL, we design four subnets. +• N-STG: Only training STG to show the results of using +GCN alone. +• N-STT: Only training STT to display the results of using +transformer alone. +• ST-GT: We remove the contrastive learning part and +add the representations learned by the STG and STT to +demonstrate the effectiveness of contrastive learning. +• ST-MGT: We replace the InfoNCE loss with the MSE +loss to illustrate that the contrastive learning plays a +crucial role in combining two different types of action +representations. +On the four subnets, we conduct experiments on X-View of +NTU-60. The results obtained by these baselines are depicted +in Table V. It can intuitively see that the recognition accuracy +can reach 89.6% when only using the STG, while the recog- +nition accuracy is only 32.5% when only using the STT. We +speculate the reason is that the transformer treats each node as +a separate unit and regards the human skeleton as a complete +graph with connections built between each joint and the rest +joints, resulting in less variation between different movements. +To verify this claim, we visualize the action representation +learned by STT in Fig. 5, where all the categories are mixed +together. This adds to the evidence that it is unreasonable to +treat the human skeleton as a complete graph. +Moreover, we adopt two different methods to demonstrate +the effectiveness of contrastive learning in Table V. In ST- +GT, we add the action representations output by STG and +STT together. In this way, the accuracy is reduced by 19.1% +compared to ConGT. Furthermore, we replace the InfoNCE +loss with MSE loss to evaluate the effect of contrastive +learning in ST-MGT and the final recognition accuracy has a +decline. In Fig. 6, we depict the results of the ST-GT, ST-MGT, +and ConGT. The blue part denotes the recognition accuracy +of ST-GT, the orange part represents the improvement of the +ST-MGT using the MSE loss, and the green part shows the +Fig. 5: The visualization of the action representation learned +by STT. In order to better show the distribution of action +representation of each category, we select the top 10 classes of +the X-View of NTU-60 dataset. Each color denotes an action +class and each point represents a skeleton sequence. +10 +20 +30 +40 +50 +60 +70 +test +Epoch +0 +20 +40 +60 +80 +Accuracy(%) +ST-GT +ST-MGT +ST-ConGT +Fig. 6: The influence of the contrastive learning paradigm. +The blue part denotes the recognition accuracy of ST-GT. The +orange part represents the improvement of the recognition ac- +curacy with the MSE loss. The green part shows the superiority +of using contrastive learning. +superiority of using contrastive learning in ConGT. We can +intuitively see that the training accuracy of ST-GT and ST- +MGT are consistently lower than ConGT. Therefore, it can be +illustrated that the contrastive learning plays a crucial role in +fusing long-distance relationships into the topology structure +of the human skeleton graph. +2) The Effect of Adaptive Graph Strategy in STG: We +demonstrate the influence of AGCN by comparing STG with +ST-GCN in the supervised manner. For a fair comparison, we +set the same number of GCN layers in STG as that in ST- +GCN. In Table VI, we can see that the recognition accuracy +of the STG outperforms 0.9% that of ST-GCN, which indicates +that the adaptive graph strategy contributes to improving the + +1 +2 +3 +40 +4 +5 +6 +7 +8 +6 +10 +20 +0 +-20 +-40 +-40 +20 +0 +20 +4011 +(a) +(b) +(c) +Fig. 7: The t-SNE visualization of action representations. Each point represents a skeleton sequence. We show the first 10 +action classes of the X-View of NTU-60 dataset, indicated by colors. (a). The STG with 6 GCN layers. (b). The STG with 9 +GCN layers. (c) The STG with 12 GCN layers. +0.01 +0.02 +0.05 +0.1 +0.2 +0.5 +1 +2 +5 +76 +78 +80 +82 +84 +86 +88 +90 +92 +Accuracy(\%) +NTU60-xview +NTU60-xsub +(a) +0.01 +0.02 +0.05 +0.1 +0.2 +0.5 +1 +2 +5 +76 +78 +80 +82 +84 +86 +88 +90 +92 +Accuracy(\%) +NTU120-xsub +NTU120-xset +(b) +0.01 +0.02 +0.05 +0.1 +0.2 +0.5 +1 +2 +5 +76 +78 +80 +82 +84 +86 +88 +90 +92 +Accuracy(\%) +NW-UCLA +(c) +Fig. 8: The influence of the magnitude of contrastive learning on (a) NTU-60, (b) NTU-120, and (c) NW-UCLA. +TABLE VI: Comparison of the performance (accuracy (%)) +on the X-View setting of the NTU-60 dataset when the STG +with or without the adjacency matrix A, the learnable matrix +L, and the embedding matrix E. wo/A denotes without A, +wo/L denotes without L, and wo/E denotes without E. +Methods +Accuracy(%) +ST-GCN +88.3 +STG +89.2 +STG wo/A +88.5 +STG wo/L +88.3 +STG wo/E +88.7 +accuracy of action recognition. Furthermore, to evaluate the +necessity of the three graphs in AGCN, we conduct an ablation +study on the three graphs. We manually delete one of the three +types of graphs and show their performance in Table VI. We +find that taking away any one of the three graphs will affect +the final recognition result negatively. When all three graphs +are simultaneously enabled, our model can achieve the best +performance. This indicates that the adaptive graph strategy is +conducive to increasing the accuracy of action recognition. +3) The Impact of the Number of GCN Layers in STG: +In addition, to demonstrate that the over-smoothing problem +occurs during the accumulation of GCN layers, we compare +TABLE VII: Recognition accuracies obtained by STG contain- +ing 6, 9, and 12 GCN layers on X-View setting of NTU-60. +Layers +Accuracy(%) +6 +89.6 +9 +89.2 +12 +86.1 +the recognition performance of STG containing 6, 9, and 12 +GCN layers on the X-View setting of NTU-60. In Table VII, +we can see that the recognition accuracy has a significant +decline when the GCN layer number is increased to 12. We +also apply t-SNE [59] to show the embedding distribution +of these three options in Fig. 7. From the visual results, we +can find that with the increase of network layers, the action +representations of classes 1, 2, 3, and 5 (circled by the red +ellipse) tend to be consistent. It also reveals that with the +increase of the number of GCN layers, the probability of the +over-smoothing problem also increases. +4) The +Impact +of +the +Contrastive +Learning +Hyper- +parameter: To justify the impact of the hyper-parameter β on +controlling the magnitude of contrastive learning, we examine +the performance of ConGT with a set of representative β +values {0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, 5}. +The performance results are shown in Fig. 8. As we can see, +our model performs best on NTU-60 when β is 0.01. While + +2 +m +40 +4 +5 +6 +7 +8 +9 +20 +10 +-20 +-40 +-40 +-20 +0 +20 +4040 +2 +4 +6 +20 +10 +20 +-40 +-60 +-40 +20 +0 +20 +40 +6060 +2 +40 +10 +20 - +0 +-20 +40 +-60 +40 +20 +0 +20 +4012 +TABLE VIII: Comparison of the top-1 test recognition ac- +curacies for Cross-Entropy Loss and Cyclical Focal Loss on +NTU-60 and NTU-120. +Datasets +Cross-Entropy Loss +Cyclical Focal Loss +NTU-60 (X-View) +91.08 +91.59 +NTU-60 (X-Sub) +84.21 +84.55 +NTU-120 (X-Set) +78.80 +80.53 +NTU-120 (X-Sub) +78.82 +79.36 +for NTU-120, the best β is 0.05. When β is 0.1, our method +achieves the best accuracy on NW-UCLA. In addition, it can +be seen that when β becomes large, the performance of ConGT +on both NTU-60, NTU-120, and NW-UCLA will decline. We +suspect it is due to that the gradient conflict between the +action recognition task and the contrastive task. Therefore, it +is necessary to select an appropriate β, when involving the +contrastive learning paradigm. +5) The Effectiveness of the Cyclical Focal Loss: In this +section, we compare the recognition results obtained by using +the cross-entropy loss and the cyclical focal loss on NTU-60 +and NTU-120 in Table VIII. It shows that when the model +is trained with the cyclical focal loss, the test accuracy is +consistently better than that using the cross-entropy loss. +0 +10 +20 +30 +40 +50 +60 +70 +Epoch +0 +20 +40 +60 +80 +Accuracy(\%) +NTU60-xviewCross +NTU60-xviewCFL +NTU60-xsubCross +NTU60-xsubCFL +NTU120-xsubCross +NTU120-xsubCFL +NTU120-xsetCross +NTU120-xsetCFL +Fig. 9: The accuracy curves of training ConGT using cross- +entropy loss and cyclical focal loss. +Fig. 9 shows the training accuracy curves of training our +network using the cross-entropy loss and the cyclical focal +loss. Although the cross-entropy loss curve and the cyclical +focal loss curve on the same dataset have strong similarities, +it is notable that training with the cyclical focal loss provides a +slightly faster learning convergence in the training. Therefore, +it can be confirmed that the cyclical focal loss better helps the +learning in the early epochs. +V. CONCLUSION +In this work, we design a novel Contrastive GCN- +Transformer Network (ConGT), which can capture the rela- +tionships between arbitrary joints in the intra- and inter- frames +more accurately while maintaining the topology structure of +human skeleton graphs. Specifically, the STG is designed to +obtain action representations maintaining the topology struc- +ture of the human skeleton graph. At the same time, the +STT is used to acquire action representations containing the +global relationships among joints. Moreover, we introduce +the contrastive learning paradigm, serving as an auxiliary +task, to maximize the mutual information between the action +representations learned via the two streams to improve the +action recognition task. In this manner, we can make up for +the weak ability of GCN to capture long-distance features on +the basis of maintaining the topology structure of the human +skeleton graph and reduce the risk of network over-smoothing. +In addition, we introduce the cyclical focal loss as the learning +objective of our model, which places heavy weights on con- +fident training samples in the first training epochs of a neural +network. Ablation studies have been performed in this work, +which verify the effectiveness of our method. Experiments on +three publicly available datasets demonstrate the superiority of +our proposed method over other methods. +ACKNOWLEDGMENT +This work was supported in part by the National Natu- +ral Science Foundation of China (No. 61976127), Shandong +Provincial Natural Science Foundation (Nos. ZR2021LZL012, +ZR2021QG004). +REFERENCES +[1] Y. Ming, F. Feng, C. Li, and J.-H. Xue, “3d-tdc: A 3d temporal dilation +convolution framework for video action recognition,” Neurocomputing, +vol. 450, pp. 362–371, 2021. +[2] I. +Rodr´ıguez-Moreno, +J. +M. +Mart´ınez-Otzeta, +I. +Goienetxea, +I. Rodriguez-Rodriguez, and B. Sierra, “Shedding light on people +action recognition in social robotics by means of common spatial +patterns,” Sensors, vol. 20, no. 8, p. 2436, 2020. +[3] Z. Xu, G. Wang, and X. Guo, “Sensor-based activity recognition of +solitary elderly via stigmergy and two-layer framework,” Engineering +Applications of Artificial Intelligence, vol. 95, p. 103859, 2020. +[4] Q. Ke, M. 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Hinton, “Visualizing data using t-sne.” Journal +of machine learning research, vol. 9, no. 11, 2008. + diff --git a/ctFJT4oBgHgl3EQfRSzM/content/tmp_files/load_file.txt b/ctFJT4oBgHgl3EQfRSzM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0af15cf72f36c60cc7f9411fb2398e5f364dde2a --- /dev/null +++ b/ctFJT4oBgHgl3EQfRSzM/content/tmp_files/load_file.txt @@ -0,0 +1,1077 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf,len=1076 +page_content='1 Skeleton-based Action Recognition through Contrasting Two-Stream Spatial-Temporal Networks Chen Pang, Xuequan Lu, Lei Lyu∗ Abstract—For pursuing accurate skeleton-based action recog- nition, most prior methods use the strategy of combining Graph Convolution Networks (GCNs) with attention-based methods in a serial way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' However, they regard the human skeleton as a complete graph, resulting in less variations between different actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=', the connection between the elbow and head in action “clapping hands”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For this, we propose a novel Contrastive GCN-Transformer Network (ConGT) which fuses the spatial and temporal modules in a parallel way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The ConGT involves two parallel streams: Spatial-Temporal Graph Convolution stream (STG) and Spatial-Temporal Transformer stream (STT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The STG is designed to obtain action representations maintaining the natural topology structure of the human skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The STT is devised to acquire action representations containing the global relationships among joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Since the action representations pro- duced from these two streams contain different characteristics, and each of them knows little information of the other, we introduce the contrastive learning paradigm to guide their output representations of the same sample to be as close as possible in a self-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Through the contrastive learning, they can learn information from each other to enrich the action features by maximizing the mutual information between the two types of action representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' To further improve action recognition accuracy, we introduce the Cyclical Focal Loss (CFL) which can focus on confident training samples in early training epochs, with an increasing focus on hard samples during the middle epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We conduct experiments on three benchmark datasets, which demonstrate that our model achieves state-of- the-art performance in action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Index Terms—Skeleton-based action recognition, Graph con- volutional network, Transformer, Contrastive learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' INTRODUCTION H UMAN action recognition has become a fundamental task in computer vision which is extensively applied in many real-world applications, such as intelligent security [1], virtual reality [2], and human–machine interaction [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Skeleton-based action recognition task has received significant attention due to its computation efficiency and robustness against viewpoints or appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The core of skeleton-based action recognition is to learn the discriminative representations of skeleton sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' At present, many deep learning based methods have achieved excellent performance by using Convolutional Neural Net- works (CNNs) [4], [5], Recurrent Neural Networks (RNNs) [6], [7] to learn the action representations based on the specific recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' However, these methods rarely consider the Chen Pang and Lei Lyu are with School of Information Science and Engineering, Shandong Normal University, Jinan 250358, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Xuequan Lu is with School of Information Technology, Deakin University, Geelong, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' ∗Corresponding author: Lei Lyu (e-mail: lvlei@sdnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='cn) co-dependency contained in body joints and ignore some important motion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' To better capture joint depen- dencies, Graph Convolutional Networks (GCNs) are exploited to aggregate information based on body structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Spatio- Temporal Graph Convolutional Network (ST-GCN) [8] is a pioneering work to model the skeleton data as a spatio- temporal graph with the joints as graph nodes and natural connections in both human body structures and time as graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Later, many variants [9]–[11] are extended based on ST-GCN, following the same strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Although GCNs have been proved to perform well on skeleton data, they still have some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' First, in GCN-based methods, the human body is represented as a predefined graph fixed over all actions, ignoring certain implicit relations between nonadjacent joints, such as the connection between hand and head during touching head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Second, with the deepening of graph convolutional layers, the probability of over-smoothing problem will increase that the representations of neighbor nodes tend to converge to the same value, which causes confusion between joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Third, in most existing GCN-based methods, the temporal connections between remote frames are underestimated since the temporal convolution operations are limited in a local neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' To cope with these defects, researchers introduced attention modules behind the GCN layers to effectively capture the long-distance relations in the supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [12] designed a decoupled spatial-temporal attention network to calculate the connections between each pair of joints without knowing their positions or mutual connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In [13], ST-TR used transformer to capture the relations of each pair of nodes, ignoring the inher- ent topology of the human skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Although the recognition accuracy can be improved by combining the GCN layers with attention modules in a serial manner, each node is treated in isolation and the human skeleton is regarded as a complete graph with connections built between each joint and the rest joints, resulting in less variations between different actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Taking the two actions of “clapping hands” and “touching nose” for example, the connections among the most joints are considered to be same, except for the stronger connection between the hands in “clapping hands” and the stronger dependence between the hands and nose in “touching nose”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' While the connection between the elbow and head should not be considered in “clapping hands”, it is helpful in “touching nose”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Therefore, it is essential to capture the long-distance relations for better action recognition while preserving the primitive human skeleton structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' To this end, we propose a novel Contrastive GCN- Transformer Network (ConGT) that considers GCN and at- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='11495v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='CV] 27 Jan 2023 2 tention model in a parallel manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Specifically, the network contains two parallel streams, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Spatial-Temporal Graph Convolution stream (STG) and Spatial-Temporal Transformer stream (STT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The STG is used to extract the joint relation- ships based on the topology of the human skeleton graph, consisting of adaptive GCN module (AGCN) and temporal convolutional network module (TCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In particular, the AGCN is designed to enforce the generated graph to reflect the relationships of joints flexibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Different from the STG, the STT is primarily responsible for accurately capturing the relationships among arbitrary joints in the intra- and inter- frames, which is comprised of spatial transformer module and temporal transformer module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Since the action representations produced from these two streams contain different character- istics and each of them knows little information of the other, we introduce the contrastive learning paradigm to guide their output representations of the same sample to be as close as possible in the embedding space in a self-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Specifically, the action representations learned by STG involve the natural topology of the human skeleton and the action representations learned by STT involve long-distance relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Through the contrastive learning paradigm, they can learn information from each other to enrich the action features by maximizing the mutual information between the two types of action representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Moreover, to further improve the performance of our model, we introduce Cyclical Focal Loss (CFL) instead of Cross-Entropy loss as our learning objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In contrast to the Cross-Entropy loss, the CFL can focus on confident training samples in early training epochs of our model, with an increasing focus on hard samples during the middle epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In summary, the main contributions of our work can be concluded as follows: We propose a novel Contrastive GCN-Transformer Net- work (ConGT), which can capture the relationships be- tween arbitrary joints in intra- and inter- frames more accurately while maintaining the topology structure of human skeleton graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We propose a Spatial-Temporal Graph Convolution stream (STG) with an adaptive graph strategy and a Spatial-Temporal Transformer stream (STT) to learn the action representations containing local and global joints relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We introduce the contrastive learning paradigm to inte- grate the information of two types of action representa- tions by maximizing mutual information between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We introduce the Cyclical Focal Loss (CFL) as the learning objective of our network to improve the action accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Action Recognition In this section, we will briefly review the related works about the three fields on skeleton-based action recognition: Convolutional Neural Networks (CNNs) based methods, Re- current Neural Networks (RNNs) based methods and Graph Convolutional Networks (GCNs) based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 1) CNN-based Methods: In the early work, the mainstream network is based on CNN and RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' CNN is mainly used to process 2D images, which can easily learn the high- level semantic features of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Thus, most CNN-based methods generally encode skeleton features to 2D pseudo images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In [4], each skeleton sequence was transformed into three clips which were correspond to every channel of the cylindrical coordinates of the skeleton sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then the frames of the clips are jointly processed by a multi-task learning network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [5] encoded the spatio-temporal information carried in skeleton sequence as joint trajectory maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [14] proposed an end-to-end hierarchical co- occurrence network to learn the con-occurrence feature with a hierarchical methodology, where different levels of contextual information is aggregated gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' They used the 3D coor- dinates of joints to generate several 2D images that are sent into the pretrained VGG-19 for action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Although the spatial features can be reserved in these pseudo images, the motion information contained in actions is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' To solve this problem, Rong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [15] used geometric algebra to learn the shape-motion representations and applied the multi- stream CNN models to fuse the complementary shape-motion representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' But 2D CNNs have a weak ability to capture the temporal and spatial features in the human skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Later, Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [16] proposed C3D network instead of 2D CNNs, which uses a heat map to denote the spatial correlations of joints and uses a stack to present the temporal sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Although this approach has yielded good results, it still ignores the motion correlation of the skeleton data which is a common challenge of CNN-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 2) RNN-based Methods: Recurrent neural networks (RNNs) [17] can process sequence data with variable lengths because its cellular states can determine which temporal states should be left and which should be forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Therefore, it has more advantages in processing temporal sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In the field of action recognition, RNN-based methods represent the skeleton data as a vector sequence that contains the location information of all joints in one frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [6] divided the human skeleton into five parts and fed them to five hierarchical RNN networks separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In [7], Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' introduced a new gating mechanism into Long Short Term Memory(LSTM) network to handle the noise and occlusion in 3D skeleton data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [18] proposed ensemble Temporal Sliding LSTM (TS-LSTM) networks containing short-term, medium-term and long-term TS-LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' They focused on the temporal correlation of various human body parts but ignored the spatial structure of the skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In most of the actions, the variation range of action in the space domain is larger than that in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' So, researchers have also been trying to design some RNN-based networks to process spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For example, in [19], Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' proposed a global-aware attention LSTM to make use of the global contextual information, which can selectively focus on the informative joints in each frame of the skeleton sequence and further enhance attention to spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' However, how to perceive the spatial correlations of the human skeleton remains a burning challenge for RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 3 3) GCN-based Methods: Because the topology of skeleton data is encoded in the form of graphs rather than two- dimensional grids or vector sequences, CNN-based methods or RNN-based methods may not be the optimal choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Recently, Graph Convolution Networks (GCNs) have achieved remark- able results in many works based on graph structure data, which can be divided into two types: spatial GCNs [7], [20], [21] and spectral GCNs [22]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For spectral GCNs, the input graphs are first transformed into the spectral domains and then operated by means of Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Spatial GCNs are applied directly to the nodes of the graph and their neighbors, which are more similar to the traditional convolution neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Our work follows the spatial GCN methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Spatial-Temporal Graph Convolutional Networks (ST-GCN) [8] is the pioneering method to model the skeleton data, which breaks the limitations that the previous method can- not effectively extract spatial and temporal features at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' ST-GCN models the joint connection and extracts correlated features as a spatio-temporal graph, where the graph convolution operates on the spatial features, and the 2D convolution operates on the temporal motion correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Recently, many works also adopt the same strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [10] combined the actional links and structural links into a generalized skeleton graph and used the actional-structural graph convolution and temporal convolution to learn the spatial-temporal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In [11], Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' proposed a spatio- temporal graph routing scheme to adaptively learn the high- order connectivity relationships for physically-apart skeleton joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Specifically, spatial graph routing aims at spatial rela- tionships based on sub-group clustering, while the temporal graph routing explores the temporal correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [9] proposed a two-stream adaptive graph convolutional network to make the value of the adjacency matrix be variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' With the adaptive strategy and the two-stream pattern, this method can model both human joints features and human bones features simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' DGNN [25] leveraged an alternating spatial aggregation scheme to update the joint and bone features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [26] proposed a disentangling and unifying graph convolutional network, including a simple disentangled multi- scale graph convolution and G3D module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The former is used to disentangle the importance of nodes in different neighborhoods, which can model the long-range relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The latter is presented to directly propagate the information across the spatial-temporal graph by leveraging the cross- spacetime edges as skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Transformer in Computer Vision Transformer [27] was proposed for the Natural Language Processing tasks to make up for the shortcoming of the RNN methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The great contribution of transformer is the self-attention, which can dynamically focus on the global context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Alexey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [28] applied a pure vision transformer to sequences of image patches, which has achieved excellent performance on the image classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In the object detection field, Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [29] proposed detection transformer reasoning about the relations of the objects and the global image context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [30] proposed Max- DeepLab for semantic segmentation, which directly predicts class-labelled mask with a mask transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [31] proposed a masked transformer for video understanding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For skeleton-based action recognition, Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [12] proposed a pure transformer network to model the correlation between joints without using the traditional skeleton graph represen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' They designed a decoupled spatial-temporal attention network to calculate the attention score between each pair of joints without knowing their positions or mutual connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Similarly, Plizzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [13] proposed a spatial and temporal transformer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The spatial self-attention module is used to capture the intra-frame correlations between human joints, while the temporal self-attention module is used to model the inter-frame relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' However, these methods have a common problem that they ignore the inherent topology of the human skeleton and overestimate the correlation between some joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In this way, the relationship that does not exist between some joints in some actions will also be forced to be imposed through the calculation of attention score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For example, in the action of “sitting down”, it is not necessary to capture the relationship between the left hand and the right hand that will bring trouble for the model to recognize actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In our work, we use the transformer structure to enhance the ability of the GCN to capture relationships between joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Different from the original transformer and its variants, the position encoding is not used in our work because of the topological invariance of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Self-Supervised Learning The intention of self-supervised learning (SSL) is to learn the internal structures of data and the feature representations from the unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In [32], it has been verified that self-supervised pre-training also can bring some assistance for supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' SSL was first used in visual representation [33], and now it has a number of applications in computer vision fields [34], [35], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' It is usually achieved by pretext task which is a hot topic of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In [37], they predicted the arrangement of multiple shuffled image patches by using SSL to learn the spatial relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [38] proposed a simple framework for contrastive learning of visual represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' They force the feature representation between positive samples to be more similar than those between negative ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For the sequential data, some methods [39], [40] learn the temporal features by predicting the sequential order of sampled frames or clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [41] proposed a video representation method via a prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For skeleton-based tasks, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [42] integrated prediction task, recognition task, and contrastive learning paradigm to learn skeleton features from different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [43] explored an unsupervised representation learning approach to compactly encode long- term global motion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [44] proposed an un- supervised encoder-decoder recurrent neural network to cluster similar movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [45] proposed an unsupervised framework based on encoder-decoder structure to extract more discriminative temporal features and explored the inherent action similarity within the action encoding by clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [46] utilized a variety of data enhancement strategies on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='unlabeled data to obtain the action representations with the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='ST&TT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='AGCN&TCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Contrastive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Softmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Temporal Convolutional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Spatial Transformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Temporal Transformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Multi-Head Self- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='\uf0c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Adaptive GCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Skeleton sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='STG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='STT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 1: The overall architecture of the proposed ConGT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The skeleton sequence is first fed into two streams, where the spatial- temporal GCN stream (STG) processes the input graph through the adaptive GCN module (AGCN) and temporal convolutional module (TCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The spatial-temporal transformer stream (STT) operates on the input graph with spatial transformer (ST) and temporal transformer (TT) modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The ST and TT modules have the same structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then the contrastive learning maximizes the mutual information across the two streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Finally, the classifier is added for action classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' And in the test phase, we choose the prediction result of the output feature of STG as the final result of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' contrastive learning paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [47] proposed a cross- view contrastive learning model by leveraging multi-view complementary supervision signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [48] proposed the contrast-reconstruction representation learning network to capture postures and motion dynamics simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In [49], Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' utilized the abundant information mining strategy to make better use of the movement patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In [50], [51], it is suggested that contrasting congruent and incongruent views of graphs with mutual information maximization can help encode rich representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Inspired by them, we also integrate con- trastive learning into the training of our network to enhance graph modelling by maximizing mutual information between two types of action representations and improve the accuracy of action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' METHOD In this section, we first present an overview of Contrastive GCN-Transformer Network (ConGT) and describe the details of each component of the ConGT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then, we show the process of enhancing ConGT with contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Finally, we depict the details of Cyclical Focal Loss (CFL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' ConGT Our proposed ConGT includes two streams: Spatial- Temporal Graph Convolution stream (STG) and Spatial- Temporal Transformer stream (STT), as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Specifically, STG is used to obtain the action features based on the topology of the human skeleton graph, consisting of adaptive graph convolutional network module (AGCN) and temporal convolutional network module (TCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The AGCN learns the topology of the graph for different layers and skeleton samples while the TCN models the temporal connec- tions between adjacent frames in time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Likewise, the STT contains spatial transformer module and temporal transformer module, which are primarily responsible for ac- curately capturing the relationships between arbitrary joints in the intra- and inter- frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Both streams can output action representations, but the generated action representations have different characteristics and each knows little information of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For this reason, we introduce the contrastive learning paradigm into ConGT to enforce the two streams learn more distinctive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Through the contrastive learning, we can maximize the interactive information from the representations learned by these two streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In this end, we can obtain the predicted class for the input skeleton graph with the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Spatial-Temporal Graph Convolution Stream Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For skeleton-based action recognition, the human action is represented as a sequence of skeleton frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In a frame, each skeleton is expressed as a graph G = (V, E), in which V = {vi|(i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' N)} denotes the collection of vertices representing N human joints and E = {ei,j|(vi, vj)} denotes the collection of edges representing the human bones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Formally, the adjacent matrix A ∈ RN×N is used to describe the structure of skeleton graph, where the value of Aij is 0 or 1 indicating whether an edge exists between joints vi and vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' And the feature tensor X ∈ RC×T ×N denotes the sequence of skeleton frames, where C represents the coordinate dimension, T denotes the total number of skeleton frames contained in the sequence, and N is the total number of human joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Graph Convolutional Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The process of updating a graph by GCN is to aggregate nodes information with 5 \uf0c4 \uf0c5 \uf0c4 Spatial Adaptive GCN Adjacency graph ෩𝑨 (CT,N) (N,CT) (N,N) (N,N) Learnable graph L Embedded graph E Skeleton graph Adaptive skeleton tensor Embedding function SoftMax layer (C,T,N) 1×1 conv Parameterized \uf0c4 Elements multiplication \uf0c5 Elements summation (C,T,N) (N,N) Skeleton tensor (N,N) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 2: The schematic diagram of the spatial adaptive graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The input consists of an adjacency matrix of the skeleton graph and a skeleton tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The learnable matrix is a parameterized matrix and is trained with the whole model, and the skeleton tensor passes through the embedding function to obtain the embedded matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then the three types of matrices are added together and multiplied with the skeleton tensor to get the output adaptive skeleton tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' edge information to generate a new graph representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For skeleton-based action recognition, the action representations can be obtained through GCN operating on the adjacent matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Let H denote the action representation which is initialized to H(0) = Xin = � X1 in, X2 in, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' , Xn in � ∈ RC×T ×N, where Xin is the graph representation of input skeleton frame, T and N are the total number of the frames and human joints, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then the graph convolution operation can be expressed as: H(l+1) t = D− 1 2 ¯AD 1 2 Hl tωl (1) where ¯A = A + I, I is the identity matrix, D is the degree matrix of ¯A, Hl t denotes the skeleton tensor in the t-th frame at the l-th layer, and ωl ∈ RCl×C(l+1) is a learnable weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In the implementation of GCN, the higher-order polynomial of the adjacency matrix A is applied to aggregate the skeleton tensor H to get the high-order neighbor information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 1 can be rewritten as: H(l+1) t = σ � Kv � k=0 � ˜AkHl t �� ⊙ ωl k (2) where Kv denotes kernel size in the spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' ˜A represents the normalized adjacency matrix, while �Ak repre- sents k-power of the normalized adjacency matrix which can represent the relationship between adjacent nodes of order k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' ωl k ∈ RCl×C(l+1) is a learnable weight matrix, ⊙ denotes the dot product, and σ(·) denotes the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Adaptive graph strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In the traditional graph convo- lution, a common way to calculate graph topological rela- tionships is using the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Each element of the adjacency matrix A has a value of 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' When A(i, j) = 0, it means there is no connection between the joints vi and vj, otherwise adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Note that even if after several multipli- cation operations between the adjacency matrix and skeleton tensor, A(i, j) = 0 will still exist, indicating that there is no relationship between joints vi and vj during the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' This will ignore the connection between some joints and further affect the recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For instance, the two hands in action “clapping hands” are not directly linked, but the interaction between them is actually very useful for recognizing this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' To this end, we employ the adaptive graph strategy [9] to adaptively reflect the relationships of joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 2, the adaptive graph strategy is imple- mented by adding the adjacency matrix, the learnable matrix, and the embedded matrix together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then the graph convolution integrated with the adaptive graph strategy can be formulated as: H(l+1) t = σ � Kv � k=0 �� �A + L + E �k Hl t �� ⊙ ωl k (3) where the normalized adjacency matrix �A ∈ RN×N denotes the original topology of the skeleton graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' L ∈ RN×N is a learnable matrix which is initialized with the adjacent matrix A to accelerate the convergence of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' As we perform actions, our joints move in groups, but the importance of each joint is different in different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Therefore, we need to define an importance weight to scale the contribution of node features to neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The learned matrix L is an attention map that indicates the importance of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In this way, the data-driven dependencies between nonadjacent joints vi and vj can be generated as the depth of the network increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The embedded matrix E ∈ RN×N learns an individual graph for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In these individual graphs, the weights of edges are calculated by measuring the similarity of graph nodes which can be obtained by the normalized embedded Gaussian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The elements of the embedded 6 matrix can represent the strength of the dependency between two joints, and the value of them ranges from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The embedded matrix E is described mathematically as follows: E = softmax �� θ1 � f (l)� δ(l) 1 �T � θ2 � f (l)� δ(l) 2 �� (4) where θ represents the embedding function that can map any two joint vectors to the same vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' δ denotes the parameters of the embedding function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' f (l) ∈ RC×T ×N denotes the feature tensor of the l-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' And it will be converted into two intermediate embeddings by embedding functions θ1 and θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then the two embeddings are multiplied to get the embedded graph E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Temporal convolution Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Unlike the spatial topol- ogy, the temporal topology of joints is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Thus, temporal relationships are usually captured by using ordinary convolu- tion operation rather than graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In general, the spatial-temporal convolution on the skeleton graph can be formulated as: H(l+1) = TCN � AGCN � H(l)�� (5) where TCN(·) is a temporal convolution with a kernel size KT × 1, and KT = 9 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' AGCN(·) denotes the adaptive graph convolution network, and H(l) denotes the action representation in the l-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Spatial-Temporal Transformer Stream Although the adaptive graph strategy can make up for the defect that A(i, j) = 0 cannot be replaced in multiplication, the long-distance connections are still easily underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Moreover, with the stacking of layers, the risk of over- smoothing of graph convolution will increase, leading to poor ability to accurately capture the relationships between joints which are depicted by the action representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For these reasons, we propose the spatial-temporal transformer stream (STT) to enhance the dependence of local neighboring joints and further effectively capture long-distance relationships in both spatial and temporal dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The work flow of the spatial-temporal transformer is de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In space, for each node nt i of the skeleton graph in the frame t, a query vector qt i, a key vector kt i, and a value vector vt i can be calculated by the trainable linear trans- formations with three parameters matrices Wq ∈ RCin×dq, Wk ∈ RCin×dk, and Wv ∈ RCin×dv, which are shared by all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' With these vectors, we use the multi-head attention to calculate the attention score to weight the value vector vt i that corresponds to the query vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In time dimension, a node in one frame will pay attention to nodes representing the same joint in other frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Similar to the spatial transformer, in temporal transformer, the first step is also to calculate a query vector qfj, a key vector kfj, and a value vector vfj for each node nfj in different frame f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then, we compute the attention score with multi-head attention, which is used to determine how much attention one node is paid to other nodes that represent the same joints along the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' As the core component of transformer, the multi-head attention is detailed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The input consists of the query vector q, the key vector k, and the value vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then, the dot products of the query with all keys are calculated to get the attention scores between all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In order to prevent the vanishing gradient in the process of backpropagation, we scale the attention scores by 1 √dk , where dk denotes the dimension of query vector and key vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Subsequently, we apply the softmax function on the attention scores to weight the value vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Finally, the attention-enhanced value vectors of nodes are aggregated in a summation manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The above process is repeated Nh times with different queries, keys and values, where Nh = 8 in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then the results of the Nh attention heads are concatenated together to constitute the output representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Formally, this process is expressed as: O(l) i = Nh � 1 � � � j∈Ni softmax � S(l) i,j √dk � v(l) j � � (6) S(l) i,j = q(l) i k(l) j (7) where O(l) i denotes the output of the multi-head attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' �Nh 1 (·) denotes the concatenation of Nh heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' q(l) i , k(l) i , and v(l) i denote the query vector, the key vector, and the value vector of node i in the frame t at the l-th layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' S(l) i,j is the attention score between nodes i and j, which is treated as a weight when aggregating the values of different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Next, the output of the multi-head attention passes through a batch normalization layer and a Multi-Layer Perceptron (MLP) block to obtain the input tensor of the next layer or the final output of STT in the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The STT can be formulated as: H(l+1) = TT � ST � H(l)�� (8) where TT(·) and ST(·) denote the temporal transformer module and the spatial transformer module, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' H(l) denotes the generated action representation in the l-th trans- former layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Enhance ConGT with Contrastive Learning The STG can learn the topology of the graph for different GCN layers and skeleton samples in an end-to-end manner, obtaining the action representations based on the topology of the human skeleton graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' However, the implicit relations between nonadjacent joints are ignored, such as the connection between hand and head during touching head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Meanwhile, the temporal connection between remote frames is underestimated due to the limitation of temporal convolution kernel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' To solve them, we bridge GCN and attention module in a parallel way with contrastive learning to enrich the action features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We design the STT based on transformer to capture the long-distance correlations between each pair of joints in both spatial and temporal dimensions without knowing their positions or mutual connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Since each stream encodes a representation that only depicts either the topology of human skeleton graph or the long-distance correlations between each pair of joints, the two types of action representations know little about each other but may mutually complement each 7 Attention Score 𝑞𝑖 𝑡 𝑘𝑖 𝑡 𝑣𝑖 𝑡 Scaling Dot Products 𝑂𝑡,𝑖 𝑙 𝑁ℎ heads S Summation Dot Products V C S Softmax C Concatenation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 3: The detail of multi-head attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Specifically, they can be the ground-truth of each other for contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then, by maximizing the mutual information between the action representations learned via the two streams through contrastive learning, our network can learn more distinctive information to enhance recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We adopt InfoNCE as the contrastive learning objective, which is defined as: Lcon = − log σ � fd � ag i , at i �� − log σ � 1 − fd � ˜ag i , at i �� (9) where ag i and at i denote the action representations obtained through the STG and STT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' �ag i represents the neg- ative samples obtained by corrupting ag i with both row-wise and column-wise shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' fd(·) is a discriminant function: Rd×Rd → R, which scores the agreement between the two in- put vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The contrastive learning objective is to maximize the agreement of two different types of representations, while minimizing the agreement with other negative representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In this way, both STG and STT can acquire information from each other and further enrich the action features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In the end, we combine the contrastive learning task with the action recognition task to form multi-task learning, where contrastive learning is the auxiliary task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We optimize our model by means of joint learning that is defined as: L = LCF L + βLcon (10) where LCF L is cyclical focal loss that is the learning objective of action recognition task and β is a hyper-parameter used to control the magnitude of the contrastive task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Cyclical Focal Loss We define the learning objective of action recognition task as the cyclical focal loss which is formed of the focal loss and the general cyclical training principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Cyclical focal loss can focus on confident predictions in the early epochs of the network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' As the number of training epochs increases, it will focus more on misclassified samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In the following, we will describe the details of the cyclical focal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Focal Loss is mostly used for binary classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' It modifies cross-entropy softmax loss to reduce the weight of easily classified samples so that the model can focus more on samples that are difficult to classify during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' It is defined as: Llc = −(1 − pt)γlog(pt) (11) where pt denotes the softmax probabilities, and γ ≥ 0 is a tunable hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For the confident training samples, the value of pt tends to be 1, and the weight (1 − pt)γ for the loss will drive the loss to zero faster than that for cross- entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Although the focus loss is beneficial in tasks with unbalanced class data, it often affects performance when the dataset is more balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Therefore, in most applications, the focus loss is not the best choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Combining the focus loss with the general cyclical training principle, the cyclical focal loss [52] is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In [53], the general cyclical training of a neural network is considered as a combination of curriculum learning in the early epochs with fine-tuning at the end of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Specifically, the easy and confident training samples are used at the start and end stages of network training, and the hard training samples are processed at the middle stage of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For this purpose, Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' [52] proposed a new loss: Lhc = − (1 + pt)γhc log (pt) (12) where pt denotes the softmax probabilities, and γhc ≥ 0 is a tunable hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In this manner, the loss can pay more attention to the confident training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' And the hard training samples can be heavily weighted via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Therefore, the cyclical focal loss can be accomplished by combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 11 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 12 in a reasonable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In [52], they used a linear schedule and defined a parameter ξ to 8 combine them that varies with the training epoch as: ξ = � 1 − fc epi epn if fc × epi ≤ epn � fc epi epn − 1 � / (fc − 1) otherwise (13) where epi denotes the number of current training epochs and epn is the total number of training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' fc denotes the cyclical factor that provides adaptability for the cyclical schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 11, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 12 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 13, the cyclical focal loss can be defined as: CFL(p, y) = ξLhc + (1 − ξ)Llc (14) In our experiments, we keep the values of the hyper- parameters consistent with those in the original work, where γlc = 2, γhc = 2, fc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' EXPERIMENTS To verify the effect of the proposed method, we conduct extensive experiments on three widely used datasets: NTU- RGBD 60 [54], NTU-RGBD 120 [55], and Northwestern- UCLA [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In this section, we first give the description of the three datasets in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Next, we will describe the experiment settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then, the comparisons between the proposed method and several state-of-the-art methods are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Finally, we investigate the contributions of each component in the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Datasets NTU-RGBD 60 (NTU-60): NTU-60 is one of the widely used datasets for skeleton-based action recognition tasks, which contains 56,880 samples of 60 different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The dataset is captured by a Microsoft Kinect V2 camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The skeleton data is formed of the 3D joint locations (X, Y, Z) of 25 joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In each video, there are no more than two persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The dataset is divided into Cross-View (X-View) Setting and Cross- Subject (X-Sub) Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In X-View, the actions are captured by three cameras with the same height in the vertical direction and different angles −45◦, 0◦, 45◦ in the horizontal direction, where the training set contains 37,920 samples and the testing set includes 18,960 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In X-Sub, the subjects of the training set and the testing set are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The training set contains 40,320 videos, and the testing set contains 16,560 videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' NTU-RGBD 120 (NTU-120): NTU-120 is an extension of NTU-60 with additional 57,367 skeleton sequences, which totally contains 114,480 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The action categories in this dataset can be divided into three major groups: 82 daily actions (eating, sitting down, standing up, etc), 12 health- related actions (falling down, blowing nose, etc), and 26 mutual actions (hugging, shaking hands, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Similar to NTU- 60, NTU-120 also has two benchmarks: 1) cross-subject (X- Sub), 2) cross-setup (X-Set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In cross-subject, the 106 subjects are equally split into the training set and the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In cross-setup, the samples whose ID is even belongs to the training set, while the samples whose ID is odd are treated as the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Northwestern-UCLA (NW-UCLA): NW-UCLA [56] con- tains 1,494 video clips covering 10 categories and is captured by three Kinect cameras simultaneously from a variety of viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Each action sample is performed by 10 different actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Following [56], we adopt the same protocol to divide the dataset that the training set is composed of the samples of the first and second viewpoints, while the testing set is made up of samples of the third viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Implementation Details The implementation of our model is conducted on the PyTorch framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We use the stochastic gradient descent (SGD) as the optimizer to train our network, where the momentum is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='9 and the weight decay is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For NTU-60 and NTU-120, we set the number of training epochs to 75 and the initial learning rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The learning rate decays at the 45th and the 55th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For the NW-UCLA dataset, the number of training epochs is set to 65, and the learning rate decays at the 50th epoch, which is initially set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In contrastive learning, the hyper-parameter β is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='01 for NTU-60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='05 for NTU-120, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1 for NW-UCLA, which is used to control the magnitude of the contrastive learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For the cyclical focal loss, we set the cyclical factor fc to 4 and both hyper-parameters γlc and γhc to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Comparison Against the State of the Art To verify the effectiveness of our network, we compare our prediction accuracy with the current state-of-the-art methods on NTU-60, NTU-120 and NW-UCLA datasets under two evaluation protocols, including linear evaluation protocol and fine-tuning protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Linear Evaluation Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' With this protocol, we eval- uate the quality of the representations learned by our method with training a linear classifier (including a fully-connected layer and a softmax layer) and freezing the parameters of the other part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We report the comparison results in Table I, Table II, and Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' TABLE I: Performance comparison on the NTU-60 dataset with linear evaluation protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Methods X-View(%) X-Sub(%) LongT GAN [43] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1 MS2L [42] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6 PCRP [45] 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='9 AS-CAL [46] 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5 CRRL [48] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='8 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6 3s-CrossSCLR [47] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='8 3s-AimCLR [49] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='9 BRL [57] 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='8 ConGT 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 Table I shows the comparisons with previous related meth- ods on NTU-60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Our method achieves the accuracy of 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='0% on X-View and 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2% on X-Sub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Compared with LongT GAN [43], MS2L [42], PCRP [45], and AS-CAL [46], our 9 method achieves an overwhelming performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' CRRL [48] is a contrast-reconstruction representation learning network that can simultaneously capture postures and motion dynamics for unsupervised skeleton-based action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' By contrast, our method performs 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2% better on X-View and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6% on X-Sub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 3s-CrosSCLR [47] also attains superior performance due to its multi-view strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Our ConGT achieves excellent performance that outperforms it 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6% on X-View and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='4% on X-Sub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 3s-AimCLR [49] is a contrastive learning framework with utilizing abundant information mining for self-supervised action representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Compared with it, the performance of our method is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2% and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='3% higher on X-View and X- Sub under Top-1 recognition accuracy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Compared to other unsupervised methods, BRL [57] gains remarkable results by using the data augmentation and multi-viewpoint sampling strategies, which achieves the accuracy of 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2% on X-View and 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='8% on X-Sub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Our method still works better than it on X-View.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' TABLE II: Performance comparison on the NTU-120 dataset with linear evaluation protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Methods X-Set(%) X-Sub(%) LongT GAN [43] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6 PCRP [45] 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='7 AS-CAL [46] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6 CRRL [48] 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 3s-CrossSCLR [47] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='9 3s-AimCLR [49] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 ISC [58] 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='9 BRL [57] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1 ConGT 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6 In Table II, we conduct the comparative experiment on NTU-120 dataset on both X-Sub and X-Set benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We also follow the standard practice in the literature, reporting the top-1 classification accuracies on both benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The competitive results in Table II verify the superiority of our proposed method over all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' TABLE III: Performance comparison on the NW-UCLA dataset with linear evaluation protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Methods Accuracy(%) LongT GAN [43] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='3 MS2L [42] 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='8 CRRL [48] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='8 ConGT 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='3 As shown in Table III, the proposed ConGT achieves the best accuracy of 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='3% on the NW-UCLA dataset, surpassing the previous state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The NW-UCLA contains ten categories of actions: pick up with on hand, pick up with two hands, drop trash, walk around, sit down, stand up, donning, doffing, throw, and carry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For each specific action, we use the boxplots to show the training accuracy of every 5 epochs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 4, where these ten classes are denoted by numbers 1-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Class 20 40 60 80 100 Accuracy(%) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 4: Different color boxes indicate the accuracy range of several categories, the black line inside each box represents the median value, boxes limits include interquartile ranges from 25% to 75% of samples, upper and lower whiskers are computed as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5 times the distance of upper and lower limits of the box, and all values outside the whiskers are considered as outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' TABLE IV: Performance comparison on the NTU-60 and NTU-120 dataset with fine-tuning protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Methods NTU-60 NTU-120 X-View X-Sub X-Set X-Sub SkeletonCLR [47] 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='9 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6 AimCLR [49] 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='4 ConGT 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='4 Fine-tuning Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Following [47], we first pre-train STG, STT and contrastive learning and then append a linear classifier to retrain the whole model, where the parameters of each layer in our network are updated with the backpropaga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We compare our model with the state-of-the-art methods in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' To make a fair comparison with SkeletonCLR [47] and AimCLR [49], we only use the bone data to compare the finetuned results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' As shown in Table IV, our ConGT defeats them both on NTU-60 and NTU-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Specifically, on X-View and X-Sub of NTU-60, our method surpasses SkeletonCLR by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='7% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='4%, respectively, and outperforms AimCLR by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='4% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' On NTU-120, compared to SkeletonCLR, the improvements reach 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='8% on X-Set and X-Sub, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Compared to AimCLR, our method surpasses it by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='8% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='0% on X-Set and X-Sub, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The results demonstrate that our model with the contrastive learning paradigm can effectively learn rich action representations of human actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Ablation Study In this section, we design ablation experiments to investigate the effectiveness of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We first validate the effectiveness of each component of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' And we 10 demonstrate that the existence of over-smoothing problem during the accumulation of GCN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Then we test the influence of the hyper-parameter β that controls the magnitude of contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Finally, we verify the effectiveness of the cyclical focal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' TABLE V: The comparison performance of ConGT with different parts on X-View of NTU-60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Subnet STG STT CL Accuracy(%) N-STG ✓ 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6 N-STT ✓ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5 ST-GT ✓ ✓ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='3 ST-MGT ✓ ✓ 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 ConGT ✓ ✓ ✓ 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='4 1) Impact of Each Component in ConGT: As three primary components of our proposed ConGT, the Spatial-Temporal Graph Convolution stream (STG), Spatial-Temporal Trans- former stream (STT), and Contrastive Learning (CL) are also the main contributions in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' To evaluate the effectiveness of STG, STT and CL, we design four subnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' N-STG: Only training STG to show the results of using GCN alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' N-STT: Only training STT to display the results of using transformer alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' ST-GT: We remove the contrastive learning part and add the representations learned by the STG and STT to demonstrate the effectiveness of contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' ST-MGT: We replace the InfoNCE loss with the MSE loss to illustrate that the contrastive learning plays a crucial role in combining two different types of action representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' On the four subnets, we conduct experiments on X-View of NTU-60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The results obtained by these baselines are depicted in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' It can intuitively see that the recognition accuracy can reach 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6% when only using the STG, while the recog- nition accuracy is only 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5% when only using the STT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We speculate the reason is that the transformer treats each node as a separate unit and regards the human skeleton as a complete graph with connections built between each joint and the rest joints, resulting in less variation between different movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' To verify this claim, we visualize the action representation learned by STT in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 5, where all the categories are mixed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' This adds to the evidence that it is unreasonable to treat the human skeleton as a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Moreover, we adopt two different methods to demonstrate the effectiveness of contrastive learning in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In ST- GT, we add the action representations output by STG and STT together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In this way, the accuracy is reduced by 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1% compared to ConGT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Furthermore, we replace the InfoNCE loss with MSE loss to evaluate the effect of contrastive learning in ST-MGT and the final recognition accuracy has a decline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 6, we depict the results of the ST-GT, ST-MGT, and ConGT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The blue part denotes the recognition accuracy of ST-GT, the orange part represents the improvement of the ST-MGT using the MSE loss, and the green part shows the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 5: The visualization of the action representation learned by STT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In order to better show the distribution of action representation of each category, we select the top 10 classes of the X-View of NTU-60 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Each color denotes an action class and each point represents a skeleton sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 10 20 30 40 50 60 70 test Epoch 0 20 40 60 80 Accuracy(%) ST-GT ST-MGT ST-ConGT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 6: The influence of the contrastive learning paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The blue part denotes the recognition accuracy of ST-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The orange part represents the improvement of the recognition ac- curacy with the MSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The green part shows the superiority of using contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' superiority of using contrastive learning in ConGT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We can intuitively see that the training accuracy of ST-GT and ST- MGT are consistently lower than ConGT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Therefore, it can be illustrated that the contrastive learning plays a crucial role in fusing long-distance relationships into the topology structure of the human skeleton graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 2) The Effect of Adaptive Graph Strategy in STG: We demonstrate the influence of AGCN by comparing STG with ST-GCN in the supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' For a fair comparison, we set the same number of GCN layers in STG as that in ST- GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In Table VI, we can see that the recognition accuracy of the STG outperforms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='9% that of ST-GCN, which indicates that the adaptive graph strategy contributes to improving the 1 2 3 40 4 5 6 7 8 6 10 20 0 20 40 40 20 0 20 4011 (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 7: The t-SNE visualization of action representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Each point represents a skeleton sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We show the first 10 action classes of the X-View of NTU-60 dataset, indicated by colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The STG with 6 GCN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The STG with 9 GCN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' (c) The STG with 12 GCN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5 1 2 5 76 78 80 82 84 86 88 90 92 Accuracy(\\%) NTU60-xview NTU60-xsub (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5 1 2 5 76 78 80 82 84 86 88 90 92 Accuracy(\\%) NTU120-xsub NTU120-xset (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5 1 2 5 76 78 80 82 84 86 88 90 92 Accuracy(\\%) NW-UCLA (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 8: The influence of the magnitude of contrastive learning on (a) NTU-60, (b) NTU-120, and (c) NW-UCLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' TABLE VI: Comparison of the performance (accuracy (%)) on the X-View setting of the NTU-60 dataset when the STG with or without the adjacency matrix A, the learnable matrix L, and the embedding matrix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' wo/A denotes without A, wo/L denotes without L, and wo/E denotes without E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Methods Accuracy(%) ST-GCN 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='3 STG 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 STG wo/A 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5 STG wo/L 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='3 STG wo/E 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='7 accuracy of action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Furthermore, to evaluate the necessity of the three graphs in AGCN, we conduct an ablation study on the three graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We manually delete one of the three types of graphs and show their performance in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We find that taking away any one of the three graphs will affect the final recognition result negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' When all three graphs are simultaneously enabled, our model can achieve the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' This indicates that the adaptive graph strategy is conducive to increasing the accuracy of action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 3) The Impact of the Number of GCN Layers in STG: In addition, to demonstrate that the over-smoothing problem occurs during the accumulation of GCN layers, we compare TABLE VII: Recognition accuracies obtained by STG contain- ing 6, 9, and 12 GCN layers on X-View setting of NTU-60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Layers Accuracy(%) 6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='6 9 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2 12 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1 the recognition performance of STG containing 6, 9, and 12 GCN layers on the X-View setting of NTU-60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In Table VII, we can see that the recognition accuracy has a significant decline when the GCN layer number is increased to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We also apply t-SNE [59] to show the embedding distribution of these three options in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' From the visual results, we can find that with the increase of network layers, the action representations of classes 1, 2, 3, and 5 (circled by the red ellipse) tend to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' It also reveals that with the increase of the number of GCN layers, the probability of the over-smoothing problem also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 4) The Impact of the Contrastive Learning Hyper- parameter: To justify the impact of the hyper-parameter β on controlling the magnitude of contrastive learning, we examine the performance of ConGT with a set of representative β values {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='5, 1, 2, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' The performance results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' As we can see, our model performs best on NTU-60 when β is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' While 2 m 40 4 5 6 7 8 9 20 10 20 40 40 20 0 20 4040 2 4 6 20 10 20 40 60 40 20 0 20 40 6060 2 40 10 20 - 0 20 40 60 40 20 0 20 4012 TABLE VIII: Comparison of the top-1 test recognition ac- curacies for Cross-Entropy Loss and Cyclical Focal Loss on NTU-60 and NTU-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Datasets Cross-Entropy Loss Cyclical Focal Loss NTU-60 (X-View) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='08 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='59 NTU-60 (X-Sub) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='21 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='55 NTU-120 (X-Set) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='80 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='53 NTU-120 (X-Sub) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='82 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='36 for NTU-120, the best β is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' When β is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content='1, our method achieves the best accuracy on NW-UCLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In addition, it can be seen that when β becomes large, the performance of ConGT on both NTU-60, NTU-120, and NW-UCLA will decline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' We suspect it is due to that the gradient conflict between the action recognition task and the contrastive task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Therefore, it is necessary to select an appropriate β, when involving the contrastive learning paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 5) The Effectiveness of the Cyclical Focal Loss: In this section, we compare the recognition results obtained by using the cross-entropy loss and the cyclical focal loss on NTU-60 and NTU-120 in Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' It shows that when the model is trained with the cyclical focal loss, the test accuracy is consistently better than that using the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 Epoch 0 20 40 60 80 Accuracy(\\%) NTU60-xviewCross NTU60-xviewCFL NTU60-xsubCross NTU60-xsubCFL NTU120-xsubCross NTU120-xsubCFL NTU120-xsetCross NTU120-xsetCFL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 9: The accuracy curves of training ConGT using cross- entropy loss and cyclical focal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' 9 shows the training accuracy curves of training our network using the cross-entropy loss and the cyclical focal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Although the cross-entropy loss curve and the cyclical focal loss curve on the same dataset have strong similarities, it is notable that training with the cyclical focal loss provides a slightly faster learning convergence in the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Therefore, it can be confirmed that the cyclical focal loss better helps the learning in the early epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' CONCLUSION In this work, we design a novel Contrastive GCN- Transformer Network (ConGT), which can capture the rela- tionships between arbitrary joints in the intra- and inter- frames more accurately while maintaining the topology structure of human skeleton graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Specifically, the STG is designed to obtain action representations maintaining the topology struc- ture of the human skeleton graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' At the same time, the STT is used to acquire action representations containing the global relationships among joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Moreover, we introduce the contrastive learning paradigm, serving as an auxiliary task, to maximize the mutual information between the action representations learned via the two streams to improve the action recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In this manner, we can make up for the weak ability of GCN to capture long-distance features on the basis of maintaining the topology structure of the human skeleton graph and reduce the risk of network over-smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' In addition, we introduce the cyclical focal loss as the learning objective of our model, which places heavy weights on con- fident training samples in the first training epochs of a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Ablation studies have been performed in this work, which verify the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' Experiments on three publicly available datasets demonstrate the superiority of our proposed method over other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFJT4oBgHgl3EQfRSzM/content/2301.11495v1.pdf'} +page_content=' ACKNOWLEDGMENT 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--- /dev/null +++ b/idE1T4oBgHgl3EQffwRJ/content/tmp_files/2301.03221v1.pdf.txt @@ -0,0 +1,1149 @@ +Representing Matroids over the Reals is ∃R-complete +Eunjung Kim, Arnaud de Mesmay, Tillmann Miltzow +January 10, 2023 +Abstract +A matroid M is an ordered pair (E, I), where E is a finite set called the ground set and a collection +I ⊂ 2E called the independent sets which satisfy the conditions: (I1) ∅ ∈ I, (I2) I′ ⊂ I ∈ I implies +I′ ∈ I, and (I3) I1, I2 ∈ I and |I1| < |I2| implies that there is an e ∈ I2 such that I1 ∪ {e} ∈ I. The rank +rk(M) of a matroid M is the maximum size of an independent set. We say that a matroid M = (E, I) +is representable over the reals if there is a map ϕ : E → Rrk(M) such that I ∈ I if and only if ϕ(I) forms +a linearly independent set. +We study the problem of Matroid R-Representability over the reals. Given a matroid M, we +ask whether there is a set of points in the Euclidean space representing M. We show that Matroid +R-Representability is ∃R-complete, already for matroids of rank 3. +The complexity class ∃R can +be defined as the family of algorithmic problems that is polynomial-time equivalent to determining if a +multivariate polynomial with integer coefficients has a real root. +Our methods are similar to previous methods from the literature. Yet, the result itself was never +pointed out and there is no proof readily available in the language of computer science. +1 +Introduction +Many articles on matroids assume that the matroid is representable [40, 15, 17]. Representability either +heavily simplifies proofs and definitions or is even essential. We show that the question of representability +over the reals is as difficult as the existential theory of the reals, that is ∃R-complete. The complexity class +∃R can be defined as the family of algorithmic problems that is polynomial-time equivalent to determining +if a multivariate polynomial (with integer coefficients) has a real root, see below for an introduction and +overview of this complexity class. +Definitions. +Before we give a general definition of a matroid, we introduce vector matroids. Given a matrix +A over a field F, we can define the corresponding vector matroid M[A] = (E, I) as follows. The ground set E +of M[A] is formed by the columns of A and we say that a subset I ⊂ E is independent in M, i.e., I ∈ I, +if the columns are linearly independent over F. A set of elements of E which is not independent is said to +be dependent. Note that any set of columns containing a zero column is dependent. The independent sets +of a vector matroid satisfy three simple properties (see below). One way to look at matroids is to see them +as abstract set systems that have those three properties. A matroid M is an ordered pair (E, I), where E +is a finite set called the ground set and a collection I ⊂ 2E called the independent sets which satisfy the +conditions: +(I1) ∅ ∈ I, +(I2) I′ ⊂ I ∈ I implies I′ ∈ I, +(I3) I1, I2 ∈ I and |I1| < |I2| implies that there is an e ∈ I2 such that I1 ∪ {e} ∈ I. +The rank rk(M) of a matroid M is the maximum size of an independent set. +We say that a matroid +M = (E, I) is representable over F if there is a matrix A over F such that M = M[A]. Note that all columns +of A live in a subspace of dimension at most rk(M[A]). Therefore, we can assume without loss of generality +that the columns of A have dimension rk(M[A]). We refer to [50] for more background on matroids. +1 +arXiv:2301.03221v1 [cs.CC] 9 Jan 2023 + +Given a matroid M, if there exists a matrix A over R such that M = M[A], we say that M is representable +over the reals. +The algorithmic problem of Matroid R-Representability is to test whether a given +matroid is representable over the reals. Since we only discuss the real case in this article, we will sometimes +say representable as a shorthand. Note that we also need to specify how the matroid M is given. In the +literature on matroids, one has often an oracle such that one can ask the oracle for each set I whether I ∈ I. +We will deviate from this practice, as it might be unclear how to describe the oracle. Instead, we will just +list all sets in I explicitly. This does not blow up the description complexity too much in our case, as we +will deal mainly with constant rank matroids. +Geometric Interpretation. +If we have a representation A ∈ R3×n of a matroid M = M[A], we can scale +every column of A by a nonzero number and it stays a valid representation. This is also the case if we scale +by −1. Furthermore, if we rotate A (i.e., multiply it on the left with an orthogonal transformation) it still +stays a valid representation. Thus, in case we have a representable rank-3 matroid over R, we can assume +that there is a representation in which all nonzero vectors have their third coordinate equal to 1. In this +way, we can consider the columns of A as a point configuration in the plane with z = 1. The property of +three vectors being dependent translates to the corresponding three points to lie on a common line. +In this geometric interpretation, we could have considered any plane different from the one with z = 1, +as long as it does not cross the origin. +This would have yielded a different point configuration. +The +resulting transformation is called a projective transformation. It maps lines to lines, except for one line +that disappears, we say that it is sent to infinity. +Conversely, for any line in the plane, there exists a +projective transformation that sends it to infinity. Throughout this article, we think of representations of +rank-3 matroids via these point configurations, and thus we will slightly abuse language by calling such a +point configuration a representation. +A motivating example. +One of the standard examples to illustrate realizability is the so-called Fano +plane. It is the matroid on seven elements whose maximal independent sets are all the triples except +{1, 4, 7}, {1, 2, 3}, {1, 5, 6}, {3, 6, 7}, {2, 5, 7}, {3, 4, 5}, {2, 4, 6}. +This can be represented pictorially as in the figure below, where the lines and the circle denote the +dependencies. +1 +2 +3 +4 +5 +6 +7 +2 + +An immediate question that arises from this picture is whether a picture exists where the circle is not +used and the dependencies are all pictured by lines. This is equivalent to asking whether the Fano matroid +is representable over the reals. It is well-known not to be [50, Proposition 6.4.8]. The problem Matroid +R-Representability addresses the general question of deciding which matroids can be represented like +that, and our main result is that this problem is ∃R-complete. +Order Types. +We saw above that matroids are an abstraction to describe point collinearities in the plane. +I.e., if we have a rank 3 matroid then every dependent set corresponds to three collinear points. Now given a +set of points, we are often also interested in the orientation of each triple: either clockwise, counter-clockwise, +or collinear. +a +b +c +d +In this example, {a, b, c} is collinear and (a, b, d), (a, c, d), and (b, c, d) are oriented counter-clockwise. +This leads to the definition of (abstract) order types, which is a pair O = (E, χ). Again, E is a finite set +called the ground set. And +χ : +�E +3 +� +→ {−1, 0, 1} +is a function called a chirotope satisfying a few simple properties that are derived from the intuition given +above. We say that a point set P ⊂ R2 represents a given order type O = (E, χ), if P has for each element +e ∈ E a corresponding point e′ ∈ P. Furthermore, for each triple a, b, c ∈ E the corresponding points +a′, b′, c′ ∈ P are oriented according to χ({a, b, c}). Note that if we lift every point in P to the plane with +z = 1 as a subset of R3, then we get the following correspondence between (p′ +x, p′ +y, 1), (q′ +x, q′ +y, 1), (r′ +x, r′ +y, 1) +and the corresponding elements p, q, r ∈ E. +sign det +� +� +p′ +x +q′ +x +r′ +x +p′ +y +q′ +y +r′ +y +1 +1 +1 +� +� = χ(p, q, r). +Note that in this specific setup a realization of a rank 3 matroid and order types are closely related. +While matroids only determine the collinearities, the order type also determines the orientation of each +triple. We want to point out that every abstract order type can be represented by a pseudoline arrangement. +A pseudoline arrangement can be defined as a collection of x-monotone curves such that any pair of curves +intersect exactly once. The orientation of a triple of pseudolines is defined by the orientation of the triangle +that they form (a degenerate triangle corresponding to a zero orientation). +a +b +c +d +Note that this pseudoline arrangement corresponds to the order type example given above. Now, the +realizability of the order types is equivalent to the stretchability of pseudoline arrangements. That is +finding a line arrangement with the same combinatorics as the pseudoline arrangement. It is one of the +central theorems in the field of the existential theory of the reals that stretchability is ∃R-complete. We +will use many ideas of that proof for our main result. +We also want to point out that the notion of an order type can be easily generalized to dimension d. The +chirotope becomes a function of all d + 1 tuples and tells us the orientation in d dimensions. To illustrate +this, if we have four points a, b, c, e in 3-space, then the points a, b, c lie one a hyperplane H. Then the +3 + +ℓ +a +b +c +d +e +f +f +a +b +c +d +e +Figure 1: As ℓ separates f from the other points, a projective transformation sending the line ℓ at infinity +will flip the orientation of the triangles involving f while keeping the other orientations unchanged. +chirotope tells us on which side of H the point e lies, for an orientation of H defined by the three points a,b +and c. +It is important to note that if we take a projective transformation of the plane, we preserve the represented +matroid. This is because lines are mapped to lines and points to points. However, projective transformations +do not preserve the order type of a point set. This is because a point may end up on the other side of some +line, as pictured in Figure 1. +In order to get a closer relationship, we work (sometimes) with matroids +endowed with a distinguished line at infinity ℓ∞. Then we consider valid representations of such matroids, +which are those where this line is at infinity (i.e., all the points lie on one side of it). This definition extends +to rank-k matroids using a hyperplane at infinity. This leads to the following definition. +Given an order type O, we say that a matroid M simulates O, if the underlying matroid of O is a subset +of M and if the following conditions are met: +• Any representation of the matroid underlying O extends to a representation of M. +• Any valid representation of M induces an point set representing O. +Note that when M simulates O, then M has a representation if and only if O has an oriented representation: +indeed, starting with a representation of M, one can always send the line at infinity to infinity using a +projective transformation and thus obtain a valid representation.. +Our results +Our main theorem is that Matroid R-Representability is complete for the existential theory of the reals. +Theorem 1. Matroid R-Representability is ∃R-complete. +We provide two proofs of Theorem 1. The first one relies on simulating arbitrary ETR-formulas using +addition and multiplication and some technical assumptions. There is a somewhat easier proof, starting from +the fact that order type realizability is ∃R-complete, and then simulating order types using normal matroids. +Theorem 2. Let k ≥ 3 be a fixed integer. Given a rank-k order type O, we can compute in linear time a +rank-k matroid M such that M simulates O. +Theorem 1 easily follows from Theorem 2 as deciding whether an order type is representable over the reals +is ∃R-complete. This follows from techniques dating back to the proof of the Mnëv Universality Theorem +(see [44, 52]). However, as explained in [44], the proof that stretchability is ∃R-hard requires a significant +number of intricate steps, some of which can be simplified in the setting of Matroid R-Representability. +Indeed, the need for different scales (see for example [44, Proof of Theorem 4.6]), which is the main difficulty +in the oriented case, can be completely circumvented in our case. Therefore, for the sake of completeness +and simplicity, we also provide a self-contained proof of Theorem 1. We think it might be educational to first +understand the proof of Theorem 1, before one tries to understand the ∃R-completeness of stretchability. +4 + +Proof Overview. +In order to give an idea of the direct proof of Theorem 1, we first describe an incorrect +proof sketch. Then we point out the issues with this first sketch and how we can fix them. +First, it is folklore that in order to prove ∃R-completeness, it is sufficient to find a way to encode variables +and some basic operations like addition (x+y = z) and multiplication (xy = z). We can force points to lie on +a specific line ℓ to represent our variables. Furthermore, using the well-known von Staudt constructions we +can simulate all the basic constraints, see Figure 2 for the construction to simulate addition. This (almost) +describes a rank three matroid M. +Furthermore, in any realization of M, we can read a valid variable +assignment. +0 +x +y +x + y +ℓ +Figure 2: Encoding addition geometrically. +The issue with this basic approach is that it could be that there is only one realization of M such that +two points coincide, or three points lie on a common line, accidentally. If we were able to anticipate this, we +could easily specify this in the description of the matroid, but in general, this is not easy. +We circumvent this general position issue with two fixes. The first fix is to reduce from a version of ETR, +where we can assume that all variable values are distinct. We call this variant Distinct-ETR, see Section 2. +The second fix is to observe that when we build the von Staudt construction, we can ensure that all helper +points have enough freedom to avoid any coincidences or collinearities with previously defined points, see +Section 3. With these two fixes, the above proof sketch works as is explained in Theorem 1. +The proof idea of Theorem 2 goes as follows. Using arithmetic operations, we can give a variable the +value y = x2 ≥ 0. Geometrically, this implies that the point representing y is on the same side of the common +line ℓ as the point representing 1. In other words, we can enforce two points to lie on the same side of a line +with respect to a given point. We lift this to half-spaces in the plane and higher dimensions. In this way, +we can enforce consistent orientations of the matroid with the given order type. +Results on Distinct-ETR. +We define the problem Distinct-ETR as a variant of ETR as follows (cf. +infra for a proper definition of ETR). We are given variables X = {x1, . . . , xn} and constraints of the form +x + y = z, +x · y = z, +x = 1, +x > 0, +for x, y, z ∈ X. Furthermore, we are promised that there is either no solution at all or there is a solution +(x1, . . . , xn) ∈ Rn such that xi ̸= xj for all i ̸= j. We show the following theorem in Section 2, which might +be of independent interest. +Theorem 3. Distinct-ETR is ∃R-complete. +While we could not find any prior proof of Theorem 1 in the literature (hence this work), there are many +related works from at least two perspectives. First, as already mentioned, when one considers order-types +instead of unoriented matroids, Theorem 1 is very well-known. Second, topological universality theorems +have been proved for the real-representability of matroids in the algebraic geometry literature, see for ex- +ample Lafforgue [38] and Lee and Vakil [39]. While our work uses tools that are similar in spirit to those +papers, it differs in that our constructions are arguably simpler and that we specifically focus on proving the +computational hardness result, which is not the point of focus of those previous works, and is not entirely +equivalent (see discussion below). Furthermore, we believe that it is worthwhile to have a complete proof +of Theorem 1 in a purely combinatorial language, as opposed to the scheme-theoretical setup of previous +works. +Shor’s proof of the Mnëv universality theorem [59, Section 4] introduces an intermediate problem called +the existential theory of totally real ordered variables, which is very similar to Distinct-ETR but features +5 + +totally ordered variables x1 < x2 . . . < xn as opposed to just requiring distinctness in our problem. This +ordering is desirable when one investigates oriented matroids, and unneeded for unoriented ones. +This +difference allows for a proof that is arguably simpler than his, or at least different. +Background and Related Work +Matroids and Greedy +A practical reason why matroids are relevant for computational purposes is that +they capture in a simple way the class of discrete objects where greedy algorithms are successful in finding +an optimal solution. +For example, the standard Kruskal and Prim algorithms to compute a Minimum +Spanning Tree in a weighted graph can be abstracted by considering the vector matroid defined by an +oriented incidence matrix of the graph (called a graphic matroid), and then generalized to compute in +polynomial time a maximum or minimum-weight basis for any matroid. This property actually characterizes +matroids, see for example Oxley [50, Section 1.8] +Some applications of representability. +For an algorithm on matroids, a suitable encoding scheme of +the input matroid is needed. A common way is to take the input matroid M = (E, I) in the form of an +independence oracle which answers whether a given subset of the ground elements E is independent or not. +There are algorithms which run with polynomial number of oracle queries to such an oracle, for example +a maximum weight independent set of a given matroid can be computed in this way. However, for many +natural matroid properties, it is known that there is no algorithm with polynomially bounded queries to +independence oracle [29] including the representability over GF(2) and the connectivity of a matroid. +A vector representation of a matroid offers a compelling alternative to an independence oracle as matroid +operations can be substantially more efficient using matrix operations. Matroid parity problem, a common +generalization of graph matching and matroid intersection problem, is solvable in polynomial time given a +vector representation [40] while super-polynomial number of calls are needed under the independence oracle +model [29]. Deciding whether the branch-width of a matroid is at most k is a common generalization of +computing the branch-width, rank-width and carving-width of a graph. While there is an algorithm with +nO(k) queries on an n element matroid for this problem [49] under the oracle model, whether the dependency +on k in the exponent can be replaced by a uniform constant is not known till now. In contrast, the branch- +width of a vector matroid can be computed in f(k) · n3 time [30] when the given representation is over a +finite field F. +Another powerful application of a vector representation can be found in the theory of kernelization in +parameterized complexity. A surprising discovery of [37] is that for many graph cut problems, compressing +the input boils down to finding a so-called representative set of a matroid. When the said matroid is a +vector matroid, a representative set of bounded size can be efficiently computed in polynomial time [42, 43]. +It turns out that solutions to graph cut problems can be encoded as independent sets in gammoids, which +form a well-known class of representable matroids and of which a vector representation can be constructed +in randomized polynomial time. +Oriented Matroids. +One might wonder why we jumped from realizability of matroids to realizability +of abstract order types, instead of using the perhaps closer notion of oriented matroids [11], for which one +can also define a realizability problem and investigate its complexity. The reason is that in our arguments, +we reason extensively with point configurations, and the geometric interpretation described above does not +adapt directly to oriented matroids, as scaling a column by a negative number could lead to a change of the +underlying oriented matroid. The correct framework to connect oriented matroids to point configurations +is to only consider acyclic oriented matroids [11, Section 1.2.b], that is, those for which the geometric +interpretation works readily without a need for rescaling by a negative number. This notion of acyclic, +oriented matroids coincides with the notion of abstract order types, and so do their realizability problems. +Note that in some of the existing literature, oriented matroids and abstract order types are sometimes +described as equivalent. Therefore, we wanted to pay attention to this subtle difference. +The existential theory of the reals. +The complexity class ∃R (pronounced as ‘ER’, ‘exists R’, or ‘ETR’) +has gained a lot of interest in recent years. It is defined via its canonical complete problem ETR (short +for Existential Theory of the Reals. ETR refers to a geometric problem and ∃R refers to the complexity +6 + +class. While there are several different variants of ETR, there is only one complexity class.) and contains +all problems that polynomial-time many-one reduce to it. In an ETR instance, we are given a sentence of +the form +∃x1, . . . , xn ∈ R : ϕ(x1, . . . , xn), +where ϕ is a well-formed and quantifier-free formula consisting of polynomial equations and inequalities in +the variables and the logical connectives {∧, ∨, ¬}. The goal is to decide whether this sentence is true. As +an example consider the formula ϕ(X, Y ) :≡ X2 + Y 2 ≤ 1 ∧ Y 2 ≥ 2X2 − 1; among (infinitely many) other +solutions, ϕ(0, 0) evaluates to true, witnessing that this is a yes-instance of ETR. We use |ϕ| to denote the +length of ϕ, that is, the number of bits necessary to write down ϕ. The solution set of an ETR-formula is +called a semi-algebraic set. The (bit)-complexity of a semi-algebraic set is the shortest length of any formula +defining the set. It is known that +NP ⊆ ∃R ⊆ PSPACE. +Here the first inclusion follows because a SAT instance can trivially be written as an equivalent ETR +instance. The second inclusion is highly non-trivial and was first proven by Canny in his seminal paper [18]. +Note that the complexity of working with continuous numbers was studied in various contexts. To avoid +confusion, let us make some remarks on the underlying machine model. The underlying machine model for +∃R (over which sentences need to be decided and where reductions are performed) is the word RAM (or +equivalently, a Turing machine) and not the real RAM [24] or the Blum-Shub-Smale model [12]. +The complexity class ∃R gains its importance by numerous important algorithmic problems that have +been shown to be complete for this class in recent years. The name ∃R was introduced by Schaefer in [52] +who also pointed out that several NP-hardness reductions from the literature actually implied ∃R-hardness. +For this reason, several important ∃R-completeness results were obtained before the need for a dedicated +complexity class became apparent. +Common features of ∃R-complete problems are their continuous solution space and the nonlinear re- +lations between their variables. +Important ∃R-completeness results include the realizability of abstract +order types [48, 59] and geometric linkages [53], as well as the recognition of geometric segment [36, 44], +unit-disk [34, 46], and ray intersection graphs [19]. +More results appeared in the graph drawing com- +munity [22, 23, 41, 54], regarding the Hausdorff distance [33], regarding polytopes [21, 51], the study of +Nash-equilibria [6, 9, 10, 25, 55], training neural networks [3, 8], matrix factorization [20, 56, 57, 58, 61], +or continuous constraint satisfaction problems [47]. In computational geometry, we would like to mention +geometric packing [4], the art gallery problem [2], and covering polygons with convex polygons [1]. +Recall that NP is usually described using a witness and a verification algorithm. The same character- +ization exists for ∃R. Instead of the witness consisting of binary words of polynomial length, we allow in +addition using real-valued numbers as a witness. Furthermore, in order to be able to use those real numbers, +we are allowed to work on the so-called real RAM model of computation. The real RAM allows arithmetic +operations with real numbers in constant time [24]. +Topological Universality. +Many results and techniques on the existential theory of the reals actually, +precede the study of this complexity class. The underlying idea was to study how complicated solution spaces +can be from a topological perspective. For example, if we want to study convex polytopes, we are often +interested in the properties of their face lattice. The face lattice is a purely combinatorial object. Therefore, +it is natural to ask which face lattices are realizable by polytopes. If there would exist an easy combinatorial +description of realizable face lattices, convex polytopes could be much better understood. Given a specific +face lattice L, we can study its suitably defined solution space S(L). As the realizability question can be +formulated as an ETR-formula, it follows that S is a semi-algebraic set. Now, let T be a different semi- +algebraic set, we wonder whether there exists a face lattice L such that S(L) is homotopy-equivalent to T. +Maybe surprisingly topological universality states that there is such an L for any semi-algebraic set T. This +type of property feels very strong, as it intuitively states that we need to encode the vast complexity of +semi-algebraic sets into the problem of realizing convex polytopes. Indeed, many of the results that establish +such topological universality results also imply ∃R-completeness [44]. +However, topological universality can also be established for NP-complete problems as has been shown +by Bertschinger, El Maalouly, Miltzow, Schnider, and Weber [7]. +As the authors showed it is sufficient +to encode the topology of simplicial complexes, as it is possible to triangulate semi-algebraic sets [27]. In +7 + +other words, the difference between the wild semi-algebraic sets and the tame simplicial complexes is not +that they emit a more complicated topological structure. The difference comes from the ability of semi- +algebraic sets to encode complicated topological spaces in a much more concise manner. (To be precise the +description complexity of a topological space might be exponentially smaller using the language of semi- +algebraic sets, compared to simplicial complexes.) ∃R-completeness can be interpreted as giving a concise +encoding of semi-algebraic sets into a different domain. To make our life easier, we don’t care about the +complete preservation of the complete topology, but merely of the property of being empty or not. Still, +in order to do so one usually also preserves topological properties. This is the reason why there is a close +connection between topological universality and ∃R-completeness. But given that NP-complete problems +may also admit universality theorems ∃R-completeness may arguably be considered the more interesting +finding. +Stronger Universality Results. +We want to point out that previous universality results often also showed +stronger results than mere topological universality. For instance, Richter-Gebert showed such a stronger +universality theorem for polytopes [51]. Specifically, let P be a polytope, then the face lattice is the family +of faces of different dimension together with their inclusion order. Given a face lattice F, we can define +the set of polytopes V (F) having face lattice F. Richter-Gebert showed that for every semi-algebraic set S +there exists the face-lattice F of a polytope such that V (F) is stably-equivalent to S. To define the notion +of stable-equivalence goes above the scope of this paper. It is interesting to note that stable-equivalence +encapsulates more than just the topology of S, but also to a degree the “geometry” of S. +Discussion on Input Matroid Encodings +In this paper, we study the R-realizability, where the input matroid is given with all bases (maximal inde- +pendent sets). This would seem unconventional at first glimpse, especially for those familiar with matroid +theory. We would like to address the subtleties around our problem setting. +Types of encodings. +For matroids, three possible descriptions are examined in the literature, namely an +explicit description of sets, a description via an oracle, and a succinct description with a matrix. +Recall that an input graph for a graph problem can be given as an adjacency matrix, or equivalently +the family of vertex subsets of size two. Similarly, an input hypergraph is typically given as a set family +with an explicit description of all hyperedges. An immediate analogue of such an hypergraph description +for a matroid is an explicit enumeration of all bases (maximal independent sets) or all circuits (minimally +dependent sets). However, explicitly stating all independent sets, bases, or circuits is unconventional; the +most common size measure of a matroid is the number of elements, which is polynomially bounded by the +size of a graph or matrix that is generalized by a matroid. On the other hand, the number of bases or circuits +can be prohibitively large in comparison to the number of elements. Specifically, it is known that the number +of distinct matroids on n elements is doubly exponential in n [35], hence in space of size polynomial in n one +cannot describe an arbitrary input matroid. For further details about explicit matroid encoding, see [45]. +When the matroids under consideration are representable over a field F, a matrix over F provides a +succinct description of a matroid. There are well-studied matroid classes that are representable such as +uniform matroids, graphic matroids, and transversal matroids. However, not all matroids are representable. +Hence, an important question is to decide whether an input matroid is representable over a specific field, or +over any field at all, and to find a representation if one exists. +Due to the limitations of the above two explicit descriptions, the most common way to encode an input +matroid without any restriction is with an oracle, often with an independence oracle. One can view an +independence oracle as a black box expressing a boolean function on n variables. The boolean function is on +n input variables and outputs 0 or 1 depending on whether the input corresponds to (a characteristic vector +of) an independent set of the said matroid. Problems with black box functions, an implicit input with oracle +access are studied in the context of learning a function with a small number of queries, e.g. Polynomial +Identity Testing, and also in the context of search problems where a graph is accessed by adjacency query. +Such a problem does not fit in the classic computational complexity, where an explicit string of numbers +is expected as an input. Moreover, learning a black box (boolean) function cannot be done efficiently. As +mentioned previously, even when the input boolean functions are restricted to be matroid oracles, deciding +8 + +whether a nontrivial matroid property holds or not requires 2Θ(n) queries [60] even for basic properties such +as connectivity and representability over F = GF(2), and even with a randomized algorithm. +Therefore, an oracle encoding for F-representability problem does not appear to be a fruitful setting +to better understand the algorithmic aspects of the problem. Moreover, we shall argue below that, with +an explicit matroid description, there is an intriguing difference in the computational complexity of F- +representability between the cases when F is finite and when F =R. +F-representability with explicit bases description. +Let us consider a matroid description that pro- +vides a matroid M as a pair (E, B), where all bases of M are stated in the collection B. In this setting, we +shall argue that the problem of deciding whether M is F-representable is in NP for a finite field F and in ∃R +for F = R. We shall also argue that F-representability is likely to be in co-NP for a finite field F. This makes +an interesting contrast with the case F = R, for which the corresponding non-representability problem is +unlikely to be in ETR due to our main result. +First, let us see that F-reprsentability is in NP for finite F. Indeed, a matrix A (whose columns are +labeled by the elements of E) over F with M[A] = M can be taken as a witness for F-representability of +M. Moreover, one can conceive a polynomial-time verification algorithm for the pair M = (E, B) and A as +follows. Let B(A) be the set of bases of M[A] and recall that M = M[A] if and only if B = B(A). Whether +B ⊆ B(A) can be easily verified in time polynomial in |B| + rk(M). For this, we first compute the column +rank of A. If it is different from rk(M), this trivially implies M[A] ̸= M. Henceforth, let us assume that the +column rank of A equals rk(M). Now, for each basis B ∈ B, one checks whether the submatrix A[E, B], the +submatrix of A consisting of all columns labeled by the elements of B, is full rank. If any B ∈ B fails the +test, we know that A is not a representation of M. +Therefore, we may assume that B ⊆ B(A). To verify whether the equality holds, we rely on the following +property, which is easily derived from the Basis Exchange Property. We include its proof in the appendix +for self-containment. +(⋆) If B ⊊ B(A), then there exists B ∈ B and B′ ∈ B(A) \ B′ such that |B△B′| = 2. +Lemma 4. Let B be the set of all bases of a matroid M and let B′ ⊊ B. Then there exist bases B′ ∈ B′ and +B ∈ B′ \ B with |B′△B| = 2. +Proof. Choose B′ ∈ B′ and B ∈ B \ B′ so that |B′ ∩ B| is maximized. Let x ∈ B′ \ B. Recall the Basis +Exchange Property: +For any distinct bases W ′, W of a matroid and an element x ∈ W ′ \ W, there exists an element +y ∈ W \ W ′ such that W ′ − x + y is a basis of the matroid. +By the Basis Exchange Property, there exists y ∈ B \ B′ such that B′′ := B′ − x + y is a basis of M. If B′′ +belongs to B′, we have B′′ ∩ B = (B′ ∩ B) + y, which contradicts the choice of B′ and B. Therefore, B′′ +belongs to B \ B′. Note that |B′ ∩ B′′| = |B′ − x| = r − 1 and (thus B′′ = B by the choice of B′ and B) and +the claim follows. +Hence, the last step of the verification algorithm tests if there exist a basis B ∈ B and two element +x ∈ B and y ∈ E − B such that B − x + y is not a basis of M but the corresponding set of columns of A is +independent, which precisely tests if B − x + y ∈ B(A) \ B. If such a triple B, x, y exists, clearly B ⊊ B(A) +and thus M[A] ̸= M. Conversely if B ⊊ B(A), there exist such a triple B, x, y by Property (∗). Therefore, +we examine all triples (B, x, y) with B ∈ B, x ∈ B and y ∈ E − B and certify that B − x + y is either in B or +dependent, in which case the verification algorithm can correctly conclude that B(A) = B, thus M[A] = M. +Otherwise, the verification algorithm concludes M[A] ̸= M and rejects the witness A. +The presented verification algorithm shows that F-representability is in NP for each finite field F. A +matrix over F as witness and the polynomial-time verification algorithm naturally extend to the case when +F = R, where the verification algorithm works on a real RAM. Therefore, R-representability is in ∃R. For +details on the characterization ∃R via a witness and a polynomial-time verification algorithm on a real RAM, +see [24], and also the discussion in the paragraph above about the existential theory of the reals. +Furthermore, it is likely that F-representability is in co-NP for each finite field F. It is known [26] that +for any prime field F, non-F-representability can be certified by evaluating the ranks of O(n2) subsets of +9 + +an n-element matroid. +Notice that when the matroid M = (E, B) is given with an explicit description +of all the bases B of M, evaluating rk(X) for X ⊆ E can be done in time polynomial in the input size +because rk(X) equals the maximum of |X ∩ B| over all B ∈ B. Therefore, F-representability is in co-NP +under the explicit bases description for each prime field F. For an arbitrary finite field F, not necessarily +prime, up to our best knowledge there is no published result which establishes that a polynomial number of +rank evaluations suffices for non-F-representability. However, it is known that a positive resolution of Rota’s +conjecture implies that only a constant, depending on |F| only, number of rank evaluations would suffice +[50] to certify that a given matroid is not F-representable. The proof of Rota’s conjecture was announced in +2014 by Geelen, Gerards and Whittle [32] although it is expected to take a few more years for the full proof +to be written for publication. +Therefore, F-representability appears to be in NP ∩ co-NP when the input is given as the exhaustive list +of bases for each finite F given the claimed proof of Rota’s conjecture. Given that, deciding the computational +complexity of F-representability for each finite F with explicit bases description is an intriguing question. For +F = GF(2), a polynomial-time algorithm is straightforward from the uniqueness of a binary representation +(up to linear transformation) and the fact that such a representation can be efficiently obtained [50]; after +constructing a matrix over GF(2), we apply the above verification algorithm for NP membership. However, +even for F = GF(3) it is not clear whether a matrix over GF(3) can be efficiently constructed although it is +known that there is a unique representation over GF(3) for a matroid representable over GF(3). As far as +we are aware, there is no efficient procedure known for constructing the representation of a matroid M with +a promise that M is representable over GF(3), when M is given with an independence oracle or even given +as a matrix over the rationals Q [28]. Getting an input matroid as explicit bases description might help to +circumvent this obstacle. +In contrast to the case of finite field, it is impossible to certify non-R-representability with a polynomial +number of rank evaluations [50]. Finally, for R-representability we showed that ∃R-complete, exhibiting +a noticeable diversion from F-representability for finite F which appears neither NP-complete nor co-NP- +complete with explicit bases description under the assumption NP ̸= co-NP. Therefore, our result highlights +the recurring contrast between representability over finite field and over the reals. +2 +Distinct-ETR +This section serves as a preparation for the later reduction. Specifically, we show ∃R-completeness of a +variant of ETR. We name this variant Distinct-ETR, as we can assume that there is a solution with all +variables holding distinct values. This property will be key for encoding into matroids. Most of this section +follows standard techniques. +Overview. +This section is dedicated to the proof of the lemma. The idea is that we first establish the +hardness of an ETR variant (STRICT-INEQ) with an open solution space. It is clear that all variables can +be assumed there to have distinct values. Then, we reduce again to a variant where we use only the basic +constraints. +The reduction goes in four steps. +1. From ETR to ETRAMI. +2. Then from ETRAMI to Feasibility. +3. Then from Feasibility to STRICT-INEQ. +4. And at last from STRICT-INEQ to Distinct-ETR. +Note that step 1,2, and 3 have already been done (among others) by Schaefer and Štefankovič [55]. We +sketch the main steps of their reduction. Specifically, we point out some properties that were not explicitly +emphasized. We start to explain a simple trick that is excessively used in those types of constructions in +order to build small, very small, and very large numbers. +10 + +Number Constructions. +Before we describe the reduction it might be useful to understand how we can +construct variables that must have specific rational values. +If we want to build integers of polynomial size, we can do this by simply repeatedly adding or subtracting +a one. +a1 = 1, +ai+1 = ai + a1, +ai−1 + a1 = ai. +It is easy to see that ai = i, for all ai that are defined in this way. +If we want to build a very large number, say 22k, the previous approach cannot be done in polynomial +time. Instead, we can use repeated squaring as follows. +x0 = a2 + a0(= 2), +xi+1 = x2 +i , +for i = 1, . . . , k. It holds inductively that xi = 22i. Similarly, we can construct very small numbers say 22−k, +as follows: +x0 + x0 = 1, +xi+1 = x2 +i , +for i = 1, . . . , k. It holds inductively that xi = 2−2i. Note that we can also use strict inequalities to build +large and small numbers. For example, +x0 > 2, +xi+1 > x2 +i , +for i = 1, . . . , k implies that xi > 22i. +We will use these standard tricks repeatedly later in the reduction. +Reduction from ETR to ETRAMI. +We define the problem ETRAMI as a variant of ETR as follows. +We are given variables X = {x1, . . . , xn} and constraints of the form +x + y = z, +x · y = z, +x = 1, +for x, y, z ∈ X. Therefore, ETRAMI is a variant of ETR without negations, inequalities, disjunctions, +conjunctions and the only constant is 1. It is folklore that ETRAMI is ∃R-complete and follows implicitly +from various papers [44, 55, 59] +Lemma 5 (folklore). ETRAMI is ∃R-complete. +∃R-membership follows from the definition. The idea of the reduction is to simplify an ETR-formula in +each step. For instance, we can remove negations by replacing ¬p > 0 by p ≤ 0. In this way, we can remove +all negations. We can replace inequalities by observing that p ≥ 0 is equivalent to ∃x : p = x2 and p > 0 +is equivalent to ∃x : px2 = 1. We can replace p = 0 ∧ q = 0 by p2 + q2 = 0. Similarly, we can replace +p = 0∨q = 0 by pq = 0. Thereafter, we end with a single polynomial equation p = 0. We construct variables +for each coefficient value. Then we replace each coefficient with an appropriate variable. At last, we replace +each occurrence of multiplication and addition inductively by introducing one more variable and one more +constraint. We end with a single equation of the form x = 0, which can be replaced by z = 1 and z + x = z. +Reduction from ETRAMI to Feasibility. +In Feasibility, we are given a single polynomial p ∈ +Z[x1, . . . , xn] of degree at most four. +We are asked if there exists some x ∈ Rn such that p(x) = 0. +Furthermore, we require each coefficient to be of absolute value at most 36n3. Below, we will show how to +achieve this upper bound. +Lemma 6. Feasibility is ∃R-complete. +Again, ∃R-membership follows from the fact that Feasibility is a special case of ETR. To show hardness +we sketch a reduction from ETRAMI that is already known [44, 55]. +Let f1 = 0, . . . , fm = 0 be the +constraints of some ETRAMI instance ϕ. (For example x + y = z becomes x + y − z = 0.) Let +p = f 2 +1 + . . . + f 2 +m. +Clearly, ϕ is satisfiable if and only if p has a zero. As each fi has a degree at most two it holds that p +has degree at most four. As there are only 3n3 possible distinct constraints in ϕ, we have m ≤ 3n3. Note +that each term f 2 +i gives rise to at most six monomials and each coefficient is at most two. (For example +(x + y − z)2 = x2 + 2xy − 2xz + y2 − 2yz + z2.) Therefore each coefficient has absolute value at most +12m = 36n3. Thus, we can rewrite p as a sum of monomials with bounded-sized coefficients, as claimed. +11 + +Reduction from Feasibility to STRICT-INEQ. +In a STRICT-INEQ instance, we are given a sen- +tence of the form +∃x1, . . . , xn ∈ R : ϕ(x1, . . . , xn), +where ϕ is a well-formed and quantifier-free formula consisting of polynomial strict-inequalities in the vari- +ables and the logical connectives {∧, ∨}. +Note that the solution space {x ∈ Rn : ϕ(x)} is always open. +Lemma 7. STRICT-INEQ is ∃R-complete. +Again membership follows from the fact that ETR is more general then STRICT-INEQ. To show ∃R- +hardness we reduce from Feasibility. The idea of the reduction is to replace p(x) = 0 by −δ < p(x) < δ, for +some sufficiently small δ. To this end, we employ two lemmas as formulated by Schaefer and Štefankovič [55]. +Note that the actual proof comes from real algebraic geometry and can be read for instance in [5] and [31]. +Lemma 8. Every non-empty semi-algebraic set in Rn of complexity at most L ≥ 4 contains a point of +distance at most 2L8n from the origin. +Lemma 9. If two semi-algebraic sets in Rn each of complexity at most L ≥ 5n have positive distance (for +example, if they are disjoint and compact), then that distance is at least 2−2L+5 . +In this context the distance between two sets A, B is defined as +d(A, B) = +inf +a∈A,b∈B ∥a − b∥. +Note that ∥ · ∥ denotes the Euclidean norm. +In order to apply Lemmas 8 and 9, we define the following three semi-algebraic sets. First, we define the +solution set for p. +S = {x ∈ Rn : p(x) = 0}. +Let R = 2L8n, where L is the bit-complexity of S. Note that L is the upper bound by the length of p. Using +Lemma 8, we know that S is empty if and only if S ∩ B(R) is empty. (We denote by B(R) the ball of radius +R around the origin.) This motivates us to define the sets +S′ +1 = {(x, z) ∈ Rn+1 : p(x) = z ∧ ∥x∥2 ≤ R2}, +and +S′ +2 = {(x, z) ∈ Rn+1 : z = 0 ∧ ∥x∥2 ≤ R2}. +Note that S′ +1 ∩ S′ +2 = (S ∩ B(R)) × {0}. +Furthermore, S′ +1 and S′ +2 are compact and thus we can apply +Lemma 9. Unfortunately, the description complexity of S′ +1 and S′ +2 are exponential, if we write R out in +binary. Therefore, we define S1 and S2 slightly differently. Namely, we add some extra variables, whose sole +purpose is to encode R using repeated squaring. Let L be the max of the bit complexity of S1 and S2. Note +that L = O(L + n log L). Let δ be as in Lemma 9 applied to S1 and S2. It holds that S = ∅ is equivalent to +S ∩ B(R) = ∅. This in turn is equivalent to S1 ∩ S2 = ∅. And this is equivalent to +p(x) ≤ −δ or δ ≤ p(x), +for all x ∈ B(R). In other words, we have +∃x : −δ < p(x) < δ and ∥x∥2 < R2 +(1) +if and only if +∃x : p(x) = 0 +This obviously also works if we would use any smaller δ. Maybe not so obviously, this also works for R +of between 2L8n and 2L8n+1. The lower bound allows us to apply Lemma 8. The upper bound allows us +to apply Lemma 9. Using repeated squaring, we create numbers a > 2L8n and b < 2L8n+1. Furthermore, +we add the inequalities a < R < b. Note that we can construct a number δ that is at most 2−2L+5. Our +STRICT-INEQ instance ϕ consists of the three inequalities from Equation (1) and some extra variables +and constraints to bound R and δ as described above. +12 + +Reduction from STRICT-INEQ to Distinct-ETR. +In this section, we show that Distinct-ETR +is ∃R-complete. We are not actually reducing from STRICT-INEQ but from the instance ϕ described +in Equation (1). Let δ, R be the given numbers, x1, . . . , xn the variables and p the polynomial as in the +previous paragraph. Recall that p has a degree at most four and the coefficients are bounded integers. We +will introduce new variables in order to construct δ, R, ∥x∥2, and p(x). Thereafter, we will argue about +distinctness. +First, we construct variables holding the values of the integers −36n3, . . . , 36n3. Those variables are meant +to represent the coefficients of p. Recall that this was an upper bound on the values of those coefficients. It +has been described above how to construct those integers. Furthermore, we add variables holding the values +R = 2L8n and δ = 2−2L+5. (As we reduce from Equation (1), we do not need to approximate the values δ +and R, but can construct them directly, as Distinct-ETR allows us to use equations.) If in this process +two variables hold the same value, we can detect this and remove one of the variables. +Now, we construct p(x) and ∥x∥2. First, we construct all possible +�n +4 +� +monomials of degree at most +four. For example, N = xyzw is constructed in three steps. N1 = xy, N2 = N1z, and N = N2w. Again, +whenever two identical monomials would appear, we would be able to notice this and make an appropriate +replacement. Let us denote +p(x) = +m +� +i=1 +aiMi. +Here, ai is the coefficient of the monomial Mi. We construct Pk = �k +i=1 aiMi inductively as follows: +P1 = a1M1 and Pk = Pk−1 + Tk, +with Tk = akMk. +We denote by P = Pm the variable holding the value of p(x). We also construct X = ∥x∥2 = x2 +1 +. . .+x2 +n +in the same way. +At last, we add the variables a, b, c, and the constraints +a + δ = P, +b + P = δ, +c + X = R. +Now we can enforce the inequalities +−δ < p(x) < δ and ∥x∥2 < R2 +by +a > 0, +b > 0, +c > 0. +To summarize, we started with the variables x1, . . . , xn. We have created variables C1, . . . , Cs that each +holds a different integer/rational number. Furthermore, we constructed some variables V1, . . . , Vt such that +each Vi is a polynomial function gi(x). Note that all the gi have a degree of at most four and small integer +coefficients. If for two variables Vi and Vj, we have that gi = gj, we can detect this and remove one of them +and replace each occurrence with the other one. This finishes the description of the reduction. We denote +this instance ψ. It remains to argue correctness. +To show correctness, we observe that ϕ has a solution if and only if ψ has a solution as well. Indeed, all +new variables and constraints only “build” the correct polynomials and the numbers δ, R. Thus it remains +to show that if ϕ has a solution if and only if ψ has a solution with all variables taking distinct values. +The backward direction is trivial. Therefore, we assume that ϕ has at least one solution x ∈ Rn. As the +solution space of ϕ is open, there is an open ball fully contained in the solution space. Clearly, the variables +C1, . . . , Cs have all fixed distinct values by construction. Every other variable Vi can be expressed as a +polynomial function gi(x). As all the gi are distinct there must be some x ∈ B such that gi(x) ̸= gj(x), for +all i ̸= j and gi(x) ̸= Cj, for all i, j. Otherwise, two of the polynomials would be identical. This finishes the +proof. +13 + +3 +Arithmetic using Matroids +In this section, we describe how to encode addition and multiplication of real numbers using rank-3 matroids. +We rely on the von Staudt constructions, which are very well-known, perhaps with the caveat that they are +usually stated for oriented matroids. But as we shall see, no orientedness is actually required to make them +work. We follow the presentation of Matoušek [44]. +The setup for both operations is as follows. We have a line ℓ containing three distinct distinguished +points called 0, 1, and ∞, and a fixed second line ℓ∞ crossing ℓ at ∞. Given the variable x, we denote +the corresponding point representing it by x in fat. This way we easily distinguish between a point and the +corresponding variable. A point x on the line ℓ can be interpreted as a real number using cross-ratios: if we +denote by d(a, b) the oriented distance between two points a and b on the line ℓ, then the quantity +(x, 1; 0, ∞) := d(x, 0) · d(1, ∞) +d(x, ∞) · d(1, 0) +is a real number invariant under projective transformations, which, by a slight abuse of notation, we simply +denote by x. Note that if ∞ is progressively sent to infinity using a projective transformation and d(0, 1) +is scaled to one in this formula, x converges to d(0, x). Thus this cross-ratio matches the geometric location +of x on ℓ under some projective transformation. +Now, given two points x and y on ℓ, we describe geometric operations to compute points on the line +x + y and x · y on ℓ representing their addition and their multiplication. +∞ +0 x +y +x + y +a +b +ℓ∞ +ℓ +0 +x +y x + y +c +d +ℓ +c +d +Figure 3: Encoding addition geometrically +Addition. +The construction is pictured in Figure 3, left. We first introduce two distinct auxiliary helper +points a and b situated anywhere on ℓ∞. The line connecting 0 to b crosses the line connecting x to a in a +point c, then the line connecting ∞ to c crosses the line connecting y to b in a point d. Similarly, the line +connecting a to d crosses ℓ in a point that we define to be x + y. The rationale behind this construction +is that by a projective transformation we can consider ℓ∞ to be a line at infinity, bringing us the Figure 3, +right. Now the line containing c and d crosses ℓ at a point at infinity, i.e., these two lines are parallel. +Likewise, the line containing c and d is parallel to ℓ. Then the parallelity of the lines immediately shows that +d(0, x + y) = d(x, y) + d(0, y). In other words, the point x + y has value x + y, justifying the notation. +∞ 0 +x +y +xy +a +b +ℓ∞ +ℓ +0 +x +y +xy +c +d +1 +ℓ +c +d +1 +Figure 4: Encoding multiplication geometrically +14 + +Multiplication. +The construction is pictured in Figure 4, left. As before, we first introduce two distinct +auxiliary helper points a and b situated anywhere on ℓ∞. The line connecting 1 and b crosses the line +connecting x and a at a point c, and the line connecting 0 and c crosses the line connecting b and y at +a point d. Finally, the line connecting a and d crosses the line ℓ at a point that we define to be xy. By +sending the line ℓ∞ at infinity using a projective transformation, we obtain Figure 4, right, where one can +readily show using the parallel lines that d(0, xy) = d(0, x)d(0, y). In other words, the point xy has value +xy, justifying the notation. +Encoding these geometric constructions using matroids. +The addition and multiplication construc- +tions defined above can be entirely encoded using matroids: the independent sets are exactly the empty set, +all the singletons, all the pairs of points, and all the triples of non-aligned points. Let us insist here that no +orientation was ever enforced during the constructions, and thus we do not need oriented matroids to de- +scribe them. Furthermore, these operations can be chained arbitrarily, allowing to encode polynomials using +matroids. However, some very important care needs to be taken here: while it is clear from the construction +which points should be aligned, we also need to make sure that points that should not be aligned can be +assumed to not be aligned. For example, it could a priori happen that a line going through two helper points +c1 and c2 somehow accidentally happens to pass through a variable x of the polynomial we are encoding. In +that case, the matroid would not properly encode the geometric construction. +This motivates the following definition: we say that the set S of helper points used during an addition or +multiplication construction is free if for any finite set of lines L and points P not involved in the construction, +the points in S can be perturbed so that: +• the incidences required by the addition or multiplication construction still hold, +• no point of S lies on a line L, and +• no pair of points of S is aligned with a point of P +(in particular, no point of S coincides with a point of P). +A key property of the addition and multiplication construction is that the four helper points that they +rely on form a free set. Indeed, the points a and b can be placed freely on the line ℓ∞, and thus can be +perturbed so as to avoid the lines and points of P and L. Such a perturbation induces a perturbation of c +and d in a two-dimensional open set, therefore allowing them to avoid lines of P and L, but also ensuring +that no line going through a pair of points in {a, b, c, d} also goes through a point in P. +This freedom will be leveraged in the proofs of Theorem 2, respectively Theorem 1, to ensure that the +matroid correctly encodes orientation predicates, respectively systems of polynomial equations. +Strict inequalities. +The multiplication construction can be leveraged to simulate a strict inequality con- +straint x > 0. Indeed, x > 0 if and only if there exists z ∈ R such that z ̸= 0 and x = z2 = zz, which can +thus be simulated using a helper point z distinct from 0 and the above multiplication construction. However, +this construction would require us to use as a helper point a fixed point z on the line ℓ, which could not +be perturbed and thus would not be free. This can be resolved by using an additional variable: we first +introduce a helper point y for which we ensure that y > 0 using the multiplication gadget. Then we use +another multiplication gadget to ensure that z > y: note that this amounts to enforcing that z lies on the +same side of y than 1 does, i.e., this can be tested by using another multiplication gadget where 0 is replaced +by y. Now, in these two multiplication gadgets, neither y nor any of the other helper points is fixed, and +therefore we can perturb them to avoid any fixed set of lines and points, showing that they form a free set +of points. +4 +Proof of Theorem 2 +Theorem 2 is proved by using the arithmetic constructions described in the previous section, in particular +the one for strict inequalities, to simulate orientation predicates. +15 + +Simulating rank 3 order types. +We will first consider the case of rank equal to 3 and treat the general +case later. Let O = (E, χ) be an order type on n elements of rank 3. (E = {e1, . . . , en}.) We construct +a matroid M from O inductively. To be more precise, we construct matroids, M3, M4, . . . , Mn = M such +that Mi simulates Oi. Here, Oi = (Ei, χi) is the order type formed by the first i elements of O. All of our +matroids Mi will feature a distinguished line ℓ∞. +The matroid M3 merely contains e1, e2, e3. Without loss of generality we assume that the triple t = +{e1, e2, e3} are independent in O. (Otherwise, all triples in O are dependent, which can trivially be simulated.) +We first add a line at infinity, on which no e1, e2 or e3 lies. We can assume that t is oriented correctly in any +representation of M3, as otherwise, we can just reflect the representation and get a correct representation. +Therefore, M3 satisfies the induction hypothesis. +Now, let us assume that we already constructed Mi−1. We construct Mi from Mi−1, by adding the +element e = ei from O. Furthermore, for each triple t = {a, b, e} ⊂ Ei, we need to ensure that t is oriented +correctly. In case that χ(t) = 0, we can encode this into Mi directly by specifying that (a, b, e) forms a +rank-2 dependent set. Thus it remains to consider the case χ(t) ̸= 0. Let t′ = {a, b, c} ⊂ Ei−1 such that +χ(t′) ̸= 0. Note that such a triple t′ must exists, as otherwise all points of Ei lie on a common line. This +would be a contradiction to the fact that M3 is formed by an independent triple. Using the orientation of +t′ we add a small constant number of helper points and we will enforce the correct orientation of t in Mi as +well. Thus it remains to show the following lemma, where we use the notion of a free set of helper points +defined in Section 3. +Lemma 10. Given the independent triple t′ = {a, b, c} in a matroid M and another point e, we can enforce +that in any valid representation of M, the triple t = {a, b, e} is oriented identically to t′, or that it is oriented +opposite to t′. We do this by adding a constant number of helper points to M. Furthermore, this set of helper +points is free. +Proof. This construction goes in essentially two steps. First, we show how we can enforce an element x on +the line ℓ = ℓ(a, b) to be on the same side of a as b. +a +b +x +It relies on standard constructions to do encode arithmetic operations as explained in Section 3. Although, +those constructions can be well described and understood, without any reference to arithmetic operations, +the language of arithmetic operations gives a better intuition. The underlying idea is that we interpret a as +zero, and b as 1. Indeed, the constraint that c lies on the same side of a as b in any representation where +the line ℓ∞ is sent to infinity amounts to enforcing that c > 0 when a is interpreting a as zero and b as one. +As explained in Section 3, this can be encoded using multiplication gadgets, in such a way that none of the +helper points accidentally lies on a previously used line or point. +In the second step, we use the previous tool to enforce that e and c lie on the same (or the opposite) side +of the line ℓ(a, b) as c. +a +d +b +c +c′ +e +Figure 5: Forcing c to be on the same side of ℓ(a, b) as e. +Half-open lines denote the strict inequality +constraints. +To this end, we first define a point c′ for which we enforce that on the line ℓ(a, c), it lies on the same side +of a as c. Then we define a point d situated on the line ℓ(a, b) and a line ℓ′ with points c′, d, e. See Figure 5 +16 + +The condition that e, c′ are on the same side of d on the line ℓ′ can be enforced using the previous gadgets. +Furthermore, it is equivalent to c′ and e being on the same side of ℓ(a, b). Lastly, by construction the triple +{a, b, c′} is oriented identically to {a, b, c}. +We can use the same tool to enforce that e and c lie on the opposite side of the line ℓ(a, b). We define c′ +as before, so that it lies on the same side of ℓ(a, b) as c, and then we simply need to enforce instead that c′ +lies on the same side of e as d. +As explained in Section 3, the strict inequalities gadgets can be directly encoded into the independent sets +of the matroid, and by construction, the helper points always have at least one degree of freedom, and thus +can form a free set of points. Therefore, the matroid Mi is entirely defined by the dependency constraints +indicated by the lines in the geometric constructions. This concludes the proof. +We can now conclude the proof of Theorem 2 for rank-3 matroids. Given an order type O of rank 3, for +which we can assume that there is at least one independent set of size 3 (otherwise the orientation predicates +are trivial), we inductively encode the orientation predicates into an matroid using the helper points provided +by Lemma 10. At each stage of the induction, the freedom of the set of helper points can be used to ensure +that the helper points do not yield any accidental dependencies with all the points and lines previously +placed. At the end of the induction, by Lemma 10, any valid representation of the resulting matroid M +induces a representation of O. Conversely, any representation of O can be extended to a representation of +M. Therefore M simulates O. All the constructions can clearly be done in linear time, which concludes the +proof. +Simulating rank k Matroids. +The construction for rank k is identical to that of rank 3 as explained +above, with two exceptions. First, the induction basis starts with a matroid on k elements instead of 3. +Second, we have to describe the simulation of an oriented k-tuple. For this, we use the same trick that forces +a point to lie on a specific side of a line, where we just replace a line with a hyperplane P, as pictured in +Figure 6. +P +d +a +a′ +x +a1 +Figure 6: Forcing x to be on the same side of P as a. +More precisely, in an inductive step where we add a point x, we need to enforce the orientation of all the +independent k-tuples t = {a1, . . . ak−1, x}. For this, we take another k-tuple t′ = {a1, . . . ak−1, a} for which +χ(t′) ̸= 0, and we devise a gadget to encode the constraint that t is oriented identically, or opposite to t′. +This will be done using the gadget described above that enforces that in any valid representation, for three +aligned points a, b and c, c lies on the same side of a as b. That gadget was defined for rank 3 matroids +and thus representations into R2, but readily works in higher dimensions: one should simply ensure using +dependency constraints that all the points involved in the gadget lie on a common plane. The hyperplane at +infinity will intersect this common plane in a line, which takes the role of the line at infinity in the gadgets. +Now, we proceed as in the rank 3 case: considering the hyperplane P generated by {a1, . . . ak−1}, we +first define a point a′ that lies on the same side as a on the line ℓ(a1, a). Then we can enforce that a′ is +on the same side of P as x by introducing the point d at the intersection of P and the line going through +a′ and x. Then we enforce that x is on the same side of d as a′. In order to enforce that a′ and x are on +opposite sides of P, we instead enforce that x and a′ are on opposite sides of d. Finally, we observe that +all the helper points that we have introduced are free, where the notion of freedom is generalized to also +17 + +disallow k-dimensional dependencies: once again this follows from the fact that all the helper points that we +introduce have some wiggle room to be perturbed. The rest of the proof proceeds identically to the rank-3 +case. +5 +Proof of Theorem 1 +The proof of Theorem 1 is by a direct reduction from the problem Distinct-ETR, using the arithmetic +constructions described in Section 3. Let us start with an instance of Distinct-ETR given by variables +X = {x1, . . . , xn} and constraints of the form +x + y = z, +x · y = z, +x = 1, +x > 0, +for x, y, z ∈ X, with the promise that if there is a solution, then there is one where all the variables are +pairwise disjoint. +As in the proof of Theorem 2, we can construct the constant 0 using a constraint x + y = x, and thus, +without loss of generality, by the distinctness assumption, we can assume that there are exactly two variables +in X equal to respectively 0 or 1, and that the others are different from 0 and 1. By a slight abuse of language, +we remove those from X and denote them directly by 0 and 1 in the rest of the description. We define a +rank-3 matroid M as follows. First, we have a line ℓ consisting of three distinguished distinct points 0, 1 +and ∞, as well as n distinct points (and distinct from {0, 1, ∞}) corresponding to the variables {x1, . . . xn}. +We also add a line at infinity ℓ∞ going through ∞ and no other point. +We then use the gadgets from Section 3 to inductively encode all the constraints. Note that there is at +most one constraint x = 1 which can be hardcoded from the start. Then let us assume inductively that +we have defined a matroid Mi encoding the first i constraints. The i + 1th constraint is an addition, a +multiplication, or a strict inequality, which can be encoded using a geometric construction as described in +Section 3. Now, the key property of these constructions is that the set of helper points is free. Therefore, the +only linear dependencies involved in the construction are those of that construction, which can be readily +encoded into a matroid Mi+1. +We now prove that the Distinct-ETR instance has a solution with distinct variables if and only if the +matroid M is representable over the reals. First, if Distinct-ETR has a solution, we obtain a representation +of M over the reals by placing all the variables x1, . . . , xn on the line ℓ at the values indicated by the solution, +and by sending the line at infinity to infinity. The geometric constructions are then represented one by one, +and since the helper points are free, by perturbing them if needed we can ensure that they are all distinct, +that no three of them are colinear, and that they do not form colinearities with previously placed points. +Therefore, this constitutes a correct representation of the matroid. Conversely, given a representation of the +matroid M over the reals, we read the values of the variables on the line ℓ using cross-ratios as explained +in Section 3 (or equivalently we send ℓ∞ to ∞ and use the oriented distance to 0). This gives us values +for the variables of the Distinct-ETR instance. The definition of the addition, multiplication and strict +inequality constructions ensures that each of the constraints will be satisfied. Furthermore, by definition of +the matroid, all of the variables are distinct. This finishes the proof. +References +[1] Mikkel Abrahamsen. Covering Polygons is Even Harder. In Nisheeth K. Vishnoi, editor, 2021 IEEE +62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 375–386, 2022. +[2] Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow. +The Art Gallery Problem is ∃R- +complete. 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Computational complexity of decomposing a symmetric +matrix as a sum of positive semidefinite and diagonal matrices. arXiv preprint arXiv:2209.05678, 2022. +22 + diff --git a/idE1T4oBgHgl3EQffwRJ/content/tmp_files/load_file.txt b/idE1T4oBgHgl3EQffwRJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..383f7dddb545022c3fcf21c3672812d7b01cba86 --- /dev/null +++ b/idE1T4oBgHgl3EQffwRJ/content/tmp_files/load_file.txt @@ -0,0 +1,978 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf,len=977 +page_content='Representing Matroids over the Reals is ∃R-complete Eunjung Kim, Arnaud de Mesmay, Tillmann Miltzow January 10, 2023 Abstract A matroid M is an ordered pair (E, I), where E is a finite set called the ground set and a collection I ⊂ 2E called the independent sets which satisfy the conditions: (I1) ∅ ∈ I, (I2) I′ ⊂ I ∈ I implies I′ ∈ I, and (I3) I1, I2 ∈ I and |I1| < |I2| implies that there is an e ∈ I2 such that I1 ∪ {e} ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The rank rk(M) of a matroid M is the maximum size of an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We say that a matroid M = (E, I) is representable over the reals if there is a map ϕ : E → Rrk(M) such that I ∈ I if and only if ϕ(I) forms a linearly independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We study the problem of Matroid R-Representability over the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Given a matroid M, we ask whether there is a set of points in the Euclidean space representing M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We show that Matroid R-Representability is ∃R-complete, already for matroids of rank 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The complexity class ∃R can be defined as the family of algorithmic problems that is polynomial-time equivalent to determining if a multivariate polynomial with integer coefficients has a real root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Our methods are similar to previous methods from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Yet, the result itself was never pointed out and there is no proof readily available in the language of computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 1 Introduction Many articles on matroids assume that the matroid is representable [40, 15, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Representability either heavily simplifies proofs and definitions or is even essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We show that the question of representability over the reals is as difficult as the existential theory of the reals, that is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The complexity class ∃R can be defined as the family of algorithmic problems that is polynomial-time equivalent to determining if a multivariate polynomial (with integer coefficients) has a real root, see below for an introduction and overview of this complexity class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Before we give a general definition of a matroid, we introduce vector matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Given a matrix A over a field F, we can define the corresponding vector matroid M[A] = (E, I) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The ground set E of M[A] is formed by the columns of A and we say that a subset I ⊂ E is independent in M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=', I ∈ I, if the columns are linearly independent over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' A set of elements of E which is not independent is said to be dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that any set of columns containing a zero column is dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The independent sets of a vector matroid satisfy three simple properties (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' One way to look at matroids is to see them as abstract set systems that have those three properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' A matroid M is an ordered pair (E, I), where E is a finite set called the ground set and a collection I ⊂ 2E called the independent sets which satisfy the conditions: (I1) ∅ ∈ I, (I2) I′ ⊂ I ∈ I implies I′ ∈ I, (I3) I1, I2 ∈ I and |I1| < |I2| implies that there is an e ∈ I2 such that I1 ∪ {e} ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The rank rk(M) of a matroid M is the maximum size of an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We say that a matroid M = (E, I) is representable over F if there is a matrix A over F such that M = M[A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that all columns of A live in a subspace of dimension at most rk(M[A]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, we can assume without loss of generality that the columns of A have dimension rk(M[A]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We refer to [50] for more background on matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='03221v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='CC] 9 Jan 2023 Given a matroid M, if there exists a matrix A over R such that M = M[A], we say that M is representable over the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The algorithmic problem of Matroid R-Representability is to test whether a given matroid is representable over the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Since we only discuss the real case in this article, we will sometimes say representable as a shorthand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that we also need to specify how the matroid M is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In the literature on matroids, one has often an oracle such that one can ask the oracle for each set I whether I ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We will deviate from this practice, as it might be unclear how to describe the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Instead, we will just list all sets in I explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This does not blow up the description complexity too much in our case, as we will deal mainly with constant rank matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Geometric Interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' If we have a representation A ∈ R3×n of a matroid M = M[A], we can scale every column of A by a nonzero number and it stays a valid representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This is also the case if we scale by −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, if we rotate A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=', multiply it on the left with an orthogonal transformation) it still stays a valid representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Thus, in case we have a representable rank-3 matroid over R, we can assume that there is a representation in which all nonzero vectors have their third coordinate equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In this way, we can consider the columns of A as a point configuration in the plane with z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The property of three vectors being dependent translates to the corresponding three points to lie on a common line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In this geometric interpretation, we could have considered any plane different from the one with z = 1, as long as it does not cross the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This would have yielded a different point configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The resulting transformation is called a projective transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It maps lines to lines, except for one line that disappears, we say that it is sent to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Conversely, for any line in the plane, there exists a projective transformation that sends it to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Throughout this article, we think of representations of rank-3 matroids via these point configurations, and thus we will slightly abuse language by calling such a point configuration a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' A motivating example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' One of the standard examples to illustrate realizability is the so-called Fano plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It is the matroid on seven elements whose maximal independent sets are all the triples except {1, 4, 7}, {1, 2, 3}, {1, 5, 6}, {3, 6, 7}, {2, 5, 7}, {3, 4, 5}, {2, 4, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This can be represented pictorially as in the figure below, where the lines and the circle denote the dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 1 2 3 4 5 6 7 2 An immediate question that arises from this picture is whether a picture exists where the circle is not used and the dependencies are all pictured by lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This is equivalent to asking whether the Fano matroid is representable over the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It is well-known not to be [50, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The problem Matroid R-Representability addresses the general question of deciding which matroids can be represented like that, and our main result is that this problem is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Order Types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We saw above that matroids are an abstraction to describe point collinearities in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=', if we have a rank 3 matroid then every dependent set corresponds to three collinear points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now given a set of points, we are often also interested in the orientation of each triple: either clockwise, counter-clockwise, or collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' a b c d In this example, {a, b, c} is collinear and (a, b, d), (a, c, d), and (b, c, d) are oriented counter-clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This leads to the definition of (abstract) order types, which is a pair O = (E, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Again, E is a finite set called the ground set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' And χ : �E 3 � → {−1, 0, 1} is a function called a chirotope satisfying a few simple properties that are derived from the intuition given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We say that a point set P ⊂ R2 represents a given order type O = (E, χ), if P has for each element e ∈ E a corresponding point e′ ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, for each triple a, b, c ∈ E the corresponding points a′, b′, c′ ∈ P are oriented according to χ({a, b, c}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that if we lift every point in P to the plane with z = 1 as a subset of R3, then we get the following correspondence between (p′ x, p′ y, 1), (q′ x, q′ y, 1), (r′ x, r′ y, 1) and the corresponding elements p, q, r ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' sign det � � p′ x q′ x r′ x p′ y q′ y r′ y 1 1 1 � � = χ(p, q, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that in this specific setup a realization of a rank 3 matroid and order types are closely related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' While matroids only determine the collinearities, the order type also determines the orientation of each triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We want to point out that every abstract order type can be represented by a pseudoline arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' A pseudoline arrangement can be defined as a collection of x-monotone curves such that any pair of curves intersect exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The orientation of a triple of pseudolines is defined by the orientation of the triangle that they form (a degenerate triangle corresponding to a zero orientation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' a b c d Note that this pseudoline arrangement corresponds to the order type example given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now, the realizability of the order types is equivalent to the stretchability of pseudoline arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' That is finding a line arrangement with the same combinatorics as the pseudoline arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It is one of the central theorems in the field of the existential theory of the reals that stretchability is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We will use many ideas of that proof for our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We also want to point out that the notion of an order type can be easily generalized to dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The chirotope becomes a function of all d + 1 tuples and tells us the orientation in d dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To illustrate this, if we have four points a, b, c, e in 3-space, then the points a, b, c lie one a hyperplane H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then the 3 ℓ a b c d e f f a b c d e Figure 1: As ℓ separates f from the other points, a projective transformation sending the line ℓ at infinity will flip the orientation of the triangles involving f while keeping the other orientations unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' chirotope tells us on which side of H the point e lies, for an orientation of H defined by the three points a,b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It is important to note that if we take a projective transformation of the plane, we preserve the represented matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This is because lines are mapped to lines and points to points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' However, projective transformations do not preserve the order type of a point set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This is because a point may end up on the other side of some line, as pictured in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In order to get a closer relationship, we work (sometimes) with matroids endowed with a distinguished line at infinity ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then we consider valid representations of such matroids, which are those where this line is at infinity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=', all the points lie on one side of it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This definition extends to rank-k matroids using a hyperplane at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This leads to the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Given an order type O, we say that a matroid M simulates O, if the underlying matroid of O is a subset of M and if the following conditions are met: Any representation of the matroid underlying O extends to a representation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Any valid representation of M induces an point set representing O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that when M simulates O, then M has a representation if and only if O has an oriented representation: indeed, starting with a representation of M, one can always send the line at infinity to infinity using a projective transformation and thus obtain a valid representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='. Our results Our main theorem is that Matroid R-Representability is complete for the existential theory of the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Matroid R-Representability is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We provide two proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The first one relies on simulating arbitrary ETR-formulas using addition and multiplication and some technical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' There is a somewhat easier proof, starting from the fact that order type realizability is ∃R-complete, and then simulating order types using normal matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let k ≥ 3 be a fixed integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Given a rank-k order type O, we can compute in linear time a rank-k matroid M such that M simulates O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Theorem 1 easily follows from Theorem 2 as deciding whether an order type is representable over the reals is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This follows from techniques dating back to the proof of the Mnëv Universality Theorem (see [44, 52]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' However, as explained in [44], the proof that stretchability is ∃R-hard requires a significant number of intricate steps, some of which can be simplified in the setting of Matroid R-Representability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Indeed, the need for different scales (see for example [44, Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='6]), which is the main difficulty in the oriented case, can be completely circumvented in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, for the sake of completeness and simplicity, we also provide a self-contained proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We think it might be educational to first understand the proof of Theorem 1, before one tries to understand the ∃R-completeness of stretchability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 4 Proof Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In order to give an idea of the direct proof of Theorem 1, we first describe an incorrect proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then we point out the issues with this first sketch and how we can fix them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' First, it is folklore that in order to prove ∃R-completeness, it is sufficient to find a way to encode variables and some basic operations like addition (x+y = z) and multiplication (xy = z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We can force points to lie on a specific line ℓ to represent our variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, using the well-known von Staudt constructions we can simulate all the basic constraints, see Figure 2 for the construction to simulate addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This (almost) describes a rank three matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, in any realization of M, we can read a valid variable assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 0 x y x + y ℓ Figure 2: Encoding addition geometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The issue with this basic approach is that it could be that there is only one realization of M such that two points coincide, or three points lie on a common line, accidentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' If we were able to anticipate this, we could easily specify this in the description of the matroid, but in general, this is not easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We circumvent this general position issue with two fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The first fix is to reduce from a version of ETR, where we can assume that all variable values are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We call this variant Distinct-ETR, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The second fix is to observe that when we build the von Staudt construction, we can ensure that all helper points have enough freedom to avoid any coincidences or collinearities with previously defined points, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' With these two fixes, the above proof sketch works as is explained in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The proof idea of Theorem 2 goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Using arithmetic operations, we can give a variable the value y = x2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Geometrically, this implies that the point representing y is on the same side of the common line ℓ as the point representing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In other words, we can enforce two points to lie on the same side of a line with respect to a given point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We lift this to half-spaces in the plane and higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In this way, we can enforce consistent orientations of the matroid with the given order type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Results on Distinct-ETR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We define the problem Distinct-ETR as a variant of ETR as follows (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' infra for a proper definition of ETR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We are given variables X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn} and constraints of the form x + y = z, x · y = z, x = 1, x > 0, for x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, we are promised that there is either no solution at all or there is a solution (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn) ∈ Rn such that xi ̸= xj for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We show the following theorem in Section 2, which might be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Distinct-ETR is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' While we could not find any prior proof of Theorem 1 in the literature (hence this work), there are many related works from at least two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' First, as already mentioned, when one considers order-types instead of unoriented matroids, Theorem 1 is very well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Second, topological universality theorems have been proved for the real-representability of matroids in the algebraic geometry literature, see for ex- ample Lafforgue [38] and Lee and Vakil [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' While our work uses tools that are similar in spirit to those papers, it differs in that our constructions are arguably simpler and that we specifically focus on proving the computational hardness result, which is not the point of focus of those previous works, and is not entirely equivalent (see discussion below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, we believe that it is worthwhile to have a complete proof of Theorem 1 in a purely combinatorial language, as opposed to the scheme-theoretical setup of previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Shor’s proof of the Mnëv universality theorem [59, Section 4] introduces an intermediate problem called the existential theory of totally real ordered variables, which is very similar to Distinct-ETR but features 5 totally ordered variables x1 < x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' < xn as opposed to just requiring distinctness in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This ordering is desirable when one investigates oriented matroids, and unneeded for unoriented ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This difference allows for a proof that is arguably simpler than his, or at least different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Background and Related Work Matroids and Greedy A practical reason why matroids are relevant for computational purposes is that they capture in a simple way the class of discrete objects where greedy algorithms are successful in finding an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For example, the standard Kruskal and Prim algorithms to compute a Minimum Spanning Tree in a weighted graph can be abstracted by considering the vector matroid defined by an oriented incidence matrix of the graph (called a graphic matroid), and then generalized to compute in polynomial time a maximum or minimum-weight basis for any matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This property actually characterizes matroids, see for example Oxley [50, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='8] Some applications of representability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For an algorithm on matroids, a suitable encoding scheme of the input matroid is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' A common way is to take the input matroid M = (E, I) in the form of an independence oracle which answers whether a given subset of the ground elements E is independent or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' There are algorithms which run with polynomial number of oracle queries to such an oracle, for example a maximum weight independent set of a given matroid can be computed in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' However, for many natural matroid properties, it is known that there is no algorithm with polynomially bounded queries to independence oracle [29] including the representability over GF(2) and the connectivity of a matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' A vector representation of a matroid offers a compelling alternative to an independence oracle as matroid operations can be substantially more efficient using matrix operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Matroid parity problem, a common generalization of graph matching and matroid intersection problem, is solvable in polynomial time given a vector representation [40] while super-polynomial number of calls are needed under the independence oracle model [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Deciding whether the branch-width of a matroid is at most k is a common generalization of computing the branch-width, rank-width and carving-width of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' While there is an algorithm with nO(k) queries on an n element matroid for this problem [49] under the oracle model, whether the dependency on k in the exponent can be replaced by a uniform constant is not known till now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In contrast, the branch- width of a vector matroid can be computed in f(k) · n3 time [30] when the given representation is over a finite field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Another powerful application of a vector representation can be found in the theory of kernelization in parameterized complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' A surprising discovery of [37] is that for many graph cut problems, compressing the input boils down to finding a so-called representative set of a matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' When the said matroid is a vector matroid, a representative set of bounded size can be efficiently computed in polynomial time [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It turns out that solutions to graph cut problems can be encoded as independent sets in gammoids, which form a well-known class of representable matroids and of which a vector representation can be constructed in randomized polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Oriented Matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' One might wonder why we jumped from realizability of matroids to realizability of abstract order types, instead of using the perhaps closer notion of oriented matroids [11], for which one can also define a realizability problem and investigate its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The reason is that in our arguments, we reason extensively with point configurations, and the geometric interpretation described above does not adapt directly to oriented matroids, as scaling a column by a negative number could lead to a change of the underlying oriented matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The correct framework to connect oriented matroids to point configurations is to only consider acyclic oriented matroids [11, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='b], that is, those for which the geometric interpretation works readily without a need for rescaling by a negative number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This notion of acyclic, oriented matroids coincides with the notion of abstract order types, and so do their realizability problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that in some of the existing literature, oriented matroids and abstract order types are sometimes described as equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, we wanted to pay attention to this subtle difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The existential theory of the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The complexity class ∃R (pronounced as ‘ER’, ‘exists R’, or ‘ETR’) has gained a lot of interest in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It is defined via its canonical complete problem ETR (short for Existential Theory of the Reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' ETR refers to a geometric problem and ∃R refers to the complexity 6 class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' While there are several different variants of ETR, there is only one complexity class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=') and contains all problems that polynomial-time many-one reduce to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In an ETR instance, we are given a sentence of the form ∃x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn ∈ R : ϕ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn), where ϕ is a well-formed and quantifier-free formula consisting of polynomial equations and inequalities in the variables and the logical connectives {∧, ∨, ¬}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The goal is to decide whether this sentence is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As an example consider the formula ϕ(X, Y ) :≡ X2 + Y 2 ≤ 1 ∧ Y 2 ≥ 2X2 − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' among (infinitely many) other solutions, ϕ(0, 0) evaluates to true, witnessing that this is a yes-instance of ETR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We use |ϕ| to denote the length of ϕ, that is, the number of bits necessary to write down ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The solution set of an ETR-formula is called a semi-algebraic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The (bit)-complexity of a semi-algebraic set is the shortest length of any formula defining the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It is known that NP ⊆ ∃R ⊆ PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Here the first inclusion follows because a SAT instance can trivially be written as an equivalent ETR instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The second inclusion is highly non-trivial and was first proven by Canny in his seminal paper [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that the complexity of working with continuous numbers was studied in various contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To avoid confusion, let us make some remarks on the underlying machine model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The underlying machine model for ∃R (over which sentences need to be decided and where reductions are performed) is the word RAM (or equivalently, a Turing machine) and not the real RAM [24] or the Blum-Shub-Smale model [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The complexity class ∃R gains its importance by numerous important algorithmic problems that have been shown to be complete for this class in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The name ∃R was introduced by Schaefer in [52] who also pointed out that several NP-hardness reductions from the literature actually implied ∃R-hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For this reason, several important ∃R-completeness results were obtained before the need for a dedicated complexity class became apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Common features of ∃R-complete problems are their continuous solution space and the nonlinear re- lations between their variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Important ∃R-completeness results include the realizability of abstract order types [48, 59] and geometric linkages [53], as well as the recognition of geometric segment [36, 44], unit-disk [34, 46], and ray intersection graphs [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' More results appeared in the graph drawing com- munity [22, 23, 41, 54], regarding the Hausdorff distance [33], regarding polytopes [21, 51], the study of Nash-equilibria [6, 9, 10, 25, 55], training neural networks [3, 8], matrix factorization [20, 56, 57, 58, 61], or continuous constraint satisfaction problems [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In computational geometry, we would like to mention geometric packing [4], the art gallery problem [2], and covering polygons with convex polygons [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Recall that NP is usually described using a witness and a verification algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The same character- ization exists for ∃R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Instead of the witness consisting of binary words of polynomial length, we allow in addition using real-valued numbers as a witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, in order to be able to use those real numbers, we are allowed to work on the so-called real RAM model of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The real RAM allows arithmetic operations with real numbers in constant time [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Topological Universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Many results and techniques on the existential theory of the reals actually, precede the study of this complexity class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The underlying idea was to study how complicated solution spaces can be from a topological perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For example, if we want to study convex polytopes, we are often interested in the properties of their face lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The face lattice is a purely combinatorial object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, it is natural to ask which face lattices are realizable by polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' If there would exist an easy combinatorial description of realizable face lattices, convex polytopes could be much better understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Given a specific face lattice L, we can study its suitably defined solution space S(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As the realizability question can be formulated as an ETR-formula, it follows that S is a semi-algebraic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now, let T be a different semi- algebraic set, we wonder whether there exists a face lattice L such that S(L) is homotopy-equivalent to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Maybe surprisingly topological universality states that there is such an L for any semi-algebraic set T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This type of property feels very strong, as it intuitively states that we need to encode the vast complexity of semi-algebraic sets into the problem of realizing convex polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Indeed, many of the results that establish such topological universality results also imply ∃R-completeness [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' However, topological universality can also be established for NP-complete problems as has been shown by Bertschinger, El Maalouly, Miltzow, Schnider, and Weber [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As the authors showed it is sufficient to encode the topology of simplicial complexes, as it is possible to triangulate semi-algebraic sets [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In 7 other words, the difference between the wild semi-algebraic sets and the tame simplicial complexes is not that they emit a more complicated topological structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The difference comes from the ability of semi- algebraic sets to encode complicated topological spaces in a much more concise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' (To be precise the description complexity of a topological space might be exponentially smaller using the language of semi- algebraic sets, compared to simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=') ∃R-completeness can be interpreted as giving a concise encoding of semi-algebraic sets into a different domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To make our life easier, we don’t care about the complete preservation of the complete topology, but merely of the property of being empty or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Still, in order to do so one usually also preserves topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This is the reason why there is a close connection between topological universality and ∃R-completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' But given that NP-complete problems may also admit universality theorems ∃R-completeness may arguably be considered the more interesting finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Stronger Universality Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We want to point out that previous universality results often also showed stronger results than mere topological universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For instance, Richter-Gebert showed such a stronger universality theorem for polytopes [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Specifically, let P be a polytope, then the face lattice is the family of faces of different dimension together with their inclusion order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Given a face lattice F, we can define the set of polytopes V (F) having face lattice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Richter-Gebert showed that for every semi-algebraic set S there exists the face-lattice F of a polytope such that V (F) is stably-equivalent to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To define the notion of stable-equivalence goes above the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It is interesting to note that stable-equivalence encapsulates more than just the topology of S, but also to a degree the “geometry” of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Discussion on Input Matroid Encodings In this paper, we study the R-realizability, where the input matroid is given with all bases (maximal inde- pendent sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This would seem unconventional at first glimpse, especially for those familiar with matroid theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We would like to address the subtleties around our problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Types of encodings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For matroids, three possible descriptions are examined in the literature, namely an explicit description of sets, a description via an oracle, and a succinct description with a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Recall that an input graph for a graph problem can be given as an adjacency matrix, or equivalently the family of vertex subsets of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Similarly, an input hypergraph is typically given as a set family with an explicit description of all hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' An immediate analogue of such an hypergraph description for a matroid is an explicit enumeration of all bases (maximal independent sets) or all circuits (minimally dependent sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' However, explicitly stating all independent sets, bases, or circuits is unconventional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' the most common size measure of a matroid is the number of elements, which is polynomially bounded by the size of a graph or matrix that is generalized by a matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' On the other hand, the number of bases or circuits can be prohibitively large in comparison to the number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Specifically, it is known that the number of distinct matroids on n elements is doubly exponential in n [35], hence in space of size polynomial in n one cannot describe an arbitrary input matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For further details about explicit matroid encoding, see [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' When the matroids under consideration are representable over a field F, a matrix over F provides a succinct description of a matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' There are well-studied matroid classes that are representable such as uniform matroids, graphic matroids, and transversal matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' However, not all matroids are representable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Hence, an important question is to decide whether an input matroid is representable over a specific field, or over any field at all, and to find a representation if one exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Due to the limitations of the above two explicit descriptions, the most common way to encode an input matroid without any restriction is with an oracle, often with an independence oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' One can view an independence oracle as a black box expressing a boolean function on n variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The boolean function is on n input variables and outputs 0 or 1 depending on whether the input corresponds to (a characteristic vector of) an independent set of the said matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Problems with black box functions, an implicit input with oracle access are studied in the context of learning a function with a small number of queries, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Polynomial Identity Testing, and also in the context of search problems where a graph is accessed by adjacency query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Such a problem does not fit in the classic computational complexity, where an explicit string of numbers is expected as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Moreover, learning a black box (boolean) function cannot be done efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As mentioned previously, even when the input boolean functions are restricted to be matroid oracles, deciding 8 whether a nontrivial matroid property holds or not requires 2Θ(n) queries [60] even for basic properties such as connectivity and representability over F = GF(2), and even with a randomized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, an oracle encoding for F-representability problem does not appear to be a fruitful setting to better understand the algorithmic aspects of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Moreover, we shall argue below that, with an explicit matroid description, there is an intriguing difference in the computational complexity of F- representability between the cases when F is finite and when F =R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' F-representability with explicit bases description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let us consider a matroid description that pro- vides a matroid M as a pair (E, B), where all bases of M are stated in the collection B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In this setting, we shall argue that the problem of deciding whether M is F-representable is in NP for a finite field F and in ∃R for F = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We shall also argue that F-representability is likely to be in co-NP for a finite field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This makes an interesting contrast with the case F = R, for which the corresponding non-representability problem is unlikely to be in ETR due to our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' First, let us see that F-reprsentability is in NP for finite F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Indeed, a matrix A (whose columns are labeled by the elements of E) over F with M[A] = M can be taken as a witness for F-representability of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Moreover, one can conceive a polynomial-time verification algorithm for the pair M = (E, B) and A as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let B(A) be the set of bases of M[A] and recall that M = M[A] if and only if B = B(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Whether B ⊆ B(A) can be easily verified in time polynomial in |B| + rk(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For this, we first compute the column rank of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' If it is different from rk(M), this trivially implies M[A] ̸= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Henceforth, let us assume that the column rank of A equals rk(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now, for each basis B ∈ B, one checks whether the submatrix A[E, B], the submatrix of A consisting of all columns labeled by the elements of B, is full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' If any B ∈ B fails the test, we know that A is not a representation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, we may assume that B ⊆ B(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To verify whether the equality holds, we rely on the following property, which is easily derived from the Basis Exchange Property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We include its proof in the appendix for self-containment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' (⋆) If B ⊊ B(A), then there exists B ∈ B and B′ ∈ B(A) \\ B′ such that |B△B′| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let B be the set of all bases of a matroid M and let B′ ⊊ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then there exist bases B′ ∈ B′ and B ∈ B′ \\ B with |B′△B| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Choose B′ ∈ B′ and B ∈ B \\ B′ so that |B′ ∩ B| is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let x ∈ B′ \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Recall the Basis Exchange Property: For any distinct bases W ′, W of a matroid and an element x ∈ W ′ \\ W, there exists an element y ∈ W \\ W ′ such that W ′ − x + y is a basis of the matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' By the Basis Exchange Property, there exists y ∈ B \\ B′ such that B′′ := B′ − x + y is a basis of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' If B′′ belongs to B′, we have B′′ ∩ B = (B′ ∩ B) + y, which contradicts the choice of B′ and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, B′′ belongs to B \\ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that |B′ ∩ B′′| = |B′ − x| = r − 1 and (thus B′′ = B by the choice of B′ and B) and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Hence, the last step of the verification algorithm tests if there exist a basis B ∈ B and two element x ∈ B and y ∈ E − B such that B − x + y is not a basis of M but the corresponding set of columns of A is independent, which precisely tests if B − x + y ∈ B(A) \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' If such a triple B, x, y exists, clearly B ⊊ B(A) and thus M[A] ̸= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Conversely if B ⊊ B(A), there exist such a triple B, x, y by Property (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, we examine all triples (B, x, y) with B ∈ B, x ∈ B and y ∈ E − B and certify that B − x + y is either in B or dependent, in which case the verification algorithm can correctly conclude that B(A) = B, thus M[A] = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Otherwise, the verification algorithm concludes M[A] ̸= M and rejects the witness A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The presented verification algorithm shows that F-representability is in NP for each finite field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' A matrix over F as witness and the polynomial-time verification algorithm naturally extend to the case when F = R, where the verification algorithm works on a real RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, R-representability is in ∃R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For details on the characterization ∃R via a witness and a polynomial-time verification algorithm on a real RAM, see [24], and also the discussion in the paragraph above about the existential theory of the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, it is likely that F-representability is in co-NP for each finite field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It is known [26] that for any prime field F, non-F-representability can be certified by evaluating the ranks of O(n2) subsets of 9 an n-element matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Notice that when the matroid M = (E, B) is given with an explicit description of all the bases B of M, evaluating rk(X) for X ⊆ E can be done in time polynomial in the input size because rk(X) equals the maximum of |X ∩ B| over all B ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, F-representability is in co-NP under the explicit bases description for each prime field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For an arbitrary finite field F, not necessarily prime, up to our best knowledge there is no published result which establishes that a polynomial number of rank evaluations suffices for non-F-representability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' However, it is known that a positive resolution of Rota’s conjecture implies that only a constant, depending on |F| only, number of rank evaluations would suffice [50] to certify that a given matroid is not F-representable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The proof of Rota’s conjecture was announced in 2014 by Geelen, Gerards and Whittle [32] although it is expected to take a few more years for the full proof to be written for publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, F-representability appears to be in NP ∩ co-NP when the input is given as the exhaustive list of bases for each finite F given the claimed proof of Rota’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Given that, deciding the computational complexity of F-representability for each finite F with explicit bases description is an intriguing question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For F = GF(2), a polynomial-time algorithm is straightforward from the uniqueness of a binary representation (up to linear transformation) and the fact that such a representation can be efficiently obtained [50];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' after constructing a matrix over GF(2), we apply the above verification algorithm for NP membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' However, even for F = GF(3) it is not clear whether a matrix over GF(3) can be efficiently constructed although it is known that there is a unique representation over GF(3) for a matroid representable over GF(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As far as we are aware, there is no efficient procedure known for constructing the representation of a matroid M with a promise that M is representable over GF(3), when M is given with an independence oracle or even given as a matrix over the rationals Q [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Getting an input matroid as explicit bases description might help to circumvent this obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In contrast to the case of finite field, it is impossible to certify non-R-representability with a polynomial number of rank evaluations [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Finally, for R-representability we showed that ∃R-complete, exhibiting a noticeable diversion from F-representability for finite F which appears neither NP-complete nor co-NP- complete with explicit bases description under the assumption NP ̸= co-NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, our result highlights the recurring contrast between representability over finite field and over the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 2 Distinct-ETR This section serves as a preparation for the later reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Specifically, we show ∃R-completeness of a variant of ETR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We name this variant Distinct-ETR, as we can assume that there is a solution with all variables holding distinct values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This property will be key for encoding into matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Most of this section follows standard techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This section is dedicated to the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The idea is that we first establish the hardness of an ETR variant (STRICT-INEQ) with an open solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It is clear that all variables can be assumed there to have distinct values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then, we reduce again to a variant where we use only the basic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The reduction goes in four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' From ETR to ETRAMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then from ETRAMI to Feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then from Feasibility to STRICT-INEQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' And at last from STRICT-INEQ to Distinct-ETR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that step 1,2, and 3 have already been done (among others) by Schaefer and Štefankovič [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We sketch the main steps of their reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Specifically, we point out some properties that were not explicitly emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We start to explain a simple trick that is excessively used in those types of constructions in order to build small, very small, and very large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 10 Number Constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Before we describe the reduction it might be useful to understand how we can construct variables that must have specific rational values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' If we want to build integers of polynomial size, we can do this by simply repeatedly adding or subtracting a one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' a1 = 1, ai+1 = ai + a1, ai−1 + a1 = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It is easy to see that ai = i, for all ai that are defined in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' If we want to build a very large number, say 22k, the previous approach cannot be done in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Instead, we can use repeated squaring as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' x0 = a2 + a0(= 2), xi+1 = x2 i , for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It holds inductively that xi = 22i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Similarly, we can construct very small numbers say 22−k, as follows: x0 + x0 = 1, xi+1 = x2 i , for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It holds inductively that xi = 2−2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that we can also use strict inequalities to build large and small numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For example, x0 > 2, xi+1 > x2 i , for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , k implies that xi > 22i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We will use these standard tricks repeatedly later in the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Reduction from ETR to ETRAMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We define the problem ETRAMI as a variant of ETR as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We are given variables X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn} and constraints of the form x + y = z, x · y = z, x = 1, for x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, ETRAMI is a variant of ETR without negations, inequalities, disjunctions, conjunctions and the only constant is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It is folklore that ETRAMI is ∃R-complete and follows implicitly from various papers [44, 55, 59] Lemma 5 (folklore).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' ETRAMI is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' ∃R-membership follows from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The idea of the reduction is to simplify an ETR-formula in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For instance, we can remove negations by replacing ¬p > 0 by p ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In this way, we can remove all negations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We can replace inequalities by observing that p ≥ 0 is equivalent to ∃x : p = x2 and p > 0 is equivalent to ∃x : px2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We can replace p = 0 ∧ q = 0 by p2 + q2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Similarly, we can replace p = 0∨q = 0 by pq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Thereafter, we end with a single polynomial equation p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We construct variables for each coefficient value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then we replace each coefficient with an appropriate variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' At last, we replace each occurrence of multiplication and addition inductively by introducing one more variable and one more constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We end with a single equation of the form x = 0, which can be replaced by z = 1 and z + x = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Reduction from ETRAMI to Feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In Feasibility, we are given a single polynomial p ∈ Z[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn] of degree at most four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We are asked if there exists some x ∈ Rn such that p(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, we require each coefficient to be of absolute value at most 36n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Below, we will show how to achieve this upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Feasibility is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Again, ∃R-membership follows from the fact that Feasibility is a special case of ETR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To show hardness we sketch a reduction from ETRAMI that is already known [44, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let f1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , fm = 0 be the constraints of some ETRAMI instance ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' (For example x + y = z becomes x + y − z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=') Let p = f 2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' + f 2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Clearly, ϕ is satisfiable if and only if p has a zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As each fi has a degree at most two it holds that p has degree at most four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As there are only 3n3 possible distinct constraints in ϕ, we have m ≤ 3n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that each term f 2 i gives rise to at most six monomials and each coefficient is at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' (For example (x + y − z)2 = x2 + 2xy − 2xz + y2 − 2yz + z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=') Therefore each coefficient has absolute value at most 12m = 36n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Thus, we can rewrite p as a sum of monomials with bounded-sized coefficients, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 11 Reduction from Feasibility to STRICT-INEQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In a STRICT-INEQ instance, we are given a sen- tence of the form ∃x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn ∈ R : ϕ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn), where ϕ is a well-formed and quantifier-free formula consisting of polynomial strict-inequalities in the vari- ables and the logical connectives {∧, ∨}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that the solution space {x ∈ Rn : ϕ(x)} is always open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' STRICT-INEQ is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Again membership follows from the fact that ETR is more general then STRICT-INEQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To show ∃R- hardness we reduce from Feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The idea of the reduction is to replace p(x) = 0 by −δ < p(x) < δ, for some sufficiently small δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To this end, we employ two lemmas as formulated by Schaefer and Štefankovič [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that the actual proof comes from real algebraic geometry and can be read for instance in [5] and [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Every non-empty semi-algebraic set in Rn of complexity at most L ≥ 4 contains a point of distance at most 2L8n from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' If two semi-algebraic sets in Rn each of complexity at most L ≥ 5n have positive distance (for example, if they are disjoint and compact), then that distance is at least 2−2L+5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In this context the distance between two sets A, B is defined as d(A, B) = inf a∈A,b∈B ∥a − b∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that ∥ · ∥ denotes the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In order to apply Lemmas 8 and 9, we define the following three semi-algebraic sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' First, we define the solution set for p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' S = {x ∈ Rn : p(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let R = 2L8n, where L is the bit-complexity of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that L is the upper bound by the length of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Using Lemma 8, we know that S is empty if and only if S ∩ B(R) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' (We denote by B(R) the ball of radius R around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=') This motivates us to define the sets S′ 1 = {(x, z) ∈ Rn+1 : p(x) = z ∧ ∥x∥2 ≤ R2}, and S′ 2 = {(x, z) ∈ Rn+1 : z = 0 ∧ ∥x∥2 ≤ R2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that S′ 1 ∩ S′ 2 = (S ∩ B(R)) × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, S′ 1 and S′ 2 are compact and thus we can apply Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Unfortunately, the description complexity of S′ 1 and S′ 2 are exponential, if we write R out in binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, we define S1 and S2 slightly differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Namely, we add some extra variables, whose sole purpose is to encode R using repeated squaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let L be the max of the bit complexity of S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that L = O(L + n log L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let δ be as in Lemma 9 applied to S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It holds that S = ∅ is equivalent to S ∩ B(R) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This in turn is equivalent to S1 ∩ S2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' And this is equivalent to p(x) ≤ −δ or δ ≤ p(x), for all x ∈ B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In other words, we have ∃x : −δ < p(x) < δ and ∥x∥2 < R2 (1) if and only if ∃x : p(x) = 0 This obviously also works if we would use any smaller δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Maybe not so obviously, this also works for R of between 2L8n and 2L8n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The lower bound allows us to apply Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The upper bound allows us to apply Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Using repeated squaring, we create numbers a > 2L8n and b < 2L8n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, we add the inequalities a < R < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that we can construct a number δ that is at most 2−2L+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Our STRICT-INEQ instance ϕ consists of the three inequalities from Equation (1) and some extra variables and constraints to bound R and δ as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 12 Reduction from STRICT-INEQ to Distinct-ETR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In this section, we show that Distinct-ETR is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We are not actually reducing from STRICT-INEQ but from the instance ϕ described in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let δ, R be the given numbers, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn the variables and p the polynomial as in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Recall that p has a degree at most four and the coefficients are bounded integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We will introduce new variables in order to construct δ, R, ∥x∥2, and p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Thereafter, we will argue about distinctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' First, we construct variables holding the values of the integers −36n3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , 36n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Those variables are meant to represent the coefficients of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Recall that this was an upper bound on the values of those coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It has been described above how to construct those integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, we add variables holding the values R = 2L8n and δ = 2−2L+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' (As we reduce from Equation (1), we do not need to approximate the values δ and R, but can construct them directly, as Distinct-ETR allows us to use equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=') If in this process two variables hold the same value, we can detect this and remove one of the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now, we construct p(x) and ∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' First, we construct all possible �n 4 � monomials of degree at most four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For example, N = xyzw is constructed in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' N1 = xy, N2 = N1z, and N = N2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Again, whenever two identical monomials would appear, we would be able to notice this and make an appropriate replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let us denote p(x) = m � i=1 aiMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Here, ai is the coefficient of the monomial Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We construct Pk = �k i=1 aiMi inductively as follows: P1 = a1M1 and Pk = Pk−1 + Tk, with Tk = akMk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We denote by P = Pm the variable holding the value of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We also construct X = ∥x∥2 = x2 1 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='+x2 n in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' At last, we add the variables a, b, c, and the constraints a + δ = P, b + P = δ, c + X = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now we can enforce the inequalities −δ < p(x) < δ and ∥x∥2 < R2 by a > 0, b > 0, c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To summarize, we started with the variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We have created variables C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , Cs that each holds a different integer/rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, we constructed some variables V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , Vt such that each Vi is a polynomial function gi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that all the gi have a degree of at most four and small integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' If for two variables Vi and Vj, we have that gi = gj, we can detect this and remove one of them and replace each occurrence with the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This finishes the description of the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We denote this instance ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' It remains to argue correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To show correctness, we observe that ϕ has a solution if and only if ψ has a solution as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Indeed, all new variables and constraints only “build” the correct polynomials and the numbers δ, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Thus it remains to show that if ϕ has a solution if and only if ψ has a solution with all variables taking distinct values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The backward direction is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, we assume that ϕ has at least one solution x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As the solution space of ϕ is open, there is an open ball fully contained in the solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Clearly, the variables C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , Cs have all fixed distinct values by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Every other variable Vi can be expressed as a polynomial function gi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As all the gi are distinct there must be some x ∈ B such that gi(x) ̸= gj(x), for all i ̸= j and gi(x) ̸= Cj, for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Otherwise, two of the polynomials would be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 13 3 Arithmetic using Matroids In this section, we describe how to encode addition and multiplication of real numbers using rank-3 matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We rely on the von Staudt constructions, which are very well-known, perhaps with the caveat that they are usually stated for oriented matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' But as we shall see, no orientedness is actually required to make them work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We follow the presentation of Matoušek [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The setup for both operations is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We have a line ℓ containing three distinct distinguished points called 0, 1, and ∞, and a fixed second line ℓ∞ crossing ℓ at ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Given the variable x, we denote the corresponding point representing it by x in fat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This way we easily distinguish between a point and the corresponding variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' A point x on the line ℓ can be interpreted as a real number using cross-ratios: if we denote by d(a, b) the oriented distance between two points a and b on the line ℓ, then the quantity (x, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 0, ∞) := d(x, 0) · d(1, ∞) d(x, ∞) · d(1, 0) is a real number invariant under projective transformations, which, by a slight abuse of notation, we simply denote by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that if ∞ is progressively sent to infinity using a projective transformation and d(0, 1) is scaled to one in this formula, x converges to d(0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Thus this cross-ratio matches the geometric location of x on ℓ under some projective transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now, given two points x and y on ℓ, we describe geometric operations to compute points on the line x + y and x · y on ℓ representing their addition and their multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' ∞ 0 x y x + y a b ℓ∞ ℓ 0 x y x + y c d ℓ c d Figure 3: Encoding addition geometrically Addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The construction is pictured in Figure 3, left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We first introduce two distinct auxiliary helper points a and b situated anywhere on ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The line connecting 0 to b crosses the line connecting x to a in a point c, then the line connecting ∞ to c crosses the line connecting y to b in a point d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Similarly, the line connecting a to d crosses ℓ in a point that we define to be x + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The rationale behind this construction is that by a projective transformation we can consider ℓ∞ to be a line at infinity, bringing us the Figure 3, right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now the line containing c and d crosses ℓ at a point at infinity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=', these two lines are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Likewise, the line containing c and d is parallel to ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then the parallelity of the lines immediately shows that d(0, x + y) = d(x, y) + d(0, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In other words, the point x + y has value x + y, justifying the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' ∞ 0 x y xy a b ℓ∞ ℓ 0 x y xy c d 1 ℓ c d 1 Figure 4: Encoding multiplication geometrically 14 Multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The construction is pictured in Figure 4, left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As before, we first introduce two distinct auxiliary helper points a and b situated anywhere on ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The line connecting 1 and b crosses the line connecting x and a at a point c, and the line connecting 0 and c crosses the line connecting b and y at a point d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Finally, the line connecting a and d crosses the line ℓ at a point that we define to be xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' By sending the line ℓ∞ at infinity using a projective transformation, we obtain Figure 4, right, where one can readily show using the parallel lines that d(0, xy) = d(0, x)d(0, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In other words, the point xy has value xy, justifying the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Encoding these geometric constructions using matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The addition and multiplication construc- tions defined above can be entirely encoded using matroids: the independent sets are exactly the empty set, all the singletons, all the pairs of points, and all the triples of non-aligned points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let us insist here that no orientation was ever enforced during the constructions, and thus we do not need oriented matroids to de- scribe them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, these operations can be chained arbitrarily, allowing to encode polynomials using matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' However, some very important care needs to be taken here: while it is clear from the construction which points should be aligned, we also need to make sure that points that should not be aligned can be assumed to not be aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For example, it could a priori happen that a line going through two helper points c1 and c2 somehow accidentally happens to pass through a variable x of the polynomial we are encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In that case, the matroid would not properly encode the geometric construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This motivates the following definition: we say that the set S of helper points used during an addition or multiplication construction is free if for any finite set of lines L and points P not involved in the construction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' the points in S can be perturbed so that: the incidences required by the addition or multiplication construction still hold,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' no point of S lies on a line L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' and no pair of points of S is aligned with a point of P (in particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' no point of S coincides with a point of P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' A key property of the addition and multiplication construction is that the four helper points that they rely on form a free set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Indeed, the points a and b can be placed freely on the line ℓ∞, and thus can be perturbed so as to avoid the lines and points of P and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Such a perturbation induces a perturbation of c and d in a two-dimensional open set, therefore allowing them to avoid lines of P and L, but also ensuring that no line going through a pair of points in {a, b, c, d} also goes through a point in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This freedom will be leveraged in the proofs of Theorem 2, respectively Theorem 1, to ensure that the matroid correctly encodes orientation predicates, respectively systems of polynomial equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Strict inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The multiplication construction can be leveraged to simulate a strict inequality con- straint x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Indeed, x > 0 if and only if there exists z ∈ R such that z ̸= 0 and x = z2 = zz, which can thus be simulated using a helper point z distinct from 0 and the above multiplication construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' However, this construction would require us to use as a helper point a fixed point z on the line ℓ, which could not be perturbed and thus would not be free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This can be resolved by using an additional variable: we first introduce a helper point y for which we ensure that y > 0 using the multiplication gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then we use another multiplication gadget to ensure that z > y: note that this amounts to enforcing that z lies on the same side of y than 1 does, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=', this can be tested by using another multiplication gadget where 0 is replaced by y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now, in these two multiplication gadgets, neither y nor any of the other helper points is fixed, and therefore we can perturb them to avoid any fixed set of lines and points, showing that they form a free set of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 4 Proof of Theorem 2 Theorem 2 is proved by using the arithmetic constructions described in the previous section, in particular the one for strict inequalities, to simulate orientation predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 15 Simulating rank 3 order types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We will first consider the case of rank equal to 3 and treat the general case later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let O = (E, χ) be an order type on n elements of rank 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' (E = {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , en}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=') We construct a matroid M from O inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To be more precise, we construct matroids, M3, M4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , Mn = M such that Mi simulates Oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Here, Oi = (Ei, χi) is the order type formed by the first i elements of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' All of our matroids Mi will feature a distinguished line ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The matroid M3 merely contains e1, e2, e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Without loss of generality we assume that the triple t = {e1, e2, e3} are independent in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' (Otherwise, all triples in O are dependent, which can trivially be simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=') We first add a line at infinity, on which no e1, e2 or e3 lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We can assume that t is oriented correctly in any representation of M3, as otherwise, we can just reflect the representation and get a correct representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, M3 satisfies the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now, let us assume that we already constructed Mi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We construct Mi from Mi−1, by adding the element e = ei from O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, for each triple t = {a, b, e} ⊂ Ei, we need to ensure that t is oriented correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In case that χ(t) = 0, we can encode this into Mi directly by specifying that (a, b, e) forms a rank-2 dependent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Thus it remains to consider the case χ(t) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let t′ = {a, b, c} ⊂ Ei−1 such that χ(t′) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that such a triple t′ must exists, as otherwise all points of Ei lie on a common line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This would be a contradiction to the fact that M3 is formed by an independent triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Using the orientation of t′ we add a small constant number of helper points and we will enforce the correct orientation of t in Mi as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Thus it remains to show the following lemma, where we use the notion of a free set of helper points defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Given the independent triple t′ = {a, b, c} in a matroid M and another point e, we can enforce that in any valid representation of M, the triple t = {a, b, e} is oriented identically to t′, or that it is oriented opposite to t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We do this by adding a constant number of helper points to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, this set of helper points is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This construction goes in essentially two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' First, we show how we can enforce an element x on the line ℓ = ℓ(a, b) to be on the same side of a as b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' a b x It relies on standard constructions to do encode arithmetic operations as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Although, those constructions can be well described and understood, without any reference to arithmetic operations, the language of arithmetic operations gives a better intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The underlying idea is that we interpret a as zero, and b as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Indeed, the constraint that c lies on the same side of a as b in any representation where the line ℓ∞ is sent to infinity amounts to enforcing that c > 0 when a is interpreting a as zero and b as one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As explained in Section 3, this can be encoded using multiplication gadgets, in such a way that none of the helper points accidentally lies on a previously used line or point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In the second step, we use the previous tool to enforce that e and c lie on the same (or the opposite) side of the line ℓ(a, b) as c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' a d b c c′ e Figure 5: Forcing c to be on the same side of ℓ(a, b) as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Half-open lines denote the strict inequality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' To this end, we first define a point c′ for which we enforce that on the line ℓ(a, c), it lies on the same side of a as c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then we define a point d situated on the line ℓ(a, b) and a line ℓ′ with points c′, d, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' See Figure 5 16 The condition that e, c′ are on the same side of d on the line ℓ′ can be enforced using the previous gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, it is equivalent to c′ and e being on the same side of ℓ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Lastly, by construction the triple {a, b, c′} is oriented identically to {a, b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We can use the same tool to enforce that e and c lie on the opposite side of the line ℓ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We define c′ as before, so that it lies on the same side of ℓ(a, b) as c, and then we simply need to enforce instead that c′ lies on the same side of e as d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As explained in Section 3, the strict inequalities gadgets can be directly encoded into the independent sets of the matroid, and by construction, the helper points always have at least one degree of freedom, and thus can form a free set of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, the matroid Mi is entirely defined by the dependency constraints indicated by the lines in the geometric constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We can now conclude the proof of Theorem 2 for rank-3 matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Given an order type O of rank 3, for which we can assume that there is at least one independent set of size 3 (otherwise the orientation predicates are trivial), we inductively encode the orientation predicates into an matroid using the helper points provided by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' At each stage of the induction, the freedom of the set of helper points can be used to ensure that the helper points do not yield any accidental dependencies with all the points and lines previously placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' At the end of the induction, by Lemma 10, any valid representation of the resulting matroid M induces a representation of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Conversely, any representation of O can be extended to a representation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore M simulates O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' All the constructions can clearly be done in linear time, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Simulating rank k Matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The construction for rank k is identical to that of rank 3 as explained above, with two exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' First, the induction basis starts with a matroid on k elements instead of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Second, we have to describe the simulation of an oriented k-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For this, we use the same trick that forces a point to lie on a specific side of a line, where we just replace a line with a hyperplane P, as pictured in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' P d a a′ x a1 Figure 6: Forcing x to be on the same side of P as a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' More precisely, in an inductive step where we add a point x, we need to enforce the orientation of all the independent k-tuples t = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' ak−1, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' For this, we take another k-tuple t′ = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' ak−1, a} for which χ(t′) ̸= 0, and we devise a gadget to encode the constraint that t is oriented identically, or opposite to t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This will be done using the gadget described above that enforces that in any valid representation, for three aligned points a, b and c, c lies on the same side of a as b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' That gadget was defined for rank 3 matroids and thus representations into R2, but readily works in higher dimensions: one should simply ensure using dependency constraints that all the points involved in the gadget lie on a common plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The hyperplane at infinity will intersect this common plane in a line, which takes the role of the line at infinity in the gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now, we proceed as in the rank 3 case: considering the hyperplane P generated by {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' ak−1}, we first define a point a′ that lies on the same side as a on the line ℓ(a1, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then we can enforce that a′ is on the same side of P as x by introducing the point d at the intersection of P and the line going through a′ and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then we enforce that x is on the same side of d as a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' In order to enforce that a′ and x are on opposite sides of P, we instead enforce that x and a′ are on opposite sides of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Finally, we observe that all the helper points that we have introduced are free, where the notion of freedom is generalized to also 17 disallow k-dimensional dependencies: once again this follows from the fact that all the helper points that we introduce have some wiggle room to be perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The rest of the proof proceeds identically to the rank-3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 5 Proof of Theorem 1 The proof of Theorem 1 is by a direct reduction from the problem Distinct-ETR, using the arithmetic constructions described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Let us start with an instance of Distinct-ETR given by variables X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn} and constraints of the form x + y = z, x · y = z, x = 1, x > 0, for x, y, z ∈ X, with the promise that if there is a solution, then there is one where all the variables are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' As in the proof of Theorem 2, we can construct the constant 0 using a constraint x + y = x, and thus, without loss of generality, by the distinctness assumption, we can assume that there are exactly two variables in X equal to respectively 0 or 1, and that the others are different from 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' By a slight abuse of language, we remove those from X and denote them directly by 0 and 1 in the rest of the description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We define a rank-3 matroid M as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' First, we have a line ℓ consisting of three distinguished distinct points 0, 1 and ∞, as well as n distinct points (and distinct from {0, 1, ∞}) corresponding to the variables {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We also add a line at infinity ℓ∞ going through ∞ and no other point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We then use the gadgets from Section 3 to inductively encode all the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Note that there is at most one constraint x = 1 which can be hardcoded from the start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Then let us assume inductively that we have defined a matroid Mi encoding the first i constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The i + 1th constraint is an addition, a multiplication, or a strict inequality, which can be encoded using a geometric construction as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Now, the key property of these constructions is that the set of helper points is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, the only linear dependencies involved in the construction are those of that construction, which can be readily encoded into a matroid Mi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' We now prove that the Distinct-ETR instance has a solution with distinct variables if and only if the matroid M is representable over the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' First, if Distinct-ETR has a solution, we obtain a representation of M over the reals by placing all the variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' , xn on the line ℓ at the values indicated by the solution, and by sending the line at infinity to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The geometric constructions are then represented one by one, and since the helper points are free, by perturbing them if needed we can ensure that they are all distinct, that no three of them are colinear, and that they do not form colinearities with previously placed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Therefore, this constitutes a correct representation of the matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Conversely, given a representation of the matroid M over the reals, we read the values of the variables on the line ℓ using cross-ratios as explained in Section 3 (or equivalently we send ℓ∞ to ∞ and use the oriented distance to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This gives us values for the variables of the Distinct-ETR instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' The definition of the addition, multiplication and strict inequality constructions ensures that each of the constraints will be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Furthermore, by definition of the matroid, all of the variables are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' References [1] Mikkel Abrahamsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' Covering Polygons is Even Harder.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content='05678, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQffwRJ/content/2301.03221v1.pdf'} diff --git a/j9E2T4oBgHgl3EQfdQfR/content/tmp_files/2301.03905v1.pdf.txt b/j9E2T4oBgHgl3EQfdQfR/content/tmp_files/2301.03905v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..841572af13d94a1bf96b2f7e173fa7a1dc69fbe2 --- /dev/null +++ b/j9E2T4oBgHgl3EQfdQfR/content/tmp_files/2301.03905v1.pdf.txt @@ -0,0 +1,1711 @@ +A new ultra low-level HPGe activity counting setup in the Felsenkeller +shallow-underground laboratory +S. Turkata, D. Bemmererb, A. Boeltzigb, A. R. Domulaa, J. Kocha,b, T. Lossina,b, M. Osswalda,b, K. Schmidtb, K. +Zubera +aTechnische Universit¨at Dresden (TU Dresden), 01069 Dresden, Germany +bHelmholtz-Zentrum Dresden-Rossendorf (HZDR), Bautzner Landstr. 400, 01328 Dresden, Germany +Abstract +A new ultra low-level counting setup has been installed in the shallow-underground laboratory Felsenkeller in Dresden, +Germany. It includes a high-purity germanium detector (HPGe) of 163 % relative efficiency within passive and active +shields. The passive shield consists of 45m rock overburden (140 meters water equivalent), 40 cm of low-activity concrete, +and a lead and copper castle enclosed by an anti-radon box. The passive shielding alone is found to reduce the background +rate to rates comparable to other shallow-underground laboratories. An additional active veto is given by five large plastic +scintillation panels surrounding the setup. It further reduces the background rate by more than one order of magnitude +down to 116(1) kg−1d−1 in an energy interval of [40 keV;2700 keV]. This low background rate is unprecedented for +shallow-underground laboratories and close to deep underground laboratories. +Keywords: +Low-background physics, Nuclear astrophysics, underground laboratory, HPGe detector, muon veto, active +shielding +1. Introduction +A variety of scientific fields call for detection systems +which are able to measure radioactive samples with ultra- +low activities in the order of µBq to mBq [1]. A case in +point is the search for rare processes such as dark matter +interactions [2, 3] or rare germanium decays [4]. +These +include neutrino-accompanied (2νββ) [5] and neutrinoless +(0νββ) [6, 7] double beta decays, among others. +The need for ultra-low background radioactivity mea- +surements is served by a number of low-background lab- +oratories which are using high-purity germanium (HPGe) +detectors in shallow or deep underground settings [8]. All +of these laboratories use massive lead and copper shields +that are sometimes complemented by additional materials +in order to suppress γ-ray background from the immedi- +ate surroundings of detector and laboratory. The effects +of radioactive radon gas are usually mitigated by placing +detector and sample in an airtight enclosure that is con- +tinually flushed with a radon-free gas, preferably N2. +Two additional background sources are more difficult +to treat, namely cosmic-ray muons and neutrons. The for- +mer usually play no role in deep-underground laboratories +[9, e.g.], where the muon flux is reduced by six or more or- +ders of magnitude compared to sea level, rendering its ef- +fects negligible for low-background activity measurements. +Email addresses: d.bemmerer@hzdr.de (D. Bemmerer), +kai.zuber@tu-dresden.de (K. Zuber) +In shallow-underground laboratories, the muon flux is usu- +ally suppressed only by one or two orders of magnitude [10, +e.g.] and may be mitigated using active veto detectors [11, +e.g.]. +Neutrons may be produced in two ways. First, neu- +tron generation by muon spallation on structural or shield- +ing materials. This process usually dominates in shallow- +underground laboratories [12]. The second neutron pro- +duction process are (α, n) reactions in the rock surround- +ing the laboratory, with the α provided by natural ra- +dioactivity from the natural decay chains. +(α, n) usu- +ally dominates the remaining, low, neutron flux in deep- +underground laboratories [13]. +An ultra-low background radioactivity counting labo- +ratory must address all of these above mentioned back- +ground sources. +In addition, it must limit the intrinsic +radioactivity of the detector by material selection [14]. +It is noted that these techniques may also benefit, among +others, experimental nuclear astrophysics. Examples in- +clude the usage of activity measurement setups to count +activation products such as 7Be [15], 18F [16], or 44Ti [17], +but also in-beam measurements in underground settings +[18]. These studies are needed for a better understanding +of solar fusion [19] and Big Bang Nucleosynthesis [20–22]. +The present work reports on a large, coaxial HPGe +detector called TU1, which is placed in the Felsenkeller +shallow-underground laboratory. This detector has been +optimized for ultra-low background activity measurements. +In previous work, the Felsenkeller laboratory has already +been characterized in several aspects: The muon flux (40× +Preprint submitted to Astroparticle Physics +arXiv:2301.03905v1 [physics.ins-det] 10 Jan 2023 + +Ion +accelerator +Concrete +V +Figure 1: Schematic layout of tunnels VIII and IX of the shallow- +underground laboratory Felsenkeller in Dresden. +The inlet shows +bunker 110, which contains two coaxial HPGes (TU1 and TU4) and +a well-type HPGe (TU2). +lower than at surface) and angular distribution have been +measured and matched by simulations [10], as well as the +neutron flux (180× lower than at surface) and energy spec- +trum [12]. +A previous study of the high-energy, Eγ > +3 MeV, part of the γ-ray background also showed a strong +reduction with respect to the surface [11]. +The present +work complements these studies [10–12] by focusing on the +low-energy, Eγ ≤ 3 MeV, part of the γ-ray background. +This work is organized as follows: Section 2 describes +the Felsenkeller shallow-underground laboratory, and sec- +tion 3 the detector and its passive and active shielding. +The data analysis and results are shown in section 4. The +sensitivity of this new setup for the example of 7Be is de- +rived in section 5. The data are discussed in section 6, and +a conclusion and outlook are offered in section 7. +2. The Felsenkeller shallow-underground laboratory +The Felsenkeller shallow-underground laboratory (Fig- +ure 1) is located in Dresden, Germany, and is shielded by +a rock overburden of 45 m (140 m.w.e.) [10]. It is part of +a former industrial site and the laboratory itself is built +into the tunnels VIII and IX of this area. The surround- +ing hornblende monzonite [23] shows naturally containing +238U and 232Th with specific activities of 170(30) Bq/kg +and 130(30) Bq/kg respectively [24]. +The laboratory hosts a 5 MV Pelletron accelerator as +well as two concrete-shielded bunkers (110 and 111). Bun- +ker 110 is used for low-background offline measurements +and is surrounded by 40 cm of low-activity concrete (see +section 3.2 below for details). +It currently hosts three +HPGe detectors (called TU1, TU2 and TU4, respectively) +for γ-ray spectrometry, as well as two silicon drift detectors +(TU3 and TU5) for X-ray spectrometry. Bunker 111 hosts +the target area of the accelerator and is used for in-beam +γ-ray spectrometry measurements on radiative capture re- +actions. +In bunker 110, the angle-integrated muon flux density +is ϕµ = 5.4(4) m−2s−1 corresponding to 140 meters of wa- +ter equivalent (m.w.e.) [10]. The neutron flux density is +ϕn = 0.61(3) m−2s−1 integrated over a broad energy in- +terval ranging from 10−9 MeV to 300 MeV [12, 24]. +3. Experimental Setup +This work concentrates on the coaxial HPGe detector +TU1. The other detectors will be described elsewhere. +3.1. The HPGe detector +The detector is a coaxial p-type high purity germa- +nium detector (HPGe) with a relative efficiency of 163 %. +It was produced by Mirion Technologies (Canberra) ac- +cording to their ultra-low background (ULB) specifications +and is of the type GX 150-250-R. The crystal has a mass +of 3.06 kg and a volume of 574 cm3 (length 90 mm and di- +ameter 90 mm). An end cap of 1.6 mm of aluminum and a +polished-off p-layer of <0.5 mm thickness enable measure- +ments down to 22 keV, well below the usual lower energy +limit for standard p-type HPGe detectors. +A photon with 22 keV induces a preamplified pulse +height of approximately 6 mV in this detector. The ob- +served background noise is ≤ 2.4 mVpp (95 % CL). The +measured resolution at 1.333 MeV γ-ray energy is 2.0 keV +full width at half maximum (FWHM). There are no signif- +icant shoulders to the peak in the pulse height spectrum, +and the ratio of full width at one fifth maximum (FWFM) +and FWHM is 2.7. +3.2. The passive shielding +The passive shielding of TU1 (Figure 2) consists of sev- +eral layers, which are listed here from the outside to the +inside. +1. The laboratory is surrounded in all directions by a +40 cm thick layer of concrete. Prior to mixing the +concrete, the solid components used had been stud- +ied individually by γ-ray spectrometry, and based on +the known composition of the concrete, specific ac- +tivities of 17(3) Bq/kg 238U and 18(2) Bq/kg 232Th +were found, assuming secular equilibrium. For 40K, +280(30) Bq/kg was found. +2. The active veto consists of a 5 cm thick layer of poly- +vinyltoluene (EJ-2001) with a density of 1.0 g/cm3. +It is listed here because it slightly attenuates inci- +dent γ-rays and moderates incident neutrons. The +active veto is described below (section 3.4). +3. A 1 cm thick layer of acrylic glass forms an anti-radon +box (section 3.3). +4. The outer lead layer has a thickness of 10 cm and a +specific 210Pb activity of 21(2) Bq/kg. +1Eljen Technology, Sweetwater, Texas, USA. +2 + +Figure 2: Left panel: Schematic drawing of the passive shielding for the TU1 detector including the lifting mechanism for the lid and the +anti-radon box. Right panel: Profile cut of the setup, from outside to inside: acrylic glass (anti-radon box), 21 Bq/kg lead, 2.5 Bq/kg lead, +and OFRP copper. The muon veto panels (Figure 3) are placed just outside the anti-radon box but omitted here for clarity. +5. The inner lead layer is 5 cm thick, with a specific +210Pb activity of 2.5(1) Bq/kg. +6. The innermost layer of the passive shielding consists +of 10 cm of oxygen-free radio-pure (OFRP) copper +(≤ 0.04 Bq/kg 238U). +For changing the sample, the lead castle can be opened +using an electric lifting mechanism (Figure 2). This mech- +Left panel (S44) +Dewar panel (S15) +Right panel (S45) +Front panel (S17) +Top panel (S16) +Figure 3: Schematic drawing of the muon veto panels surrounding +the anti-radon box (figure 2). The top panel (S16) has four holes for +the lifting mechanism, and the dewar-side panel (S15) has a recess +for the cold finger and additional cutouts for nitrogen supply and +overflow. +anism lifts a cutout of the upper copper and lead bricks, +as well as a part of the anti-radon box and the whole scin- +tillation panel S16 (figure 3). +3.3. The anti-radon box +The air in the bunkers of the Felsenkeller underground +laboratory is exchanged four times per hour by an auto- +matic ventilation system that is continuously active day +and night. Part of the incoming air is fresh air brought +in from the outside, and the remainder dried and recir- +culated. Prior to the construction of the laboratory and +its mechanical ventilation, a radon concentration of 0- +300 Bq/m3 was measured in tunnel VIII and IX, depending +on weather conditions. After completion of the laboratory, +and with the automatic ventilation system running, the +radon concentration in bunker 110 has been remeasured. +In 14 days of measurements, it ranged from 0-53 Bq/m3, +with an average of 11(7) Bq/m3. +In order to minimize the impact [14, 25] of the remain- +ing radon on the background rate of TU1, an airtight anti- +radon box has been installed. The box is constructed from +acrylic glass and flushed with radon-free nitrogen from the +boil-off of the HPGe dewar, monitored by a bubbler. +3.4. The active muon veto +The active shielding for TU1 is composed of five large +plastic (EJ200) scintillation panels from Scionix (figure +3) with 5 cm thickness and a size ranging between 0.5 m2 +(S16) and 1 m2 (S44 & S45). The panels cover the anti- +radon box from each side except from the bottom. EJ200 +is a type of polyvinyltoluene with a light output of 10,000 +photons/MeVee (MeV electron equivalent), a light atten- +uation length of 380 cm, a rise time of 0.9 ns and a decay +3 + +time constant of 2.1 ns, making it suitable for timing mea- +surements in the 0.1 ns range even in scintillating detectors +as large as several meters [26]. +The scintillation light is read out by ET9900 photo- +multiplier tubes2 (PMTs) that are integrated within the +scintillation panels and that have a 2π sensitive solid angle. +Also the high voltage supplies of the PMTs are included +in the panel, so that only two connections are required +on one side of each panel: an input for low-voltage power +(LEMO 00), and the output for the PMT anode signal +(BNC). The scintillating panels are coated with a reflector +and wrapped in lightproof vinyl. +3.5. Electronics and data acquisition +The signals from the six detectors (TU1 and five veto +panels) are recorded in list mode using a CAEN DT5725S +digitizer. This module includes eight channels with 14 bit +resolution at 250 MS/s sampling rate. All channels share +the same internal clock, but each channel is separately +triggered by the digital trigger included in the DT5725S +device. The recorded waveforms are converted to a time +stamp and pulse height with a trapezoidal filter by the +CAEN pulse height firmware, version DPP-PHA 4.22 139.130, +directly inside the DT5725S unit. The event-by-event time +stamp, pulse height, and flags further characterizing the +event as possible pile-up and overflow are saved in a buffer, +transferred via USB cable to the computer and saved on +hard disk. +For the scintillating panels, typical trigger rates of 9- +140 s−1 are observed per panel. +For the TU1 detector, +the typical trigger rate without sample is 5 s−1. +Given +the typical time lengths of the trapezoidal filter of 4.4 µs +(TU1) and 5.0 µs (scintillators), dead time and pile-up ef- +fects are expected to be insignificant. The data analysis is +performed offline. +4. Data analysis +In this section, the offline data analysis is described, +including the optimization of the active veto parameters. +4.1. Pulse height spectrum in TU1 using only the passive +shield +The pulse height spectrum of the TU1 detector, mea- +sured within bunker 110 without any additional shielding, +displays many γ lines due to the natural decay chains (Fig- +ure 4, black spectrum). The 40K and 208Tl lines emerge +prominently. In addition, there are a number of neutron- +induced features due to the remaining neutron background +[12] that can be picked out due to their triangular shapes +and above 2.6 MeV, there are a number of weak branches +from 208Tl. The γ-ray energy resolution at the 40K peak +(1460.8 keV) is 3.1 keV (full width at half maximum, FWHM) +2ET Enterprises, Ltd., Uxbridge, UK. +in the adopted digitizer-based DAQ scheme, worse than +the 2.2 keV FWHM measured with the same detector and +analog electronics. +When applying the full passive shielding, the contin- +uum below 2.6 MeV is reduced by three to five orders of +magnitude, and the neutron features are no longer appar- +ent (Figure 4, blue spectrum). Instead, a wide continuum +appears that is only slightly energy dependent, showing +the effects of the energy loss of the remaining cosmic-ray +induced muon flux. +4.2. Active muon veto +In a first step, the pulse height spectrum of the scin- +tillating panels has been calibrated in energy using the +Compton edges of radionuclide standards made of 137Cs, +60Co, and 88Y. These point-like standards were placed +on the center of one of the large sides of each scintillat- +ing panel, in turn. As a second step, the trigger thresh- +old for the scintillating panels was increased to approxi- +mately 2 MeV, in order to avoid unnecessarily recording +radionuclide-induced pulses detected in the scintillators. +The resulting energy-calibrated spectrum (Figure 5, +black curve) shows a peak near 2 MeVee that is given by +the above mentioned trigger condition and a broad peak +near 6 MeVee that is assumed to be due to muons. +The observed pulse height in the histogram is propor- +tional to the number of scintillation photons detected by +the PMT. Due to the finite reflectivity of the reflective +wrapping, as well as the light attenuation in the crystal +itself, not all of the emitted photons are detected. This is +even enhanced for panels S15 and S16 which also include +holes and a recess. The effect has been studied using a +60Co radionuclide standard placed at several places across +the scintillating panel. Using the light yield at the center +of the panel as reference, >90% of the area of the panel +shows observed pulse heights between 0.7 and 1.2 times the +reference pulse height. The remaining <10% are located +very close to the PMT and show higher light yields. It is +up to 5 times the reference light yield when the 60Co source +is placed directly atop the sensitive area of the PMT, so +that it may see scintillation light from both sides. +At the depth of the Felsenkeller lab, the muon energy +spectrum peaks at approximately 30 GeV, and the energy +loss of muons is about 2 MeVee per cm of EJ-200 mate- +rial traversed [27]. Qualitatively, the large cross section +of the panels is expected to lead to muons going per- +pendicularly through the panel dominating the spectrum. +They would lose approximately 10 MeVee, and applying +the above mentioned factor of 0.7 for less than ideal light +collection, would register near 7 MeVee, consistent with +the observed peak at 6 MeVee. +In addition, many muon paths are possible that are sig- +nificantly longer than 5 cm. As a consequence, the muon- +induced spectrum in S45 extends to rather high energy, up +to dozens of MeVee (Figure 5, black curve). +In the following two subsections, by offline analysis of +background runs lasting several days, the two main pa- +4 + +500 +1000 +1500 +2000 +2500 +3000 + [keV] +TU1 +E +2 +− +10 +1 +− +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +] +-1 + d) +⋅ + kg +⋅ +Counting rate [(keV +Figure 4: Pulse height spectrum for the TU1 detector, normalized to running time, bin width, and detector mass: Bunker 110, without any +shielding except for the concrete walls (black curve, 6.8 days running time). Full passive shielding and anti-radon box (blue curve, 90.0 days). +The same spectrum, but applying also the active shield (red curve). The γ-lines identified in the red spectrum are listed in Table 2. See text +for details. +rameters for the active muon veto are optimized: First, +the timing coincidence condition (section 4.3), and second, +the pulse height threshold (section 4.4). A third subsec- +tion (4.5) then discusses additional information gained by +combining the two cuts. +4.3. Time coincidence condition +The time difference between the trigger times tTU1 in +TU1 and those of the scintillating panels ti (with i ∈ +{S15, S16, S17, S44, S45}) shows a sharp peak that is con- +tained in the time interval [-5 µs,+5 µs]. A typical exam- +ple for this time difference spectrum is given in Figure 6. +The mutual time differences between individual scintilla- +tor panels are not shown but display a similar pattern. +The sharp peak in the [-5 µs,+5 µs] coincidence window +of figure 6 is caused by muons that pass both S45 and TU1. +The peak width is given by the time resolution of TU1, +which is approximately on the level of the 2 µs binning used +in Figure 6. A second, longer time coincidence window of +[-150 µs,+5 µs] will be introduced below. +4.4. Energy threshold in the scintillator panels +Even with a trigger threshold near 2 MeV (section 4.2), +there is still a significant number of radionuclide-induced +events in the pulse height spectrum of the scintillator. This +can be seen by comparing the free-running pulse height +spectrum of one scintillator panel (Figure 5, black curve) +with the same spectrum when requiring time coincidence, +within a “short” coincidence window of [-5 µs,+5 µs], with +any of the other detectors (Figure 5, blue curve). +In order to reduce the impact of these remaining radio- +nuclide-induced events, an additional threshold energy +Ethresh,i (with i ∈ {S15, S16, S17, S44, S45}) is defined. +This threshold was carefully optimized. Too low Ethresh,i +would allow a too large impact of radionuclide-induced +events, whereas a too high Ethresh,i would reduce the muon +veto efficiency. +This optimization was carried by analyzing a one day +long run with a 2 kBq 7Be source mounted in close ge- +ometry. Starting from the local minimum to the left of +the muon peak (Emin, S45 = 3.6 MeV for the blue curve +in Figure 5), the energy threshold was varied in the of- +fline analysis. Then, for each given value of the energy +threshold, two quantities were determined. First, the cut +survival probability Pcut survival for events in the 7Be peak +area in the TU1 detector given by +Pcut survival = +NTU1(7Be)|cut +NTU1(7Be)|free running +(1) +The second quantity was the no-source specific background +rate ˙N40−2700 in the 40-2700 keV energy region. It is de- +termined by analyzing a 30-day run without any source +and with the same running conditions as the 7Be run with +the same energy threshold values and requiring anticoinci- +dence with all the scintillator panels within a time window +[-150 µs;+5 µs]. See section 4.5 below for the justification +for this somewhat enlarged time coincidence window. +The optimal energy threshold was then defined to be +the energy for which Pcut survival = 0.995 was found. This +was typically at 0.8× the local minimum energy (red dashed +line at Ethresh, S45 = 2.9 MeV in Figure 5). For the other +scintillators, not shown in the Figure, similar energy thresh- +olds of 2-3 MeV were determined. +The impact of the choice of the energy threshold is +5 + +0 +10 +20 +30 +40 +50 +60 +70 + [MeV] +S45 +E +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +Counts / 5keV +S45 +S45 and any other detector +S45 and TU1 + +1 +10 +2 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +Figure 5: Pulse height spectrum of scintillation panel S45, 29.5 days +running time. +Black curve: raw spectrum. +Blue curve: requiring +coincidence within [-5 µs,+5 µs] with any of {S15, S16, S17, S44} +or TU1 (ETU1 ≥ 0.04 MeV). Red curve: requiring coincidence with +TU1 (ETU1 ≥ 0.04 MeV). Red dashed line: energy cut for this panel +(2.9 MeV, section 4.4). See text for details. +Table 1: Impact of the variation of the energy threshold by a factor +f on the cut survival probability Pcut survival and on the specific +background rate ˙N40−2700. See text for details. +f +Pcut survival +˙N40−2700 +Remark +0.4 +0.958(1) +112(1) +0.6 +0.980(1) +113(1) +0.8 +0.995(1) +116(1) +Adopted value +1.0 +0.998(1) +120(1) +1.2 +0.998(1) +124(1) +illustrated in Table 3 where all energy thresholds are varied +together by a constant factor f = Ethresh,i/Emin,i, and for +each value of f the cut survival probability and background +count rate are shown. +It is clear from Table 1 that the background count- +ing rate changes by only a few percent in the region stud- +ied, while the cut survival probability is rather sensitive to +the choice of the energy threshold. A value of Pcut survival +=0.995 was adopted. This value introduces a small sys- +tematic shift, no more than -0.5%, which is lower than +the typical geometry-dependent detection efficiency uncer- +tainty of 1-3%. This adopted value does not significantly +worsen the precision of the activity determination. +As a final check for the adopted threshold energy, the +ratio of counts in the S45 pulse height spectrum requir- +ing coincidence with TU1 (red curve in Figure 5) with +the S45 free running spectrum (black curve in Figure 5) +is computed. For ES45 < Ethresh, S45, the observed ratio is +4.9×10−5. This is consistent with expectation for random +coincidences, given the 10 µs width of the “short” time co- +500 +− +400 +− +300 +− +200 +− +100 +− +0 +100 +200 +300 +400 +500 +s] +µ + [ +TU1 +t + - +S45 +t +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +s +µ + +Counts / 2 +Figure 6: +Black curve: +Event by event time difference between +events in scintillation panel S45 (tS45) and in TU1 (tTU1), 29.5 days. +Red curve, in addition ETU1 ≥ 0.04 MeV and ES45 ≥ Ethresh, S45 +(Ethresh, S45 = 2.9 MeV) are required. +The grey box shows the +“long” coincidence timing window of [-150 µs,+5 µs]. +See text for +details. +incidence window and the TU1 trigger rate of 4.9 s−1. For +ES45 > Ethresh, S45, instead, the observed ratio is much +larger, 4.3×10−3. In conclusion, S45-TU1 coincident events +can be fully explained by random coincidences for ES45 < +Ethresh, S45, while only 1% of S45-TU1 coincident events +with ES45 > Ethresh, S45 can be explained by random coin- +cidences. Thus, it also confirms that the threshold value +Ethresh, S45 is appropriately chosen. +4.5. Combination of timing and energy cuts +As a next step, the newly determined energy threshold +is adopted, and the time difference spectrum between TU1 +and S45 is re-determined, now requiring ES45 ≥ Ethresh, S45 +(Ethresh, S45 and ETU1 ≥ 0.04 MeV). As expected, the +resultant spectrum (Figure 6, red curve) still shows the +prompt muon coincidence peak at -5 µs ≤ (tS45 − tTU1) ≤ +5µs. +In addition, now two distinct features emerge that had +previously been covered by the radionuclide-induced events, +namely at time differences (tS45 − tTU1) ≈ -100 µs and +(tS45 − tTU1) ≈ -10 µs, respectively. +The first feature may be due to muons which first pass +the scintillator panel and subsequently scatter on the lead +shield to create a free neutron. +This (µ,n) process has +previously been found to dominate the neutron flux, es- +pecially at high neutron energies, in the Felsenkeller lab- +oratory [12]. The delay of ∼100 µs between the passage +of the muon through the scintillator panel and the signal +in the HPGe detector may be due to free (µ,n) neutrons +that, after creation, are first scattered a few times, and +6 + +slightly moderated, in the lead shield before they give rise +to a signal in the HPGe detector. +The second feature, with a delay of typically 10 µs or +less between the passage of the muon through the scintil- +lator, may be due to the decay of stopped muons (mean +lifetime 2.2 µs) in the lead shield, and subsequent detection +of secondary electrons, positrons, or annihilation radiation +from the decay products. +The specific rate of events within the extended part of +the anticoincidence window, i.e. [-150 µs;-5 µs], that also +satisfies the energy cuts for TU1 and the scintillators is +20(1) kg−1d−1. Thus, the no-source background rate (Ta- +ble 1) of 116 kg−1d−1 for the full anticoincidence window +of [-150 µs;+5 µs] would increase to 136 kg−1d−1, if a more +narrow anticoincidence window of [-5 µs;+5 µs] would be +used. +4.6. Rates of coincident events +The coincidence rates (ETU1 ≥ 40 keV, Epanel ≥ Ethresh, +time coincidence window tcut=[-150 µs,+5 µs]) between the +scintillation panels and TU1 are R = 0.0207(1)s−1 (TU1∧S15), +0.0337(1)s−1 (TU1∧S16), R = 0.0208(1)s−1 (TU1∧S17), +0.0298(1)s−1 (TU1∧S44), and 0.0273(1)s−1 (TU1∧S45), +respectively. +As expected, the coincidence ratios for the panels on +opposite sides are close: The opposing panels S44 and S45 +differ by just 9%, and the opposing panels S15 and S17 +by <1%. The highest absolute coincidence rate is found +for TU1∧S16, consistent with expectation based on the +measured muon angular distribution in this area [10]. +The total rate of vetoed events in TU1 is 0.0763(2)s−1 +for ETU1 ≥ 0.04 MeV and the “long” anticoincidence win- +dow of [-150 µs,+5 µs]. +This is consistent with the ex- +pected rate of muons penetrating TU1 of 0.08 s−1. This +rate has been estimated using the muon flux of ϕµ = +5.4(4)m−2s−1 that has previously been measured at this +location [10] and an estimated muon-exposed cross section +of TU1 of 145 cm2. +4.7. Preamplified signals of opposite polarity +An uncommon characteristic of the TU1 detector is +the fact that there is a significant rate, 6 s−1, of signals +with unphysical polarity, opposite to the usual one, at the +preamplifier output. These pulses have the same general +shape and amplitude as the correctly preamplified pulses, +but opposite polarity. However, they are not correlated in +time to any other events, either from TU1 or from any of +the scintillating panels. For safety, these unphysical events +are studied via a dedicated channel of the data acquisition +(self-triggered just as the other channels) and stored for +possible future analysis. The observed pulse height spec- +trum does not show the characteristic features of an HPGe +spectrum, but rather resembles a Landau distribution. +The opposite polarity events are believed to be caused +by an electronic noise source in the preamplifier unit. Given +their rate and random distribution in time, there is a +6×10−6 probability for them to affect a true physical event, +which is negligible in the present, low-background appli- +cation. +4.8. Dead time +The total dead time td,tot of TU1 is given by the dead +time td,raw due to the ∼10 µs processing time of the dig- +itizer and an additional cut efficiency εcut (section 4.4), +which takes into account the loss of counts due to random +anticoincidences. +The digitizer dead time has been estimated based on +the count rate of the resulting spectrum (output count +rate, OCR) and the 1 µs trigger hold-off time during which +the digitizer unit does not accept new triggers. In a pro- +cedure similar to the one recommended by CAEN [28], +the unknown input count rate (ICR) is determined in an +iterative procedure. From a starting ICR and assuming +Poisson-distributed events, a predicted OCR is determined +and then compared to the observed OCR in order to ob- +tain an improved ICR estimate for the next iteration. This +iteration converges typically after five cycles. +The TU1 channel of the digitizer self-triggers on the ab- +solute height of the second derivative of the pulse, which +triggers both for negative (physical) and positive (unphys- +ical) pulses. In the TU1 channel, the unphysical events are +converted to channel 0 of the pulse height histogram and +fully taken into account during the dead time determina- +tion. They contribute a relative dead time of 6×10−6. In +addition, the unphysical events are converted and stored +in a separate channel of the digitizer, see section 4.7 above. +In a typical run, about 2% of the detected events in +the TU1 channel do not follow a Poisson distribution in +time, but instead follow directly a preceding event. This +behaviour is consistent with electronic noise or mechani- +cal vibrations. Due to their limited number, these non- +Poissonian events do not have a significant impact on the +precision of the dead time determination. +4.9. Remaining background in TU1 +The remaining background after the construction of the +passive shielding and the application of the active veto is +shown as an orange histogram in figure 4. The remaining +γ-lines, as well as their origins and counting rates, are +listed in table 2, and will be discussed in the following. +4.9.1. Neutron induced lines in germanium +The four low energetic γ-lines at 52-56, 64-67, 140, +and 198 keV are due to isomeric states in several germa- +nium isotopes. These states in 71mGe (Ex = 198.4 keV, +20 ms half-life), 73mGe (66.7 keV, 0.5 s half-life), and 75mGe +(139.7 keV, 48 s half-life) are populated by neutron cap- +tures, (n, 2n) reactions, or (n, n′γ) reactions within the +germanium crystal itself. All three have half-lives that are +too long for an effective active veto. +The peaks at 52-56 and 64-67 keV (Table 2) are due to +the decay of the Ex = 67 keV metastable second excited +7 + +Table 2: Energy and counting rates of the remaining γ-lines in the +TU1 detector, after applying the active veto. See text for details. +Energy +Nuclide +Origin +Rate +[keV] +[kg−1d−1] +52.0-55.6 +73mGe +72Ge(n,γ), 74Ge(n,2n) +1.28(19) +63.9-66.7 +73mGe +72Ge(n,γ), 74Ge(n,2n) +0.38(17) +72.8-75.0 +Pb X +Lead Kα & Kβ X-rays +0.33(12) +139.7 +75mGe +74Ge(n,γ), 76Ge(n,2n) +1.34(16) +198.4 +71mGe +70Ge(n,γ), 72Ge(n,2n) +1.00(15) +351.9 +214Pb +238U decay chain +0.29(11) +511 +e+e− +Annihilation +2.64(13) +583.2 +208Tl +232Th decay chain +0.14(8) +609.3 +214Bi +238U decay chain +0.21(8) +834.8 +54Mn +54Fe(n, p)54Mn +0.17(7) +1173.2 +60Co +63Cu(n, α)60Co +0.18(5) +1274.5 +22Na +Lab-specific nuclide +0.20(5) +1332.5 +60Co +63Cu(n, α)60Co +0.16(5) +1460.8 +40K +Natural radioactivity +0.49(6) +2614.5 +208Tl +232Th decay chain +0.22(4) +state of 73Ge [29]. The half-life of the first excited state +(T1/2 = 2.9 µs) is comparable to the length of the timing +window for the HPGe, giving rise to partial summation +effects. If the decay of the first excited state takes enough +time, the lower-energy cluster of peaks in the TU1 spec- +trum will be fed by the 53.4 keV photon (or the 52.0 keV- +53.4 keV conversion electron respectively), which can be +separated by the software from the subsequent decay. If +this decay takes a shorter amount of time though, both +peak clusters in the TU1 spectrum can be fed due to sum- +mation effects. The lower-energy one via the 53.4 keV pho- +ton (or the 52.0 keV-53.4 keV conversion electron respec- +tively) and their summation with the subsequently emitted +conversion electron of 2.2 keV. The higher-energy cluster +then originates from the summation of the 53.4 keV photon +(or the 52.0 keV-53.4 keV conversion electron respectively) +and their subsequently emitted conversion electron with +11.9 keV-13.3 keV. +The 139.7 keV γ ray is due to the decay of the 75mGe +state at the same energy to the ground state of 75Ge. The +rate of this γ line can be used to approximate the ther- +mal neutron flux density inside the passive shielding, re- +sulting in ϕ = (3.2 ± 1.0) × 10−5 cm−2s−1 using a con- +version factor from literature [30]. +During a dedicated +campaign to measure the neutron flux in the Felsenkel- +ler laboratory [12], the thermal flux inside room 110 has +been measured. The thermal flux obtained with an un- +moderated 3He tube during a run previous to the instal- +lation of TU1, i.e. in an empty bunker 110, with its lead +shield is (2.3 ± 0.2) × 10−5 cm−2s−1 [24, 31]. It is known +that large passive lead shields can enhance the neutron +flux in a shallow-underground laboratory such as Felsen- +keller [24]. This is mainly caused by (µ, n) reactions in +the high atomic charge shielding material. Neutrons of ∼ +MeV kinetic energy are produced in these reactions [12, 24] +and subsequently moderated, for example, in the 200 kg of +EJ-200 organic scintillator forming the muon veto (section +3.4). +Finally, the 198.4 keV peak is caused by the decay of +the metastable level of 71Ge [29, 32, 33], which has a half- +life of 20 ms. +Its decay scheme suggests a chain emis- +sion of, first, a 12-22 keV conversion electron and, second +a gamma-ray of E = 175.0 keV [34]. +The peak in the +spectrum is a summation line, which benefits from both +a negligible escaping probability of the emitted conversion +electrons and the short half-life of the first excited state of +71Ge at Ex = 175 keV (T1/2 = 79 ns) with respect to the +comparatively longer timing window of the HPGe (section +4.3). +4.9.2. Neutron activation lines +The spectrum shows three lines from the decay of long- +lived radionuclides that may be attributed to neutron acti- +vation: 54Mn (Eγ = 834.8 keV, T1/2 = 312 d) may be pro- +duced via the 54Fe(n,p)54Mn reaction, and 60Co (1173.2 +and 1332.5 keV, T1/2 = 1925 d) via the 63Cu(n, α)60Co +reaction. +These nuclides are typical contaminations in +low-background HPGe detectors. They are likely due to +cosmic-ray activation on shielding or structural materials. +It is expected that these contaminations slowly decay +out in the next years and stabilize at a much lower level, +consistent with the neutron flux in Felsenkeller that is 180 +times lower than at surface [12]. +4.9.3. Annihilation peak at 511 keV +The annihilation peak at 511 keV is attenuated by a +factor of 11 by the active veto. The veto thus removes +most of the muon-induced effects, including the decay of +stopped muons. +The remaining 511 keV counting rate is largely due to +the annihilation of positrons emitted in β+ decays in the +detector and shielding material. It is noted that there is +a small contribution (∼0.04 kg−1d−1) due to photons with +Eγ = 510.77 keV from the decay of 208Tl, which also causes +the 583 and 2615 keV lines. +4.9.4. Radon-induced effects +The low specific radon activity of 11(7) Bq/m3 in the +measurement bunker (section 3.3) may lead to small amounts +of radioactive 222Rn (T1/2 = 3.8 d) to diffuse into the sen- +sitive volume near TU1. This effect is mitigated by the +anti-radon box and its flushing. Nevertheless, γ rays from +the radon daughters 214Pb and 214Bi appear in the TU1 +spectrum at 352 and 609 keV, respectively. An additional +weak feature at 295 keV is not 2σ significant. +After a typical sample change, the integral counting +rate is saturated at its final value after 10000 s, which is +significantly shorter than the physical half-life of 222Rn, +showing the desired effects of the active flushing of the +anti-radon box. +8 + +4.9.5. Naturally occurring radionuclides +The remaining rates in the γ-lines at 1460.8 keV (from +40K) and 2614.5 keV (from 208Tl) are 0.49(6) kg−1d−1 and +0.22(4) kg−1d−1 respectively (table 2). Using the observed +rates of these two γ rays without shielding and applying +a calculated attenuation based on the weakest spot of the +passive shielding, counting rates of 0.3 kg−1d−1 are found +for these two lines. +For 208Tl, based on these values there is no reason to +assume a significant internal contamination of TU1 within +the shielding materials or the detector. +Also for 40K, the effect of the attenuation is compara- +ble to the expectation. However, the attenuation is still +smaller than conservatively assumed. The remaining rate +of 40K within TU1 may therefore also include contamina- +tions within the shielding and the detector. In fact, 40K +has previously been found to be a remaining contaminant +in high-purity copper samples [35]. +5. Experimental study on 7Be +As an illustration for the capabilities of the TU1 detec- +tor for activation studies for nuclear astrophysics, a study +of weak activated 7Be samples produced during the inves- +tigation of the 3He(α, γ)7Be reaction at the Felsenkeller +5 MV Pelletron accelerator is discussed here. +5.1. Efficiency calibration +7Be has a half-life of T1/2 = 53.22(6) d and emits a γ- +ray of E = 478 keV with an emission probability of η = +10.44(4) %. +In order to determine the absolute full-energy peak ef- +ficiency at E = 478 keV, three calibration samples of 7Be +have been produced at the cyclotron accelerator of the +Institute for Nuclear Research in Debrecen (ATOMKI), +Hungary. To this end, LiF has been evaporated on tan- +talum disks (0.22 mm thickness and 27 mm diameter) and +subsequently bombarded with 5.5 MeV protons, resulting +in 7Be activities of 0.3, 22, and 48 kBq, respectively. The +geometry of the tantalum disks and proton beam spot has +been chosen to conform to the geometry of the much less +radioactive 7Be samples from the 3He(α, γ)7Be irradiation. +The activity of the three ATOMKI 7Be samples has +then determined in far geometry, using arrays of several +other well-calibrated radionuclide standards, independently +at ATOMKI and at HZDR. Using these three calibration +sources, the absolute full-energy peak detection efficiency +for the TU1 detector, when using a point-like sample in +close geometry, has subsequently been determined to be +ε = 14.45(19) %. +5.2. Activation analysis +For the ongoing campaign on the 3He(α, γ)7Be reaction +at Felsenkeller, a tantalum disk (hereafter called sample +ST8) of 0.22 mm thickness and 27 mm diameter has been +implanted with 1018 cm−2 3He at the HZDR Ion Beam +400 +420 +440 +460 +480 +500 +520 +540 + [keV] +TU1 +E +2 +− +10 +1 +− +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +] +-1 + d) +⋅ + kg +⋅ +Counting rate [(keV + +Figure 7: Pulse height spectrum in the region of the 478 keV line +of 7Be. Black spectrum: +7Be calibration source with 2 kBq. Blue: +Activated 7Be sample (0.7 Bq), without active veto. Red: Activated +7Be sample, with active veto. Light blue: background without sam- +ple, with active veto. Shaded light blue area: 10 keV interval for the +determination of the detection limit LD. See text for details. +Table 3: Detection limit LD and activity Lsignal=noise for signal equal +to noise for 50 d counting time and a 10 keV wide region of interest +for 7Be (478 keV) and 181Hf (482 keV). See text for details. +Method +7Be +181Hf +[mBq] +[mBq] +Detection limit LD at 90% CL +0.81 +0.11 +Lsignal=noise +2.20 +0.31 +Center. ST8 was then irradiated with a 5-10 µA beam of +4He+ (Eα = 0.8 − 1.0 MeV) at Felsenkeller to investigate +the 3He(α, γ)7Be reaction, applying a charge of Q = 2.3 C. +The resulting pulse height spectrum near 478 keV for +the activated sample ST8 is shown in Figure 7: blue curve +without active veto, red curve with active veto. From the +comparison of these two spectra, the cut survival proba- +bility of Pcut survival = 0.995 is found (Section 4.4). The +data are compared with the spectrum of the calibration +sample (black curve) and the background without sample +(light blue curve). +5.3. Sensitivity for 7Be detection +The nuclide 7Be is not only relevant for nuclear astro- +physics [15, 19]. Due to its cosmogenic origin, it may also +be used to monitor atmospherical and geological processes +[36–38]. +The present background spectrum (light blue curve in +Figure 7) and detection efficiency (section 5.1), as well +as literature data on decay branching ratio and half-life, +9 + +are now used in order to derive the sensitivity of TU1 for +detecting 7Be. +The detection limit LD in Bq, using a coverage factor +of kα = 1.282 for 90% confidence level, a branching ra- +tio β, an absolute full-enery peak detection efficiency ε, a +background rate ˙NBG, and a counting time t is given by +[39] +LD = k2 +α + 2kα +� +2 ˙NBGt +ε × β × Cdecay × t. +(2) +This value has been determined for a 10 keV wide region +of interest and two typical nuclides: In addition to 7Be, +also for 181Hf, which has a similar half-life (42.39 d) but +a higher branching ratio, β = 80.5% for its 482 keV γ +ray. Due to the similar half-lives, the decay corrections +for 50 d counting time are also similar, Cdecay = 0.73 (7Be) +and 0.68 (181Hf). Detection efficiency and background dif- +fer only negligibly for the two relevant energies, 478 and +482 keV. +For comparison, for both cases also another value has +been computed, namely the activity Lsignal=noise for which +the net count rate (signal) is equal to the background +(noise). The data show that the TU1 detector can eas- +ily reach the µBq range, if a suitably long counting time +is adopted (Table 3). +The same background data was then used to calculate +the maximum half-life T max +1/2 +that can be determined in a +sample with NA (1 mol) decaying nuclei, neglecting the +decay and self-absorption corrections. Again 50 d count- +ing time, a 10 keV wide region of interest near 478 keV, +and a detection efficiency of 14.45% are used, and now a +branching of β = 100%. +T max +1/2 += +ln 2 +λmax = ln 2 × NA +LD += +ln 2 × NA × ε × β × Cdecay × t +k2α + 2kα +� +2 ˙NBGt +(3) += +2.1 × 1020y +This value is in the range of double-beta decay half-lives +with the emission of neutrinos that have recently been de- +termined [4]. +6. Discussion +In this section, the new data are interpreted and com- +pared with literature data. +6.1. Interpretation of the present data +The passive shielding suppresses the integrated count- +ing rate between 40 keV and 2700 keV by a factor of 4300 +(Figures 4 and 8), showing its high effectivity. In the pas- +sively shielded spectrum (blue curve in Figure 8), only +two peaks from the natural background remain, namely +40K and 208Tl. The rest of the passively shielded spec- +trum is dominated by muon-induced events, both in the +continuum and in the 511 keV annihilation peak. +Furthermore, the passively shielded spectrum is very +similar to a previously published spectrum [14] that has +been obtained in the VKTA laboratory in the nearby Fel- +senkeller tunnel IV, which has the same rock overburden +and muon flux as tunnel VIII studied here [10]. +There +are only two significant differences between these spectra +(black and blue curves in Figure 8). +First, a slightly lower 40K rate in the present spec- +trum. This is remarkable, given that there is a significant +amount of 40K in the laboratory walls surrounding TU1 +(section 3.2). In the case of tunnel IV, the 40K γ rays are +already attenuated outside the detector shield itself by a +measurement chamber made of ancient steel (MK2) that +hosts several detectors [14]. The comparison seems to in- +dicate that the present passive shield, 15 cm of radiopure +lead and 10 cm of copper, improves upon the previous one +(10 cm normal lead, 5 cm radiopure lead, 5 cm radiopure +copper) [14]. Importantly, it shows that the omission of a +measurement chamber like MK2 is more than compensated +by the present, additional 5 cm of copper and, possibly, by +the usage of radiopure lead for the outer shielding. The +second difference between the black and blue spectra is a +65Zn contamination (Eγ = 1115 keV, T1/2 = 244 d) in the +VKTA spectrum, which has been measured several years +ago. While this contamination used to be significant, in +the meantime it decayed. +The present active shielding further reduces the inte- +grated counting rate by a factor of 17, bringing the total +background suppression to a factor of 73000, almost five +orders of magnitude. At the same time, random coinci- +dences reduce the peak detection efficiency by just 0.47%. +This small reduction is acceptable given the fact that the +setup is designed for low counting rate experiments that +will be limited by the statistical, not the systematical un- +certainty. +The active shield reveals a number of γ rays that had +been covered by the muon-induced continuum in the spec- +trum without active shield. +It is possible that a future +slight increase of the lead shield thickness, by up to an +additional 5 cm, might further attenuate the remaining ra- +dionuclide lines from 40K and 208Tl, at the expense of a +somewhat higher (µ, n) rate [12] that would increase the +neutron-induced lines at 198 keV and below (sections 4.9.1 +and 4.9.2). +The actively shielded spectrum still shows a significant +continuum, without any clearly discernible Compton edges +(Figure 8, red spectrum). This may indicate that correct- +ing the small imperfections in the active muon veto, for +example the holes for the lifting mechanism of the lid, +may lead to a further background reduction. In another +shallow-underground laboratory, the muon veto reduced +the 40-2700 keV counting rate by a factor of 89 [43], indi- +cating that correcting the small imperfections in the muon +veto may yield another factor of five reduction in counting +rate. +10 + +500 +1000 +1500 +2000 +2500 + [keV] +E +2 +− +10 +1 +− +10 +1 +10 +] +-1 + d) +⋅ + kg +⋅ +Counting rate [(keV + +Figure 8: Pulse height spectrum of the TU1 detector without active veto (blue spectrum) and with active veto (red spectrum), normalized +to bin width and detector size and shown for the typical background region of interest within [40 keV;2700 keV]. For comparison, a previous +spectrum from the D6 detector of the VKTA lab in Felsenkeller tunnel IV [14] is shown in black. +6.2. Comparison with other setups reported in the litera- +ture +In order to further extend the comparison and discus- +sion, data from a selection of low-background HPGe de- +tectors in underground laboratories is listed in table 4. +Their integrated counting rates within the energy inter- +val of [40 keV,2700 keV] (for the Gran Sasso detectors, a +slightly different range of [100 keV,2700 keV] is used) are +plotted in figure 9 as a function of their corresponding rock +overburden. +The setups used for comparison span a broad range +of rock overburden and generally reflect the state of the +art, with expert adjustments being made for the relevant +depths. +Two general trends are apparent (Figure 9): +First, +the counting rate generally decreases with increasing rock +overburden. Second, the scatter of the data points from +different laboratories decreases with increasing rock over- +burden. Both of these effects are due to cosmic-ray in- +duced muons. Once the radionuclide-induced background +has been strongly attenuated, the muon flux remains the +main variable affecting the observed integral background +rate. While at high rock overburden, the muon flux is at- +tenuated so much that it plays no major role any more, at +medium-low rock overburden, it is important to optimize +the treatment of muon-induced effects, including an active +veto. The differences in the particular treatment of the +muon background in various laboratories are reflected in +the spread of counting rates at similar depths (Figure 9). +It is apparent that the background rate in the present +setup is in the same league with much deeper underground +detectors, and that it is lower than the background of some +setups with even greater rock overburden. +7. Summary and outlook +The present work describes a recently installed new +HPGe detector called TU1 in the shallow-underground +laboratory Felsenkeller (Dresden, Germany). Using a so- +phisticated passive and active shield, the integrated back- +ground counting rate in the detector is reduced to 116 kg−1d−1 +in the [40 keV;2700 keV] interval, an unprecedented low +value for a shallow-underground laboratory. This detector +is now the most sensitive radioactivity-measurement setup +in Germany. +A detailed study of the remaining background is pre- +sented, and for the examples of point-like sources of 7Be +and 181Hf, the detection limits are derived for typical count- +ing times. The data are compared to relevant similar lab- +oratories. +Comprehensive simulation studies are currently ongo- +ing and with more statistics in the coming years, it will +be possible to explore the correlations between incoming +muons and their signature in the shielded germanium de- +tector in greater detail. At the same time, it is planned to +further improve muon veto efficiency by closing remaining +small gaps in the active scintillator shield. +Acknowledgments +The authors are indebted to Tam´as Sz¨ucs (ATOMKI +Debrecen) for providing the 7Be calibration samples, to +Detlev Degering (VKTA) for valuable discussions, and to +Toralf D¨oring, Maik G¨orler, Andreas Hartmann, Bernd +Rimarzig (HZDR), and Martin Siegel (TU Dresden) for +technical support. — Financial support by Deutsche For- +schungsgemeinschaft DFG (INST 269/631-1 FUGG, TU +11 + +Table 4: Background count rates for selected detectors, normalized to detector mass. The energy interval is [40 keV;2700 keV], except for the +Gator detector at LNGS where the energy interval starts only at 60 keV. +Detector/ +Location +Depth +Count rate +Count rate +Reference +Laboratory +[m.w.e] +without veto +with veto +[kg−1 d−1] +[kg−1 d−1] +D4 +Seibersdorf Laboratory, Austria +≈1 +85100±400 +8110±40 +[40, 41] +DLB +TU Dortmund, Germany +10 +34400±60 +2900±6 +[42] +GIOVE +MPIK Heidelberg, Germany +15 +31027±48 +348±3 +[43, 44] +CAVE +IAEA, Monaco +543 +840±50 +[8] +D6 +Felsenkeller (VKTA), Germany +110 +2938±5 +[14] +TU1 +Felsenkeller (TU Dresden and HZDR), Germany +140 +1982±3 +116±1 +present work +Ge-14 +HADES, Belgium +500 +208±4 +178±8 +[45, 46] +GeMSE +La Vue des Alpes Laboratory, Switzerland +620 +91±1 +[47, 48] +GeOroel +LSC, Canfranc, Spain +2450 +142 +[49] +GeCRIS +LNGS, Gran Sasso, Italy +3800 +111±1 +[50] +GeMPI +LNGS, Gran Sasso, Italy +3800 +59±1 +[50] +Gator +LNGS, Gran Sasso, Italy – [60 keV;2700 keV] +3800 +89.0±0.7 +[51] +OBELIX +LSM, Modane, France +4800 +68±1 +[5] +Dresden Institutional Strategy ”support the best”, ZU123/21- +1, and BE4100/4-1), by the Konrad-Adenauer-Stiftung, +and by the European Union (ChETEC-INFRA, project +no. 101008324) is gratefully acknowledged. +References +[1] P. 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Ludwig, Underground measurements and simulations on the +muon intensity and 12C-induced nuclear reactions at low ener- +gies, Ph.D. thesis, TU Dresden (2021). +14 + diff --git a/j9E2T4oBgHgl3EQfdQfR/content/tmp_files/load_file.txt b/j9E2T4oBgHgl3EQfdQfR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1788db63337cbaae0ecff4025beade3df17a1a57 --- /dev/null +++ b/j9E2T4oBgHgl3EQfdQfR/content/tmp_files/load_file.txt @@ -0,0 +1,1257 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf,len=1256 +page_content='A new ultra low-level HPGe activity counting setup in the Felsenkeller shallow-underground laboratory S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Turkata, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Bemmererb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Boeltzigb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Domulaa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Kocha,b, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Lossina,b, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Osswalda,b, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Schmidtb, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Zubera aTechnische Universit¨at Dresden (TU Dresden), 01069 Dresden, Germany bHelmholtz-Zentrum Dresden-Rossendorf (HZDR), Bautzner Landstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 400, 01328 Dresden, Germany Abstract A new ultra low-level counting setup has been installed in the shallow-underground laboratory Felsenkeller in Dresden, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It includes a high-purity germanium detector (HPGe) of 163 % relative efficiency within passive and active shields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The passive shield consists of 45m rock overburden (140 meters water equivalent), 40 cm of low-activity concrete, and a lead and copper castle enclosed by an anti-radon box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The passive shielding alone is found to reduce the background rate to rates comparable to other shallow-underground laboratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' An additional active veto is given by five large plastic scintillation panels surrounding the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It further reduces the background rate by more than one order of magnitude down to 116(1) kg−1d−1 in an energy interval of [40 keV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2700 keV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This low background rate is unprecedented for shallow-underground laboratories and close to deep underground laboratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Keywords: Low-background physics, Nuclear astrophysics, underground laboratory, HPGe detector, muon veto, active shielding 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Introduction A variety of scientific fields call for detection systems which are able to measure radioactive samples with ultra- low activities in the order of µBq to mBq [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' A case in point is the search for rare processes such as dark matter interactions [2, 3] or rare germanium decays [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' These include neutrino-accompanied (2νββ) [5] and neutrinoless (0νββ) [6, 7] double beta decays, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The need for ultra-low background radioactivity mea- surements is served by a number of low-background lab- oratories which are using high-purity germanium (HPGe) detectors in shallow or deep underground settings [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' All of these laboratories use massive lead and copper shields that are sometimes complemented by additional materials in order to suppress γ-ray background from the immedi- ate surroundings of detector and laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The effects of radioactive radon gas are usually mitigated by placing detector and sample in an airtight enclosure that is con- tinually flushed with a radon-free gas, preferably N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Two additional background sources are more difficult to treat, namely cosmic-ray muons and neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The for- mer usually play no role in deep-underground laboratories [9, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' ], where the muon flux is reduced by six or more or- ders of magnitude compared to sea level, rendering its ef- fects negligible for low-background activity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Email addresses: d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='bemmerer@hzdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='de (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Bemmerer), kai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='zuber@tu-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='de (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Zuber) In shallow-underground laboratories, the muon flux is usu- ally suppressed only by one or two orders of magnitude [10, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='] and may be mitigated using active veto detectors [11, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Neutrons may be produced in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' First, neu- tron generation by muon spallation on structural or shield- ing materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This process usually dominates in shallow- underground laboratories [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The second neutron pro- duction process are (α, n) reactions in the rock surround- ing the laboratory, with the α provided by natural ra- dioactivity from the natural decay chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' (α, n) usu- ally dominates the remaining, low, neutron flux in deep- underground laboratories [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' An ultra-low background radioactivity counting labo- ratory must address all of these above mentioned back- ground sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In addition, it must limit the intrinsic radioactivity of the detector by material selection [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It is noted that these techniques may also benefit, among others, experimental nuclear astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Examples in- clude the usage of activity measurement setups to count activation products such as 7Be [15], 18F [16], or 44Ti [17], but also in-beam measurements in underground settings [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' These studies are needed for a better understanding of solar fusion [19] and Big Bang Nucleosynthesis [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The present work reports on a large, coaxial HPGe detector called TU1, which is placed in the Felsenkeller shallow-underground laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This detector has been optimized for ultra-low background activity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In previous work, the Felsenkeller laboratory has already been characterized in several aspects: The muon flux (40× Preprint submitted to Astroparticle Physics arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='03905v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='ins-det] 10 Jan 2023 Ion accelerator Concrete V Figure 1: Schematic layout of tunnels VIII and IX of the shallow- underground laboratory Felsenkeller in Dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The inlet shows bunker 110, which contains two coaxial HPGes (TU1 and TU4) and a well-type HPGe (TU2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' lower than at surface) and angular distribution have been measured and matched by simulations [10], as well as the neutron flux (180× lower than at surface) and energy spec- trum [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' A previous study of the high-energy, Eγ > 3 MeV, part of the γ-ray background also showed a strong reduction with respect to the surface [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The present work complements these studies [10–12] by focusing on the low-energy, Eγ ≤ 3 MeV, part of the γ-ray background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This work is organized as follows: Section 2 describes the Felsenkeller shallow-underground laboratory, and sec- tion 3 the detector and its passive and active shielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The data analysis and results are shown in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The sensitivity of this new setup for the example of 7Be is de- rived in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The data are discussed in section 6, and a conclusion and outlook are offered in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The Felsenkeller shallow-underground laboratory The Felsenkeller shallow-underground laboratory (Fig- ure 1) is located in Dresden, Germany, and is shielded by a rock overburden of 45 m (140 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=') [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It is part of a former industrial site and the laboratory itself is built into the tunnels VIII and IX of this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The surround- ing hornblende monzonite [23] shows naturally containing 238U and 232Th with specific activities of 170(30) Bq/kg and 130(30) Bq/kg respectively [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The laboratory hosts a 5 MV Pelletron accelerator as well as two concrete-shielded bunkers (110 and 111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Bun- ker 110 is used for low-background offline measurements and is surrounded by 40 cm of low-activity concrete (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2 below for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It currently hosts three HPGe detectors (called TU1, TU2 and TU4, respectively) for γ-ray spectrometry, as well as two silicon drift detectors (TU3 and TU5) for X-ray spectrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Bunker 111 hosts the target area of the accelerator and is used for in-beam γ-ray spectrometry measurements on radiative capture re- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In bunker 110, the angle-integrated muon flux density is ϕµ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4(4) m−2s−1 corresponding to 140 meters of wa- ter equivalent (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=') [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The neutron flux density is ϕn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='61(3) m−2s−1 integrated over a broad energy in- terval ranging from 10−9 MeV to 300 MeV [12, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Experimental Setup This work concentrates on the coaxial HPGe detector TU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The other detectors will be described elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The HPGe detector The detector is a coaxial p-type high purity germa- nium detector (HPGe) with a relative efficiency of 163 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It was produced by Mirion Technologies (Canberra) ac- cording to their ultra-low background (ULB) specifications and is of the type GX 150-250-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The crystal has a mass of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='06 kg and a volume of 574 cm3 (length 90 mm and di- ameter 90 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' An end cap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='6 mm of aluminum and a polished-off p-layer of <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 mm thickness enable measure- ments down to 22 keV, well below the usual lower energy limit for standard p-type HPGe detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' A photon with 22 keV induces a preamplified pulse height of approximately 6 mV in this detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The ob- served background noise is ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 mVpp (95 % CL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The measured resolution at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='333 MeV γ-ray energy is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0 keV full width at half maximum (FWHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' There are no signif- icant shoulders to the peak in the pulse height spectrum, and the ratio of full width at one fifth maximum (FWFM) and FWHM is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The passive shielding The passive shielding of TU1 (Figure 2) consists of sev- eral layers, which are listed here from the outside to the inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The laboratory is surrounded in all directions by a 40 cm thick layer of concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Prior to mixing the concrete, the solid components used had been stud- ied individually by γ-ray spectrometry, and based on the known composition of the concrete, specific ac- tivities of 17(3) Bq/kg 238U and 18(2) Bq/kg 232Th were found, assuming secular equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' For 40K, 280(30) Bq/kg was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The active veto consists of a 5 cm thick layer of poly- vinyltoluene (EJ-2001) with a density of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0 g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It is listed here because it slightly attenuates inci- dent γ-rays and moderates incident neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The active veto is described below (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' A 1 cm thick layer of acrylic glass forms an anti-radon box (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The outer lead layer has a thickness of 10 cm and a specific 210Pb activity of 21(2) Bq/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 1Eljen Technology, Sweetwater, Texas, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 2 Figure 2: Left panel: Schematic drawing of the passive shielding for the TU1 detector including the lifting mechanism for the lid and the anti-radon box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Right panel: Profile cut of the setup, from outside to inside: acrylic glass (anti-radon box), 21 Bq/kg lead, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 Bq/kg lead, and OFRP copper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The muon veto panels (Figure 3) are placed just outside the anti-radon box but omitted here for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The inner lead layer is 5 cm thick, with a specific 210Pb activity of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5(1) Bq/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The innermost layer of the passive shielding consists of 10 cm of oxygen-free radio-pure (OFRP) copper (≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='04 Bq/kg 238U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' For changing the sample, the lead castle can be opened using an electric lifting mechanism (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This mech- Left panel (S44) Dewar panel (S15) Right panel (S45) Front panel (S17) Top panel (S16) Figure 3: Schematic drawing of the muon veto panels surrounding the anti-radon box (figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The top panel (S16) has four holes for the lifting mechanism, and the dewar-side panel (S15) has a recess for the cold finger and additional cutouts for nitrogen supply and overflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' anism lifts a cutout of the upper copper and lead bricks, as well as a part of the anti-radon box and the whole scin- tillation panel S16 (figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The anti-radon box The air in the bunkers of the Felsenkeller underground laboratory is exchanged four times per hour by an auto- matic ventilation system that is continuously active day and night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Part of the incoming air is fresh air brought in from the outside, and the remainder dried and recir- culated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Prior to the construction of the laboratory and its mechanical ventilation, a radon concentration of 0- 300 Bq/m3 was measured in tunnel VIII and IX, depending on weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' After completion of the laboratory, and with the automatic ventilation system running, the radon concentration in bunker 110 has been remeasured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In 14 days of measurements, it ranged from 0-53 Bq/m3, with an average of 11(7) Bq/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In order to minimize the impact [14, 25] of the remain- ing radon on the background rate of TU1, an airtight anti- radon box has been installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The box is constructed from acrylic glass and flushed with radon-free nitrogen from the boil-off of the HPGe dewar, monitored by a bubbler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The active muon veto The active shielding for TU1 is composed of five large plastic (EJ200) scintillation panels from Scionix (figure 3) with 5 cm thickness and a size ranging between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 m2 (S16) and 1 m2 (S44 & S45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The panels cover the anti- radon box from each side except from the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' EJ200 is a type of polyvinyltoluene with a light output of 10,000 photons/MeVee (MeV electron equivalent), a light atten- uation length of 380 cm, a rise time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9 ns and a decay 3 time constant of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='1 ns, making it suitable for timing mea- surements in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='1 ns range even in scintillating detectors as large as several meters [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The scintillation light is read out by ET9900 photo- multiplier tubes2 (PMTs) that are integrated within the scintillation panels and that have a 2π sensitive solid angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Also the high voltage supplies of the PMTs are included in the panel, so that only two connections are required on one side of each panel: an input for low-voltage power (LEMO 00), and the output for the PMT anode signal (BNC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The scintillating panels are coated with a reflector and wrapped in lightproof vinyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Electronics and data acquisition The signals from the six detectors (TU1 and five veto panels) are recorded in list mode using a CAEN DT5725S digitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This module includes eight channels with 14 bit resolution at 250 MS/s sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' All channels share the same internal clock, but each channel is separately triggered by the digital trigger included in the DT5725S device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The recorded waveforms are converted to a time stamp and pulse height with a trapezoidal filter by the CAEN pulse height firmware, version DPP-PHA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='22 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='130, directly inside the DT5725S unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The event-by-event time stamp, pulse height, and flags further characterizing the event as possible pile-up and overflow are saved in a buffer, transferred via USB cable to the computer and saved on hard disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' For the scintillating panels, typical trigger rates of 9- 140 s−1 are observed per panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' For the TU1 detector, the typical trigger rate without sample is 5 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Given the typical time lengths of the trapezoidal filter of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 µs (TU1) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0 µs (scintillators), dead time and pile-up ef- fects are expected to be insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The data analysis is performed offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Data analysis In this section, the offline data analysis is described, including the optimization of the active veto parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Pulse height spectrum in TU1 using only the passive shield The pulse height spectrum of the TU1 detector, mea- sured within bunker 110 without any additional shielding, displays many γ lines due to the natural decay chains (Fig- ure 4, black spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The 40K and 208Tl lines emerge prominently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In addition, there are a number of neutron- induced features due to the remaining neutron background [12] that can be picked out due to their triangular shapes and above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='6 MeV, there are a number of weak branches from 208Tl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The γ-ray energy resolution at the 40K peak (1460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8 keV) is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='1 keV (full width at half maximum, FWHM) 2ET Enterprises, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=', Uxbridge, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' in the adopted digitizer-based DAQ scheme, worse than the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2 keV FWHM measured with the same detector and analog electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' When applying the full passive shielding, the contin- uum below 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='6 MeV is reduced by three to five orders of magnitude, and the neutron features are no longer appar- ent (Figure 4, blue spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Instead, a wide continuum appears that is only slightly energy dependent, showing the effects of the energy loss of the remaining cosmic-ray induced muon flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Active muon veto In a first step, the pulse height spectrum of the scin- tillating panels has been calibrated in energy using the Compton edges of radionuclide standards made of 137Cs, 60Co, and 88Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' These point-like standards were placed on the center of one of the large sides of each scintillat- ing panel, in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' As a second step, the trigger thresh- old for the scintillating panels was increased to approxi- mately 2 MeV, in order to avoid unnecessarily recording radionuclide-induced pulses detected in the scintillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The resulting energy-calibrated spectrum (Figure 5, black curve) shows a peak near 2 MeVee that is given by the above mentioned trigger condition and a broad peak near 6 MeVee that is assumed to be due to muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The observed pulse height in the histogram is propor- tional to the number of scintillation photons detected by the PMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Due to the finite reflectivity of the reflective wrapping, as well as the light attenuation in the crystal itself, not all of the emitted photons are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This is even enhanced for panels S15 and S16 which also include holes and a recess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The effect has been studied using a 60Co radionuclide standard placed at several places across the scintillating panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Using the light yield at the center of the panel as reference, >90% of the area of the panel shows observed pulse heights between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2 times the reference pulse height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The remaining <10% are located very close to the PMT and show higher light yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It is up to 5 times the reference light yield when the 60Co source is placed directly atop the sensitive area of the PMT, so that it may see scintillation light from both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' At the depth of the Felsenkeller lab, the muon energy spectrum peaks at approximately 30 GeV, and the energy loss of muons is about 2 MeVee per cm of EJ-200 mate- rial traversed [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Qualitatively, the large cross section of the panels is expected to lead to muons going per- pendicularly through the panel dominating the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' They would lose approximately 10 MeVee, and applying the above mentioned factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7 for less than ideal light collection, would register near 7 MeVee, consistent with the observed peak at 6 MeVee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In addition, many muon paths are possible that are sig- nificantly longer than 5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' As a consequence, the muon- induced spectrum in S45 extends to rather high energy, up to dozens of MeVee (Figure 5, black curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In the following two subsections, by offline analysis of background runs lasting several days, the two main pa- 4 500 1000 1500 2000 2500 3000 [keV] TU1 E 2 − 10 1 − 10 1 10 2 10 3 10 4 10 5 10 6 10 ] 1 d) ⋅ kg ⋅ Counting rate [(keV Figure 4: Pulse height spectrum for the TU1 detector, normalized to running time, bin width, and detector mass: Bunker 110, without any shielding except for the concrete walls (black curve, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8 days running time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Full passive shielding and anti-radon box (blue curve, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The same spectrum, but applying also the active shield (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The γ-lines identified in the red spectrum are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' rameters for the active muon veto are optimized: First, the timing coincidence condition (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3), and second, the pulse height threshold (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' A third subsec- tion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5) then discusses additional information gained by combining the two cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Time coincidence condition The time difference between the trigger times tTU1 in TU1 and those of the scintillating panels ti (with i ∈ {S15, S16, S17, S44, S45}) shows a sharp peak that is con- tained in the time interval [-5 µs,+5 µs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' A typical exam- ple for this time difference spectrum is given in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The mutual time differences between individual scintilla- tor panels are not shown but display a similar pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The sharp peak in the [-5 µs,+5 µs] coincidence window of figure 6 is caused by muons that pass both S45 and TU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The peak width is given by the time resolution of TU1, which is approximately on the level of the 2 µs binning used in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' A second, longer time coincidence window of [-150 µs,+5 µs] will be introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Energy threshold in the scintillator panels Even with a trigger threshold near 2 MeV (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2), there is still a significant number of radionuclide-induced events in the pulse height spectrum of the scintillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This can be seen by comparing the free-running pulse height spectrum of one scintillator panel (Figure 5, black curve) with the same spectrum when requiring time coincidence, within a “short” coincidence window of [-5 µs,+5 µs], with any of the other detectors (Figure 5, blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In order to reduce the impact of these remaining radio- nuclide-induced events, an additional threshold energy Ethresh,i (with i ∈ {S15, S16, S17, S44, S45}) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This threshold was carefully optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Too low Ethresh,i would allow a too large impact of radionuclide-induced events, whereas a too high Ethresh,i would reduce the muon veto efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This optimization was carried by analyzing a one day long run with a 2 kBq 7Be source mounted in close ge- ometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Starting from the local minimum to the left of the muon peak (Emin, S45 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='6 MeV for the blue curve in Figure 5), the energy threshold was varied in the of- fline analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Then, for each given value of the energy threshold, two quantities were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' First, the cut survival probability Pcut survival for events in the 7Be peak area in the TU1 detector given by Pcut survival = NTU1(7Be)|cut NTU1(7Be)|free running (1) The second quantity was the no-source specific background rate ˙N40−2700 in the 40-2700 keV energy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It is de- termined by analyzing a 30-day run without any source and with the same running conditions as the 7Be run with the same energy threshold values and requiring anticoinci- dence with all the scintillator panels within a time window [-150 µs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='+5 µs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' See section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 below for the justification for this somewhat enlarged time coincidence window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The optimal energy threshold was then defined to be the energy for which Pcut survival = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='995 was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This was typically at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8× the local minimum energy (red dashed line at Ethresh, S45 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9 MeV in Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' For the other scintillators, not shown in the Figure, similar energy thresh- olds of 2-3 MeV were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The impact of the choice of the energy threshold is 5 0 10 20 30 40 50 60 70 [MeV] S45 E 1 10 2 10 3 10 4 10 5 10 6 10 Counts / 5keV S45 S45 and any other detector S45 and TU1 1 10 2 10 1 10 2 10 3 10 4 10 5 10 6 10 Figure 5: Pulse height spectrum of scintillation panel S45, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 days running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Black curve: raw spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Blue curve: requiring coincidence within [-5 µs,+5 µs] with any of {S15, S16, S17, S44} or TU1 (ETU1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='04 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Red curve: requiring coincidence with TU1 (ETU1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='04 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Red dashed line: energy cut for this panel (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9 MeV, section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Table 1: Impact of the variation of the energy threshold by a factor f on the cut survival probability Pcut survival and on the specific background rate ˙N40−2700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' f Pcut survival ˙N40−2700 Remark 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='958(1) 112(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='980(1) 113(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='995(1) 116(1) Adopted value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='998(1) 120(1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='998(1) 124(1) illustrated in Table 3 where all energy thresholds are varied together by a constant factor f = Ethresh,i/Emin,i, and for each value of f the cut survival probability and background count rate are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It is clear from Table 1 that the background count- ing rate changes by only a few percent in the region stud- ied, while the cut survival probability is rather sensitive to the choice of the energy threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' A value of Pcut survival =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='995 was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This value introduces a small sys- tematic shift, no more than -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5%, which is lower than the typical geometry-dependent detection efficiency uncer- tainty of 1-3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This adopted value does not significantly worsen the precision of the activity determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' As a final check for the adopted threshold energy, the ratio of counts in the S45 pulse height spectrum requir- ing coincidence with TU1 (red curve in Figure 5) with the S45 free running spectrum (black curve in Figure 5) is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' For ES45 < Ethresh, S45, the observed ratio is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9×10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This is consistent with expectation for random coincidences, given the 10 µs width of the “short” time co- 500 − 400 − 300 − 200 − 100 − 0 100 200 300 400 500 s] µ [ TU1 t S45 t 1 10 2 10 3 10 4 10 5 10 6 10 s µ Counts / 2 Figure 6: Black curve: Event by event time difference between events in scintillation panel S45 (tS45) and in TU1 (tTU1), 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Red curve, in addition ETU1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='04 MeV and ES45 ≥ Ethresh, S45 (Ethresh, S45 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9 MeV) are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The grey box shows the “long” coincidence timing window of [-150 µs,+5 µs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' incidence window and the TU1 trigger rate of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' For ES45 > Ethresh, S45, instead, the observed ratio is much larger, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3×10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In conclusion, S45-TU1 coincident events can be fully explained by random coincidences for ES45 < Ethresh, S45, while only 1% of S45-TU1 coincident events with ES45 > Ethresh, S45 can be explained by random coin- cidences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Thus, it also confirms that the threshold value Ethresh, S45 is appropriately chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Combination of timing and energy cuts As a next step, the newly determined energy threshold is adopted, and the time difference spectrum between TU1 and S45 is re-determined, now requiring ES45 ≥ Ethresh, S45 (Ethresh, S45 and ETU1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='04 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' As expected, the resultant spectrum (Figure 6, red curve) still shows the prompt muon coincidence peak at -5 µs ≤ (tS45 − tTU1) ≤ 5µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In addition, now two distinct features emerge that had previously been covered by the radionuclide-induced events, namely at time differences (tS45 − tTU1) ≈ -100 µs and (tS45 − tTU1) ≈ -10 µs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The first feature may be due to muons which first pass the scintillator panel and subsequently scatter on the lead shield to create a free neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This (µ,n) process has previously been found to dominate the neutron flux, es- pecially at high neutron energies, in the Felsenkeller lab- oratory [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The delay of ∼100 µs between the passage of the muon through the scintillator panel and the signal in the HPGe detector may be due to free (µ,n) neutrons that, after creation, are first scattered a few times, and 6 slightly moderated, in the lead shield before they give rise to a signal in the HPGe detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The second feature, with a delay of typically 10 µs or less between the passage of the muon through the scintil- lator, may be due to the decay of stopped muons (mean lifetime 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2 µs) in the lead shield, and subsequent detection of secondary electrons, positrons, or annihilation radiation from the decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The specific rate of events within the extended part of the anticoincidence window, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' [-150 µs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='-5 µs], that also satisfies the energy cuts for TU1 and the scintillators is 20(1) kg−1d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Thus, the no-source background rate (Ta- ble 1) of 116 kg−1d−1 for the full anticoincidence window of [-150 µs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='+5 µs] would increase to 136 kg−1d−1, if a more narrow anticoincidence window of [-5 µs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='+5 µs] would be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Rates of coincident events The coincidence rates (ETU1 ≥ 40 keV, Epanel ≥ Ethresh, time coincidence window tcut=[-150 µs,+5 µs]) between the scintillation panels and TU1 are R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0207(1)s−1 (TU1∧S15), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0337(1)s−1 (TU1∧S16), R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0208(1)s−1 (TU1∧S17), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0298(1)s−1 (TU1∧S44), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0273(1)s−1 (TU1∧S45), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' As expected, the coincidence ratios for the panels on opposite sides are close: The opposing panels S44 and S45 differ by just 9%, and the opposing panels S15 and S17 by <1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The highest absolute coincidence rate is found for TU1∧S16, consistent with expectation based on the measured muon angular distribution in this area [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The total rate of vetoed events in TU1 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0763(2)s−1 for ETU1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='04 MeV and the “long” anticoincidence win- dow of [-150 µs,+5 µs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This is consistent with the ex- pected rate of muons penetrating TU1 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='08 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This rate has been estimated using the muon flux of ϕµ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4(4)m−2s−1 that has previously been measured at this location [10] and an estimated muon-exposed cross section of TU1 of 145 cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Preamplified signals of opposite polarity An uncommon characteristic of the TU1 detector is the fact that there is a significant rate, 6 s−1, of signals with unphysical polarity, opposite to the usual one, at the preamplifier output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' These pulses have the same general shape and amplitude as the correctly preamplified pulses, but opposite polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' However, they are not correlated in time to any other events, either from TU1 or from any of the scintillating panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' For safety, these unphysical events are studied via a dedicated channel of the data acquisition (self-triggered just as the other channels) and stored for possible future analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The observed pulse height spec- trum does not show the characteristic features of an HPGe spectrum, but rather resembles a Landau distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The opposite polarity events are believed to be caused by an electronic noise source in the preamplifier unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Given their rate and random distribution in time, there is a 6×10−6 probability for them to affect a true physical event, which is negligible in the present, low-background appli- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Dead time The total dead time td,tot of TU1 is given by the dead time td,raw due to the ∼10 µs processing time of the dig- itizer and an additional cut efficiency εcut (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4), which takes into account the loss of counts due to random anticoincidences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The digitizer dead time has been estimated based on the count rate of the resulting spectrum (output count rate, OCR) and the 1 µs trigger hold-off time during which the digitizer unit does not accept new triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In a pro- cedure similar to the one recommended by CAEN [28], the unknown input count rate (ICR) is determined in an iterative procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' From a starting ICR and assuming Poisson-distributed events, a predicted OCR is determined and then compared to the observed OCR in order to ob- tain an improved ICR estimate for the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This iteration converges typically after five cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The TU1 channel of the digitizer self-triggers on the ab- solute height of the second derivative of the pulse, which triggers both for negative (physical) and positive (unphys- ical) pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In the TU1 channel, the unphysical events are converted to channel 0 of the pulse height histogram and fully taken into account during the dead time determina- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' They contribute a relative dead time of 6×10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In addition, the unphysical events are converted and stored in a separate channel of the digitizer, see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In a typical run, about 2% of the detected events in the TU1 channel do not follow a Poisson distribution in time, but instead follow directly a preceding event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This behaviour is consistent with electronic noise or mechani- cal vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Due to their limited number, these non- Poissonian events do not have a significant impact on the precision of the dead time determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Remaining background in TU1 The remaining background after the construction of the passive shielding and the application of the active veto is shown as an orange histogram in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The remaining γ-lines, as well as their origins and counting rates, are listed in table 2, and will be discussed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Neutron induced lines in germanium The four low energetic γ-lines at 52-56, 64-67, 140, and 198 keV are due to isomeric states in several germa- nium isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' These states in 71mGe (Ex = 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 keV, 20 ms half-life), 73mGe (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7 keV, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 s half-life), and 75mGe (139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7 keV, 48 s half-life) are populated by neutron cap- tures, (n, 2n) reactions, or (n, n′γ) reactions within the germanium crystal itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' All three have half-lives that are too long for an effective active veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The peaks at 52-56 and 64-67 keV (Table 2) are due to the decay of the Ex = 67 keV metastable second excited 7 Table 2: Energy and counting rates of the remaining γ-lines in the TU1 detector, after applying the active veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Energy Nuclide Origin Rate [keV] [kg−1d−1] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0-55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='6 73mGe 72Ge(n,γ), 74Ge(n,2n) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='28(19) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9-66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7 73mGe 72Ge(n,γ), 74Ge(n,2n) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='38(17) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8-75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0 Pb X Lead Kα & Kβ X-rays 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='33(12) 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7 75mGe 74Ge(n,γ), 76Ge(n,2n) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='34(16) 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 71mGe 70Ge(n,γ), 72Ge(n,2n) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='00(15) 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9 214Pb 238U decay chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='29(11) 511 e+e− Annihilation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='64(13) 583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2 208Tl 232Th decay chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='14(8) 609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3 214Bi 238U decay chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='21(8) 834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8 54Mn 54Fe(n, p)54Mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='17(7) 1173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2 60Co 63Cu(n, α)60Co 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='18(5) 1274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 22Na Lab-specific nuclide 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='20(5) 1332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 60Co 63Cu(n, α)60Co 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='16(5) 1460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8 40K Natural radioactivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='49(6) 2614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 208Tl 232Th decay chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='22(4) state of 73Ge [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The half-life of the first excited state (T1/2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9 µs) is comparable to the length of the timing window for the HPGe, giving rise to partial summation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' If the decay of the first excited state takes enough time, the lower-energy cluster of peaks in the TU1 spec- trum will be fed by the 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 keV photon (or the 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0 keV- 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 keV conversion electron respectively), which can be separated by the software from the subsequent decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' If this decay takes a shorter amount of time though, both peak clusters in the TU1 spectrum can be fed due to sum- mation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The lower-energy one via the 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 keV pho- ton (or the 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0 keV-53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 keV conversion electron respec- tively) and their summation with the subsequently emitted conversion electron of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The higher-energy cluster then originates from the summation of the 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 keV photon (or the 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0 keV-53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 keV conversion electron respectively) and their subsequently emitted conversion electron with 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9 keV-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7 keV γ ray is due to the decay of the 75mGe state at the same energy to the ground state of 75Ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The rate of this γ line can be used to approximate the ther- mal neutron flux density inside the passive shielding, re- sulting in ϕ = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0) × 10−5 cm−2s−1 using a con- version factor from literature [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' During a dedicated campaign to measure the neutron flux in the Felsenkel- ler laboratory [12], the thermal flux inside room 110 has been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The thermal flux obtained with an un- moderated 3He tube during a run previous to the instal- lation of TU1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' in an empty bunker 110, with its lead shield is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2) × 10−5 cm−2s−1 [24, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It is known that large passive lead shields can enhance the neutron flux in a shallow-underground laboratory such as Felsen- keller [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This is mainly caused by (µ, n) reactions in the high atomic charge shielding material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Neutrons of ∼ MeV kinetic energy are produced in these reactions [12, 24] and subsequently moderated, for example, in the 200 kg of EJ-200 organic scintillator forming the muon veto (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Finally, the 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4 keV peak is caused by the decay of the metastable level of 71Ge [29, 32, 33], which has a half- life of 20 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Its decay scheme suggests a chain emis- sion of, first, a 12-22 keV conversion electron and, second a gamma-ray of E = 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0 keV [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The peak in the spectrum is a summation line, which benefits from both a negligible escaping probability of the emitted conversion electrons and the short half-life of the first excited state of 71Ge at Ex = 175 keV (T1/2 = 79 ns) with respect to the comparatively longer timing window of the HPGe (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Neutron activation lines The spectrum shows three lines from the decay of long- lived radionuclides that may be attributed to neutron acti- vation: 54Mn (Eγ = 834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8 keV, T1/2 = 312 d) may be pro- duced via the 54Fe(n,p)54Mn reaction, and 60Co (1173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2 and 1332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 keV, T1/2 = 1925 d) via the 63Cu(n, α)60Co reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' These nuclides are typical contaminations in low-background HPGe detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' They are likely due to cosmic-ray activation on shielding or structural materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It is expected that these contaminations slowly decay out in the next years and stabilize at a much lower level, consistent with the neutron flux in Felsenkeller that is 180 times lower than at surface [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Annihilation peak at 511 keV The annihilation peak at 511 keV is attenuated by a factor of 11 by the active veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The veto thus removes most of the muon-induced effects, including the decay of stopped muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The remaining 511 keV counting rate is largely due to the annihilation of positrons emitted in β+ decays in the detector and shielding material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It is noted that there is a small contribution (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='04 kg−1d−1) due to photons with Eγ = 510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='77 keV from the decay of 208Tl, which also causes the 583 and 2615 keV lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Radon-induced effects The low specific radon activity of 11(7) Bq/m3 in the measurement bunker (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3) may lead to small amounts of radioactive 222Rn (T1/2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8 d) to diffuse into the sen- sitive volume near TU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This effect is mitigated by the anti-radon box and its flushing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Nevertheless, γ rays from the radon daughters 214Pb and 214Bi appear in the TU1 spectrum at 352 and 609 keV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' An additional weak feature at 295 keV is not 2σ significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' After a typical sample change, the integral counting rate is saturated at its final value after 10000 s, which is significantly shorter than the physical half-life of 222Rn, showing the desired effects of the active flushing of the anti-radon box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Naturally occurring radionuclides The remaining rates in the γ-lines at 1460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8 keV (from 40K) and 2614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 keV (from 208Tl) are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='49(6) kg−1d−1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='22(4) kg−1d−1 respectively (table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Using the observed rates of these two γ rays without shielding and applying a calculated attenuation based on the weakest spot of the passive shielding, counting rates of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3 kg−1d−1 are found for these two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' For 208Tl, based on these values there is no reason to assume a significant internal contamination of TU1 within the shielding materials or the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Also for 40K, the effect of the attenuation is compara- ble to the expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' However, the attenuation is still smaller than conservatively assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The remaining rate of 40K within TU1 may therefore also include contamina- tions within the shielding and the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In fact, 40K has previously been found to be a remaining contaminant in high-purity copper samples [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Experimental study on 7Be As an illustration for the capabilities of the TU1 detec- tor for activation studies for nuclear astrophysics, a study of weak activated 7Be samples produced during the inves- tigation of the 3He(α, γ)7Be reaction at the Felsenkeller 5 MV Pelletron accelerator is discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Efficiency calibration 7Be has a half-life of T1/2 = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='22(6) d and emits a γ- ray of E = 478 keV with an emission probability of η = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='44(4) %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In order to determine the absolute full-energy peak ef- ficiency at E = 478 keV, three calibration samples of 7Be have been produced at the cyclotron accelerator of the Institute for Nuclear Research in Debrecen (ATOMKI), Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' To this end, LiF has been evaporated on tan- talum disks (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='22 mm thickness and 27 mm diameter) and subsequently bombarded with 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5 MeV protons, resulting in 7Be activities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3, 22, and 48 kBq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The geometry of the tantalum disks and proton beam spot has been chosen to conform to the geometry of the much less radioactive 7Be samples from the 3He(α, γ)7Be irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The activity of the three ATOMKI 7Be samples has then determined in far geometry, using arrays of several other well-calibrated radionuclide standards, independently at ATOMKI and at HZDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Using these three calibration sources, the absolute full-energy peak detection efficiency for the TU1 detector, when using a point-like sample in close geometry, has subsequently been determined to be ε = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='45(19) %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Activation analysis For the ongoing campaign on the 3He(α, γ)7Be reaction at Felsenkeller, a tantalum disk (hereafter called sample ST8) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='22 mm thickness and 27 mm diameter has been implanted with 1018 cm−2 3He at the HZDR Ion Beam 400 420 440 460 480 500 520 540 [keV] TU1 E 2 − 10 1 − 10 1 10 2 10 3 10 4 10 5 10 6 10 ] 1 d) ⋅ kg ⋅ Counting rate [(keV Figure 7: Pulse height spectrum in the region of the 478 keV line of 7Be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Black spectrum: 7Be calibration source with 2 kBq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Blue: Activated 7Be sample (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7 Bq), without active veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Red: Activated 7Be sample, with active veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Light blue: background without sam- ple, with active veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Shaded light blue area: 10 keV interval for the determination of the detection limit LD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Table 3: Detection limit LD and activity Lsignal=noise for signal equal to noise for 50 d counting time and a 10 keV wide region of interest for 7Be (478 keV) and 181Hf (482 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Method 7Be 181Hf [mBq] [mBq] Detection limit LD at 90% CL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='11 Lsignal=noise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='31 Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' ST8 was then irradiated with a 5-10 µA beam of 4He+ (Eα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='8 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0 MeV) at Felsenkeller to investigate the 3He(α, γ)7Be reaction, applying a charge of Q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The resulting pulse height spectrum near 478 keV for the activated sample ST8 is shown in Figure 7: blue curve without active veto, red curve with active veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' From the comparison of these two spectra, the cut survival proba- bility of Pcut survival = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='995 is found (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The data are compared with the spectrum of the calibration sample (black curve) and the background without sample (light blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Sensitivity for 7Be detection The nuclide 7Be is not only relevant for nuclear astro- physics [15, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Due to its cosmogenic origin, it may also be used to monitor atmospherical and geological processes [36–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The present background spectrum (light blue curve in Figure 7) and detection efficiency (section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='1), as well as literature data on decay branching ratio and half-life, 9 are now used in order to derive the sensitivity of TU1 for detecting 7Be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The detection limit LD in Bq, using a coverage factor of kα = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='282 for 90% confidence level, a branching ra- tio β, an absolute full-enery peak detection efficiency ε, a background rate ˙NBG, and a counting time t is given by [39] LD = k2 α + 2kα � 2 ˙NBGt ε × β × Cdecay × t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' (2) This value has been determined for a 10 keV wide region of interest and two typical nuclides: In addition to 7Be, also for 181Hf, which has a similar half-life (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='39 d) but a higher branching ratio, β = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='5% for its 482 keV γ ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Due to the similar half-lives, the decay corrections for 50 d counting time are also similar, Cdecay = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='73 (7Be) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='68 (181Hf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Detection efficiency and background dif- fer only negligibly for the two relevant energies, 478 and 482 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' For comparison, for both cases also another value has been computed, namely the activity Lsignal=noise for which the net count rate (signal) is equal to the background (noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The data show that the TU1 detector can eas- ily reach the µBq range, if a suitably long counting time is adopted (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The same background data was then used to calculate the maximum half-life T max 1/2 that can be determined in a sample with NA (1 mol) decaying nuclei, neglecting the decay and self-absorption corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Again 50 d count- ing time, a 10 keV wide region of interest near 478 keV, and a detection efficiency of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='45% are used, and now a branching of β = 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' T max 1/2 = ln 2 λmax = ln 2 × NA LD = ln 2 × NA × ε × β × Cdecay × t k2α + 2kα � 2 ˙NBGt (3) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='1 × 1020y This value is in the range of double-beta decay half-lives with the emission of neutrinos that have recently been de- termined [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Discussion In this section, the new data are interpreted and com- pared with literature data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Interpretation of the present data The passive shielding suppresses the integrated count- ing rate between 40 keV and 2700 keV by a factor of 4300 (Figures 4 and 8), showing its high effectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In the pas- sively shielded spectrum (blue curve in Figure 8), only two peaks from the natural background remain, namely 40K and 208Tl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The rest of the passively shielded spec- trum is dominated by muon-induced events, both in the continuum and in the 511 keV annihilation peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Furthermore, the passively shielded spectrum is very similar to a previously published spectrum [14] that has been obtained in the VKTA laboratory in the nearby Fel- senkeller tunnel IV, which has the same rock overburden and muon flux as tunnel VIII studied here [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' There are only two significant differences between these spectra (black and blue curves in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' First, a slightly lower 40K rate in the present spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This is remarkable, given that there is a significant amount of 40K in the laboratory walls surrounding TU1 (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In the case of tunnel IV, the 40K γ rays are already attenuated outside the detector shield itself by a measurement chamber made of ancient steel (MK2) that hosts several detectors [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The comparison seems to in- dicate that the present passive shield, 15 cm of radiopure lead and 10 cm of copper, improves upon the previous one (10 cm normal lead, 5 cm radiopure lead, 5 cm radiopure copper) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Importantly, it shows that the omission of a measurement chamber like MK2 is more than compensated by the present, additional 5 cm of copper and, possibly, by the usage of radiopure lead for the outer shielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The second difference between the black and blue spectra is a 65Zn contamination (Eγ = 1115 keV, T1/2 = 244 d) in the VKTA spectrum, which has been measured several years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' While this contamination used to be significant, in the meantime it decayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The present active shielding further reduces the inte- grated counting rate by a factor of 17, bringing the total background suppression to a factor of 73000, almost five orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' At the same time, random coinci- dences reduce the peak detection efficiency by just 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='47%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This small reduction is acceptable given the fact that the setup is designed for low counting rate experiments that will be limited by the statistical, not the systematical un- certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The active shield reveals a number of γ rays that had been covered by the muon-induced continuum in the spec- trum without active shield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It is possible that a future slight increase of the lead shield thickness, by up to an additional 5 cm, might further attenuate the remaining ra- dionuclide lines from 40K and 208Tl, at the expense of a somewhat higher (µ, n) rate [12] that would increase the neutron-induced lines at 198 keV and below (sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The actively shielded spectrum still shows a significant continuum, without any clearly discernible Compton edges (Figure 8, red spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This may indicate that correct- ing the small imperfections in the active muon veto, for example the holes for the lifting mechanism of the lid, may lead to a further background reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' In another shallow-underground laboratory, the muon veto reduced the 40-2700 keV counting rate by a factor of 89 [43], indi- cating that correcting the small imperfections in the muon veto may yield another factor of five reduction in counting rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 10 500 1000 1500 2000 2500 [keV] E 2 − 10 1 − 10 1 10 ] 1 d) ⋅ kg ⋅ Counting rate [(keV Figure 8: Pulse height spectrum of the TU1 detector without active veto (blue spectrum) and with active veto (red spectrum), normalized to bin width and detector size and shown for the typical background region of interest within [40 keV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2700 keV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' For comparison, a previous spectrum from the D6 detector of the VKTA lab in Felsenkeller tunnel IV [14] is shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Comparison with other setups reported in the litera- ture In order to further extend the comparison and discus- sion, data from a selection of low-background HPGe de- tectors in underground laboratories is listed in table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Their integrated counting rates within the energy inter- val of [40 keV,2700 keV] (for the Gran Sasso detectors, a slightly different range of [100 keV,2700 keV] is used) are plotted in figure 9 as a function of their corresponding rock overburden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The setups used for comparison span a broad range of rock overburden and generally reflect the state of the art, with expert adjustments being made for the relevant depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Two general trends are apparent (Figure 9): First, the counting rate generally decreases with increasing rock overburden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Second, the scatter of the data points from different laboratories decreases with increasing rock over- burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Both of these effects are due to cosmic-ray in- duced muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Once the radionuclide-induced background has been strongly attenuated, the muon flux remains the main variable affecting the observed integral background rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' While at high rock overburden, the muon flux is at- tenuated so much that it plays no major role any more, at medium-low rock overburden, it is important to optimize the treatment of muon-induced effects, including an active veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The differences in the particular treatment of the muon background in various laboratories are reflected in the spread of counting rates at similar depths (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' It is apparent that the background rate in the present setup is in the same league with much deeper underground detectors, and that it is lower than the background of some setups with even greater rock overburden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Summary and outlook The present work describes a recently installed new HPGe detector called TU1 in the shallow-underground laboratory Felsenkeller (Dresden, Germany).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Using a so- phisticated passive and active shield, the integrated back- ground counting rate in the detector is reduced to 116 kg−1d−1 in the [40 keV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2700 keV] interval, an unprecedented low value for a shallow-underground laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' This detector is now the most sensitive radioactivity-measurement setup in Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' A detailed study of the remaining background is pre- sented, and for the examples of point-like sources of 7Be and 181Hf, the detection limits are derived for typical count- ing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The data are compared to relevant similar lab- oratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Comprehensive simulation studies are currently ongo- ing and with more statistics in the coming years, it will be possible to explore the correlations between incoming muons and their signature in the shielded germanium de- tector in greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' At the same time, it is planned to further improve muon veto efficiency by closing remaining small gaps in the active scintillator shield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Acknowledgments The authors are indebted to Tam´as Sz¨ucs (ATOMKI Debrecen) for providing the 7Be calibration samples, to Detlev Degering (VKTA) for valuable discussions, and to Toralf D¨oring, Maik G¨orler, Andreas Hartmann, Bernd Rimarzig (HZDR), and Martin Siegel (TU Dresden) for technical support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' — Financial support by Deutsche For- schungsgemeinschaft DFG (INST 269/631-1 FUGG, TU 11 Table 4: Background count rates for selected detectors, normalized to detector mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The energy interval is [40 keV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2700 keV], except for the Gator detector at LNGS where the energy interval starts only at 60 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Detector/ Location Depth Count rate Count rate Reference Laboratory [m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='e] without veto with veto [kg−1 d−1] [kg−1 d−1] D4 Seibersdorf Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Austria ≈1 85100±400 8110±40 [40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 41] DLB TU Dortmund,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Germany 10 34400±60 2900±6 [42] GIOVE MPIK Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Germany 15 31027±48 348±3 [43,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 44] CAVE IAEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Monaco 543 840±50 [8] D6 Felsenkeller (VKTA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Germany 110 2938±5 [14] TU1 Felsenkeller (TU Dresden and HZDR),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Germany 140 1982±3 116±1 present work Ge-14 HADES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Belgium 500 208±4 178±8 [45,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 46] GeMSE La Vue des Alpes Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Switzerland 620 91±1 [47,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 48] GeOroel LSC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Canfranc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Spain 2450 142 [49] GeCRIS LNGS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Gran Sasso,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Italy 3800 111±1 [50] GeMPI LNGS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Gran Sasso,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Italy 3800 59±1 [50] Gator LNGS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Gran Sasso,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Italy – [60 keV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2700 keV] 3800 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='7 [51] OBELIX LSM, Modane, France 4800 68±1 [5] Dresden Institutional Strategy ”support the best”, ZU123/21- 1, and BE4100/4-1), by the Konrad-Adenauer-Stiftung, and by the European Union (ChETEC-INFRA, project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 101008324) is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Povinec (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' ), Analysis of Environmental Radionuclides (Ra- dioactivity in the Environment), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' 11, Elsevier Science, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' [2] R.' metadata={'source': 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M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Lam- pert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Hult, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Arnold, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Neumaier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Reyss, A new low- level γ-ray spectrometry system for environmental radioactivity 12 1 10 2 10 3 10 4 10 Depth [m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='] 2 10 3 10 4 10 ] 1 d 1 Integrated counting rate for 40-2700 keV [kg Seibersdorf, AT Dortmund, DE Heidelberg, DE IAEA, Monaco Felsenkeller VKTA, DE HADES, BE Bern, CH Canfranc, ES Gran Sasso, IT Modane, FR TU1, Felsenkeller, DE, present work 8 − 10 7 − 10 6 − 10 5 − 10 4 − 10 3 − 10 2 − 10 1 − 10 1 ] 1 sr 1 s 2 (Vertical) flux intensity [cm Muon component Neutron component Figure 9: Integrated counting rates in the [40 keV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content='2700 keV] region for different low-background detectors as a function of the rock over- burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' Data from Table 4 and with GeMPI plotted in case of the three LNGS detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' The vertical flux intensity of muons and the flux intensity neutrons are estimated and added on the second y-axis [24, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E2T4oBgHgl3EQfdQfR/content/2301.03905v1.pdf'} +page_content=' at the underground laboratory 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0000000000000000000000000000000000000000..15f0ead9374953bc13e036662a530f25d81751a2 --- /dev/null +++ b/ktE1T4oBgHgl3EQfNgPF/content/tmp_files/2301.03004v1.pdf.txt @@ -0,0 +1,583 @@ +Joule-Thomson effect of AdS black holes in conformal gravity +Yang Guo∗, Hao Xie†, and Yan-Gang Miao‡ +School of Physics, Nankai University, Tianjin 300071, China +Abstract +We investigate the Joule-Thomson (JT) effect of AdS black holes in conformal gravity. We +derive the JT coefficient in terms of the relevant thermodynamic quantities and then make a +similar derivation via a direct way. We analyze the JT coefficient and find that the JT coefficients +obtained from two different approaches are equivalent. Moreover, we present a novel isenthalpic +process in which the inversion temperature is minimal and it separates the corresponding heating- +cooling phase. We analyze the inversion temperature and its corresponding inversion curve that +separates the regions for the JT effect to be allowable or forbidden, where such an effect can only +be observed in the allowable region. We also discuss the effects of two important parameters on +the inversion curves. +∗guoy@mail.nankai.edu.cn +†xieh@mail.nankai.edu.cn +‡Corresponding author: miaoyg@nankai.edu.cn +arXiv:2301.03004v1 [gr-qc] 8 Jan 2023 + +Contents +1 +Introduction +1 +2 +AdS black holes and thermodynamics in conformal gravity +2 +3 +Joule-Thomson expansion +3 +3.1 +Two approaches to derive Joule-Thomson coefficients . . . . . . . . . . . . . . . . . . +3 +3.1.1 +A derivation based on the relations between the JT coefficient and thermo- +dynamic quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +3.1.2 +A direct derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +3.2 +A novel isenthalpic process and inversion curves on T − P plane +. . . . . . . . . . . +5 +4 +Conclusions and discussions +7 +1 +Introduction +The black hole thermodynamics has always been a topic of great concerns and challenges over the +past five decades. A black hole is no longer a mechanical system described formally by an analogy +with thermodynamics but actually a thermodynamic system with temperature and entropy since +the important discoveries, including the Hawking area theorem [1], the Bekenstein entropy [2], and +the Hawking radiation [3], were made. In the extended phase space, the cosmological constant is +viewed [4, 5, 6] as a thermodynamic variable and identified as the thermodynamic pressure. This +has led to a wide study in black hole thermodynamics called ‘Black Hole Chemistry’, where many +phenomenological aspects of thermodynamics are introduced into black hole physics. +A number of phenomenological studies show that black holes behave in many ways quite anal- +ogous to common chemical phenomena, such as the liquid/gas phase transition [7, 8], the van der +Waals fluid behavior [9, 10, 11], the triple point [12, 13, 14], and the heat engines [15, 16, 17, 18], +etc. Furthermore, we can consider the adiabatic expansion in the field of black hole chemistry in +which the cosmological constant is treated as a thermodynamic pressure. In the common ther- +modynamics, the adiabatic expansion usually refers to the well-known Joule-Thomson (JT) effect, +which describes the change in temperature of a gas or liquid when the gas or liquid passes through +a throttle, like a porous plug or a small valve. This process is known as throttling or JT expansion, +in which the enthalpy remains unchanged. In black hole thermodynamics, we propose the JT effect +of AdS black holes in conformal gravity and study it in detail because the cosmological constant +plays the role of a thermodynamic pressure. +When a black hole is dealt with as a thermodynamic system, its further investigations provide +insights into the fundamental relationship among gravity, thermodynamics, and quantum physics. +In this paper, we analyze the JT expansion of AdS black holes in the conformal gravity [19, 20] +in which the mass can be introduced by employing [21, 22] the Noether charge associated with a +time-like Killing vector. However, this parameter is not explicitly contained in a metric function +but is equivalent to a specific combination [23, 24] of parameters in a metric function. This feature +distinguishes the conformal gravity from the other theories of gravity in which the mass is simple +and well-formed in a metric function. We consider the JT effect of AdS black holes, including the +heating-cooling effect and inversion temperature, in conformal gravity. On the one hand, we want +to verify whether the mass and the other thermodynamic quantities of AdS black holes in conformal +gravity are well-defined during the JT process. On the other hand, we want to reveal some effects of +1 + +conformal gravity on the JT expansion. More importantly, the novel phase behavior of AdS black +holes in conformal gravity can extend the burgeoning field of black hole chemistry. +Our paper is organized as follows. In Sec. 2 we make a brief review for AdS black holes in +conformal gravity and the relevant thermodynamics. In Sec. 3.1 we derive the analytic expression +of JT coefficients in two different ways, where one is based on the relevant thermodynamic quantities +and the other is a direct derivation. In Sec. 3.2 we present a novel isenthalpic process and discuss +the effects of two important parameters on the inversion curves. Finally, we give our conclusions +and discussions in Sec. 4 +2 +AdS black holes and thermodynamics in conformal gravity +A conformal gravity, constructed from the quadratic Weyl term, may be an alternative theory of +gravitation, and its Lagrangian reads [19, 20] +L = 1 +2αCµνρσCµνρσ, +(1) +where Cµνρσ is the Weyl tensor and the sign choice of coupling constant α is critical [25] though α +is independent of the equations of motion. The conformal gravity admits [21] the following general +static black hole solutions, +ds2 = −f(r)dt2 + dr2 +f(r) + r2dΩ2 +2,ϵ, +(2) +f(r) = c1r + c0 + d +r − Λr2 +3 , +(3) +where dΩ2 +2,ϵ is the line element of a unit hyperbolic two-space H2, or a torus T 2, or a sphere +S2, which corresponds to ϵ = −1, 0, 1, respectively, and the parameters c0, c1 and d are arbitrary +constants constrained by +3c1d + ϵ2 = c2 +0. +(4) +The mass of AdS black holes in conformal gravity can be calculated [21, 22] by defining the +conserved charge associated with the time-like Killing vector, which should be identified with the +enthalpy, +H = +α +24πr+ +�� +−c0 + ϵ + 2Λr2 ++ +� +d +r+ ++ (c0 − ϵ) +� +−3c0 + Λr2 ++ +� +3 +� +, +(5) +where r+ is the largest root of f(r) = 0, and the Hawking temperature can thus be obtained, +T = −3c0r+ − 6d + 8πPr3 ++ +12πr2 ++ +, +(6) +where the pressure is introduced, +P = − Λ +8π, +(7) +2 + +in the extended phase space. Therefore, we can write the first law of black hole thermodynamics, +dH = TdS + ΨdΞ + V dP, +(8) +where Ξ = c1 and its conjugate Ψ is viewed as a massive spin-2 hair. According to the explicit +expressions of enthalpy H and temperature T given in Eqs. (5) and (6), we can proceed to calculate +the heat capacity at constant pressure by employing the standard thermodynamic method, +CP = +�∂H +∂T +� +P,Ξ += α (c0 − ϵ) +� +3c0r+ + 6d − 8πPr3 ++ +� +6 +� +3c0r+ + 12d + 8πPr3 ++ +� +. +(9) +3 +Joule-Thomson expansion +In the common thermodynamics the well-known Joule-Thomson (Joule-Kelvin) effect describes the +change of temperature of a gas or liquid when the gas or liquid passes through a throttle, such as +a porous plug or a small valve, in a thermally insulated environment. This procedure is called the +throttling process or Joule-Thomson expansion and the enthalpy H remains unchanged in such a +process. In this section, we generalize the Joule-Thomson effect to the thermodynamics of AdS black +holes in conformal gravity and investigate the peculiar behavior of the Joule-Thomson expansion of +AdS black holes in the conformal gravity governed by Eq. (1). As in the common thermodynamics, +a physical effect can usually be depicted by a partial derivative, so one defines +µ = +�∂T +∂P +� +H +(10) +to represent the rate of change of temperature T with respect to pressure P at constant enthalpy +H, where µ is called the Joule-Thomson coefficient. +3.1 +Two approaches to derive Joule-Thomson coefficients +3.1.1 +A derivation based on the relations between the JT coefficient and thermody- +namic quantities +The entropy S is a state function and can be viewed as the function of the pressure P, Hawking +temperature T and massive spin-2 hair Ξ. So its total differential formula takes the form, +dS = +� ∂S +∂P +� +T,Ξ +dP + +�∂S +∂T +� +P,Ξ +dT + +�∂S +∂Ξ +� +T,P +dΞ. +(11) +Substituting this expression into Eq. (8) and using the condition dΞ = 0, we find +dH = T +�∂S +∂T +� +P,Ξ +dT + +�� +T ∂S +∂P +� +T,Ξ ++ V +� +dP, +(12) +which can be rearranged to give +CP = +�∂H +∂T +� +P,Ξ += T +�∂S +∂T +� +P,Ξ +, +(13) +�∂H +∂P +� +T,Ξ += T +� ∂S +∂P +� +T,Ξ ++ V. +(14) +3 + +As a result, we obtain the heat capacity at constant pressure from Eq. (13). +Moreover, using +Maxwell’s relation, +� ∂S +∂P +� +T,Ξ += − +�∂V +∂T +� +P,Ξ +, +(15) +we rewrite Eq. (14) as +�∂H +∂P +� +T,Ξ += −T +�∂V +∂T +� +P,Ξ ++ V. +(16) +We note that the JT coefficient depends on the three variables (T, P, H) and the enthalpy +as a state function can be expressed as H = H(T, P). By applying the cyclic rule we obtain the +following useful formula, +�∂T +∂P +� +H +� ∂P +∂H +� +T,Ξ +�∂H +∂T +� +P,Ξ += −1. +(17) +Finally, we work out the following expression of the JT coefficient by using Eq. (13), Eq. (16), and +Eq. (17), +µ = +�∂T +∂P +� +H += +1 +CP +� +T +�∂V +∂T +� +P,Ξ +− V +� +, +(18) +which shows a clear relationship between the JT coefficient and thermodynamic quantities. Taking +the thermodynamic quantities into account in Eq. (18), we can immediately obtain the exact JT +coefficient once the heat capacity CP and the other thermodynamic quantities are identified. Thus, +Eq. (18) is a very helpful formula to express the JT coefficient in terms of the other thermodynamic +quantities that can conveniently be measured. +3.1.2 +A direct derivation +In this subsection, we compute the JT coefficient alternatively by using a direct way. We first write +the pressure P and Hawking temperature T as the function of entropy H and event horizon r+, +respectively, +P(H, r+) = 3 +� +αc0d + αc2 +0r+ − αc0r+ϵ − αdϵ + 24πHr2 ++ +� +8παr2 ++ (−c0r+ − 6d + r+ϵ) +, +(19) +T(H, r+) = αc0r+ (2c0r+ + 9d − 2r+ϵ) + 3αd (4d − r+ϵ) + 24πHr3 ++ +4παr2 ++ (−c0r+ − 6d + r+ϵ) +. +(20) +With the aid of the above two equations, we then give the JT coefficient directly, +µ = +�∂T +∂P +� +H += −4c0r+ (c0r+ + 6d − r+ϵ) − 4d +� +12d + 8πPr3 ++ − 3r+ϵ +� +(ϵ − c0) +� +3c0r+ + 6d − 8πPr3 ++ +� +. +(21) +We can see that the JT coefficient is independent of coupling constant α. It is also easy to check +that the JT coefficients depicted by Eq. (18) and Eq. (21) are equivalent to each other, where what +we need to do is to substitute the required thermodynamic quantities, such as the heat capacity +and temperature, into Eq. (18). +4 + +3.2 +A novel isenthalpic process and inversion curves on T − P plane +The Joule-Thomson expansion occurs in a thermally insulated environment. During this expansion, +the enthalpy H remains unchanged. Utilizing Eqs. (19) and (20), we present an isenthalpic process +in Fig. 1, which allows us to give the corresponding heating-cooling phase. The JT coefficient µ +corresponds to the slope on the isenthalpic curve. On the right half of the curve (µ > 0), the +temperature drops when the system we consider, e.g. an AdS black hole, goes through a throttling +process. Therefore, the right half part is called cooling region because the temperature is decreasing +with a decrease of the pressure. In contrast, on the left half of the curve (µ < 0), the temperature +of AdS black holes rises after throttling and this half part is called heating region. The inversion +point, (Pinv, Tinv) at µ = 0, separates the cooling and heating regions where Tinv is the inversion +temperature given by +Tinv = V +� ∂T +∂V +� +P +. +(22) +2.5 +2.6 +2.7 +2.8 +2.9 +3.0 +3.1 +1.5 +1.6 +1.7 +1.8 +1.9 +2.0 +Figure 1: The isenthalpic process for AdS black holes in conformal gravity with d = −1, c0 = 1/10, +ϵ = 1, and H = 2. The black point on the curve corresponds to the inversion point, (Pinv, Tinv) = +(2.811, 1.545), in the isenthalpic process with µ = 0. +We emphasize that Tinv is the minimum temperature that the AdS black hole of conformal grav- +ity can reach in the Joule-Thomson process, while the corresponding temperature is the maximum +during the Joule-Thomson expansion in those black hole models [26, 27, 28, 29] and the van der +Waals fluid [30, 31]. This shows the peculiar behavior of the AdS black hole model of conformal +gravity in the aspect of inversion temperature. +Next, we analyze the other peculiar behavior in the aspect of inversion curves. An inversion +curve with the setting, µ = 0, can be obtained from Eq. (22), which is shown in Fig. 2 (red curve) +and Fig. 3 (black curve). The isenthalpic curves are shown as colored curves for various values +of enthalpy, see Fig. 3, where the inversion point, (Pinv, Tinv), rises monotonically as the enthalpy +increases. The inversion curve is depicted by the locations of isenthalpic curves that have a vanishing +5 + +slope. Here we can observe a significant feature that the Joule-Thompson expansion of AdS black +holes in conformal gravity experiences a cooling process at first, and then a heating process if a large +enough initial pressure is given. Additionally, the inversion curve no longer crosses these isenthalpic +curves but makes a tangency to them, which presents a different behavior from that of the charged +AdS black holes [30]. Our phenomenon suggests that each point on the inversion curve marks the +boundary, above which the JT expansion can occur but below which the JT expansion cannot occur. +Thus, our inversion curve denotes the dividing curve between the allowable and forbidden regions +for the JT expansion as shown in Fig. 2, while the corresponding inversion curve means the dividing +curve between the cooling and heating regions for those black hole models [32, 33, 34]. As a result, +the Joule-Thomson expansion will only occur in the parameter space above the inversion curve for +our model. +0 +2 +4 +6 +8 +0 +1 +2 +3 +4 +5 +Figure 2: The inversion curve for AdS black holes in conformal gravity with d = −1, ϵ = 1, and +c0 = 1/10. The JT expansion occurs in the allowable region. +0 +2 +4 +6 +8 +0 +1 +2 +3 +4 +5 +Figure 3: The isenthalpic and inversion curves for AdS black holes in conformal gravity with d = −1, +ϵ = 1, and c0 = 1/10. +6 + +The relationship between the inversion curve and parameter d for a fixed parameter c0 = 1/2 is +depicted by the left diagram of Fig. 4. The inversion temperature increases with a decrease of the +parameter d, which indicates that the smaller the parameter d is, the earlier the transition between +the cooling and heating processes is achieved in the Joule-Thompson process of conformal gravity. +The right diagram of Fig. 4 shows the relationship between the inversion curve and parameter c0 +with a fixed parameter d = −1. The inversion temperature drops with an increase of the parameter +c0, which shows that the larger the parameter c0 is, the later the transition between the cooling and +heating processes is realized in the Joule-Thompson process. +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.00 +0.05 +0.10 +0.15 +Figure 4: The dependence of the inversion curves on the parameters d and c0. The left panel gives +the inversion curves for d = −1, −2, −3, −4, −5 with a fixed c0 = 1/2. The right panel presents the +inversion curves for c0 = 0.1, 0.3, 0.5, 0.7, 0.9 with a fixed d = −1. +4 +Conclusions and discussions +In this work, we investigate the Joule-Thompson expansion for AdS black holes in conformal gravity. +We derive the expression of JT coefficients in terms of the relevant thermodynamic quantities +at first, and then we make the derivation via a direct way. We show that the results from two +different approaches are equivalent to each other, which indicates that the enthalpy and first law of +thermodynamics are well-defined for the AdS black holes in conformal gravity. The critical gravity +depends [25] on the sign choice of coupling constant α and the Einstein gravity emerges [35] from +the conformal gravity with α = 1/2 in the infrared limit. Our results show a clear feature that the +JT coefficient is independent of coupling constant α that appears in the Lagrangian of conformal +gravity. +On the one hand, we find a novel isenthalpic process in which Tinv is the minimum temperature +of the Joule-Thompson expansion that separates the corresponding heating-cooling phase. On the +other hand, the inversion curve separates the allowable and forbidden regions for the JT effect to be +observed. These peculiar behaviors of AdS black holes depend on the characteristics of conformal +gravity and its thermodynamics, where the mass of AdS black holes is the combination of the other +parameters, see Eq. (5), rather than being explicitly contained in the metric, see Eqs. (2) and +(3), and the Hawking temperature as a function of the event horizon takes the specific form, see +Eq. (6). The emergence of the novel isenthalpic process is deeply related to these characteristics +of the conformal gravity and its corresponding thermodynamics. Moreover, we analyze the effects +of two important parameters d and c0 on the inversion curves, and obtain that the two parameters +7 + +have similar effects, i.e., the inversion temperature will drop with an increase of d and c0, which +indicates a delay in the transition from a cooling to heating phase. +Acknowledgments +This work was supported in part by the National Natural Science Foundation +of China under Grant No. 12175108. +References +[1] S.W. Hawking, Black holes in general relativity, Commun. Math. Phys. 25 (1972) 152. +[2] J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333. +[3] S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199. +[4] M.M. Caldarelli, G. Cognola and D. Klemm, Thermodynamics of Kerr-Newman-AdS black +holes and conformal field theories, Class. Quant. Grav. 17 (2000) 399 [hep-th/9908022]. +[5] D. Kastor, S. Ray and J. Traschen, Enthalpy and the Mechanics of AdS Black Holes, Class. +Quant. 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Maldacena, Einstein Gravity from Conformal Gravity, 1105.5632. +9 + diff --git a/ndFQT4oBgHgl3EQfpzbH/content/tmp_files/2301.13378v1.pdf.txt b/ndFQT4oBgHgl3EQfpzbH/content/tmp_files/2301.13378v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..02f3eb2228f9303c39c4d450c00223e05d414ee3 --- /dev/null +++ b/ndFQT4oBgHgl3EQfpzbH/content/tmp_files/2301.13378v1.pdf.txt @@ -0,0 +1,2120 @@ +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS +OF FRACTIONAL GAUSSIAN NOISE AS THE FUNCTIONS +OF HURST INDEX +ANATOLIY MALYARENKO1, YULIYA MISHURA1,2, KOSTIANTYN RALCHENKO2,3, +AND SERGIY SHKLYAR2 +Abstract. This paper is devoted to the study of the properties of entropy as +a function of the Hurst index, which corresponds to the fractional Gaussian +noise. Since the entropy of the Gaussian vector depends on the determinant +of the covariance matrix, and the behavior of this determinant as a function +of the Hurst index is rather difficult to study analytically at high dimensions, +we also consider simple alternative entropy functionals, whose behavior, on +the one hand, mimics the behavior of entropy and, on the other hand, is not +difficult to study. Asymptotic behavior of the normalized entropy (so called +entropy rate) is also studied for the entropy and for the alternative functionals. +1. Introduction +The concept of entropy for a random variable was introduced by Shannon [17] +to characterize the irreducible complexity of a particular sort of randomness. By +definition, for a random variable ξ with probability density function pξ(x), the +entropy (that is sometimes called differential entropy, see e.g. [13]) is given by the +formula +H(ξ) = −E log pξ(ξ) = − +� +R +pξ(x) log pξ(x) dx. +Entropy of Gaussian vector was in detail studied in the book [19]. +It is not +difficult, therefore, to write formulas for the entropy of a stationary Gaussian pro- +cess with discrete time. A particular, but rather important and interesting case of +a stationary Gaussian process with discrete time is the fractional Gaussian noise +1 Division of Mathematics and Physics, M¨alardalen University, 721 23 V¨aster˚as, +Sweden +2 Department of Probability Theory, Statistics and Actuarial Mathematics, Taras +Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, 01601 Kyiv, +Ukraine +3 Sydney Mathematical Research Institute, The University of Sydney, Sydney NSW +2006, Australia +E-mail addresses: anatoliy.malyarenko@mdu.se, yuliyamishura@knu.ua, +kostiantynralchenko@knu.ua, shklyar@univ.kiev.ua. +2020 Mathematics Subject Classification. 60G22, 60G10, 60G15, 94A17. +Key words and phrases. Fractional Gaussian noise, Hurst index, entropy, entropy functionals, +entropy rate. +The second author was supported by The Swedish Foundation for Strategic Research, grant Nr. +UKR22-0017. The third author was supported by the Sydney Mathematical Research Institute +under Ukrainian Visitors Program. The second and the third authors acknowledge that the present +research is carried through within the frame and support of the ToppForsk project nr. 274410 of +the Research Council of Norway with title STORM: Stochastics for Time-Space Risk Models. +1 +arXiv:2301.13378v1 [math.PR] 31 Jan 2023 + +2 +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +with the Hurst index H ∈ (0, 1). On the one hand, it is not hard to produce the +formula for the entropy of fractional Gaussian noise from formulas (5.4.5)–(5.4.6) in +[19]. In the present paper we provide the corresponding expression for the entropy +of this process, see (2.5)–(2.6). +On the other hand, note that the behavior of the fractional Gaussian noise +substantially depends on its Hurst parameter H. In particular, it has long memory +property for H ∈ (1/2, 1), and in the case H ∈ (0, 1/2) it is the process with +short memory, see e. g., the book [15] and the papers [1, 2, 3, 9, 16]. Of course, +these properties are closely connected to the properties of corresponding fractional +operators: fractional integrals and derivatives that convert the Wiener process into +the fractional Brownian motion. The properties of these operators are the subject +of thousands of books and papers, let us mention only the recent general paper [12] +and references therein. In our paper the properties of fractional operators will be +reflected indirectly in a certain sense, through the properties of the corresponding +random processes and their numerical characteristics. +However, a natural question about the behavior of the entropy of the fractional +Gaussian noise as a function of H ∈ (0, 1) has not been resolved, it has not even +been raised. Apparently, the reason is the fact that the formula for the entropy of a +Gaussian vector contains the determinant of the covariance matrix, and the behav- +ior of this determinant at high dimensions is rather difficult to study analytically +whatever method is used, for example, the Cholesky decomposition or expansion +using eigenvalues. By studying the behavior of entropy numerically, we noticed +the effect that the entropy of fractional Gaussian noise increases with increasing H +from 0 to 1/2 and decreases with increasing H from 1/2 to 1. This is quite natural, +since H = 1/2 corresponds to the sequence of independent random variables, and +therefore its entropy is the greatest. This is our main hypothesis, we confirm it +analytically for small n and numerically for large ones. +The paper is organized as follows. Section 2 is devoted to the behavior of the +entropy of fractional Gaussian noise as a function of the Hurst parameter H for +fixed n. We start with the definition of the entropy and exact formulas for it in +the case of fractional Gaussian noise. We present the entropy as a surface of H +and n which clearly show the behavior of the determinant itself, its logarithm and, +as a consequence, the entropy as the functions of H and n. +Then we study in +detail two particular cases, namely n = 2 and n = 3 which support analytically the +hypothesis that the entropy of fractional Gaussian noise increases with increasing +H from 0 to 1/2 and decreases with increasing H from 1/2 to 1. In Section 3 +we are interested in the behavior of the entropy as n → ∞. We derive the lower +bounds for the entropy and for its limiting value known as entropy rate. Moreover, +we give the exact formula for the entropy rate via spectral density. In Section 4 +we introduce two alternative entropy functionals which depend on the elements of +the covariance matrix, mimic the behavior of real entropy and, at the same time, +are quite easy for analytical study. +The asymptotic behavior of the alternative +functionals as n → ∞ is studied in subsection 4.2. Auxiliary results concerning +stationary Gaussian processes and their entropy are collected in the Appendix. + +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +3 +2. Entropy of fractional Gaussian noise as a function of H +2.1. Entropy of Gaussian vector. Recall again that the entropy of absolutely +continuous random variable with probability density function pξ(x) is defined by +H(ξ) = −E log pξ(ξ) = − +� +R +pξ(x) log pξ(x) dx, +see [19, Eq. (1.6.2)]. Similarly, one can define the entropy of n-dimensional ab- +solutely continuous random vector, using the joint density of its components. In +particular, if n-dimensional random vector ξ has a multivariate Gaussian distribu- +tion N(µn, Σn) with mean µn and covariance matrix Σn, then the logarithm of its +density equals +log pξ(x) = −1 +2(x − µn)⊤Σ−1 +n (x − µn) − n +2 log(2π) − 1 +2 log(det Σn), +x ∈ Rn. +Hence, the entropy of ξ ∼ N(µn, Σn) is given by +H(ξ) = n +2 (1 + log(2π)) + 1 +2 log(det Σn). +(2.1) +This is a well-known formula, see [10, Theorem 8.4.1] or [19, Eq. (5.4.6)]. +Remark 2.1. 1. We use natural logarithm log = loge in the definition of the entropy. +Note that in the information theory (see, e. g., [10]) the entropy is sometimes defined +using log2 instead of log (this is motivated by measurements in bits). In this case +the formula (2.1) is written as follows: +H(ξ) = 1 +2 log2 +� +(2πe)n det Σn +� +. +2. For Gaussian vectors, Stratonovich in [19] introduced the alternative definition +of the entropy, namely the entropy with respect to the measure ν(dξ1, . . . , dξn) = +(2πe)−n/2dξ1 . . . dξn. This approach leads to the following simplified version of (2.1): +�H(ξ) = 1 +2 log(det Σn). +(2.2) +Remark 2.2. As we shall see below, the behavior of both versions of entropy, H(ξ) +and �H(ξ), as the function of Hurst index are the same and coincides with the +behavior of det Σn: all of them increase in H when H increases from 0 to 1/2 and +decrease when H increases from 1/2 to 1. Their behavior in n is different: det Σn +and consequently �H(ξ) decrease in n for any fixed H, however, H(ξ) increases in +n, due to the linear term n +2 (1 + log(2π)). +2.2. Fractional Gaussian noise. Consider fractional Gaussian noise starting from +zero. Let BH = +� +BH +t , t ≥ 0 +� +be a fractional Brownian motion (fBm) with Hurst +index H ∈ (0, 1), i.e., a centered Gaussian process with covariance function of the +form +EBH +t BH +s = 1 +2 +� +t2H + s2H − |t − s|2H� +. +(2.3) +Let us consider the following discrete-time process: +GH +k = BH +k − BH +k−1, +k = 1, 2, 3, . . . . +It is well known that the process BH has stationary increments, which implies +that +� +GH +k , k ≥ 1 +� +is a stationary Gaussian sequence (known as fractional Gaussian + +4 +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +noise). It follows from (2.3) that its autocovariance function is given by +ρ0(H) = 1, ρk(H) = EGH +1 GH +k+1 = 1 +2 +� +(k + 1)2H − 2k2H + (k − 1)2H� +, k ≥ 1. +(2.4) +Therefore, according to (2.1), the entropy of (GH +1 , . . . GH +n ) equals +H(GH +1 , . . . GH +n ) = n +2 (1 + log(2π)) + 1 +2 log(det Σn(H)), +(2.5) +where +Σn(H) = cov(GH +1 , . . . GH +n ) = +� +� +� +� +� +1 +ρ1(H) +ρ2(H) +. . . +ρn−1(H) +ρ1(H) +1 +ρ1(H) +. . . +ρn−2(H) +... +... +... +... +... +ρn−1(H) +ρn−2(H) +ρn−3(H) +. . . +1 +� +� +� +� +� +(2.6) +Formula (2.2) is transformed to +�H(GH +1 , . . . GH +n ) = 1 +2 log(det Σn(H)). +Remark 2.3. Let us mention several particular cases, when the determinant +det Σn(H) can be calculated explicitly. +Let H = 1 +2. Then all ρk( 1 +2) = 0, k ≥ 1, and ρ0( 1 +2) = 1. Therefore, det Σn( 1 +2) = 1, +n ≥ 1, and consequently log(det Σn( 1 +2)) = 0. +Let H = 1. Then BH +t += ξt, where ξ ∼ N(0, 1). Therefore, GH +k = ξ, k ≥ 0, and +ρk(1) = 1, k ≥ 0. This means that for any n ≥ 2 det Σn(1) = 0, and consequently +log(det Σn(1)) = −∞. Moreover, +ρk(H) = 1 +2 +� +(k + 1)2H + (k − 1)2H − 2k2H� +→ 1 +2 +� +(k + 1)2 + (k − 1)2 − 2k2� += 1, +as H ↑ 1. +Let H = 0. Then the situation is a bit more involved. Namely, in this case B0 +t +is a white noise of the form B0 +t = ξt−ξ0 +√ +2 , where {ξt, t ≥ 0} are N(0, 1) independent +random variables [7]. Therefore +ρ0(0) = 1, +ρ1(0) = 1 +2E(ξ1 − ξ0)(ξ2 − ξ1) = −1 +2 +and +ρk(0) = 0, k ≥ 2. +Moreover, +ρ1(H) = 1 +2 +� +22H − 2 +� +→ −1 +2 = ρ1(0) and ρk(H) → 0, H ↓ 0, k ≥ 2. +Consider +Σn(0) = +� +� +� +� +� +� +� +1 +− 1 +2 +· · · +0 +0 +− 1 +2 +1 +· · · +0 +0 +... +... +... +... +... +0 +0 +· · · +1 +− 1 +2 +0 +0 +· · · +− 1 +2 +1 +� +� +� +� +� +� +� +Determinant det Σn(0) of this tridiagonal matrix is calculated by the formula +det Σn(0) = det Σn−1(0) − 1 +4 det Σn−2(0) = . . . += k + 1 +2k +det Σn−k(0) − +k +2k+1 det Σn−k−1(0), + +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +5 +Figure 1. det Σn(H) as a function of H and n +where det Σ0(0) = 1, det Σ−1(0) = 0. Therefore +det Σn(0) = n + 1 +2n +, +n ≥ 1, +and consequently log(det Σn(0)) = log(n + 1) − n log 2. Obviously, both det Σn(0) +and log(det Σn(0)) decrease in n and tend to zero and −∞, respectively. +It is quite difficult to prove the monotonic properties of det Σn(H) and its loga- +rithm analytically in general case. Therefore our main conjecture +(A) det Σn(H) and log(det Σn(H)) increase from n+1 +2n to 1 and from log(n+1)− +n log 2 to 0, respectively, when H increases from 0 to 1 +2, and decrease from 1 +to 0 and from 0 to −∞, respectively when H increases from 1 +2 to 1, decreasing +in n for any fixed H +is in general checked numerically. +The surface of det Σn(H) as a function of H and n is presented at Figure 1. We +observe that for any fixed n ≥ 2 det Σn(H) increases in H ∈ (0, 1 +2) and decreases +in H ∈ ( 1 +2, 1). Also, it decreases in n for any H ∈ (0, 1). Figures 2 and 3 present +entropies �H(GH +1 , . . . GH +n ) and H(GH +1 , . . . GH +n ), respectively. +It is more logical to +arrange these entropies surfaces in this order, see Remark 2.2. +However, below we study in more detail two particular cases, namely n = 2 and +n = 3 and prove that they increase when H increases from 0 to 1 +2 and decrease +when H increases from 1 +2 to 1. As we shall see, even in the case n = 3 the proof of +monotonicity requires a lot of technical work. + +Thedeterminantofcovariancematrixoffractional Gaussiannoise +1 +0.8 +(H)") +0.6 +0.4 +0.2 +0 +600 +500 +0.8 +0.6 +400 +0.4 +0.2 +n +300 +0 +The Hurst index H6 +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +Figure 2. log det Σn(H) as a function of H and n +2.3. Cases n = 2 and n = 3. Consider the determinants for n = 2 and n = 3 in +the spirit of their monotonicity in H. +Lemma 2.4 (Case n = 2). The determinant det Σ2(H) increases from 3 +4 to 1 when +H increases from 0 to 1 +2 and decreases from 1 to 0 when H increases from 1 +2 to 1. +Consequently, log(det Σ2(H)) increases from log 3 − 2 log 2 to 0 when H increases +from 0 to 1 +2 and decreases from 0 to −∞ when H increases from 1 +2 to 1. +Proof. For n = 2, we have +det Σ2(H) = +���� +1 +ρ1(H) +ρ1(H) +1 +���� = 1 − ρ2 +1(H), +(2.7) +where +ρ1(H) = 1 +2 +� +22H − 2 +� += 22H−1 − 1. +So, +det Σ2(H) = 1 − +� +22H−1 − 1 +�2 = 1 − 24H−2 + 22H − 1 = −24H−2 + 22H. +Consider function +ϕ2(H) = −24H−2 + 22H, +H ∈ (0, 1). +Its derivative equals +ϕ′ +2(H) = −4 · 24H−2 log 2 + 2 · 22H log 2 = 22H+1 log 2 +� +1 − 22H−1� +, +and ϕ′ +2(H) > 0 for H ∈ (0, 1 +2), ϕ′ +2(H) < 0 for H ∈ ( 1 +2, 1). +□ + +The logarithm of determinantof covariancematrixof fractional Gaussian noise +0 +-100 +Indet(2n(H)) +200 +300 +400 +-500 +-600 +600 +500 +1 +0.8 +0.6 +400 +0.4 +0.2 +n +300 +0 +TheHurstindexENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +7 +Figure 3. H(GH +1 , . . . GH +n ) as a function of H and n +Lemma 2.5 (Case n = 3). The determinant det Σ3(H) increases from 1 +2 to 1 when +H increases from 0 to 1 +2 and decreases from 1 to 0 when H increases from 1 +2 to 1. +Consequently, log(det Σ3(H)) increases from − log 2 to 0 when H increases from 0 +to 1 +2 and decreases from 0 to −∞ when H increases from 1 +2 to 1. +Proof. The value of the determinant equals +det Σ3(H) = +������ +1 +ρ1(H) +ρ2(H) +ρ1(H) +1 +ρ1(H) +ρ2(H) +ρ1(H) +1 +������ += 1+2ρ2 +1(H)ρ2(H)−ρ2 +2(H)−2ρ2 +1(H), (2.8) +where +ρ2(H) = 1 +2 +� +32H − 22H+1 + 1 +� +. +Consider function +ϕ3(H) = 1 + 2x2y − y2 − 2x2, where x = ρ1(H), y = ρ2(H), +and calculate its derivative in H: +ϕ′ +3(H) = 4xx′ +Hy − 2yy′ +H − 4xx′ +H + 2x2y′ +H += 2 +� +x(2yx′ +H + xy′ +H) − (yy′ +H + 2xx′ +H) +� +. +First, let H ∈ ( 1 +2, 1]. Then +x′ +H = 22H log 2 > 0, +y′ +H = 32H log 3 − 2 · 22H log 2. + +The entropy of (GH,...,GH) with linear term +800 +700 +009 +400 +300 +200 +600 +500 +1 +0.8 +0.6 +400 +0.4 +0.2 +n +300 +0 +The Hurst index H8 +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +Let us prove that y′ +H > 0. Indeed, +y′ +H = 22H+1 log 2 +��3 +2 +�2H log 3 +log 4 − 1 +� +. +If H = 1 +2, then +�3 +2 +�2H log 3 +log 4 − 1 = log 27 +log 16 − 1 > 0. +Since y′ +H evidently increases in H, it is strictly positive. Note that x ≤ 1. Therefore +for H ∈ ( 1 +2, 1] +ϕ′ +3(H) < 2 +� +2yx′ +H + xy′ +H − yy′ +H − 2xx′ +H +� += 2(x − y)(y′ +H − 2x′ +H). +Further, +x − y = 22H−1 − 1 − 1 +2 · 32H + 22H − 1 +2 = 3 +2 +� +22H − 32H−1 − 1 +� +=: ψ(H). +It is easy to see that ψ( 1 +2) = ψ(1) = 0. Its second derivative equals +ψ′′(H) = 6 +� +22H log2 2 − 32H−1 log2 3 +� += 6 · 22H log2 3 +� +log2 2 +log2 3 − +�3 +2 +�2H +· 1 +3 +� +. +Let H = 1 +2. Then +log2 2 +log2 3 − 1 +2 ≈ 0.6932 +1.0992 − 1 +2 ≈ 480249 +1207801 − 1 +2 < 0. +It means that ψ′′(H) < 0 on the interval [ 1 +2, 1]. Moreover, +ψ′( 1 +2) = 3 (2 log 2 − log 3) > 0. +It means that on the interval [ 1 +2, 1] +ψ(H) = x − y > 0. +Let us analyze +ζ(H) = y′ +H − 2x′ +H = 32H log 3 − 4 · 22H log 2 = 22H log 3 +��3 +2 +�2H +− log 16 +log 3 +� +. +If H = 1, then +�3 +2 +�2H +− log 16 +log 3 = 9 +4 − log 16 +log 3 ≈ 9 +4 − 2.2 < 0. +Consequently, ζ(H) < 0, and ϕ′ +3(H) < 0 that is equivalent to decreasing of the +determinant det Σ3(H) on [1/2, 1]. +Now, let H ∈ [0, 1 +2). While x′ +H > 0, the situation with y′ +H is more involved. +Denote H0 = 0.2868143617175754, the unique root of the equation +32H log 3 − 2 · 22H log 2 = 0. +Then y′ +H < 0 on [0, H0) and y′ +H > 0 on (H0, 1 +2]. If H ∈ [H0, 1 +2], then in the formula +for ϕ′ +3(H) we have +x ≤ 0, +y ≤ 0, +x′ +H ≥ 0, +y′ +H ≥ 0, +whence +xyx′ +H ≥ 0, +−2yy′ +H ≥ 0, +−4xx′ +H ≥ 0, +2x2y′ +H ≥ 0, + +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +9 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.4 +0.6 +0.8 +1.0 +det Σ2(H) +det Σ3(H) +Figure 4. Graphs of det Σ2(H) (blue) and det Σ3(H) (orange) +i. e. ϕ′ +3(H) ≥ 0. +Now, let H ∈ [0, H0]. Transform ϕ′ +3(H) as follows: +ϕ′ +3(H) = 2 +� +2xyx′ +H − yy′ +H − 2xx′ +H + x2y′ +H +� += 2 +� +2x′ +Hx(y − 1) − y′ +H +� +y − x2�� +. +Further, |x| < 1, therefore y − x2 > y − 1, and on [0, H0] +(−y′ +H) +� +y − x2� +> (−y′ +H) (y − 1). +So, +ϕ′ +3(H) > 2(y − 1) (2xx′ +H − y′ +H) . +Obviously, y − 1 < 0. Consider +2xx′ +H − y′ +H = 2 +� +22H−1 − 1 +� +· 22H log 2 − 32H log 3 + 2 · 22H log 2 += 24H log 2 − 2 · 22H log 2 − 32H log 3 + 2 · 22H log 2 += 32H log 2 +��4 +3 +�2H +− log 3 +log 2 +� +< 0 +for any H ∈ [0, H0] (in fact, for any H ∈ [0, 1 +2]). Therefore, ϕ′ +3(H) > 0 that is +equivalent to increasing of the determinant det Σ3(H) on [0, 1/2]. +□ +Remark 2.6. For all H ∈ (0, 1), det Σ2(H) ≥ det Σ3(H) (where the equality is +achieved only for H = 1 +2 and for H ↑ 1). Indeed, by (2.7) and (2.8), we get +det Σ2(H) − det Σ3(H) = ρ2 +1(H) − 2ρ2 +1(H)ρ2(H) + ρ2 +2(H) += +� +ρ1(H) − ρ2(H) +�2 + 2ρ1(H)ρ2(H) +� +1 − ρ1(H) +� +≥ 0, +since ρ1(H) ≤ 1, and ρ1(H) and ρ2(H) have the same sign (they both are negative +for H ∈ (0, 1 +2) and positive for H ∈ ( 1 +2, 1)). +Figure 4 contains the graphs of +det Σ2(H) and det Σ3(H). +In the general case, the monotonicity of det Σn(H) as a function of n can be +proved by representing it as a product of conditional variances, see Remark A.5 in +the appendix. + +10 +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +3. Entropy, entropy rate and innovation variance. Lower bound for +innovation variance +3.1. Fractional Gaussian noise on the whole axis. Until now, we have consid- +ered the entropy of stationary fractional Gaussian noise starting from zero. How- +ever, quite often stationary processes start from −∞, especially if the question of +their regularity and some other properties are being investigated. Therefore, we +recall how we can construct fractional Gaussian noise starting from −∞. For this +purpose we use the Mandelbrot–van Ness representation of the fractional Brownian +motion. Let us briefly recall the concepts related to this object. +Standard two-sided Brownian motion is a process W = {Wt, t ∈ R} constructed +as a couple of two independent Brownian motions {W−t, t ≥ 0} and {Wt, t ≥ 0}, +one with the time reflected. Two-sided fractional Brownian motion is a zero-mean +Gaussian process BH = {BH +t , t ∈ R} with covariance function +EBH +s BH +t = 1 +2(|s|2H + |t|2H − |s − t|2H). +It admits the Mandelbrot–van Ness representation +BH +t = cH +� t +−∞ +� +(t − s) +H− 1 +2 ++ +− (−s) +H− 1 +2 ++ +� +dWs, +(3.1) +where cH = (2H sin(πH)Γ(2H))1/2 +Γ(H+1/2) += +� +2HΓ(3/2−H) +Γ(H+1/2)Γ(2−2H) +�1/2 +. Obviously, process BH +has stationary increments BH +s − BH +s−1, s ∈ R, whose covariance equals +E +� +BH +s − BH +s−1 +� � +BH +t − BH +t−1 +� += 1 +2 +� +|s − t − 1|2H − 2|s − t|2H + |s − t + 1|2H� +, +s, t ∈ R. +3.2. Lower bound for the innovation variance. According to Proposition A.3 +in the appendix, the entropy of a stationary Gaussian process {Xk, k = 1, 2, . . . } +can be expressed in terms of the following conditional variances: +r(k) = var[Xk | X1, . . . , Xk−1], +(3.2) +see formula (A.6). The values r(k) are deterministic, nonnegative and decreasing, +hence, there exists the finite limit +σ2 +inov(X) = lim +n→∞ r(n) ≥ 0, +(3.3) +which is called innovation variance. +Furthermore, for a stationary Gaussian process we have +σ2 +inov(X) = lim +n→∞ r(n) = lim +n→∞ var[Xn | Xn−1, . . . , X1] += lim +n→∞ var[Xt | Xt−1, . . . , Xt−n+1] += var[Xt | Xt−1, Xt−2, . . .] +for all t ∈ R. +(3.4) +It turns out that for fractional Gaussian noise GH the limit (3.3) is strictly +positive for all H, and moreover, it admits the following lower bound. +Theorem 3.1 (Lower bound for the innovation variance). For all H ∈ (0, 1), +σ2 +inov(GH) ≥ +Γ +� 3 +2 − H +� +Γ +� +H + 1 +2 +� +Γ(2 − 2H) =: σ2 +H. +(3.5) + +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +11 +Proof. As a particular case of (3.4), +σ2 +inov +� +GH� += var +� +GH +1 | GH +0 , GH +1 , GH +−1, . . . +� +. +Notice that GH +1 = BH +1 , and all GH +t += BH +t − BH +t−1, t ≤ 0, can be represented as +integrals w.r.t. the Brownian motion {Wt, t ≤ 0} with use of (3.1), whence +σ(GH +0 , GH +−1, GH +−2, . . .) ⊂ σ(Ws, s ≤ 0). +By the partitioning of conditional variance, see (A.3), +σ2 +inov +� +GH� += var[BH +1 | GH +0 , GH +−1, GH +−2, . . .] ≥ var[BH +1 | Ws, s ≤ 0]. +(3.6) +Finally, since the process {BH +t , t > 0} is a Volterra Gaussian process with the +representation (3.1), we see that the conditional variance in the right-hand side of +(3.6) can be calculated by the formula (A.2) as follows +var[BH +1 | Ws, s ≤ 0] = +� 1 +0 +c2 +H(1 − s)2H−1 ds = c2 +H +2H = +Γ +� 3 +2 − H +� +Γ +� +H + 1 +2 +� +Γ(2 − 2H). +□ +3.3. Lower bound for the entropy and the entropy rate. Taking (3.1) into +account, let us study the asymptotic behavior of the entropy of fractional Gaussian +noise as n → ∞. We start with the definition of entropy rate, see [10, Eq. (4.2)]. +Definition 3.2. The entropy rate of a discrete-time stochastic process X is +H∞(X) = lim +n→∞ +H(X1, . . . , Xn) +n +if this limit exists. +For the case of Gaussian process X, we may define also +�H∞(X) = lim +n→∞ +�H(X1, . . . , Xn) +n +, +where �H(X1, . . . , Xn) is introduced in (2.2). +Let X be a stationary Gaussian process. Then, applying Proposition A.3 from +the appendix, we obtain that its entropy rate equals +H∞(X) = 1 + log(2π) +2 ++ 1 +2 lim +n→∞ +n +� +k=1 +log r(k), +where r(k) is defined by (3.2). If σ2 +inov(X) > 0, then +lim +n→∞ +1 +n +n +� +k=1 +log r(k) = lim +k→∞ log r(k) = log(σ2 +inov(X)), +hence, +H∞(X) = 1 + log(2π) +2 ++ log σinov(X). +(3.7) +If σ2 +inov(X) = 0, then the entropy rate of the process X is infinite: H∞(X) = −∞. +Using the results of previous subsection, we can see that for the fractional Gauss- +ian noise GH, the entropy rate exists and moreover, it admits a finite lower bound. +Namely, we have the following result. + +12 +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +Theorem 3.3 (Lower bounds for the entropy and entropy rate). The entropy and +the entropy rate of fractional Gaussian noise satisfy inequalities: +H(GH +1 , . . . , GH +n ) ≥ n +2 +� +1 + log(2π) + log σ2 +H +� +, +H∞(GH) ≥ 1 + log(2π) +2 ++ log σH, +(3.8) +where σ2 +H is defined in (3.5). +Proof. Since GH is a stationary Gaussian process, we have by Proposition A.3, +H(GH +1 , . . . , GH +n ) = n + n log(2π) +2 ++ 1 +2 +n +� +k=1 +log r(k) ≥ n +2 +� +1 + log(2π) + log σ2 +H +� +, +since r(k) ≥ σ2 +inov ≥ σ2 +H for all k, see Theorem 3.1. The inequality (3.8) follows +immediately from the representation (3.7) and the lower bound (3.5). +□ +3.4. Calculation of the entropy rate via spectral density. According to [19, +Eq. (5.5.17)], the entropy rate of the stationary Gaussian process X can be ex- +pressed in the form +H∞(X) = 1 + log(2π) +2 ++ 1 +2 +� 1 +0 +log ϕ(µ)dµ = 1 + log(2π) +2 ++ 1 +2 +� 1/2 +−1/2 +log ϕ(µ)dµ, +(3.9) +where ϕ(µ) = �∞ +k=−∞ γ(k)e−2πiµk. In particular, for fractional Gaussian noise, +this approach leads to the following result. +Lemma 3.4. The entropy rate of the fractional Gaussian noise admits the following +representation: +H∞(GH) = 1 +2 +� +1 + log +� +sin(πH)Γ(2H + 1)(2π)−2H�� ++ 1 +2 +� 1/2 +−1/2 +log +� +∞ +� +k=−∞ +|µ + k|−2H−1 +� +dµ. +(3.10) +Proof. According to [5, Proposition 2.1] the spectral density of fractional Gaussian +noise GH is given by +f(λ) = 1 +2π ++∞ +� +k=−∞ +ρk(H)eikλ += 1 +π sin(πH)Γ(2H + 1)(1 − cos λ) ++∞ +� +k=−∞ +|λ + 2πk|−2H−1, +−π ≤ λ ≤ π. +Therefore, it follows from (3.9) that the entropy rate can be calculated as follows +H∞(GH) = 1 + log(2π) +2 ++ 1 +2 +� 1/2 +−1/2 +log +� +2πf(2πµ) +� +dµ += 1 +2 +� +1 + log +� +2 sin(πH)Γ(2H + 1)(2π)−2H�� ++ 1 +2 +� +1 +2 +− 1 +2 +log +� +1 − cos(2πµ) +� +dµ + 1 +2 +� 1/2 +−1/2 +log +� +∞ +� +k=−∞ +|µ + k|−2H−1 +� +dµ. + +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +13 +0.2 +0.4 +0.6 +0.8 +1.0 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +1.4 +n = 10 +n = 50 +n = 100 +H∞(GH) +lower bound +Figure 5. The normalized entropy H(GH +1 , . . . , GH +n )/n for n = 10, +50, and 100, the entropy rate H∞(GH), and the lower bound (3.8) +It is not hard to compute +� 1 +2 +− 1 +2 log +� +1−cos(2πµ) +� +dµ = − log 2, whence (3.10) follows. +□ +Remark 3.5. For computational reasons, it may be convenient to express the infinite +sum from (3.10) as ++∞ +� +k=−∞ +|µ + k|−2H−1 = ζ(2H + 1, µ) + ζ(2H + 1, −µ) − |µ|−2H−1 +where ζ(s, a) = �∞ +k=0 |a + k|−s denotes the Hurwitz zeta function. +Figure 5 contains the graphs of +1 +nH(GH +1 , . . . , GH +n ) for n = 10, 50, and 100 to- +gether with the entropy rate H∞(GH) (computed by the formula (3.10)) and the +lower bound (3.8). From one hand, it confirms the convergence of the normalized +entropies to the entropy rate. From the other hand, we see that formula (3.8) gives +rather accurate lower bound for all values of H. Moreover, the graph of H∞(GH) +confirms the following theoretical values for particular cases (see Remark 2.3). +H = 0: H∞ +� +G0� += lim +n→∞ +1 +2 +� +1 + log π + 1 +n log(n + 1) +� += 1 +2 (1 + log π) ≈ 1.07236; +H = 1 +2 : H∞ +� +G +1 +2 +� += 1 +2 +� +1 + log(2π) +� +≈ 1.41894; +H = 1: H∞ +� +G1� += −∞. +4. Entropy functionals +4.1. Definition and the main properties of entropy functionals. Taking into +account two facts: +(i) Standard entropy is related to the determinant of covariance matrix; +(ii) It is impossible (or at least rather difficult) to study the properties of the +determinant consequently of the entropy as the function of H for the high +values of n, + +14 +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +let us introduce two alternative entropy functionals that are based on the elements +of covariance matrix in the following way: the first functional is proportional to the +sum of squares of all different elements of covariance matrix for H ∈ (0, 1): +E1 +H(N) = −(H − 1/2)2 +1 − H +F 1 +H(N) = −(H − 1/2)2 +1 − H +N +� +k=1 +(2ρk(H))2 += −(H − 1/2)2 +1 − H +� N +� +k=2 +� +(k + 1)2H + (k − 1)2H − 2k2H�2 + +� +22H − 2 +�2 +� +, +and the second functional is related to the permanent of covariance matrix as +follows: +E2 +H(N) = −(H − 1/2)2 +1 − H +F 2 +H(N) = −(H − 1/2)2 +1 − H +N +� +k=1 +(N − k + 1) |2ρk(H)| += −(H − 1/2)2 +1 − H +� N +� +k=2 +(N − k + 1) +��(k + 1)2H + (k − 1)2H − 2k2H�� ++ N +��22H − 2 +�� +� +. +Remark 4.1. In both cases we separated the term 22H − 2 that corresponds to +k = 1 because we intend to study the behaviour of both functionals as functions +of H ∈ [0, 1], and its behaviour differs from other terms. +Recall also that for +H ∈ [1/2, 1] the absolute values in E2 +H(N) can be omitted. +Theorem 4.2. Both functionals E1 +H(N) and E2 +H(N) for any fixed N ≥ 2 have +the following behaviour as the functions of H ∈ [0, 1]: they increase in H ∈ [0, 1 +2], +are zero for H = 1 +2 and decrease in H ∈ [ 1 +2, 1]. Functional E1 +H(N) increases from +−1/4 to 0 and decreases from 0 to −∞, and E2 +H(N) increases from −N/4 to 0 and +decreases from 0 to −∞. +Proof. Note that the function φ(H) = (H−1/2)2 +1−H +has a derivative +φ′(H) = (H − 1/2)(3/2 − H) +(1 − H)2 +, +therefore it decreases on [0, 1/2] and increases on [1/2, 1] being nonnegative. There- +fore it is sufficient to establish that F i +H(N), i = 1, 2 decrease in H when H increases +from 0 to 1/2 and increase in H when H increases from 1/2 to 1. First, consider +H ∈ ( 1 +2, 1]. Then (k + 1)2H + (k − 1)2H − 2k2H > 0 and +∂F 1 +H(N) +∂H += 4 +N +� +k=2 +� +(k + 1)2H + (k − 1)2H − 2k2H� +× +� +(k + 1)2H log(k + 1) + (k − 1)2H log(k − 1) − 2k2H log k +� ++ 4 +� +22H − 2 +� +22H log 2; +∂F 2 +H(N) +∂H += 2 +N +� +k=2 +(N − k + 1) +� +(k + 1)2H log(k + 1) + (k − 1)2H log(k − 1) − 2k2H log k +� ++ 2N22H log 2. + +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +15 +Let us analyze the value +ζ(k, H) = (k + 1)2H log(k + 1) + (k − 1)2H log(k − 1) − 2k2H log k. +Consider the function +ϕ(x) = x2H log x, +x ≥ 1, 2H > 1. +Its second derivative equals +ϕ′′(x) = x2H−2� +2H(2H − 1) log x + 4H − 1 +� +, +(4.1) +and for x ≥ 1 ϕ(x) = x2H log x > 0. It means that ϕ is convex for x ≥ 1, whence +ζ(k, H) > 0 for k ≥ 2, H > 1 +2. Obviously, both additional terms 4 +� +22H − 2 +� +22H log 2 +and 2N22H log 2 are strictly positive. So, both derivatives, ∂F i +H(N) +∂H +> 0, i = 1, 2, +H ∈ ( 1 +2, 1], and so F 1 +H(N) and F 2 +H(N) are strictly increasing in H from 0 to +F 1 +1 (N) = 22N and F 2 +1 (N) = N(N + 1). +Second, consider H ∈ [0, 1 +2). In this case (k + 1)2H + (k − 1)2H − 2k2H < 0 for +k ≥ 2, therefore, it is more convenient to rewrite ∂F 1 +H(N) +∂H +as +∂F 1 +H(N) +∂H += 4 +N +� +k=2 +� +2k2H − (k + 1)2H − (k − 1)2H� +× +� +2k2H log k − (k + 1)2H log(k + 1) − (k − 1)2H log(k − 1) +� ++ 4 log 2 · 22H � +22H − 2 +� +. +(4.2) +Let us analyze the behaviour of all terms in (4.2). Consider again function ϕ from +(4.1). Its second derivative is negative for such x that log x > +4H−1 +2H(1−2H) and is +positive if log x < +4H−1 +2H(1−2H). Since we consider x ≥ 1, for H ≤ 1 +4 we have that +ϕ′′(x) < 0 for all x ≥ 1, and for H ∈ ( 1 +4, 1 +2) ϕ′′(x) > 0 for x ∈ (1, x0) and ϕ′′(x) < 0 +for x ∈ (x0, ∞), where x0 = exp +� +4H−1 +2H(1−2H) +� +. Put N0 = ⌊x0⌋. Then +∂F 1 +H(N) +∂H +< 4 +N +� +k=N0 +� +2k2H − (k + 1)2H − (k − 1)2H� +× +� +2k2H log k − (k + 1)2H log(k + 1) − (k − 1)2H log(k − 1) +� ++ 4 log 2 · 22H � +22H − 2 +� +. + +16 +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +For any fixed H ∈ (0, 1 +2) ψ(k) = 2k2H − (k + 1)2H − (k − 1)2H has a derivative +∂ψ +∂k (k) = 2H +� +2k2H−1 − (k + 1)2H−1 − (k − 1)2H−1� +< 0, therefore, +∂F 1 +H(N) +∂H +< 4 +� +2N 2H +0 +− (N0 + 1)2H − (N0 − 1)2H� +× +N +� +k=N0 +� +2k2H log k − (k + 1)2H log(k + 1) − (k − 1)2H log(k − 1) +� ++ 4 log 2 · 22H � +22H − 2 +� += 4 +� +2N 2H +0 +− (N0 + 1)2H − (N0 − 1)2H� � +N 2H log N +− (N + 1)2H log(N + 1) + N 2H +0 +log N0 − (N0 − 1)2H log(N0 − 1) +� ++ 4 log 2 · 22H � +22H − 2 +� +< 4 +� +2N 2H +0 +− (N0 + 1)2H − (N0 − 1)2H� � +N 2H +0 +log N0 − (N0 − 1)2H log(N0 − 1) +� ++ 4 log 2 · 22H � +22H − 2 +� +< 4 +� +2 − 22H� � +N 2H +0 +log N0 − (N0 − 1)2H log(N0 − 1) − 22H log 2 +� +. +Again, for fixed H consider function +ζ(x) = x2H log x − (x − 1)2H log(x − 1), +x ≥ N0. +Its derivative equals +ζ′(x) = (2H log x + 1)x2H−1 − (x − 1)2H−1(2H log(x − 1) + 1), +x ≥ N0 +and function δ(x) = x2H−1(2H log x+1) has δ′(x) = ϕ′′(x) < 0, x ≥ N0. Therefore, +ζ′(x) < 0, x ≥ N0, and +N 2H +0 +log N0 − (N0 − 1)2H log(N0 − 1) − 22H log 2 < 22H log 2 − 22H log 2 = 0. (4.3) +Concerning F 2 +H(N), for H ∈ [0, 1 +2) it equals +F 2 +H(N) = +N +� +k=2 +(N − k + 1) +� +2k2H − (k + 1)2H − (k − 1)2H� ++ N +� +2 − 22H� +and +∂F 2 +H(N) +∂H += 2 +N +� +k=2 +(N − k + 1) +� +2k2H log k − (k + 1)2H log(k + 1) − (k − 1)2H log(k − 1) +� +− 2N22H log 2 +< 2N +N +� +k=N0 +� +2k2H log k − (k + 1)2H log(k + 1) − (k − 1)2H log(k − 1) +� +− 2N22H log 2 +≤ 2N +� +N 2H log N − (N + 1)2H log(N + 1) + N 2H +0 +log N0 +− (N0 − 1)2H log(N0 − 1) − 22H log 2 +� +< 0 +due to (4.3). +□ + +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +17 +4.2. Entropy rate for entropy functionals. It is very easy to see from formula +(2.4) that ρk(H) decrease in k for H ∈ (1/2, 1) being positive and increase in k for +H ∈ (0, 1/2) being negative, therefore all the summands in (2ρk(H))2 in E1 +H(N) +decrease in k. Moreover, +2ρk(H) = k2H +�� +1 + 1 +k +�2H ++ +� +1 − 1 +k +�2H +− 2 +� +∼ 2k2H 2H(2H − 1) +2k2 += 2H(2H − 1)k2H−2, +as k → ∞, +therefore +� +2ρk(H) +�2 ∼ 4H2(2H − 1)2k4H−4 as k → ∞. It means that entropy +functional E1 +H(N) has the following asymptotic properties. +Lemma 4.3. +(i) Let H ∈ (0, 3 +4). Then the series �∞ +k=1 +� +2ρk(H) +�2 converges, +and +E1 +H(N) → E1 +H(∞) = −(H − 1 +2)2 +1 − H +∞ +� +k=1 +� +2ρk(H) +�2 +as N → ∞. +(ii) Let H = 3 +4. Then +lim +N→∞ +E1 +H(N) +log N += − 9 +16. +(iii) Let H ∈ ( 3 +4, 1). Then +lim +N→∞ +E1 +H(N) +N 4H−3 = − 4H2(2H − 1)4 +(1 − H)(4H − 3). +Proof. Item (i) is evident. +(ii) Indeed, with H = 3/4 +lim +N→∞ +E1 +H(N) +log N += lim +N→∞ +E1 +3/4(N) +log N += −(H − 1 +2)2 +1 − H +lim +N→∞ +� +2ρk( 3 +4) +�2 +1 +N += −(H − 1 +2)2 +1 − H +· 42H2(2H − 1)2N −1 +N −1 += −4H2(2H − 1)4 +1 − H += − 9 +16. +(iii) Indeed, +lim +N→∞ +E1 +H(N) +N 4H−3 = −(H − 1 +2)2 +1 − H +lim +N→∞ +42H2(2H − 1)2N 4H−4 +(4H − 3)N 4H−4 += − 4H2(2H − 1)4 +(1 − H)(4H − 3). +□ +Lemma 4.4. +(i) Let H ∈ (0, 1 +2). Then +lim +N→∞ E2 +H(N) = − +∞ +� +k=1 +|ρk(H)| (H − 1 +2)2 +1 − H +. +(ii) Let H = 1 +2. Then E2 +H(N) = 0, N ≥ 1, and its limit equals zero. +(iii) Let H ∈ ( 1 +2, 1). Then +lim +N→∞ +E2 +H(N) +N 2H += −(H − 1 +2)2 +1 − H +. + +18 +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +Proof. Consider separately +S1 +N = N +N +� +k=2 +��(k + 1)2H + (k − 1)2H − 2k2H�� +and +S2 +N = +N +� +k=2 +(k − 1) +��(k + 1)2H + (k − 1)2H − 2k2H�� . +(i) Let H ∈ (0, 1 +2). Then +��(k + 1)2H + (k − 1)2H − 2k2H�� ∼ k2H−22H(1 − 2H), +and �∞ +k=2 +��(k + 1)2H + (k − 1)2H − 2k2H�� < ∞. Therefore +S1 +N +N → +∞ +� +k=2 +��(k + 1)2H + (k − 1)2H − 2k2H�� , +as N → ∞. +Further, (k − 1) +��(k + 1)2H + (k − 1)2H − 2k2H�� ∼ k2H−12H(1 − 2H). Therefore +lim +N→∞ +S2 +N +N +lim +N→∞ N 2H−12H(1 − 2H) = 0. +(iii) Let H ∈ ( 1 +2, 1). Then +S1 +N +N 2H ∼ N 2H−22H(2H − 1) +(2H − 1)N 2H−2 +, +so +lim +N→∞ +S1 +N +N 2H = 2H. +Further, +S2 +N +N 2H ∼ N 2H−12H(2H − 1) +2HN 2H−1 +, +so +lim +N→∞ +S2 +N +N 2H = 2H − 1, +whence the proof follows. +□ +Appendix A. Some results on stationary Gaussian processes +A.1. Partitioning of conditional variance. Let (Ω, F, P) be the probability +space. The conditional variance of the random variable X with EX2 < ∞ given +the σ-field A ⊂ F is defined as +var[X | A] = E[(X − E[X | A])2 | A] = E[X2 | A] − (E[X | A])2. +Let A and B be two σ-fields, A ⊂ B ⊂ F, and X be a random variable with +EX2 < ∞. Then +var[X | A] = E[var[X | B] | A] + var[E[X | B] | A] +(A.1) +and +E var[X | A] = E var[X | B] + E var[E[X | B] | A]. +In general case, the conditional variance var[X | B] is a B-measurable random +variable. In particular cases, var[X | B] is deterministic. For example, the condi- +tional variance of a component of a Gaussian random vector given other components +is nonrandom. The conditional variance of an observation Xt of a Gaussian pro- +cess X given observation of the process on some set I is nonrandom. The same +holds true for a linear functional of the process X. More specifically, the following +holds true: if X = {Xs, s ∈ T} is a Gaussian process, t ∈ T and I ⊂ T, then + +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +19 +var[Xt | Xs, s ∈ I] is nonrandom. Furthermore, if W = {Ws, s ∈ R} is a two-sided +Wiener process, φ ∈ L2(R) is a deterministic function, and I ⊂ R, then +var +�� ∞ +−∞ +φ(s) dWs +���� Ws, s ∈ I +� +is nonrandom. +For the Volterra Gaussian process +Xt = +� t +−∞ +K(t, s) dWs, +t ≥ 0, +where K(t, · ) ∈ L2((−∞, t]) for all t > 0, the following relation holds +var[Xt | F0] = +� t +0 +K(t, s)2 ds, +t > 0, +(A.2) +where F0 = σ(Ws, s ≤ 0). Indeed, +E(Xt | F0) = E +�� 0 +−∞ +K(t, s) dWs + +� t +0 +K(t, s) dWs +���� F0 +� += +� 0 +−∞ +K(t, s) dWs, +and similarly +var[Xt | F0] = E[X2 +t | F0] − (E[X | F0])2 += E +��� 0 +−∞ +K(t, s) dWs +�2 ++ +�� t +0 +K(t, s) dWs +�2 ++ 2 +� 0 +−∞ +K(t, s) dWs +� t +0 +K(t, s) dWs +���� F0 +� +− +�� 0 +−∞ +K(t, s) dWs +�2 += E +��� t +0 +K(t, s) dWs +�2� += +� t +0 +K(t, s)2 ds. +If, in (A.1), the conditional expectation var[X | B] is nonrandom (or, more +generally, if var[X | B] is an A-measurable random variable), then (A.1) takes the +form +var[X | A] = var[X | B] + var[E[X | B] | A], +whence +var[X | A] ≥ var[X | B]. +(A.3) +The equality holds in (A.3) if and only if var[E[X | B] | A] = 0. The sufficient +condition for equality in (A.3) is E[X | A] = E[X | B] almost surely. The sufficient +conditions for strict inequality in (A.3) are that both E[X | A] and E[X | B] are +nonrandom and P(E[X | A] ̸= E[X | B]) > 0. +A.2. Entropy of a stationary Gaussian process. Let X = {Xt, t=1, 2, . . .} be +a stationary Gaussian process with the autocovariance function γ(h) = cov(Xt+h, Xt); +The covariance matrix of n consecutive observations of the process X is denoted +Γn; it is a symmetric Toeplitz matrix: +Γn = +� +� +� +� +γ(0) +γ(1) +. . . +γ(n − 1) +γ(1) +γ(0) +. . . +γ(n − 2) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +γ(n − 1) +γ(n − 2) +. . . +γ(0) +� +� +� +� + +20 +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +Assumption A.1. For all n ∈ N the matrix Γn is nonsingular. +Remark A.2. If X is a fractional Gaussian noise, then Assumption A.1 is satisfied, +see [4, Theorem 1]. +Under Assumption A.1, the entropy of n consecutive observations of the process +X is equal to +H(X1, . . . , Xn) = H(Xt+1, . . . , Xt+n) = n + n log(2π) +2 ++ 1 +2 log(det Γn). +(A.4) +The goal of this subsection is to express this entropy in terms of the following +quantities: +r(1) = var(X1), +r(k) = var[Xk | X1, . . . , Xk−1], +k = 2, 3, . . . +(A.5) +Recall that r(k) is nonrandom for any k, since the process X is Gaussian, see +subsection A.1. +Proposition A.3. Let X = {Xk, k = 1, 2, . . .} be a stationary Gaussian process, +whose covariance matrix satisfies Assumption A.1. Then +1. The sequence {r(k), k ∈ N}, defined by (A.5), is deterministic, non- +negative and decreasing; hence, it is convergent. +2. The entropy of n consecutive observations of the process X is equal to +H(X1, . . . , Xn) = n + n log(2π) +2 ++ 1 +2 +n +� +k=1 +log r(k). +(A.6) +3. The determinant of the covariance matrix Γn is expressed in the following +form: +det Γn = +n +� +k=1 +r(k). +(A.7) +Proof. 1. The monotonicity of r(k) follows from (A.3) and (A.5). Indeed, +r(k) = var[Xk | X1, . . . , Xk−1] = var[Xk+1 | X2, . . . , Xk] +≥ var[Xk+1 | X1, X2, . . . , Xk] = r(k + 1). +2. Due to the chain rule for the entropies [10, Theorem 8.6.2] +H(X1, . . . , Xn) = H(X1) + +n +� +k=2 +H(Xk | X1, . . . , Xk−1) +(A.8) +where H(Xk | X1, . . . , Xk−1) is the conditional entropy, +H(Xk | X1, . . . , Xk−1) = − +� +· · · +�� +pX1,...,Xk−1,Xk(x1, . . . , xk−1, xk) +× log pXk | X1=x1,...,Xk−1=xk−1(xk) dx1 . . . dxk−1 dxk, +and H(X1) = 1 +2 log(2eπr(1)), since X1 ∼ N(0, r(1)). +The conditional distribution of Xk given X1, . . . , Xk−1 is Gaussian, with fixed +variance: +[Xk | X1=x1, . . . , Xk−1=xk−1] ∼ N(E[Xk | X1=x1, . . . , Xk−1=xk−1], r(k)). + +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +21 +Thus, by (2.1), the entropy of the conditional distribution [Xk | X1 = x1, . . . , +Xk−1 = xk−1] is +H(Xk | X1=x1, . . . , Xk−1=xk−1) = 1 +2 log(2eπr(k)). +(A.9) +Note that the right-hand side of (A.9) does not depend on x1, . . . , xk−1. The condi- +tional entropy can be expressed through the entropy of the underlying conditional +distribution: +H(Xk | X1, . . . , Xk−1) = +� +· · · +� +pX1,...,Xk−1(x1, . . . , xk−1) × +× H(Xk | X1=x1, . . . , Xk−1=xk−1) dx1 . . . dxk−1. +Thus, +H(Xk | X1, . . . , Xk−1) += +� +· · · +� +pX1,...,Xk−1(x1, . . . , xk−1) 1 +2 log(2eπr(k)) dx1 . . . dxk−1 += 1 +2 log(2eπr(k)). +By the chain rule (A.8), +H(X1, . . . , Xn) = +n +� +k=1 +1 +2 log(2eπr(k)) = n +2 log(2eπ) + 1 +2 +n +� +k=1 +log r(k), +which coincides with (A.6). +3. Comparing (A.4) and (A.6) we immediately get the representation (A.7). +□ +Remark A.4. 1. The first statement of Proposition A.3 is known; it can be found, +e.g., in [11, Theorem 2.10.1]. +2. The formula (A.7) can be proved also with the help of the Cholesky decom- +position of the covariance matrix Γn. Namely, Γn can be represented as +Γn = +� +� +� +� +ℓ1,1 +0 +. . . +0 +ℓ2,1 +ℓ2,2 +. . . +0 +. . . . . . . . . . . . . . . . . . . . +ℓn,1 +ℓn,2 +. . . +ℓn,n +� +� +� +� +� +� +� +� +ℓ1,1 +ℓ2,1 +. . . +ℓn,1 +0 +ℓ2,2 +. . . +ℓn,2 +. . . . . . . . . . . . . . . . . . . . +0 +0 +. . . +ℓn,n +� +� +� +� +where ℓ2 +k,k = r(k), see [14, Eq. (A19)]. Hence, det Γn = �n +k=1 ℓ2 +k,k = �n +k=1 r(k). +Let us mention that a similar method (based on so called LDL⊤-decomposition of +the covariance matrix) is described in [8, § 8.6]. +Remark A.5. The representation (A.7) implies that the determinant of the covari- +ance matrix Σn(H) of the fractional Gaussian noise GH decreases as a function of +n, since in this case +r(k) ≤ r(1) = var +� +GH +1 +� += 1, +for all k. + +22 +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +A.3. Entropy rate and the nondeterminism of stationary process. Denote +by +Mt(X) = span(Xt, Xt−1, Xt−2, . . .) +the smallest closed linear subspace of the Hilbert space L2(Ω, F, P) that contains +random variables Xt, , Xt−1, Xt−2, . . . Let +M−∞(X) = +∞ +� +t=−∞ +Mt(X), +M(X) = span(Xt, t ∈ Z). +Definition A.6. A centered wide-sense stationary process {Xt, t ∈ Z} is called +deterministic if M−∞(X) = M(X), i. e., Ms(X) = Mt(X) for all s, t ∈ Z. The +process X is called completely non-deterministic if M−∞(X) = 0. +Lemma A.7. A centered stationary Gaussian process {Xt, t ∈ Z} is deterministic +if and only if +var[Xt | Xt−1, Xt−2, Xt−3, . . .] = 0 +(here the left-hand side does not depend on t due to stationarity). +It is well known that a stationary mean-zero process X = {X(t), t ∈ Z} ad- +mits the Wold’s representation as a sum of two orthogonal processes: X(t) = +M(t) + N(t), where M = {M(t), t ∈ Z} is deterministic and N = {N(t), t ∈ Z} is +completely non-deterministic, see, e. g. [18, Appendix B.4] or [6, Section 7.1]. +In view of (3.4), a stationary Gaussian process X is deterministic if and only +if for this process σ2 +inov(X) = 0. Taking into account (3.7), we get the following +result. +Proposition A.8. Under Assumption A.1, a stationary Gaussian process has a +finite entropy rate if and only if it is non-deterministic. +References +[1] E. Al`os, J. A. Le´on, and D. Nualart. Stochastic Stratonovich calculus fBm for fractional +Brownian motion with Hurst parameter less than 1/2. Taiwanese J. Math., 5(3):609–632, +2001. +[2] E. Al`os, O. Mazet, and D. Nualart. Stochastic calculus with respect to fractional Brownian +motion with Hurst parameter lesser than 1 +2 . Stochastic Process. Appl., 86(1):121–139, 2000. +[3] V. V. Anh and A. Inoue. Prediction of fractional Brownian motion with Hurst index less than +1/2. Bull. Austral. Math. Soc., 70(2):321–328, 2004. +[4] O. Banna, Y. Mishura, K. Ralchenko, and S. Shklyar. 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Wiley, Hoboken NJ, 2006. +[11] W. A. Fuller. Introduction to statistical time series. Wiley Series in Probability and Statistics: +Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1996. + +ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN +23 +[12] Y. Luchko. Operational calculus for the general fractional derivative and its applications. +Fract. Calc. Appl. Anal., 24(2):338–375, 2021. +[13] J. V. Michalowicz, J. M. Nichols, and F. Bucholtz. Handbook of differential entropy. CRC +Press, Boca Raton, FL, 2014. +[14] Y. Mishura, K. Ralchenko, and S. Shklyar. General conditions of weak convergence of discrete- +time multiplicative scheme to asset price with memory. Risks, 8(1):11, 2020. +[15] Y. S. Mishura. Stochastic calculus for fractional Brownian motion and related processes, +volume 1929 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2008. +[16] I. Norros, E. Valkeila, and J. Virtamo. An elementary approach to a Girsanov formula and +other analytical results on fractional Brownian motions. Bernoulli, 5(4):571–587, 1999. +[17] C. E. Shannon. A mathematical theory of communication. Bell System Tech. J., 27:379–423, +623–656, 1948. +[18] R. H. Shumway and D. S. Stoffer. Time Series Analysis and Its Applications, With R Ex- +amples. Springer, Cham, 2017. +[19] R. L. Stratonovich. Theory of information and its value. Springer, Cham, 2020. + diff --git a/ndFQT4oBgHgl3EQfpzbH/content/tmp_files/load_file.txt b/ndFQT4oBgHgl3EQfpzbH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4b21b95e23876ffa4d3c11b5c3c31001815d4278 --- /dev/null +++ b/ndFQT4oBgHgl3EQfpzbH/content/tmp_files/load_file.txt @@ -0,0 +1,1116 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf,len=1115 +page_content='ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FRACTIONAL GAUSSIAN NOISE AS THE FUNCTIONS OF HURST INDEX ANATOLIY MALYARENKO1, YULIYA MISHURA1,2, KOSTIANTYN RALCHENKO2,3, AND SERGIY SHKLYAR2 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analytically at high dimensions, we also consider simple alternative entropy functionals, whose behavior, on the one hand, mimics the behavior of entropy and, on the other hand, is not difficult to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Asymptotic behavior of the normalized entropy (so called entropy rate) is also studied for the entropy and for the alternative functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Introduction The concept of entropy for a random variable was introduced by Shannon [17] to characterize the irreducible complexity of a particular sort of randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' By definition, for a random variable ξ with probability density function pξ(x), the entropy (that is sometimes called differential entropy, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' [13]) is given by the formula H(ξ) = −E log pξ(ξ) = − � R pξ(x) log pξ(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Entropy of Gaussian vector was in detail studied in the book [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It is not difficult, therefore, to write formulas for the entropy of a stationary Gaussian pro- cess with discrete time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' A particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' but rather important and interesting case of a stationary Gaussian process with discrete time is the fractional Gaussian noise 1 Division of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' M¨alardalen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 721 23 V¨aster˚as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Sweden 2 Department of Probability Theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Statistics and Actuarial Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Taras Shevchenko National University of Kyiv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 64/13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Volodymyrska Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 01601 Kyiv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Ukraine 3 Sydney Mathematical Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The University of Sydney,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Sydney NSW 2006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Australia E-mail addresses: anatoliy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='malyarenko@mdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='se, yuliyamishura@knu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='ua, kostiantynralchenko@knu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='ua, shklyar@univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='kiev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='ua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 60G22, 60G10, 60G15, 94A17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Fractional Gaussian noise, Hurst index, entropy, entropy functionals, entropy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The second author was supported by The Swedish Foundation for Strategic Research, grant Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' UKR22-0017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The third author was supported by the Sydney Mathematical Research Institute under Ukrainian Visitors Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The second and the third authors acknowledge that the present research is carried through within the frame and support of the ToppForsk project nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 274410 of the Research Council of Norway with title STORM: Stochastics for Time-Space Risk Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='13378v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='PR] 31 Jan 2023 2 ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN with the Hurst index H ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' On the one hand, it is not hard to produce the formula for the entropy of fractional Gaussian noise from formulas (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6) in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In the present paper we provide the corresponding expression for the entropy of this process, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' On the other hand, note that the behavior of the fractional Gaussian noise substantially depends on its Hurst parameter H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In particular, it has long memory property for H ∈ (1/2, 1), and in the case H ∈ (0, 1/2) it is the process with short memory, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=', the book [15] and the papers [1, 2, 3, 9, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Of course, these properties are closely connected to the properties of corresponding fractional operators: fractional integrals and derivatives that convert the Wiener process into the fractional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The properties of these operators are the subject of thousands of books and papers, let us mention only the recent general paper [12] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In our paper the properties of fractional operators will be reflected indirectly in a certain sense, through the properties of the corresponding random processes and their numerical characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' However, a natural question about the behavior of the entropy of the fractional Gaussian noise as a function of H ∈ (0, 1) has not been resolved, it has not even been raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Apparently, the reason is the fact that the formula for the entropy of a Gaussian vector contains the determinant of the covariance matrix, and the behav- ior of this determinant at high dimensions is rather difficult to study analytically whatever method is used, for example, the Cholesky decomposition or expansion using eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' By studying the behavior of entropy numerically, we noticed the effect that the entropy of fractional Gaussian noise increases with increasing H from 0 to 1/2 and decreases with increasing H from 1/2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' This is quite natural, since H = 1/2 corresponds to the sequence of independent random variables, and therefore its entropy is the greatest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' This is our main hypothesis, we confirm it analytically for small n and numerically for large ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Section 2 is devoted to the behavior of the entropy of fractional Gaussian noise as a function of the Hurst parameter H for fixed n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' We start with the definition of the entropy and exact formulas for it in the case of fractional Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' We present the entropy as a surface of H and n which clearly show the behavior of the determinant itself, its logarithm and, as a consequence, the entropy as the functions of H and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then we study in detail two particular cases, namely n = 2 and n = 3 which support analytically the hypothesis that the entropy of fractional Gaussian noise increases with increasing H from 0 to 1/2 and decreases with increasing H from 1/2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In Section 3 we are interested in the behavior of the entropy as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' We derive the lower bounds for the entropy and for its limiting value known as entropy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Moreover, we give the exact formula for the entropy rate via spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In Section 4 we introduce two alternative entropy functionals which depend on the elements of the covariance matrix, mimic the behavior of real entropy and, at the same time, are quite easy for analytical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The asymptotic behavior of the alternative functionals as n → ∞ is studied in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Auxiliary results concerning stationary Gaussian processes and their entropy are collected in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Entropy of fractional Gaussian noise as a function of H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Entropy of Gaussian vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Recall again that the entropy of absolutely continuous random variable with probability density function pξ(x) is defined by H(ξ) = −E log pξ(ξ) = − � R pξ(x) log pξ(x) dx, see [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Similarly, one can define the entropy of n-dimensional ab- solutely continuous random vector, using the joint density of its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In particular, if n-dimensional random vector ξ has a multivariate Gaussian distribu- tion N(µn, Σn) with mean µn and covariance matrix Σn, then the logarithm of its density equals log pξ(x) = −1 2(x − µn)⊤Σ−1 n (x − µn) − n 2 log(2π) − 1 2 log(det Σn), x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Hence, the entropy of ξ ∼ N(µn, Σn) is given by H(ξ) = n 2 (1 + log(2π)) + 1 2 log(det Σn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1) This is a well-known formula, see [10, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1] or [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' We use natural logarithm log = loge in the definition of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Note that in the information theory (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=', [10]) the entropy is sometimes defined using log2 instead of log (this is motivated by measurements in bits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In this case the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1) is written as follows: H(ξ) = 1 2 log2 � (2πe)n det Σn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' For Gaussian vectors, Stratonovich in [19] introduced the alternative definition of the entropy, namely the entropy with respect to the measure ν(dξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , dξn) = (2πe)−n/2dξ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' dξn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' This approach leads to the following simplified version of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1): �H(ξ) = 1 2 log(det Σn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' As we shall see below, the behavior of both versions of entropy, H(ξ) and �H(ξ), as the function of Hurst index are the same and coincides with the behavior of det Σn: all of them increase in H when H increases from 0 to 1/2 and decrease when H increases from 1/2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Their behavior in n is different: det Σn and consequently �H(ξ) decrease in n for any fixed H, however, H(ξ) increases in n, due to the linear term n 2 (1 + log(2π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Fractional Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consider fractional Gaussian noise starting from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let BH = � BH t , t ≥ 0 � be a fractional Brownian motion (fBm) with Hurst index H ∈ (0, 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=', a centered Gaussian process with covariance function of the form EBH t BH s = 1 2 � t2H + s2H − |t − s|2H� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3) Let us consider the following discrete-time process: GH k = BH k − BH k−1, k = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It is well known that the process BH has stationary increments, which implies that � GH k , k ≥ 1 � is a stationary Gaussian sequence (known as fractional Gaussian 4 ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3) that its autocovariance function is given by ρ0(H) = 1, ρk(H) = EGH 1 GH k+1 = 1 2 � (k + 1)2H − 2k2H + (k − 1)2H� , k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4) Therefore, according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1), the entropy of (GH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' GH n ) equals H(GH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' GH n ) = n 2 (1 + log(2π)) + 1 2 log(det Σn(H)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5) where Σn(H) = cov(GH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' GH n ) = � � � � � 1 ρ1(H) ρ2(H) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ρn−1(H) ρ1(H) 1 ρ1(H) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ρn−2(H) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ρn−1(H) ρn−2(H) ρn−3(H) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 1 � � � � � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6) Formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2) is transformed to �H(GH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' GH n ) = 1 2 log(det Σn(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let us mention several particular cases, when the determinant det Σn(H) can be calculated explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let H = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then all ρk( 1 2) = 0, k ≥ 1, and ρ0( 1 2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore, det Σn( 1 2) = 1, n ≥ 1, and consequently log(det Σn( 1 2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then BH t = ξt, where ξ ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore, GH k = ξ, k ≥ 0, and ρk(1) = 1, k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' This means that for any n ≥ 2 det Σn(1) = 0, and consequently log(det Σn(1)) = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Moreover, ρk(H) = 1 2 � (k + 1)2H + (k − 1)2H − 2k2H� → 1 2 � (k + 1)2 + (k − 1)2 − 2k2� = 1, as H ↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then the situation is a bit more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Namely, in this case B0 t is a white noise of the form B0 t = ξt−ξ0 √ 2 , where {ξt, t ≥ 0} are N(0, 1) independent random variables [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore ρ0(0) = 1, ρ1(0) = 1 2E(ξ1 − ξ0)(ξ2 − ξ1) = −1 2 and ρk(0) = 0, k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Moreover, ρ1(H) = 1 2 � 22H − 2 � → −1 2 = ρ1(0) and ρk(H) → 0, H ↓ 0, k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consider Σn(0) = � � � � � � � 1 − 1 2 · · 0 0 − 1 2 1 · · 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 0 0 · · 1 − 1 2 0 0 · · − 1 2 1 � � � � � � � Determinant det Σn(0) of this tridiagonal matrix is calculated by the formula det Σn(0) = det Σn−1(0) − 1 4 det Σn−2(0) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' = k + 1 2k det Σn−k(0) − k 2k+1 det Σn−k−1(0), ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' det Σn(H) as a function of H and n where det Σ0(0) = 1, det Σ−1(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore det Σn(0) = n + 1 2n , n ≥ 1, and consequently log(det Σn(0)) = log(n + 1) − n log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Obviously, both det Σn(0) and log(det Σn(0)) decrease in n and tend to zero and −∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It is quite difficult to prove the monotonic properties of det Σn(H) and its loga- rithm analytically in general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore our main conjecture (A) det Σn(H) and log(det Σn(H)) increase from n+1 2n to 1 and from log(n+1)− n log 2 to 0, respectively, when H increases from 0 to 1 2, and decrease from 1 to 0 and from 0 to −∞, respectively when H increases from 1 2 to 1, decreasing in n for any fixed H is in general checked numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The surface of det Σn(H) as a function of H and n is presented at Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' We observe that for any fixed n ≥ 2 det Σn(H) increases in H ∈ (0, 1 2) and decreases in H ∈ ( 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Also, it decreases in n for any H ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Figures 2 and 3 present entropies �H(GH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' GH n ) and H(GH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' GH n ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It is more logical to arrange these entropies surfaces in this order, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' However, below we study in more detail two particular cases, namely n = 2 and n = 3 and prove that they increase when H increases from 0 to 1 2 and decrease when H increases from 1 2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' As we shall see, even in the case n = 3 the proof of monotonicity requires a lot of technical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Thedeterminantofcovariancematrixoffractional Gaussiannoise 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8 (H)") 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2 0 600 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2 n 300 0 The Hurst index H6 ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' log det Σn(H) as a function of H and n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Cases n = 2 and n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consider the determinants for n = 2 and n = 3 in the spirit of their monotonicity in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4 (Case n = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The determinant det Σ2(H) increases from 3 4 to 1 when H increases from 0 to 1 2 and decreases from 1 to 0 when H increases from 1 2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consequently, log(det Σ2(H)) increases from log 3 − 2 log 2 to 0 when H increases from 0 to 1 2 and decreases from 0 to −∞ when H increases from 1 2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' For n = 2, we have det Σ2(H) = ���� 1 ρ1(H) ρ1(H) 1 ���� = 1 − ρ2 1(H), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='7) where ρ1(H) = 1 2 � 22H − 2 � = 22H−1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' So, det Σ2(H) = 1 − � 22H−1 − 1 �2 = 1 − 24H−2 + 22H − 1 = −24H−2 + 22H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consider function ϕ2(H) = −24H−2 + 22H, H ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Its derivative equals ϕ′ 2(H) = −4 · 24H−2 log 2 + 2 · 22H log 2 = 22H+1 log 2 � 1 − 22H−1� , and ϕ′ 2(H) > 0 for H ∈ (0, 1 2), ϕ′ 2(H) < 0 for H ∈ ( 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' □ The logarithm of determinantof covariancematrixof fractional Gaussian noise 0 100 Indet(2n(H)) 200 300 400 500 600 600 500 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2 n 300 0 TheHurstindexENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' H(GH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' GH n ) as a function of H and n Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5 (Case n = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The determinant det Σ3(H) increases from 1 2 to 1 when H increases from 0 to 1 2 and decreases from 1 to 0 when H increases from 1 2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consequently, log(det Σ3(H)) increases from − log 2 to 0 when H increases from 0 to 1 2 and decreases from 0 to −∞ when H increases from 1 2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The value of the determinant equals det Σ3(H) = ������ 1 ρ1(H) ρ2(H) ρ1(H) 1 ρ1(H) ρ2(H) ρ1(H) 1 ������ = 1+2ρ2 1(H)ρ2(H)−ρ2 2(H)−2ρ2 1(H), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8) where ρ2(H) = 1 2 � 32H − 22H+1 + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consider function ϕ3(H) = 1 + 2x2y − y2 − 2x2, where x = ρ1(H), y = ρ2(H), and calculate its derivative in H: ϕ′ 3(H) = 4xx′ Hy − 2yy′ H − 4xx′ H + 2x2y′ H = 2 � x(2yx′ H + xy′ H) − (yy′ H + 2xx′ H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' First, let H ∈ ( 1 2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then x′ H = 22H log 2 > 0, y′ H = 32H log 3 − 2 · 22H log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The entropy of (GH,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=',GH) with linear term 800 700 009 400 300 200 600 500 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2 n 300 0 The Hurst index H8 ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN Let us prove that y′ H > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Indeed, y′ H = 22H+1 log 2 ��3 2 �2H log 3 log 4 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' If H = 1 2, then �3 2 �2H log 3 log 4 − 1 = log 27 log 16 − 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Since y′ H evidently increases in H, it is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Note that x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore for H ∈ ( 1 2, 1] ϕ′ 3(H) < 2 � 2yx′ H + xy′ H − yy′ H − 2xx′ H � = 2(x − y)(y′ H − 2x′ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Further, x − y = 22H−1 − 1 − 1 2 · 32H + 22H − 1 2 = 3 2 � 22H − 32H−1 − 1 � =: ψ(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It is easy to see that ψ( 1 2) = ψ(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Its second derivative equals ψ′′(H) = 6 � 22H log2 2 − 32H−1 log2 3 � = 6 · 22H log2 3 � log2 2 log2 3 − �3 2 �2H 1 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let H = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then log2 2 log2 3 − 1 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6932 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='0992 − 1 2 ≈ 480249 1207801 − 1 2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It means that ψ′′(H) < 0 on the interval [ 1 2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Moreover, ψ′( 1 2) = 3 (2 log 2 − log 3) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It means that on the interval [ 1 2, 1] ψ(H) = x − y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let us analyze ζ(H) = y′ H − 2x′ H = 32H log 3 − 4 · 22H log 2 = 22H log 3 ��3 2 �2H − log 16 log 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' If H = 1, then �3 2 �2H − log 16 log 3 = 9 4 − log 16 log 3 ≈ 9 4 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consequently, ζ(H) < 0, and ϕ′ 3(H) < 0 that is equivalent to decreasing of the determinant det Σ3(H) on [1/2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Now, let H ∈ [0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' While x′ H > 0, the situation with y′ H is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Denote H0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2868143617175754, the unique root of the equation 32H log 3 − 2 · 22H log 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then y′ H < 0 on [0, H0) and y′ H > 0 on (H0, 1 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' If H ∈ [H0, 1 2], then in the formula for ϕ′ 3(H) we have x ≤ 0, y ≤ 0, x′ H ≥ 0, y′ H ≥ 0, whence xyx′ H ≥ 0, −2yy′ H ≥ 0, −4xx′ H ≥ 0, 2x2y′ H ≥ 0, ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='0 det Σ2(H) det Σ3(H) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Graphs of det Σ2(H) (blue) and det Σ3(H) (orange) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ϕ′ 3(H) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Now, let H ∈ [0, H0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Transform ϕ′ 3(H) as follows: ϕ′ 3(H) = 2 � 2xyx′ H − yy′ H − 2xx′ H + x2y′ H � = 2 � 2x′ Hx(y − 1) − y′ H � y − x2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Further, |x| < 1, therefore y − x2 > y − 1, and on [0, H0] (−y′ H) � y − x2� > (−y′ H) (y − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' So, ϕ′ 3(H) > 2(y − 1) (2xx′ H − y′ H) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Obviously, y − 1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consider 2xx′ H − y′ H = 2 � 22H−1 − 1 � 22H log 2 − 32H log 3 + 2 · 22H log 2 = 24H log 2 − 2 · 22H log 2 − 32H log 3 + 2 · 22H log 2 = 32H log 2 ��4 3 �2H − log 3 log 2 � < 0 for any H ∈ [0, H0] (in fact, for any H ∈ [0, 1 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore, ϕ′ 3(H) > 0 that is equivalent to increasing of the determinant det Σ3(H) on [0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' For all H ∈ (0, 1), det Σ2(H) ≥ det Σ3(H) (where the equality is achieved only for H = 1 2 and for H ↑ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Indeed, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8), we get det Σ2(H) − det Σ3(H) = ρ2 1(H) − 2ρ2 1(H)ρ2(H) + ρ2 2(H) = � ρ1(H) − ρ2(H) �2 + 2ρ1(H)ρ2(H) � 1 − ρ1(H) � ≥ 0, since ρ1(H) ≤ 1, and ρ1(H) and ρ2(H) have the same sign (they both are negative for H ∈ (0, 1 2) and positive for H ∈ ( 1 2, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Figure 4 contains the graphs of det Σ2(H) and det Σ3(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In the general case, the monotonicity of det Σn(H) as a function of n can be proved by representing it as a product of conditional variances, see Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 10 ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Entropy, entropy rate and innovation variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Lower bound for innovation variance 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Fractional Gaussian noise on the whole axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Until now, we have consid- ered the entropy of stationary fractional Gaussian noise starting from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' How- ever, quite often stationary processes start from −∞, especially if the question of their regularity and some other properties are being investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore, we recall how we can construct fractional Gaussian noise starting from −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' For this purpose we use the Mandelbrot–van Ness representation of the fractional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let us briefly recall the concepts related to this object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Standard two-sided Brownian motion is a process W = {Wt, t ∈ R} constructed as a couple of two independent Brownian motions {W−t, t ≥ 0} and {Wt, t ≥ 0}, one with the time reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Two-sided fractional Brownian motion is a zero-mean Gaussian process BH = {BH t , t ∈ R} with covariance function EBH s BH t = 1 2(|s|2H + |t|2H − |s − t|2H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It admits the Mandelbrot–van Ness representation BH t = cH � t −∞ � (t − s) H− 1 2 + − (−s) H− 1 2 + � dWs, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1) where cH = (2H sin(πH)Γ(2H))1/2 Γ(H+1/2) = � 2HΓ(3/2−H) Γ(H+1/2)Γ(2−2H) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Obviously, process BH has stationary increments BH s − BH s−1, s ∈ R, whose covariance equals E � BH s − BH s−1 � � BH t − BH t−1 � = 1 2 � |s − t − 1|2H − 2|s − t|2H + |s − t + 1|2H� , s, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Lower bound for the innovation variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' According to Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3 in the appendix, the entropy of a stationary Gaussian process {Xk, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' } can be expressed in terms of the following conditional variances: r(k) = var[Xk | X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2) see formula (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The values r(k) are deterministic, nonnegative and decreasing, hence, there exists the finite limit σ2 inov(X) = lim n→∞ r(n) ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3) which is called innovation variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Furthermore, for a stationary Gaussian process we have σ2 inov(X) = lim n→∞ r(n) = lim n→∞ var[Xn | Xn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , X1] = lim n→∞ var[Xt | Xt−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xt−n+1] = var[Xt | Xt−1, Xt−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='] for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4) It turns out that for fractional Gaussian noise GH the limit (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3) is strictly positive for all H, and moreover, it admits the following lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1 (Lower bound for the innovation variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' For all H ∈ (0, 1), σ2 inov(GH) ≥ Γ � 3 2 − H � Γ � H + 1 2 � Γ(2 − 2H) =: σ2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5) ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' As a particular case of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4), σ2 inov � GH� = var � GH 1 | GH 0 , GH 1 , GH −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Notice that GH 1 = BH 1 , and all GH t = BH t − BH t−1, t ≤ 0, can be represented as integrals w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' the Brownian motion {Wt, t ≤ 0} with use of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1), whence σ(GH 0 , GH −1, GH −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=') ⊂ σ(Ws, s ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' By the partitioning of conditional variance, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3), σ2 inov � GH� = var[BH 1 | GH 0 , GH −1, GH −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='] ≥ var[BH 1 | Ws, s ≤ 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6) Finally, since the process {BH t , t > 0} is a Volterra Gaussian process with the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1), we see that the conditional variance in the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6) can be calculated by the formula (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2) as follows var[BH 1 | Ws, s ≤ 0] = � 1 0 c2 H(1 − s)2H−1 ds = c2 H 2H = Γ � 3 2 − H � Γ � H + 1 2 � Γ(2 − 2H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Lower bound for the entropy and the entropy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Taking (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1) into account, let us study the asymptotic behavior of the entropy of fractional Gaussian noise as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' We start with the definition of entropy rate, see [10, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The entropy rate of a discrete-time stochastic process X is H∞(X) = lim n→∞ H(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xn) n if this limit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' For the case of Gaussian process X, we may define also �H∞(X) = lim n→∞ �H(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xn) n , where �H(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xn) is introduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let X be a stationary Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then, applying Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3 from the appendix, we obtain that its entropy rate equals H∞(X) = 1 + log(2π) 2 + 1 2 lim n→∞ n � k=1 log r(k), where r(k) is defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' If σ2 inov(X) > 0, then lim n→∞ 1 n n � k=1 log r(k) = lim k→∞ log r(k) = log(σ2 inov(X)), hence, H∞(X) = 1 + log(2π) 2 + log σinov(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='7) If σ2 inov(X) = 0, then the entropy rate of the process X is infinite: H∞(X) = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Using the results of previous subsection, we can see that for the fractional Gauss- ian noise GH, the entropy rate exists and moreover, it admits a finite lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Namely, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 12 ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3 (Lower bounds for the entropy and entropy rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The entropy and the entropy rate of fractional Gaussian noise satisfy inequalities: H(GH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , GH n ) ≥ n 2 � 1 + log(2π) + log σ2 H � , H∞(GH) ≥ 1 + log(2π) 2 + log σH, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8) where σ2 H is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Since GH is a stationary Gaussian process, we have by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3, H(GH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , GH n ) = n + n log(2π) 2 + 1 2 n � k=1 log r(k) ≥ n 2 � 1 + log(2π) + log σ2 H � , since r(k) ≥ σ2 inov ≥ σ2 H for all k, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8) follows immediately from the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='7) and the lower bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Calculation of the entropy rate via spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' According to [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='17)], the entropy rate of the stationary Gaussian process X can be ex- pressed in the form H∞(X) = 1 + log(2π) 2 + 1 2 � 1 0 log ϕ(µ)dµ = 1 + log(2π) 2 + 1 2 � 1/2 −1/2 log ϕ(µ)dµ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='9) where ϕ(µ) = �∞ k=−∞ γ(k)e−2πiµk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In particular, for fractional Gaussian noise, this approach leads to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The entropy rate of the fractional Gaussian noise admits the following representation: H∞(GH) = 1 2 � 1 + log � sin(πH)Γ(2H + 1)(2π)−2H�� + 1 2 � 1/2 −1/2 log � +∞ � k=−∞ |µ + k|−2H−1 � dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' According to [5, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1] the spectral density of fractional Gaussian noise GH is given by f(λ) = 1 2π +∞ � k=−∞ ρk(H)eikλ = 1 π sin(πH)Γ(2H + 1)(1 − cos λ) +∞ � k=−∞ |λ + 2πk|−2H−1, −π ≤ λ ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='9) that the entropy rate can be calculated as follows H∞(GH) = 1 + log(2π) 2 + 1 2 � 1/2 −1/2 log � 2πf(2πµ) � dµ = 1 2 � 1 + log � 2 sin(πH)Γ(2H + 1)(2π)−2H�� + 1 2 � 1 2 − 1 2 log � 1 − cos(2πµ) � dµ + 1 2 � 1/2 −1/2 log � +∞ � k=−∞ |µ + k|−2H−1 � dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4 n = 10 n = 50 n = 100 H∞(GH) lower bound Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The normalized entropy H(GH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , GH n )/n for n = 10, 50, and 100, the entropy rate H∞(GH), and the lower bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8) It is not hard to compute � 1 2 − 1 2 log � 1−cos(2πµ) � dµ = − log 2, whence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='10) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' For computational reasons, it may be convenient to express the infinite sum from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='10) as +∞ � k=−∞ |µ + k|−2H−1 = ζ(2H + 1, µ) + ζ(2H + 1, −µ) − |µ|−2H−1 where ζ(s, a) = �∞ k=0 |a + k|−s denotes the Hurwitz zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Figure 5 contains the graphs of 1 nH(GH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , GH n ) for n = 10, 50, and 100 to- gether with the entropy rate H∞(GH) (computed by the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='10)) and the lower bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' From one hand, it confirms the convergence of the normalized entropies to the entropy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' From the other hand, we see that formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8) gives rather accurate lower bound for all values of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Moreover, the graph of H∞(GH) confirms the following theoretical values for particular cases (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' H = 0: H∞ � G0� = lim n→∞ 1 2 � 1 + log π + 1 n log(n + 1) � = 1 2 (1 + log π) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='07236;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' H = 1 2 : H∞ � G 1 2 � = 1 2 � 1 + log(2π) � ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='41894;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' H = 1: H∞ � G1� = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Entropy functionals 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Definition and the main properties of entropy functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Taking into account two facts: (i) Standard entropy is related to the determinant of covariance matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (ii) It is impossible (or at least rather difficult) to study the properties of the determinant consequently of the entropy as the function of H for the high values of n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 14 ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN let us introduce two alternative entropy functionals that are based on the elements of covariance matrix in the following way: the first functional is proportional to the sum of squares of all different elements of covariance matrix for H ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 1): E1 H(N) = −(H − 1/2)2 1 − H F 1 H(N) = −(H − 1/2)2 1 − H N � k=1 (2ρk(H))2 = −(H − 1/2)2 1 − H � N � k=2 � (k + 1)2H + (k − 1)2H − 2k2H�2 + � 22H − 2 �2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' and the second functional is related to the permanent of covariance matrix as follows: E2 H(N) = −(H − 1/2)2 1 − H F 2 H(N) = −(H − 1/2)2 1 − H N � k=1 (N − k + 1) |2ρk(H)| = −(H − 1/2)2 1 − H � N � k=2 (N − k + 1) ��(k + 1)2H + (k − 1)2H − 2k2H�� + N ��22H − 2 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In both cases we separated the term 22H − 2 that corresponds to k = 1 because we intend to study the behaviour of both functionals as functions of H ∈ [0, 1], and its behaviour differs from other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Recall also that for H ∈ [1/2, 1] the absolute values in E2 H(N) can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Both functionals E1 H(N) and E2 H(N) for any fixed N ≥ 2 have the following behaviour as the functions of H ∈ [0, 1]: they increase in H ∈ [0, 1 2], are zero for H = 1 2 and decrease in H ∈ [ 1 2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Functional E1 H(N) increases from −1/4 to 0 and decreases from 0 to −∞, and E2 H(N) increases from −N/4 to 0 and decreases from 0 to −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Note that the function φ(H) = (H−1/2)2 1−H has a derivative φ′(H) = (H − 1/2)(3/2 − H) (1 − H)2 , therefore it decreases on [0, 1/2] and increases on [1/2, 1] being nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' There- fore it is sufficient to establish that F i H(N), i = 1, 2 decrease in H when H increases from 0 to 1/2 and increase in H when H increases from 1/2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' First, consider H ∈ ( 1 2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then (k + 1)2H + (k − 1)2H − 2k2H > 0 and ∂F 1 H(N) ∂H = 4 N � k=2 � (k + 1)2H + (k − 1)2H − 2k2H� × � (k + 1)2H log(k + 1) + (k − 1)2H log(k − 1) − 2k2H log k � + 4 � 22H − 2 � 22H log 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ∂F 2 H(N) ∂H = 2 N � k=2 (N − k + 1) � (k + 1)2H log(k + 1) + (k − 1)2H log(k − 1) − 2k2H log k � + 2N22H log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN 15 Let us analyze the value ζ(k, H) = (k + 1)2H log(k + 1) + (k − 1)2H log(k − 1) − 2k2H log k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consider the function ϕ(x) = x2H log x, x ≥ 1, 2H > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Its second derivative equals ϕ′′(x) = x2H−2� 2H(2H − 1) log x + 4H − 1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1) and for x ≥ 1 ϕ(x) = x2H log x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It means that ϕ is convex for x ≥ 1, whence ζ(k, H) > 0 for k ≥ 2, H > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Obviously, both additional terms 4 � 22H − 2 � 22H log 2 and 2N22H log 2 are strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' So, both derivatives, ∂F i H(N) ∂H > 0, i = 1, 2, H ∈ ( 1 2, 1], and so F 1 H(N) and F 2 H(N) are strictly increasing in H from 0 to F 1 1 (N) = 22N and F 2 1 (N) = N(N + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Second, consider H ∈ [0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In this case (k + 1)2H + (k − 1)2H − 2k2H < 0 for k ≥ 2, therefore, it is more convenient to rewrite ∂F 1 H(N) ∂H as ∂F 1 H(N) ∂H = 4 N � k=2 � 2k2H − (k + 1)2H − (k − 1)2H� × � 2k2H log k − (k + 1)2H log(k + 1) − (k − 1)2H log(k − 1) � + 4 log 2 · 22H � 22H − 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2) Let us analyze the behaviour of all terms in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consider again function ϕ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Its second derivative is negative for such x that log x > 4H−1 2H(1−2H) and is positive if log x < 4H−1 2H(1−2H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Since we consider x ≥ 1, for H ≤ 1 4 we have that ϕ′′(x) < 0 for all x ≥ 1, and for H ∈ ( 1 4, 1 2) ϕ′′(x) > 0 for x ∈ (1, x0) and ϕ′′(x) < 0 for x ∈ (x0, ∞), where x0 = exp � 4H−1 2H(1−2H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Put N0 = ⌊x0⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then ∂F 1 H(N) ∂H < 4 N � k=N0 � 2k2H − (k + 1)2H − (k − 1)2H� × � 2k2H log k − (k + 1)2H log(k + 1) − (k − 1)2H log(k − 1) � + 4 log 2 · 22H � 22H − 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 16 ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN For any fixed H ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 1 2) ψ(k) = 2k2H − (k + 1)2H − (k − 1)2H has a derivative ∂ψ ∂k (k) = 2H � 2k2H−1 − (k + 1)2H−1 − (k − 1)2H−1� < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='∂F 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='H(N) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='∂H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='< 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2N 2H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='− (N0 + 1)2H − (N0 − 1)2H� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='k=N0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2k2H log k − (k + 1)2H log(k + 1) − (k − 1)2H log(k − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='+ 4 log 2 · 22H � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='22H − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='= 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2N 2H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='− (N0 + 1)2H − (N0 − 1)2H� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='N 2H log N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='− (N + 1)2H log(N + 1) + N 2H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='log N0 − (N0 − 1)2H log(N0 − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='+ 4 log 2 · 22H � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='22H − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='< 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2N 2H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='− (N0 + 1)2H − (N0 − 1)2H� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='N 2H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='log N0 − (N0 − 1)2H log(N0 − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='+ 4 log 2 · 22H � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='22H − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='< 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2 − 22H� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='N 2H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='log N0 − (N0 − 1)2H log(N0 − 1) − 22H log 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Again, for fixed H consider function ζ(x) = x2H log x − (x − 1)2H log(x − 1), x ≥ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Its derivative equals ζ′(x) = (2H log x + 1)x2H−1 − (x − 1)2H−1(2H log(x − 1) + 1), x ≥ N0 and function δ(x) = x2H−1(2H log x+1) has δ′(x) = ϕ′′(x) < 0, x ≥ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore, ζ′(x) < 0, x ≥ N0, and N 2H 0 log N0 − (N0 − 1)2H log(N0 − 1) − 22H log 2 < 22H log 2 − 22H log 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3) Concerning F 2 H(N), for H ∈ [0, 1 2) it equals F 2 H(N) = N � k=2 (N − k + 1) � 2k2H − (k + 1)2H − (k − 1)2H� + N � 2 − 22H� and ∂F 2 H(N) ∂H = 2 N � k=2 (N − k + 1) � 2k2H log k − (k + 1)2H log(k + 1) − (k − 1)2H log(k − 1) � − 2N22H log 2 < 2N N � k=N0 � 2k2H log k − (k + 1)2H log(k + 1) − (k − 1)2H log(k − 1) � − 2N22H log 2 ≤ 2N � N 2H log N − (N + 1)2H log(N + 1) + N 2H 0 log N0 − (N0 − 1)2H log(N0 − 1) − 22H log 2 � < 0 due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' □ ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Entropy rate for entropy functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It is very easy to see from formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4) that ρk(H) decrease in k for H ∈ (1/2, 1) being positive and increase in k for H ∈ (0, 1/2) being negative, therefore all the summands in (2ρk(H))2 in E1 H(N) decrease in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Moreover, 2ρk(H) = k2H �� 1 + 1 k �2H + � 1 − 1 k �2H − 2 � ∼ 2k2H 2H(2H − 1) 2k2 = 2H(2H − 1)k2H−2, as k → ∞, therefore � 2ρk(H) �2 ∼ 4H2(2H − 1)2k4H−4 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It means that entropy functional E1 H(N) has the following asymptotic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (i) Let H ∈ (0, 3 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then the series �∞ k=1 � 2ρk(H) �2 converges, and E1 H(N) → E1 H(∞) = −(H − 1 2)2 1 − H ∞ � k=1 � 2ρk(H) �2 as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (ii) Let H = 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then lim N→∞ E1 H(N) log N = − 9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (iii) Let H ∈ ( 3 4, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then lim N→∞ E1 H(N) N 4H−3 = − 4H2(2H − 1)4 (1 − H)(4H − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Item (i) is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (ii) Indeed, with H = 3/4 lim N→∞ E1 H(N) log N = lim N→∞ E1 3/4(N) log N = −(H − 1 2)2 1 − H lim N→∞ � 2ρk( 3 4) �2 1 N = −(H − 1 2)2 1 − H 42H2(2H − 1)2N −1 N −1 = −4H2(2H − 1)4 1 − H = − 9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (iii) Indeed, lim N→∞ E1 H(N) N 4H−3 = −(H − 1 2)2 1 − H lim N→∞ 42H2(2H − 1)2N 4H−4 (4H − 3)N 4H−4 = − 4H2(2H − 1)4 (1 − H)(4H − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (i) Let H ∈ (0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then lim N→∞ E2 H(N) = − ∞ � k=1 |ρk(H)| (H − 1 2)2 1 − H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (ii) Let H = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then E2 H(N) = 0, N ≥ 1, and its limit equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (iii) Let H ∈ ( 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then lim N→∞ E2 H(N) N 2H = −(H − 1 2)2 1 − H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 18 ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Consider separately S1 N = N N � k=2 ��(k + 1)2H + (k − 1)2H − 2k2H�� and S2 N = N � k=2 (k − 1) ��(k + 1)2H + (k − 1)2H − 2k2H�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (i) Let H ∈ (0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then ��(k + 1)2H + (k − 1)2H − 2k2H�� ∼ k2H−22H(1 − 2H), and �∞ k=2 ��(k + 1)2H + (k − 1)2H − 2k2H�� < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore S1 N N → ∞ � k=2 ��(k + 1)2H + (k − 1)2H − 2k2H�� , as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Further, (k − 1) ��(k + 1)2H + (k − 1)2H − 2k2H�� ∼ k2H−12H(1 − 2H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Therefore lim N→∞ S2 N N lim N→∞ N 2H−12H(1 − 2H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (iii) Let H ∈ ( 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then S1 N N 2H ∼ N 2H−22H(2H − 1) (2H − 1)N 2H−2 , so lim N→∞ S1 N N 2H = 2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Further, S2 N N 2H ∼ N 2H−12H(2H − 1) 2HN 2H−1 , so lim N→∞ S2 N N 2H = 2H − 1, whence the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Some results on stationary Gaussian processes A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Partitioning of conditional variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let (Ω, F, P) be the probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The conditional variance of the random variable X with EX2 < ∞ given the σ-field A ⊂ F is defined as var[X | A] = E[(X − E[X | A])2 | A] = E[X2 | A] − (E[X | A])2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let A and B be two σ-fields, A ⊂ B ⊂ F, and X be a random variable with EX2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then var[X | A] = E[var[X | B] | A] + var[E[X | B] | A] (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1) and E var[X | A] = E var[X | B] + E var[E[X | B] | A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In general case, the conditional variance var[X | B] is a B-measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In particular cases, var[X | B] is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' For example, the condi- tional variance of a component of a Gaussian random vector given other components is nonrandom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The conditional variance of an observation Xt of a Gaussian pro- cess X given observation of the process on some set I is nonrandom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The same holds true for a linear functional of the process X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' More specifically, the following holds true: if X = {Xs, s ∈ T} is a Gaussian process, t ∈ T and I ⊂ T, then ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN 19 var[Xt | Xs, s ∈ I] is nonrandom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Furthermore, if W = {Ws, s ∈ R} is a two-sided Wiener process, φ ∈ L2(R) is a deterministic function, and I ⊂ R, then var �� ∞ −∞ φ(s) dWs ���� Ws, s ∈ I � is nonrandom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' For the Volterra Gaussian process Xt = � t −∞ K(t, s) dWs, t ≥ 0, where K(t, · ) ∈ L2((−∞, t]) for all t > 0, the following relation holds var[Xt | F0] = � t 0 K(t, s)2 ds, t > 0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2) where F0 = σ(Ws, s ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Indeed, E(Xt | F0) = E �� 0 −∞ K(t, s) dWs + � t 0 K(t, s) dWs ���� F0 � = � 0 −∞ K(t, s) dWs, and similarly var[Xt | F0] = E[X2 t | F0] − (E[X | F0])2 = E ��� 0 −∞ K(t, s) dWs �2 + �� t 0 K(t, s) dWs �2 + 2 � 0 −∞ K(t, s) dWs � t 0 K(t, s) dWs ���� F0 � − �� 0 −∞ K(t, s) dWs �2 = E ��� t 0 K(t, s) dWs �2� = � t 0 K(t, s)2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' If, in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1), the conditional expectation var[X | B] is nonrandom (or, more generally, if var[X | B] is an A-measurable random variable), then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1) takes the form var[X | A] = var[X | B] + var[E[X | B] | A], whence var[X | A] ≥ var[X | B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3) The equality holds in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3) if and only if var[E[X | B] | A] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The sufficient condition for equality in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3) is E[X | A] = E[X | B] almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The sufficient conditions for strict inequality in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3) are that both E[X | A] and E[X | B] are nonrandom and P(E[X | A] ̸= E[X | B]) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Entropy of a stationary Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let X = {Xt, t=1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='} be a stationary Gaussian process with the autocovariance function γ(h) = cov(Xt+h, Xt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The covariance matrix of n consecutive observations of the process X is denoted Γn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' it is a symmetric Toeplitz matrix: Γn = � � � � γ(0) γ(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' γ(n − 1) γ(1) γ(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' γ(n − 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' γ(n − 1) γ(n − 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' γ(0) � � � � 20 ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' For all n ∈ N the matrix Γn is nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' If X is a fractional Gaussian noise, then Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1 is satisfied, see [4, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Under Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1, the entropy of n consecutive observations of the process X is equal to H(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xn) = H(Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xt+n) = n + n log(2π) 2 + 1 2 log(det Γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4) The goal of this subsection is to express this entropy in terms of the following quantities: r(1) = var(X1), r(k) = var[Xk | X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1], k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5) Recall that r(k) is nonrandom for any k, since the process X is Gaussian, see subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let X = {Xk, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='} be a stationary Gaussian process, whose covariance matrix satisfies Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The sequence {r(k), k ∈ N}, defined by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5), is deterministic, non- negative and decreasing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' hence, it is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The entropy of n consecutive observations of the process X is equal to H(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xn) = n + n log(2π) 2 + 1 2 n � k=1 log r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The determinant of the covariance matrix Γn is expressed in the following form: det Γn = n � k=1 r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The monotonicity of r(k) follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Indeed, r(k) = var[Xk | X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1] = var[Xk+1 | X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk] ≥ var[Xk+1 | X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk] = r(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Due to the chain rule for the entropies [10, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='2] H(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xn) = H(X1) + n � k=2 H(Xk | X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8) where H(Xk | X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1) is the conditional entropy, H(Xk | X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1) = − � · · �� pX1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=',Xk−1,Xk(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , xk−1, xk) × log pXk | X1=x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=',Xk−1=xk−1(xk) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' dxk−1 dxk, and H(X1) = 1 2 log(2eπr(1)), since X1 ∼ N(0, r(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The conditional distribution of Xk given X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1 is Gaussian, with fixed variance: [Xk | X1=x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1=xk−1] ∼ N(E[Xk | X1=x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1=xk−1], r(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN 21 Thus, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1), the entropy of the conditional distribution [Xk | X1 = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1 = xk−1] is H(Xk | X1=x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1=xk−1) = 1 2 log(2eπr(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='9) Note that the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='9) does not depend on x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , xk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The condi- tional entropy can be expressed through the entropy of the underlying conditional distribution: H(Xk | X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1) = � · · � pX1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=',Xk−1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , xk−1) × × H(Xk | X1=x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1=xk−1) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' dxk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Thus, H(Xk | X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xk−1) = � · · � pX1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=',Xk−1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , xk−1) 1 2 log(2eπr(k)) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' dxk−1 = 1 2 log(2eπr(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' By the chain rule (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8), H(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' , Xn) = n � k=1 1 2 log(2eπr(k)) = n 2 log(2eπ) + 1 2 n � k=1 log r(k), which coincides with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Comparing (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6) we immediately get the representation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' □ Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The first statement of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3 is known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' it can be found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=', in [11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The formula (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='7) can be proved also with the help of the Cholesky decom- position of the covariance matrix Γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Namely, Γn can be represented as Γn = � � � � ℓ1,1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 0 ℓ2,1 ℓ2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ℓn,1 ℓn,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ℓn,n � � � � � � � � ℓ1,1 ℓ2,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ℓn,1 0 ℓ2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ℓn,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' ℓn,n � � � � where ℓ2 k,k = r(k), see [14, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' (A19)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Hence, det Γn = �n k=1 ℓ2 k,k = �n k=1 r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let us mention that a similar method (based on so called LDL⊤-decomposition of the covariance matrix) is described in [8, § 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The representation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='7) implies that the determinant of the covari- ance matrix Σn(H) of the fractional Gaussian noise GH decreases as a function of n, since in this case r(k) ≤ r(1) = var � GH 1 � = 1, for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' 22 ENTROPY AND ALTERNATIVE ENTROPY FUNCTIONALS OF FGN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Entropy rate and the nondeterminism of stationary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Denote by Mt(X) = span(Xt, Xt−1, Xt−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=') the smallest closed linear subspace of the Hilbert space L2(Ω, F, P) that contains random variables Xt, , Xt−1, Xt−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Let M−∞(X) = ∞ � t=−∞ Mt(X), M(X) = span(Xt, t ∈ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' A centered wide-sense stationary process {Xt, t ∈ Z} is called deterministic if M−∞(X) = M(X), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=', Ms(X) = Mt(X) for all s, t ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' The process X is called completely non-deterministic if M−∞(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' A centered stationary Gaussian process {Xt, t ∈ Z} is deterministic if and only if var[Xt | Xt−1, Xt−2, Xt−3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='] = 0 (here the left-hand side does not depend on t due to stationarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' It is well known that a stationary mean-zero process X = {X(t), t ∈ Z} ad- mits the Wold’s representation as a sum of two orthogonal processes: X(t) = M(t) + N(t), where M = {M(t), t ∈ Z} is deterministic and N = {N(t), t ∈ Z} is completely non-deterministic, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' [18, Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4] or [6, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='4), a stationary Gaussian process X is deterministic if and only if for this process σ2 inov(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Taking into account (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='7), we get the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Under Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content='1, a stationary Gaussian process has a finite entropy rate if and only if it is non-deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Al`os, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Le´on, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Nualart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Stochastic Stratonovich calculus fBm for fractional Brownian motion with Hurst parameter less than 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Taiwanese J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=', 5(3):609–632, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Al`os, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Mazet, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Nualart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Stochastic Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=', 86(1):121–139, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' [3] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Anh and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFQT4oBgHgl3EQfpzbH/content/2301.13378v1.pdf'} +page_content=' Inoue.' 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China +YUN MA, Peking University, China +GANG HUANG, Peking University, China +XUANZHE LIU, Peking University, China +WebAssembly (abbreviated WASM) has emerged as a promising language of the Web and also been used for a wide spectrum of software +applications such as mobile applications and desktop applications. These applications, named as WASM applications, commonly run in +WASM runtimes. Bugs in WASM runtimes are frequently reported by developers and cause the crash of WASM applications. However, +these bugs have not been well studied. To fill in the knowledge gap, we present a systematic study to characterize and detect bugs in +WASM runtimes. We first harvest a dataset of 311 real-world bugs from hundreds of related posts on GitHub. Based on the collected +high-quality bug reports, we distill 31 bug categories of WASM runtimes and summarize their common fix strategies. Furthermore, we +develop a pattern-based bug detection framework to automatically detect bugs in WASM runtimes. We apply the detection framework +to five popular WASM runtimes and successfully uncover 53 bugs that have never been reported previously, among which 14 have +been confirmed and 6 have been fixed by runtime developers. +CCS Concepts: • Software and its engineering → Software notations and tools. +Additional Key Words and Phrases: WebAssembly, WebAssembly runtime +ACM Reference Format: +Yixuan Zhang, Shangtong Cao, Haoyu Wang, Zhenpeng Chen, Xiapu Luo, Dongliang Mu, Yun Ma, Gang Huang, and Xuanzhe Liu. 2023. +Characterizing and Detecting WebAssembly Runtime Bugs. 1, 1 (January 2023), 25 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +WebAssembly (abbreviated WASM) has quickly emerged as a promising language of the Web in recent years [39]. +WASM is a binary instruction specification [39, 44, 55] for a stack-based virtual machine and provides developers with +Authors’ addresses: Yixuan Zhang, Peking University, Beijing, China, zhangyixuan.6290@pku.edu.cn; Shangtong Cao, Beijing University of Posts +and Telecommunications, Beijing, China, shangtongcao@bupt.edu.cn; Haoyu Wang, Huazhong University of Science and Technology, Wuhan, China, +haoyuwang@hust.edu.cn; Zhenpeng Chen, University College London, London, UK, zp.chen@ucl.ac.uk; Xiapu Luo, The Hong Kong Polytechnic University, +Hong Kong, China, csxluo@comp.polyu.edu.hk; Dongliang Mu, Huazhong University of Science and Technology, Wuhan, China, dzm91@hust.edu.cn; +Yun Ma, Peking University, Beijing, China, mayun@pku.edu.cn; Gang Huang, Peking University, Beijing, China, hg@pku.edu.cn; Xuanzhe Liu, Peking +University, Beijing, China, liuxuanzhe@pku.edu.cn. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not +made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components +of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to +redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2023 Association for Computing Machinery. +Manuscript submitted to ACM +Manuscript submitted to ACM +1 +arXiv:2301.12102v1 [cs.SE] 28 Jan 2023 + +2 +Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al. +an equivalent textual format [22] for reading, testing, learning instructions, and debugging, etc. Although WASM was +initially proposed for Web applications [53, 56], it is moving fast towards a much wider spectrum of domains, including +desktop applications [13, 23], mobile applications [13], IoT [47, 58], blockchain [4, 6, 29], serverless computing [13, 38], +and edge computing [37? ], etc.. To develop these applications (named WASM applications), developers can compile +high-level programming languages to WASM binaries or convert the equivalent manually-written textual format to +WASM binaries. WASM binaries are commonly executed in WASM runtimes. A WASM runtime provides an efficient, +memory-safe, sandboxed execution environment for WASM applications [26]. However, a great variety of WASM +runtime specific bugs have been reported by developers, inevitably impeding the development of the WASM application +ecosystem. Despite this, WASM runtime bugs have not been systematically studied by our community. Therefore, there +is a general lack of an understanding of these bugs, including their root causes, fix patterns, and how to detect these +bugs in emerging WASM runtimes. +This Work. To fill in the knowledge gap, we present the first comprehensive study on characterizing and detecting +bugs in WASM runtimes. We focus our study on three most popular and representative WASM runtimes, including +wasmtime [20], wasmer [17], and wasm-micro-runtime (WAMR) [24]. We first collect 903 bug-related posts from GitHub, +a commonly-used data sources for studying software bugs, and make an effort to identify 311 real-world bugs of these +WASM runtimes (see 3). Based on the collected bugs, we manually construct a taxonomy of 31 bug categories (see 4), +indicating the diversity of WASM runtime bugs. Moreover, we summarize common fix patterns for each bug category +(see 5). These empirical results provide a high-level categorization that can serve as a guide for developers to resolve +common faults and for researchers to develop tools for detecting and fixing common WASM runtime bugs. +Furthermore, we develop a pattern-based bug detection framework based on the knowledge summarized from the +bug taxonomy, to test the presence of bugs in WASM runtimes (see 6). To evaluate the generalizability of our study, +beyond the three analyzed WASM runtimes, we further consider two emerging WASM runtimes (wasm3 and WASMEdge) +for bug detection. We have successfully identified 53 previously-unknown bugs. We report these bugs to the developers +of corresponding WASM runtimes. By the time of this writing, 14 bugs have been confirmed by the developers, and 6 of +them have been fixed based on our suggestions. +To summarize, this paper makes the following contributions: +• We conduct the first systematic study of bugs in WASM runtimes. We summarize common bug categories and +their corresponding fix strategies. Our results can help understand and characterize bugs in WASM runtimes +while shedding lights on future WASM related studies. +• We develop a pattern-based bug detection framework based on the knowledge summarized from bug categories +we created to automatically detect bugs in WASM runtimes. By applying the detection framework to real-world +WASM runtimes, it shows that our proposed framework can effectively detect bugs and provide useful information +to facilitate bug diagnosis and fixing. +• We will make the scripts, datasets, and bug detector available to the research community for other researchers to +replicate and build upon. +2 +BACKGROUND +2.1 +WASM binaries +WASM is a low-level assembly-like language that is designed for efficient execution and compact representation. +The WASM binary file is compact like Java class files and is saved with the .wasm suffix [61]. The WASM specification +Manuscript submitted to ACM + +Characterizing and Detecting WebAssembly Runtime Bugs +3 +defines a conceptual stack virtual machine for most WASM instructions to work on, performing numbers’ pop and +push and leaving the result on the stack. A pretty-printed textual format (i.e., .wat) [22] is also provided for developers, +which can be used to learn the syntax, understand the WASM module, test WASM program, optimize applications, +debug code, and write WASM programs by hand, etc. +Developers and users can use the wabt [10] tool to translate WASM binaries to WASM textual format or vice versa. +High-level language +WASM binaries +WASM compiler +WASM textual +format +wabt +WASM runtime +Operating system +WASM binaries +Hardware +Fig. 1. The execution process of WASM binaries. +2.2 +Execution of WASM binaries +As a binary instruction format, WASM is designed as a portable compilation target for high-level programming +languages [26]. As shown in Figure 1, developers can use WASM compilers to translate high-level language programs +to WASM binaries. There are dozens of compilers available to compile different source language programs to WASM +binaries, such as AssemblyScript, Emscripten, Rustc/WASM-Bindgen, etc [52]. WASM can be executed at native speed +[39] on a wide range of platforms. The tool for this critical process is a WASM runtime, an intermediate layer between +the WASM binaries and the hardware platforms. A WASM runtime should consider the structure, operating system, +and other differences between various platforms and provide a relatively secure execution environment for the WASM +binaries. As shown in Figure 1, developers can create applications in high-level languages, compile them into WASM +binaries [3, 52], and execute WASM binaries in WASM runtimes. Alternatively, they could develop simple WASM +programs in the textual format, convert them to WASM binaries through wabt [10], and execute the binaries in WASM +runtimes. +2.3 +WASM Runtime Architecture +Based on the implementation of well known WASM runtimes [12, 14, 15, 17, 20, 24? ], we have summarized the +general architecture of WASM runtimes in Figure 2, which can be divided into six major components. +Backend compiler. WASM runtimes support executing WASM binaries in the following modes: interpreter mode, +Ahead-of-Time compilation mode (AoT), and Just-in-Time compilation mode (JIT). WASM runtimes support compiling +WASM binaries into native code before executing it locally using AoT compilers. To speed up the execution efficiency, +some WASM runtimes use the just-in-time compilation of hot code through JIT compilers. JIT compilers and AoT +compilers are considered backend compilers in the WebAssembly workflow. +Interpreter. Some WASM runtimes provide interpretive execution on the WASM binaries. +Manuscript submitted to ACM + +4 +Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al. +WebAssembly System Interface +High-level +language API +Auxiliary +tools +Interpreter +Backend +compiler +System calls +Hardware +Operating system +Runtime environment +WASM runtime +Fig. 2. The general architecture of a WASM runtime. +Runtime environment. The runtime environment supports allocating memory, performing stack operations, +reporting execution error messages, and other features. +High-level language API. The WASM runtimes can be embedded in different high-level languages (e.g., C/C++, +Java, Python, Rust, etc.) as a library to allow users to use WASM in any scenarios with various languages. +WebAssembly system interface. WASM runtimes provide WASM applications with WebAssembly system interface +(WASI) [25] as a modular system interface [12], focusing on security and portability. WASI is the bridge between the +sandbox environment and operating systems. WASI is an API that provides access to several OS-like features, including +file operation and clock. +Auxiliary tools. WASM runtimes also provide handy little tools for the users, such as WASM module cache, WASM +textual file format validation, etc. +3 +CHARACTERIZATION METHODOLOGY +We first perform an empirical study to characterize WASM runtime bugs. Specifically, we seek to investigate: 1) the +taxonomy of bugs, i.e., the reasons leading to the bugs, and 2) the fix strategies, i.e., how to address these bugs. +To approach the answer, we collect and analyze the bug reports posted on Github and Stack Overflow, following the +traditional empirical methods in the SE community [32, 36, 41, 51, 52, 54, 57, 60, 62, 63]. Figure 3 shows the overview of +our study methodology. +3.1 +Collection of WASM Runtime Bugs +3.1.1 +Selecting WASM Runtimes. As shown in Table 1, we select three most popular WASM runtimes as target, including +wasmer [17], wasmtime [20], and wasm-micro-runtime (WAMR) [24]. We believe they are the most representative WASM +runtimes for us to characterize real-world WASM runtime bugs across different implementations, as 1) all of them +are mature projects (with over 100,000 LOC) that have gained the thousands of stars on GitHub, 2) they have covered +different kinds of execution modes (i.e., Interpreter, JIT and AoT), and 3) they have implemented in different languages +(i.e., Rust and C/C++). +Manuscript submitted to ACM + +Characterizing and Detecting WebAssembly Runtime Bugs +5 +Data Collection form GitHub issues. +Data Collection form SO questions. +Refined +dataset. +RQ1 : Faults taxonomy. +RQ2 : Fix strategies. +Analyze data. +Fig. 3. Overview of the methodology. +Table 1. Statistics of our harvested dataset. +Runtime +Stars Commits GitHub issues SO posts +Total +wasmer +12,026 +11,332 +403 (179) +41 (0) +444 (179) +wasmtime +7,360 +9,754 +167 (94) +52 (0) +219 (94) +WAMR +2,720 +686 +333 (38) +4 (0) +337 (38) +Total +903 (311) +97 (0) +1000 (311) +∗The refined numbers are in the parentheses. +3.1.2 +Data Collection from GitHub. Following previous work [32, 51, 52, 57, 62, 63], we extract issues in the official +GitHub repositories of the selected WASM runtimes. GitHub issues contain many bug information, including source +code, detailed reports, and contributors’ discussions [34]. These characteristics make GitHub issues suitable for analyzing +bug root causes and summarizing fix strategies. For details, we use the GitHub search API [5] to extract the related +issues on May 14, 2022. GitHub issues include various topics, including bug reports, feature requests, documentation +updates, etc. Thus, to highlight the purposes of bugs, we take advantage of the bug issue label to identify related issues. +We collect issues related to wasmer and wasmtime by filtering labels with “bug”. Due to all the issues from WAMR are +not labeled, we extract all the issues from WAMR for further analysis. Overall, we obtained 403 issues from wasmer, 167 +issues from wasmtime, and 333 from WAMR. +3.1.3 +Data Collection from SO. Initially, we also considered posts from Stack Overflow. +Each SO question has at least one tag based on its topics. We extract the posts related to the selected WASM runtimes +on May 14, 2022. As a result, we obtain 41 posts for wasmer, 52 posts for wasmtime, and 4 posts for WAMR. Table 1 shows +the collected raw data. +3.1.4 +Refining the Dataset. We perform manual investigation on the collected data. First, we filtered out issues and +posts with no definite answers, to ensure the accuracy and certainty of bugs and fix strategies. Second, we exclude +installation/build bugs, documentation bugs, and other issues and posts unrelated to WASM binaries’ execution from +the source data. Finally, as shown in Table 1, the total number of WASM runtime bugs is 311. The scale of this dataset +is comparable and more extensive than those used in existing bug-related studies [28, 30, 32, 34, 52, 60, 62] that also +require manual inspection. All the 311 issues are from GitHub because all the reports we collected from SO are not +related to the WASM binaries execution in WASM runtimes. It is probably because there are few WASM experts on +Manuscript submitted to ACM + +6 +Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al. +SO since WASM is an emerging language. Therefore, WASM developers tend to report the bugs they encounter to the +official WASM runtime repositories to seek immediate help. +3.2 +Labelling Bugs and Fix Strategies +The refined 311 bug reports are used for distilling features and fix strategies through manual labelling by two authors +and an intercessor. +3.2.1 +Pilot Labelling. First, we randomly sample 50% of the posts (𝑁 = 155) from the selected WASM runtimes for +pilot labeling. The first two authors of the paper jointly participate in the process. According to the WASM runtime +architecture and the root causes, they create the bug categories and fix strategies by analyzing the GitHub issues. +3.2.2 +Reliability Analysis. For reliability analysis, the first two authors independently label the remaining 40% issues +based on the taxonomy constructed in the prior stage. In detail, they mark each issue with the posted bug, fix strategy +categories, and the issues that cannot be classified into the current taxonomies as a new category. To measure the +reliability during the independent labelling, we employ the widely used Cohen’s Kappa indicator (𝜅) for bug and fix +strategies of 0.921 and 0.915, indicating almost perfect agreement [33]. The agreement levels demonstrate the reliability +of our labelling. +The divergence in the labelling process is then discussed and settled after the labeling process. For the newly added +categories by the first two authors, we discuss them with the intercessor. As a result, we add two new categories +to the bug taxonomy and three new categories into the fix strategy taxonomy. Furthermore, the first two authors +independently label the remaining 10% issues. During this process, no more bug taxonomy or fix strategy is added, +indicating saturation of the taxonomy. After finishing the whole labelling stage, the Cohen’s Kappa indicator (𝜅) +for bug and fix strategies is 0.929 and 0.925, showing almost perfect agreement [33]. Additionally, the three authors +involved in the taxonomy check the final labeling result together. +We will detail the bugs and fix patterns in the following sections. +4 +TAXONOMY OF WASM RUNTIME BUGS +We present the hierarchical taxonomy of WASM runtime bugs according to the WASM runtime architecture (see 2). +As shown in Figure 4, the taxonomy is organized into three-level categories, including a root category (WASM Runtime +Bugs), four inner categories linked to different components in a WASM runtime (e.g., Backend Compilation), and 31 +specific leaf categories (e.g., Register allocation error). +The backend compilers (JIT compilers and AoT compilers) of the architecture are summarized into one inner bug +category, called Backend Compilation (A), which converts WASM binaries into native code. The bugs in the lowest +part in a WASM runtime are called WASI Robustness (B). The handy little tools in WASM runtimes are called Auxiliary +Tools (D). Other bugs that occur while running WASM binaries are classified into Runtime Environment (C), including +memory allocation, calling host functions, and so on. It is worth mentioning that bugs that occur while using high-level +language API are either divided into Backend Compilation (A) or Runtime Environment (C). WASM users could use the +API to compile WASM binaries and take advantage of functionalities in the Runtime Environment, and it is an interface +for users to make good use of a WASM runtime. Moreover, the interpreter part is merged in a leaf category of Backend +Compilation (A), as only WAMR provides an interpreter, and only one bug is found in the interpreter. +Manuscript submitted to ACM + +Characterizing and Detecting WebAssembly Runtime Bugs +7 +WASM Runtime Bugs +311 +[B] WASI Robustness +54 +[A.1] Incompatible +infrastructure version +3 +[A.2] Incorrect +compilation +28 +[A.3] Compilation +failure +[A.4] Register +allocation error  +15 +[A.5] Incomplete +operating system +support  +9 +[A.6] Incomplete +hardware support +7 +[A.7] Unsupported +data operation  +14 +[A.8] Validation error +7 +[A.9] WASM debugging informatino error  +[A.10] Others +8 +[B.1] File operation +error +14 +[B.2] Import error +3 +[B.3] Unsupported +operation +5 +[B.4] Input and +output stream error +7 +[B.5] operating +system support +error +4 +[B.6] WASI +version error +4 +[B.7] Other +counterpart error +5 +[B.8] Clock bugs +3 +[B.9] Others +9 +[C] Runtime Environment +120 +[C.1] Module +instantiation bugs +16 +[C.2] Module import +error +8 +[C.3] Calling host +functions +12 +[C.7] Thread safety +issue +[C.8] Stack issue +4 +[C.9] Entry point +error +3 +[C.4] Memory issue +19 +[C.10] Unhandled +error +9 +[C.11] Data type +conversion +4 +[C.6] Unsupported +features +4 +[C.12] Others +18 +[D] Auxiliary Tools +18 +20 +[A] Backend Compilation +119 +[C.5] Trap error +9 +12 +11 +8 +Fig. 4. Taxonomy of bug symptoms. The number in the top right corner indicates the number of bugs for each category. +Highlight 1: We construct a taxonomy of 31 leaf bug symptom categories in WASM runtimes, indicating the root +causes and the diversity. +4.1 +Backend Compilation +As the first stage of executing WASM binaries, backend compilation is used to translate WASM binaries into native +code. +In general, backend compilers convert WASM binaries into their intermediate representation (IR), allocate registers +and optimize the code. Note that backend compilers could convert WASM binaries into the IR proposed in other +compilation framework infrastructures (e.g., LLVM). The whole process needs to support various OSes and CPU +architectures. We observe 119 bugs in this category, accounting for 38.3% of all the classified bugs and covering 10 leaf +categories. +Various backend compilers use their own IR as the intermediate step to translate WASM instructions to native code. +During the process of compiling, compilers could generate incorrect IR or incorrect native code during the translation +of WASM binaries. Besides, optimizing the code could also lead to an error. These bugs are summarized as Incorrect +compilation (A.2). A compiler may raise an exception when generating native code or even fail to generate the native +node (A.3), which accounts for 16.8% bugs in Backend Compilation (A). Moreover, some WASM runtimes rely on the +existing compilation framework, such as LLVM. Thus, Using the incorrect version of infrastructure (A.1) could lead to +unexpected results, accounting for 2.5% bugs in Backend Compilation (A). +Besides converting WASM instructions into native machine instructions, the backend compilers must allocate +registers. However, they may result in the Incorrect register allocation (A.4), including incorrectly using special registers, +loading data from an unexpected register and exhausting registers. These bugs account for 7.6% of bugs in Backend +Compilation (A). As shown in Example (a) (Figure 5), the backend compiler in wasmtime gets saved and restored in r15 +Manuscript submitted to ACM + +8 +Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al. +as a CSR (control and status register), which is expected to be used as a pinned register. The allocation of r15 poses a +bug. As another example in Example b) (Figure 6), wasmtime allocates registers for the given wat file. However, during +lowering SIMD instructions, the allocation of registers shows a bug. The movdqa instruction moves out of v6, but v6 is +never set. This kind of bugs will cause panic during the execution of WASM binaries, and the execution process cannot +be completed. Most WASM runtimes only support JIT or AoT compilation, while WAMR also provides an interpreter +to deal with WASM. There is only one bug in the interpreter. The interpreter could not correctly pass parameters to +submodules, leading to an incorrect result. Moreover, this is summarized into Others (A.10). +Fault description: As shown in the Assembly code converting from wasmtime IR +, r15, the pinned register, gets saved and restored as a CSR, making it impossible to use as a +pinned register. +Fault symptom: Register allocation error +Assembly code: + 0: 55 push rbp + 1: 48 89 e5 mov rbp, rsp + 4: 48 83 ec 10 sub rsp, 0x10 + 8: 4c 89 3c 24 mov qword ptr [rsp], r15 + c: 4c 8b 0f mov r9, qword ptr [rdi] + f: 49 83 c7 01 add r15, 1 + 13: 4c 8b 3c 24 mov r15, qword ptr [rsp] + 17: 48 83 c4 10 add rsp, 0x10 + 1b: 48 89 ec mov rsp, rbp + 1e: 5d pop rbp + 1f: c3 ret +Fig. 5. Example (a) - GitHub wasmtime issue #4170 +In the compilation process, WASM runtimes run the WASM file across various operating systems (A.5). They account +for 7.6% of bugs in the current inner category. The backend compilers encounter problems only caused by specific +operating systems and lack consideration for their particular circumstances. With its architecture and instruction set, +WASM runtimes also run the WASM files across different CPUs. Some problems are only present in specific CPUs or +specific architecture machines. These problems are summarized as Incomplete hardware support (A.6) which account for +5.9% bugs in Backend Compilation. +Further, we also observe that a portion (i.e., 11.8%) of bugs in Unsupported data operation (A.7). For example, in +the IR used by the backend compiler in wasmtime, the srem.8 and srem.16 are not supported. Besides, wasmtime +converts the WASM binaries into cranelift IR before execution. It lacks the design of supporting the data operation in +big endianness machines (e.g., GitHub wasmtime issue #3288). Executing the clif file on s390 hardware shows wrong +results only for i16, i32, and i64 types, while i8 passes these tests. The s390 architecture is big-endian, while the data +operation in wasmtime was taken from the lower bites. Thus, the data operation of i16, i32, and i64 was not supported +in the big-endianness machine. This kind of bug could pose different execution results on or execution exceptions. +This bug can result in inconsistent results or execution exceptions for the same WASM binaries executed on different +machines. Besides, WASM specification introduced Single Instruction Multiple Data (SIMD) instructions to improve the +execution efficiency. Backend compilers may lack the support for data operation related to SIMD instructions, such as +the operation of the v128 data type. +Manuscript submitted to ACM + +Characterizing and Detecting WebAssembly Runtime Bugs +9 +Fault description: Wasmtime convert the wat file into Vcode, as shown below. The movdqa +instruction moves out of v6, which is never set. +Fault symptom: Register allocation error +Wat file: +(module + (type (;0;) (func)) + (func (;0;) (type 0) + v128.const i32x4 0x00000000 0x00000000 0x00000000 0x00000000 + i64x2.extend_low_i32x4_u + v128.const i32x4 0x00000000 0x00000000 0x00000000 0x00000000 + i64x2.mul + i32x4.all_true + i64.load offset=1 align=1 + drop + unreachable) + (func (;1;) (type 0) + nop) + (memory (;0;) 5613 17832)) +Vcode: +VCode_ShowWithRRU {{ + Entry block: 0 +Block 0: + (original IR block: block0) + (instruction range: 0 .. 13) + Inst 0: movq %rdi, %v0J + Inst 1: movq %rsi, %v1J + Inst 2: movdqa %v6V, %v7V + Inst 3: pxor %v14V, %v14V + Inst 4: pcmpeqd %v7V, %v14V + Inst 5: ptest %v14V, %v14V + Inst 6: setz %v8Jb + Inst 7: movq %v8J, %v9J + Inst 8: andq $1, %v9J + Inst 9: movl %v9Jl, %v10Jl + Inst 10: movq 36(%v0J), %v11J + Inst 11: movq 1(%v11J,%v10J,1), %v13J + Inst 12: ud2 unreachable +}} +Fig. 6. Example (b) - GitHub wasmtime issue #3337 +During the compilation process, the verifier must validate the legalization of IR, WASM instructions, and the +temporary files emitted by AoT compilers. The incorrectness or strictness of validation (A.8) causes errors in the backend +compilation process, accounting for 5.9% bugs in Backend Compilation (A). To make it easier for the users to utilize +the WASM runtimes and learn about the code error, the WASM runtime developers should consider the debugging +information during the backend compilation. The WASM debugging information (A.9) bugs lead to several consequences, +such as failing to provide debugging information or even influencing the compilation of WASM information, accounting +for 6.7% bugs in Backend Compilation (A). +Manuscript submitted to ACM + +10 +Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al. +Highlight 2: Bugs in Backend Compilation account for 38.3% of WASM runtime bugs, covering 10 leaf categories. +A large proportion (23.5%) of these bugs are thrown with incorrect compilation. +4.2 +WASI Robustness +WASI is the fundamental part of a WASM runtime, allowing WASM to run outside the web. WASI supports WASM +with several OS-like features, including files, sockets and the clock. +Each WASM runtime could implement its specific features. Bugs in this part account for 17.4% of our dataset. +WASI allows the WASM binaries to perform file operations, including making a new directory, writing and reading +files, deleting files, and so on. The most common bug related to WASI is File operation error (B.1), which accounts for +27.8%. For example, users failed to rename a file through WASI by applying wasmer in the wasmer issue #2297. Besides, +Input and output stream error (B.4) and Clock Bugs (B.8) are also found in this part, accounting for 13.0% and 5.6% of +bugs in WASI Robustness individually. +Different WASM runtimes implement their own WASI, which may lack the support for some operations (B.3). 9.3% of +bugs in WASI Robustness are triggered due to unsupported operations. +In addition to the basic functionalities, WASI relies on different WASI modules and versions. Frontend compilers +convert the high-level language into WASM binaries which may include a few WASI modules and different versions. +WASM should import different WASI modules (B.2) to support specific functionalities. These imports encounter a few +bugs, such as module not found, incompatible version, etc. These bugs account for 5.6% in WASI Robustness. Due to the +updated versions of runtimes, new WASI versions are continuously provided by the runtimes. Using the incompatible +WASI version (B.6) in WASM runtimes could be unsupported and account for 7.4% of bugs in WASI Robustness. +Moreover, WASI is the bridge between WASM and the OS, which should support different OSes. The diverse in these +OSes can result in Operating system support error (B.5), accounting for 5.6% bugs in this category. +Furthermore, all the WASM runtimes should support WASI to interact with the low-level system, and wasmer also +provides another application binary interface (ABI) to do this. +Bugs in this part are regarded as Other counterpart error (B.7) that account for 9.3% of bugs in this category. +Highlight 3: 17.4% of bugs are related to WASI implementation, covering 9 symptom categories. In particular, +46.3% of the bugs in this category are related to the basic functionalities of WASI (i.e., B.1, B.2, and B.4). +4.3 +Runtime Environment +After compiling WASM to native code, WASM runtimes support the WASM with an execution environment. The +runtime environment supports WASM with module import, trap message tracking, metering computing cost, and other +functionalities. Bugs happen in the Runtime Environment (C) account for 38.6% of bugs in total. +We observe a significant proportion (20%) of bugs about module operation in Runtime Environment, including +Module instantiation bugs (C.1) and Module import error (C.2). Specifically, 13.3% of bugs in this category are related +to WASM module instantiation. WASM module has to be instantiated before execution. These bugs are related to the +instance allocator, module loading, multiple instantiation error, etc. High-level language API makes it easier for WASM +Manuscript submitted to ACM + +Characterizing and Detecting WebAssembly Runtime Bugs +11 +developers to utilize WASM runtimes. The API could import modules from the host environment or other WASM +modules. These bugs are about unknown imports, calling host functions, etc. +Functions in WASM could call the host functions defined in high-level language (C.3) and account for 10% bugs in +Runtime Environment (C). This process contains bugs of parameter passing, finding host functions, etc. +Memory issue (C.4) is a common kind of bugs when executing the WASM binaries, accounting for 15.8% bugs +in this category. These bugs are about memory management, including memory allocation, multi-memory support, +out-of-memory error, memory release and memory growth. +When executing WASM binaries, WASM runtime could encounter bugs in dealing with traps (C.5) and lead to an +abortion, accounting for a total of 7.5% of bugs in this category. These bugs are related to the process of the unreachable +instructions in WASM binaries. Besides, WASM runtimes sometimes do nothing with the errors, and the errors were +not carefully reported to users. The WASM runtime should generate an exception or return a well-defined error (C.10). +In executing native code generated from WASM, many users encounter Thread safety issue (C.7) and Stack issue (C.8). +Thread safety issue (C.7) refers to the thread safety when executing WASM binaries. Stack issue (C.8) refers to the bugs +about the stack, such as match rules for popping when calling WASM functions. These bugs account for a total of 13.3% +of bugs in Runtime Environment (C). +We also observe three bugs about the entry point of a WASM module (C.9). +The functions named “_main”, “_start”, “main”, and “start” are regarded as entry points of a WASM module. A WASM +runtime will call the entry point function by default without setting the function name through the command line or +high-level language API. However, some WASM runtimes require each WASM module to hold an entry point which is +too strict. These bugs are summarized as Entry point error (C.9). +Furthermore, when developers use high-level language API to do some operations of a WASM module, they may +encounter data type conversion problems (C.11), accounting for 3.3% of bugs in the current inner bug category. +Besides, the Runtime Environment can not meet all expectations of functionalities from users. Runtime Environment +lacks support for some features (C.6) that users need, which account for 3.3% of bugs on its own. +Highlight 4: Most (i.e., 38.6%) bugs occur in the Runtime Environment, covering a broad spectrum of symptoms +(i.e., 12 leaf categories). Among them, memory issue is the most common, accounting for 15.8% of bugs in this +category. +4.4 +Auxiliary tools +Besides executing WASM files, WASM runtimes provide users with handy little tools related to WASM, including +validating the format of WASM files, WASM module cache, Wat and WASM files conversion and package manager. As +different WASM runtimes differ significantly in this respect, this category is not classified into leaf categories. This +category accounts for 5.8% of all the classified bugs. For example, in wasmer issue 2028, when passing environment +variables into wasmer run via the –env flag, the program will fail if the environment variable contains an ‘=,’ which +should be allowed. Moreover, when validating the format of WASM binaries, wasmer uses command wasmer validate to +do this. Although setting the parameter –enable-simd, it incorrectly reports an error when validating a WASM module +with SIMD. +Manuscript submitted to ACM + +12 +Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al. +5 +FIX STRATEGIES OF WASM RUNTIME BUGS +To figure out how developers fix various types of bugs, we distill their fix strategies in this section for each inner bug +category. Due to bugs in categories Auxiliary tools are either too specific or irrelevant to WASM runtime themselves, +and they only account for 5.8% of bugs, we do not study the fix strategies for them. We have summarized the general fix +strategies for the remaining three inner symptom categories. As shown in Figure 7, 9 and 10, the X axis shows each leaf +bug category in Figure 4, and the Y axis represents the corresponding fix strategies following with their totally used +frequency under the inner category. We elaborate on the summarized fix strategies of their frequent symptoms and +demonstrate some examples of bugs and corresponding fixes in the real world. +5.1 +Fix Strategies for Backend Compilation +We summarize eight systematic fix strategies for bugs in Backend Compilation and illustrate the distribution of these +strategies on leaf categories in Figure 7. +A.1 +A.2 +A.3 +A.4 +A.5 +A.6 +A.7 +A.8 +A.9 +Bug leaf category +Fix register allocation(18) +Eliminate unreasonable operation(5) +Fix compilation rules(44) +Add compilation functionality for SIMD instructions(3) +Fix data operation(10) +Supplement validation rules(9) +Using the correct infrastructure version(3) +Fix debug information(8) +Others(11) +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Fig. 7. Distribution of fix strategies for Backend Compilation. +Fix compilation rules. 39.6% of bugs in Backend Compilation can be solved by modifying compilation rules in +different backend compilers. Compilation rules are the guide for translating WASM instructions to native code. For +example, wasmtime developers modify the inst.isle file to change the rules of emitting native code. Wasmer fixes +the emitter file for different CPU architectures to emit native code. This fix strategy covers five bug symptoms and is +especially frequently adopted in the Incorrect compilation (A.2) and Compilation failure (A.3) bug categories. For example, +71.4% of Incorrect compilation (A.2) bugs are fixed after modifying the compilation rules in the backend compilers +to support more reasonable translation. As for Compilation failure (A.3 bugs, the backend compilers may encounter +unexpected exceptions and abortion due to the unreasonable compilation rules. Therefore, developers fix these bugs by +changing the compilation rules to meet the actual requirements and support emitting correct native code in most cases. +As shown in Example (c) (Figure 8), a developer reports that the i64˙rotr instruction in WASM is incorrectly compiled +with LLVM in wasmer when given a rotate amount of 0. The corresponding fix is to modify the translation process of this +instruction in the LLVM backend compiler in the wasmer. Compilation rules such as lowering rules guide translating +WASM instructions to native code. +Even worse than emitting incorrect native code is that the backend compilers fail to compile some instructions or +the whole WASM module. For example (wasmtime issue #2347), a user reports that the backend compiler in wasmtime +(V0.20 and main branch) fails to compile a WASM module. The wasmtime developers fix it by modifying the compilation +rules. In detail, they do block manipulation in the wasmtime translation of some table-related instructions and explicitly +call the ensure_inserted_block(). +Manuscript submitted to ACM + +Characterizing and Detecting WebAssembly Runtime Bugs +13 +Fault description: The i64.rotr instruction is mis-compiled with +LLVM backend compiler in wasmer when given a rotate amount of 0. +Fault symptom: Incorrect compilation +Fix strategy: Fix compilation rules +Wat file code: + (func (;0;) (result i64) + i64.const 4 + i64.const 0 + i64.rotr) + (export "_main" (func 0))) +Fig. 8. Example (c) - GitHub wasmer issue #2215 +Fix register allocation. 16.2% of bugs in Backend Compilation, involving three frequent bug categories, can be +fixed by changing the register allocation. As aforementioned in Section 4, while generating native code from WASM +instructions, the backend compilers are expected to allocate registers. Nevertheless, they may use incorrect registers, load +data from an unexpected register, exhaust registers, etc. Various WASM instructions and instruction set architectures +(ISA) make it hard for the backend compilers to allocate suitable registers. +Fix data operation. The fix strategy is used for 9.0% of bugs in Backend Compilation, covering a wide range of bug +categories, including Incorrect compilation (A.2), Compilation failure(A.3), Incomplete operating system support (A.5) and +Unsupported data operation (A.7). The strategies include fixing data alignment, adding support for i8, i16, fixing byte +order, dealing with undefined upper bits, converting data types, returning multi-value data, supporting v128 data type, +etc. +Supplement validation rules. Supplement rules of verifier will tackle the problems in validation, repairing 8.1% of +bugs in Backend Compilation, and mainly fix the Validation error (A.8). For example (wasmer issues #2187), a developer +reports that he can not get the memory page in WASM of 65536, although the user sets memory minimum and maximum +sizes range 0..65536 inclusive by WASM instructions. The verifier is expected to block 65537 and higher. However, +wasmer only works on 65535 and lower. This corresponding fix strategy is to modify its validation rules. +Fix debug information. This fix strategy repairs 7.2% of bugs in Backend Compilation (A), dealing with 87.5% bugs +in WASM debugging information error (A.9).WASM debugging information error (A.9 may lead to the failure of providing +incorrect debug information that misleads the users. By fixing debug information, these bugs could be well settled. +Eliminate unreasonable operation. Some functionalities in backend compilers of WASM runtimes lead to an +unexpected consequence. These functionalities are meaningless and need to be limited. For example (wasmtime +issues #2883), wasmtime users try to use ssub_sat with two I64 values. They use cranelift-object with a triple in the +intermediate presentation (IR) in the wasmtime backend compiler. It is worth mentioning that ssub_sat is a vector +command, but it is used with a scalar. Moreover, the developers eliminate the unreasonable operation, limiting saturating +arithmetic instructions: uadd_sat, sadd_sat, usub_sat, and ssub_sat, and applying them only to vector types. This +kind of fix strategy fixes 4.5% of bugs in Backend Compilation. +Manuscript submitted to ACM + +14 +Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al. +Add compilation functionality for SIMD instructions. Some WASM runtimes do not support the intact func- +tionalities to deal with SIMD instructions. Adding the support will address the bugs. This strategy fixes 3 bugs in +Incorrect compilation (A.2) or Compilation failure (A.3) related to SIMD instructions. +Using the correct version of the infrastructure. Since some backend compilers in WASM runtimes rely on +existing frameworks such as LLVM, adjusting the LLVM’s version can handle some problems. This fix strategy fixes all +the bugs in Incompatible infrastructure version (A.1). +Highlight 5: We identify eight systematic fix strategies for bugs in Backend Compilation. The three most +common strategies are fixing compilation rules, registering allocation, and data operation, resolving 39.6%, +16.2%, and 9.0% of bugs in this category, respectively. +5.2 +Fix Strategies for WASI Robustness +As illustrated in Figure 9, we identify seven frequent fix strategies for bugs in WASI Robustness. +B.1 +B.2 +B.3 +B.4 +B.5 +B.6 +B.7 +B.8 +Bug leaf category +Supplement features(8) +Fix the file operation(16) +Fix intput and output stream error(6) +Fix WASI import(4) +Fix counterpart error(3) +Fix clock error(3) +Fix WASI version(4) +Other(1) +0 +2 +4 +6 +8 +10 +12 +14 +Fig. 9. Distribution of fix strategies for WASI Robustness. +Fix the file operation. This strategy fixes 35.6% bugs in WASI Robustness, including all the bugs in File operation +error (B.1) and half of the bugs in Operating system support error (B.5). For example (wasmer issue #2297), a user reports +that renaming a temporary file through a WASM file fails and prints unable to rename temporary. The developers +fix the implementation of WASI to allow this operation. +Supplement features. WASM runtimes are expected to implement the necessary features. +However, some WASM runtimes do not fulfill this expectation. In such cases, WASM runtime developers need to +implement the expected functionalities in WASI and thus the Unsupported operation (B.3) bugs can be fixed. +Fix input and output stream error. When the required message is not successfully printed, or the necessary +information is not correctly imported into WASM, the Input and output stream error (B.4) occurs. In these cases, +developers need to fix the input and output streams. This fix strategy fixes 13.3% bugs in WASI Robustness. +Fix WASI import & Fix the WASI version. The two strategies are mainly used to tackle the Import error (B.2) and +WASI version error (B.6) which occur when using different modules from WASI. The developers fix WASI import to +use the suitable WASI module to support specific functionalities, and fix WASI version to make the WASM binaries +compatible with the current circumstances. The two strategies all fix 8.9% of bugs in WASI Robustness. +Manuscript submitted to ACM + +Characterizing and Detecting WebAssembly Runtime Bugs +15 +Fix counterpart error. This fix strategy can resolve the Other counterpart error (B.7), accounting for 6.7% of bugs in +WASI Robustness, which could be regarded as the repair method for all the counterparts ABI. +Fix clock error. The fix strategy only focuses on the Clock bugs (B.8) in WASI Robustness and reslove 6.7% bugs. +Highlight 6: We distill seven systematic strategies for bugs in WASI Robustness. The most common one is +fixing the file operation, which resolves 35.6% of bugs in this category. +5.3 +Fix Strategies for Runtime Environment. +We identify eleven frequent fix strategies for bugs in Runtime Environment and Figure 10 shows the distribution. +C.1 C.2 C.3 C.4 C.5 C.6 C.7 C.8 C.9 C.10C.11 +Bug leaf category +Complement unimplemented features(12) +Fix memory allocation(13) +Fix memory leak(14) +Fix thread operation(10) +Fix memory release(3) +Fix trap issue(8) +Fix entry point detecting(3) +Fix parameters and return values for host functions(4) +Fix error message(12) +Repair data operation(9) +Fix stack operation(4) +Others(10) +0 +2 +4 +6 +8 +10 +Fig. 10. Distribution of fix strategies for Runtime Environment. +Fix memory allocation & Fix memory leak & Fix memory release. 29.4% of bugs in Runtime Environment can +be resolved by the three fix strategies for memory management. The three strategies mainly fix the Module instantiation +bugs (C.1) and Memory issues (C.4). After compiling the WASM binaries into native code, WASM runtimes need to +instantiate the current WASM module. Errors about instance allocation and module loading errors could happen. +Besides, other bugs related to multi-memory support, out-of-memory, and memory growth could occur. The developers +mainly use these methods to deal with such cases. +Fix error message. This fix strategy is to modify the error message to make it more reasonable or catch unexpected +failures with an error message. This strategy fixes 11.8% bugs in Runtime Environment. For example, a user reports that +(wasmer issues #868) the WASM runtime gives the wrong message about the error in the execution. In another case, the +user reports that (wasmer issue #830) the WASM runtime should throw an error instead of the occurrence of the panic +when giving a string instead of a digit in the WASM program. Both two cases need to be fixed by modifying the error +message. +Complement unimplemented features. The fix strategy resolves a wide range of bugs in the Runtime Environ- +ment, including Module instantiation bugs (C.1), Module import error (C.2), Calling host functions (C.3), Memory +issue (C.4), Trap error (C.5), Unsupported features (C.6) and Thread safety issue (C.7), accounting for 11.8% of the total +number. +Fix thread operation. 83.3% of the bugs in the Thread safety issue (C.7) are fixed by this resolution. For example +(WAMR issue #1144), a user reports that the wasm_runtime_atomic_wait is not thread-safe and one of subthreads call +wasm_runtime_spawn_exec_env return nullptr. The developers use this strategy to fix atomic wait to be thread-safe by +a lock. +Manuscript submitted to ACM + +16 +Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al. +Fix trap issue. This strategy is entirely used to fix trap error (C.5), accounting for 7.8% of bugs in Runtime Environ- +ment, including fixing trap catch, complementing the missing trap information, and so on. +Repair data operation. 8.9% of bugs in Runtime Environment are fixed by repairing data operation, which mainly +address the data type mismatch between WASM and high-level language API. Besides, it also fixes data alignment. All +the Data type conversion (C.11) bugs are fixed by repairing data operation. +Fix parameters and return values for host functions. As suggested in Figure 10, 25% of bugs in Calling host +functions (C.3) are fixed by modifying the parameters and return values because there is a mismatch or passing error +between the parameters and return values. +Fix entry point detecting. 2.9% of bugs in Runtime Environment are resolved by fixing the detection of the default +entry point. The WASM runtime needs to check the existence of the specially named entry function before execution. +Fix stack operation. All the bugs in Stack issues (C.8) are resolved by this strategy. The WASM runtime is expected +to fix the order of the items when popped from the stack to match the WASM invocation rules. +Highlight 7: The fix strategies for bugs in Runtime Environment are diverse. +WAMR is the runtime with the best thread support of the three. Developers use WAMR to apply threads more often +than wasmer and wasmtime, thus revealing more problems in WAMR of the thread safety bugs. +6 +PATTERN-BASED BUG DETECTOR FOR WASM RUNTIMES +Our aforementioned analysis suggests that most bugs have specific patterns and share similarities across different +WASM runtimes. Thus, in this section, we seek to develop a pattern-based bug detection framework to identify bugs +in WASM runtimes. Our key idea is to construct test cases that can trigger various kinds of bugs we summarized in +Section 4. Specifically, we seek to construct one or more test cases to trigger each of the summarized bug category. Note +that, the constructed test cases are either re-constructed from the bug reports or we create the from scratch according +to the bug patterns. +Next, we will present the details of our bug-triggering test cases for the three major bug categories: Backend +compilation (A), WASI Robustness (B) and Runtime envrionment (C). Note that, D. Auxiliary Tools is an additional part +provided by WASM runtimes, and the tools provided by WASM runtimes in this part vary considerably. Thus, the +category Auxiliary Tools (D) is not considered in this section. Further, for some leaf categories, it is hard for us to +construct test cases, which thus are not covered by our detection framework. In total, our detection framework is +constituted of the following 19 bug detectors. +6.1 +Bug detectors for Backend Compilation +[A.2] Incorrect compilation. SIMD instructions are the newly introduced features for WASM binaries. WASM runtimes +show the bug pattern when compiling specific SIMD instructions, such as using i64x2 and i32x4 to simulate the v128 +type and the optimization for them. To identify such bugs, we select typical WASM binaries with these instructions to +do the detection. +[A.3] Compilation failure. Even worse, WASM runtimes could fail to generate native code for some instructions, +especially those with v128 as parameters or the large WASM modules. We construct the bug detector to detect the +select with v128 as the parameters, a large WASM module such as Ti database with all the backend compilers in WASM +runtimes. +Manuscript submitted to ACM + +Characterizing and Detecting WebAssembly Runtime Bugs +17 +[A.4] Register allocation error. WASM runtimes could load data from an undefined register or get fused with other +instructions when compiling the specific instructions, such as i64x2.extend_low_i32x4_u and f64x2.replace_lane. To +identify such bugs, we select typical WASM binaries with these instructions to do the detection. +We extract the specific OS-related bug-triggering WASM modules for [A.5] Incomplete operating system support and +unsupported data operation such as alignment of SIMD for [A.7] Unsupported data operation. +We use the max value linear memory to detect the [A.8] Validation error and WASM binaries fromm bug reports +which easily trigger debugging information to detect the [A.9] WASM debugging information error. +6.2 +Bug detectors for WASI Robustness +[B.1] File operation error. Different WASM runtimes show similar bug patterns about file operation errors. These +easily bug-triggering file operations include renaming, moving, counting, and mapping. Thus, based on the shared file +operation bug types mentioned in the bug issues, we design the bug detector to detect these bug types. For example, we +test whether WASM runtimes could rename a file or report error information when the file does not exist. Besides, the +detector could test whether WASM runtimes can move a file, count the file number in a directory, or do a mapping +operation. +[B.2] Import error. The most commonly found bug about import is that some WASM runtimes could not support im- +porting multiple WASI versions in one WASM module. Thus, we import both wasi_snapshot_preview1 and wasi_unstable, +the most used WASI versions, in one WASM module to detect this bug. +We extract the WASM binaries, including the unsupported pre-opened directories with / and ./ for WASI, to detect +the bug in [B.3] Unsupported operation. +[B.4] Input and output stream error. To support detecting bugs about standard input and output stream, we use C++ +and compile the C++ program into WASM binaries by emscripten[27] to see the rights and types in __wasi_filestat_t. If +the WASM runtime could not successfully print the expected result, it could be a bug. For example, wasmtime print OS +error when detecting this kind of bug, which the developer confirms. +[B.5] Operating system support error. There are two OS-specific parts of WASI implementation: clocks and polling. +Due to the difference among OSes, the same operation could fail in a specific OS. For example, the QuickJS engine based +on WASM binaries only fails in windows due to the differences between POSIX and windows async APIs. We extract +the QuickJS engine from bug issue to detect this bug. +6.3 +Bug detectors for Runtime environment +[C.1] Module instantiation faults. WASM runtimea provide various high-level language APIs for users to execute +WASM binaries embedded in different applications. When running WASM binaries in a high-level language, the first +step is to load the WASM module from a file or directly load the textual format WASM module in a string variable. And +then, instantiate the WASM module, including validating the WASM module, compiling the WASM binaries with the +appointed backend compiler, allocating the memory allocation for the table, global, etc. However, WASM runtimes +could not support the instantiation for an empty module. Some WASM runtimes will encounter memory leaks when +instantiating multiple WASM modules in a short time. Thus, we use the bug detector to detect whether WASM runtimes +support instantiates an empty WASM module and whether it will lead to memory leaks when instantiating multiple +WASM modules in a short period. +Manuscript submitted to ACM + +18 +Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al. +[C.2] Module import error. We observe that some WASM runtimes omit the step to check the index of imported items, +such as skipping to report the error of ‘index out of bounds‘ errors when import_global_index is greater than imports. +globals length. We extract the related WASM binaries from the raw bug report to detect this bug by the bug detector. +[C.3] Calling host functions. We use the bugs detector for this kind of bug to detect whether WASM runtimes could +support importing a self-defined module, not only from ’env.’ Besides, some WASM runtimes show the bug pattern +about mis-mapping multiple host functions. We use the bug detector to test whether WASM runtimes could successfully +run the functions by importing them in the correct order or if the runtime could inspect the mapping by importing +them in the wrong order and report the error message. +[C.4] Memory issue. By the bug detector, we detect whether WASM runtimes could grow the linear memory dynami- +cally. We extract the WASM module from bug issues and modify it to grow the memory using memory.grow instruction +and using memory.size instruction to check the linear memory size after the growth. +[C.5] Trap error. These bugs are related to the process of the unreachable instructions in WASM modules. By the bug +detector, we use a WASM module with unreachable instructions to test whether WASM runtimes could successfully +break the execution and report the information in the location where unreachable is. +[C.9] Entry point error. WASM runtimes are expected to regard the function labeled with ’start’ or ’_start’ as the entry +point and execute this function default and allow the WASM module without an entry point. WASM runtimes show a +similar bug pattern about the entry point: do not run the entry point function or reject the WASM modules without an +entry point. We construct the WASM module without or with an entry point to detect this kind of bug. +[C.10] Unhandled error. Some WASM runtimes usually encounter panic directly without any operation to avoid it by +reporting the error information. The most commonly found are unhandled errors with unsupported operation and +invalid access to the data section. We extract typical WASM module examples to detect this bug. +6.4 +Reliability of the bug detectors +As a portion of the bug detectors are curated by ourselves based on the code snippets and bug description provided in +the bug reports, we first need to evaluate the reliability of the constructed bug detector. Note that, we already have the +ground truth, i.e., WASM runtimes (with specific version) have some kinds of bugs. Thus, we make effort to reconstruct +the environment (i.e., OS, WASM runtime version, configuration, etc.) to replicate the reported bug for each category. +At last, the bug detector can trigger the reported bugs, which suggest the reliability of our detection framework. +6.5 +Detecting new bugs +As the bug detector we create was constructed based on the knowledge summarized from wasmer, wasmtime, and +WAMR, we further apply it to different WASM runtimes, seeking to identify new bugs. +Experimental Setting. In this experiment, besides the studied WASM runtimes (wasmer, wasmtime and WAMR,), we +further consider two unexplored runtimes, i.e., wasm3 and WASMEdge, to investigate the generalizability of our study. +The bug detector is applied to the following WASM runtimes: wasmer 2.3.0, wasmtime 0.38.0, WAMR 05-18-2022, +wasm3 0.5.0, WASMEdge 0.9.1 on different execution modes (interpreter, AoT, JIT) and across three different operating +systems (macOS 10.15, Ubuntu 20.04, and Windows 11). +Result. As shown in Table 2, we find 53 new bugs, covering all the tested WASM runtimes. By the time of this +submission, 14 of them have been confirmed by the developers, with 6 already been fixed in the main branch based on +our suggestions. +Manuscript submitted to ACM + +Characterizing and Detecting WebAssembly Runtime Bugs +19 +Table 2. The experiment result of the bug detector. We mark a leaf category on a WASM runtime as ✓if it passes all the execution +modes across all the OS platforms. Otherwise, it is marked with the number of detected bugs. +Leaf category +wasmer wasmtime +WAMR +wasm3 +WasmEdge +[A.2] Incorrect compilation +1 +✓ +3 +1 +2 +[A.3] Compilation failure +1 +1 +1 +2 +3 +[A.4] Register allocation error +✓ +✓ +1 +✓ +✓ +[A.5] Incomplete operating system support +✓ +✓ +1 +✓ +1 +[A.7] Unsupported data operation +✓ +✓ +1 +✓ +1 +[A.8] Validation error +✓ +✓ +1 +1 +✓ +[A.9] WASM debugging information error +1 +✓ +1 +✓ +2 +[B.1] File operation error +✓ +✓ +3 +4 +2 +[B.2] Import error +✓ +✓ +✓ +✓ +1 +[B.3] Unsupported operation +✓ +✓ +✓ +1 +1 +[B.4] Input and output stream error +✓ +1 +1 +1 +1 +[B.5] Operating system support error +✓ +✓ +✓ +✓ +✓ +[C.1] Module instantiation faults +✓ +✓ +✓ +✓ +3 +[C.2] Module import error +✓ +✓ +✓ +1 +✓ +[C.3] Calling host functions +✓ +✓ +✓ +✓ +1 +[C.4] Memory issue +✓ +✓ +1 +✓ +1 +[C.5] Trap error +✓ +✓ +✓ +1 +✓ +[C.9] Entry point error +✓ +✓ +✓ +✓ +✓ +[C.10] Unhandled error +✓ +✓ +✓ +2 +1 +Test target: This test case tests whether a Wasm runtime could correctly +compile the rotr instruction in Wasm binaries. +Wat file code: +(module + (func (;0;) (result i64) + i64.const 4 + i64.const 0 + i64.rotr) + (export "_main" (func 0))) +Expected result: +4 +Fig. 11. Case study of [A.3] Incorrect compilation +Case Studies. As shown in Figure 11, it is expected to print the number 4 when testing the rotr instruction for +WASM binaries. However, the actual output in WAMR is a random number. Every time executing, it leads to a different +output. The developers have confirmed it is a bug and fixed it in the main branch, dealing with the parameter 0 separately. +This bug belongs to Incorrect compilation (A.3). +An additional example is shown in Figure 12. It is expected to print the correct directory number 203 when testing +WASI in the runtime. However, WasmEdge prints 147 as a result, which is already confirmed as a new bug by the +Manuscript submitted to ACM + +20 +Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al. +Test target: +This test case tests the read dir functions for WASI. +Wasm file code: +The Wasm file is more than 2000 lines and is limited to show here. +It is provided in the artifact. +Expected result: +203 +Fig. 12. Case study of [B.1] File operation error +developers. Once the number of files is larger than 147, it will be truncated in WasmEdge. And the file renaming belongs +to [B.1] File operation error fails in wasm3, which is also confirmed. +As shown in Figure 13, it is expected to allocate the linear memory to the max value. However, the allocation fails in +WAMR and wasm3. This bug belongs to Validation error (A.8) since the max value is not permitted by the validator. The +developers in WAMR updated the max memory page value in the interpreter, and the developers from wasm3 updated the +max linear memory pages from 32768 to 65535 in the commit fbbacefeaf28e019244bbfa281fc4dea3dbdedc9. Besides, it is +expected to print the v128 data type to support the SIMD instructions. However, it prints nothing in WasmEdge. This bug +belongs to Unsupported data operation (A.8). The developers have confirmed it is a bug and fixed it in the main branch. +Test target: When allocating WASM linear memory with the max value, +memory allocation failed. This is rejected by the validator in the backend +compiler. +Wat file code: +(module +  (memory (;0;) 65536) +) +Expected result: +Successfully allocate WASM linear memory. +Fig. 13. Case study of [A.8] Validation error +Interestingly, we found that the test cases in our detection framework can trigger more than one types of bugs. For +Example, the WASM module in Figure 14 is used to test whether WASM runtimes could successfully compile the div +and copysign instructions for float data ([A.3] Compilation failure). Beyond this, we found that it can identify bugs +that belong to [C.9] Entry point error in wasm3. The WASM module could be successfully compiled in wasm3. However, +‘_start’ is not considered the entry point in wasm3, although other WASM runtimes do. The developer confirmed it and +considered fixing it by checking the return type of ’_start’ and acting according to it. Moreover, the WASM module +in Figure 15 is used to test whether WASM runtimes could successfully compile the select instruction with two v128 +parameters ([A.3] Compilation failure). It detects a bug in wasm3 which should be summarized to [A.7] Unsupported +data operation. Because WasmEdge could compile the module, it does not support printing v128 data. The developer +confirmed that they only support print i32, i64, f32, and f64, which posed a bug, and they would fix it in the future. +Manuscript submitted to ACM + +Characterizing and Detecting WebAssembly Runtime Bugs +21 +Test target: This test case tests whether a Wasm runtime could +successfully compile the div instruction and copysign instruction for +float type and execute the start function by default. +Wat file code: +(module + (type (;0;) (func (result f64))) + (func (;0;) (type 0) (result f64) + f64.const 0x0p+0 (;=0;) + f64.const 0x0p+0 (;=0;) + f64.const 0x0p+0 (;=0;) + f64.div + f64.copysign) + (export "_start" (func 0))) +Expected result: +0 +Fig. 14. Case study of [C.9] Entry point error +Test target: This test case tests whether a Wasm runtime could successfully compile the +select instruction with two v128 parameters and execute the start function by default. +Wat file code: +(module + (func (result v128) + v128.const i32x4 0x00000009 0x00000000 0x00000000 0x00000000 + v128.const i32x4 0x00000007 0x00000000 0x00000000 0x00000000 + i32.const 0 + select) + (export "func1" (func 0))) +Expected result: +79228162514264337593543950336 +Fig. 15. Case study of [A.7] Unsupported data operation +Highlight 8: Our crafted bug detector can effectively detect bugs in real-world WASM runtimes and provide +helpful information to facilitate bug diagnosis and fixing. Interestingly, we found that the test cases in our +detection framework can trigger more than one types of bugs. It further suggests that the summarized bugs +show similar patterns among different WASM runtimes. +Manuscript submitted to ACM + +22 +Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al. +7 +DISCUSSION +7.1 +Implications +Given the rapidly increasing popularity of WASM, our study has timely and practical implications for both WASM +runtime developers and researchers. First, our contribution could help developers dive into and resolve common bugs +in WASM runtimes more efficiently. Our proposed bug detection framework could effectively detect bugs and provide +useful information to facilitate bug fixing. Second, as an emerging research direction, our study sheds lights on future +studies on WASM, including automated testing of WASM runtimes, and developing more advance techniques to fix +bugs, etc. +7.2 +Threats to Validity +First, our analysis pipeline involves a manual analysis of bugs, which might introduce bias to our observations. +To lower the influence of subjective threat, three authors take part in the analysis of bug and fix strategy analysis, +discussing the inconsistent issues until reaching an agreement. Second, our empirical study only targets the most +popular WASM runtimes, while there are many WASM runtimes in the wild, and they may pose other kinds of bugs +that we did not cover in this paper. Nevertheless, we believe the selected three projects are representative enough for us +to characterize common kinds of runtime bugs. Third, it is difficult to ensure that our crafted bug detectors are sound +and can cover all the bug patterns of WASM runtimes. To deal with the problem, we perform a reliability evaluation +of the bug detector and show that they can indeed trigger the known WASM runtime bugs. Nevertheless, for some +bug reports, we cannot reproduce them to trigger the bugs the authors mentioned. We believe that some advanced +techniques like fuzzing and differential testing can be adopted for complementary. +8 +RELATED WORK +WebAssembly Runtime. WASM runtime has been used in a wide spectrum of applications. Ménétrey et al. proposed +WebAssembly trusted runtime, TWINE [50], to execute unmodified, language-independent applications. They leverage +Intel SGX to build the runtime environment. Gadepalli et al. [37] proposed a light-weight WASM runtime, Sledge, for the +edge computing. Wen et al. propose Wasmachine [58], an OS aiming to efficiently and securely execute WebAssembly +applications in IoT and Fog devices with constrained resources. WASM runtime is the fundamental part of various +applications. However, there are no studies about bugs in WASM runtimes. We present the first comprehensive study +on characterizing and detecting bugs in WASM runtimes. +Other WASM Related Studies. WASM is a promising and newly emerged area. There have been studies on several +aspects of WASM, including the WASM execution efficiency [39, 40, 43, 56], WASM compilers [1, 42, 52], WASM binary +security [1, 31, 41, 44, 45, 48], etc. As WASM runtime is one of the fundamental components, our study provides timely +insights to all stakeholders in the ecosystem. +Empirical Study on bugs. There have been a large number of empirical studies focusing on software bugs across a +wide range of applications. For example, Chen et al. [32] studied the faults related to the deployment of DL models on +mobile devices; Zhang et al. [62] conducted an empirical study of TensorFlow program bugs, Lu et al. [46] provided the +comprehensive real-world concurrency bug characteristic study, Di Franco et al. [35] presented the first comprehensive +study of real-world numerical bugs. Recently, the rapid development of WebAssembly has inspired empirical studies +on WebAsssmebly binaries and compilers. For example, Romano et al. [52] conducted an empirical study of bugs in +WebAssembly compilers. They investigated 146 bug reports in Emscripten related to the unique challenges WebAssembly +Manuscript submitted to ACM + +Characterizing and Detecting WebAssembly Runtime Bugs +23 +compilers encounter compared with traditional compilers. Moreover, Hilbig et al. [41] presented a comprehensive +empirical study of 8,461 unique WebAssembly binaries. Following the widely used bug-studying method in the prior +studies, we apply these bug characterization methods to the bugs in a different domain, i.e., WebAssembly runtime. +Furthermore, we construct a bug detector for these summarized bugs based on the characterization and find 14 confirmed +bugs. +9 +CONCLUSION +This paper has presented the first comprehensive study of bugs and the corresponding fix strategies of WASM +runtimes. By manually analyzing 311 real-world bugs extracted from the most popular WASM runtimes, we have +constructed a taxonomy of bug symptoms with 31 categories, and distilled the fix strategies for them. Based on the +knowledge extracted, we further develop a pattern-based bug detection framework to automatically detect bugs across +WASM runtimes. 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Journal of Systems and Software 174 (2021), +110884. +Manuscript submitted to ACM + diff --git a/o9FLT4oBgHgl3EQfhS9Z/content/tmp_files/load_file.txt b/o9FLT4oBgHgl3EQfhS9Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5b52eeeab93bb6cd3972c8f12c4ac02a2d7ce83 --- /dev/null +++ b/o9FLT4oBgHgl3EQfhS9Z/content/tmp_files/load_file.txt @@ -0,0 +1,1393 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf,len=1392 +page_content='Characterizing and Detecting WebAssembly Runtime Bugs YIXUAN ZHANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' China SHANGTONG CAO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Beijing University of Posts and Telecommunications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' China HAOYU WANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Huazhong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' China ZHENPENG CHEN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' UK XIAPU LUO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The Hong Kong Polytechnic University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' China DONGLIANG MU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Huazhong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' China YUN MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' China GANG HUANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' China XUANZHE LIU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' China WebAssembly (abbreviated WASM) has emerged as a promising language of the Web and also been used for a wide spectrum of software applications such as mobile applications and desktop applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These applications, named as WASM applications, commonly run in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Bugs in WASM runtimes are frequently reported by developers and cause the crash of WASM applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, these bugs have not been well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To fill in the knowledge gap, we present a systematic study to characterize and detect bugs in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We first harvest a dataset of 311 real-world bugs from hundreds of related posts on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Based on the collected high-quality bug reports, we distill 31 bug categories of WASM runtimes and summarize their common fix strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Furthermore, we develop a pattern-based bug detection framework to automatically detect bugs in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We apply the detection framework to five popular WASM runtimes and successfully uncover 53 bugs that have never been reported previously, among which 14 have been confirmed and 6 have been fixed by runtime developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' CCS Concepts: • Software and its engineering → Software notations and tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Additional Key Words and Phrases: WebAssembly, WebAssembly runtime ACM Reference Format: Yixuan Zhang, Shangtong Cao, Haoyu Wang, Zhenpeng Chen, Xiapu Luo, Dongliang Mu, Yun Ma, Gang Huang, and Xuanzhe Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Characterizing and Detecting WebAssembly Runtime Bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 1, 1 (January 2023), 25 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION WebAssembly (abbreviated WASM) has quickly emerged as a promising language of the Web in recent years [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM is a binary instruction specification [39, 44, 55] for a stack-based virtual machine and provides developers with Authors’ addresses: Yixuan Zhang, Peking University, Beijing, China, zhangyixuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6290@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Shangtong Cao, Beijing University of Posts and Telecommunications, Beijing, China, shangtongcao@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Haoyu Wang, Huazhong University of Science and Technology, Wuhan, China, haoyuwang@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Zhenpeng Chen, University College London, London, UK, zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='chen@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Xiapu Luo, The Hong Kong Polytechnic University, Hong Kong, China, csxluo@comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='polyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Dongliang Mu, Huazhong University of Science and Technology, Wuhan, China, dzm91@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Yun Ma, Peking University, Beijing, China, mayun@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Gang Huang, Peking University, Beijing, China, hg@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Xuanzhe Liu, Peking University, Beijing, China, liuxuanzhe@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM Manuscript submitted to ACM 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='12102v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='SE] 28 Jan 2023 2 Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' an equivalent textual format [22] for reading, testing, learning instructions, and debugging, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Although WASM was initially proposed for Web applications [53, 56], it is moving fast towards a much wider spectrum of domains, including desktop applications [13, 23], mobile applications [13], IoT [47, 58], blockchain [4, 6, 29], serverless computing [13, 38], and edge computing [37?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' ], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='. To develop these applications (named WASM applications), developers can compile high-level programming languages to WASM binaries or convert the equivalent manually-written textual format to WASM binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM binaries are commonly executed in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' A WASM runtime provides an efficient, memory-safe, sandboxed execution environment for WASM applications [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, a great variety of WASM runtime specific bugs have been reported by developers, inevitably impeding the development of the WASM application ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Despite this, WASM runtime bugs have not been systematically studied by our community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Therefore, there is a general lack of an understanding of these bugs, including their root causes, fix patterns, and how to detect these bugs in emerging WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To fill in the knowledge gap, we present the first comprehensive study on characterizing and detecting bugs in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We focus our study on three most popular and representative WASM runtimes, including wasmtime [20], wasmer [17], and wasm-micro-runtime (WAMR) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We first collect 903 bug-related posts from GitHub, a commonly-used data sources for studying software bugs, and make an effort to identify 311 real-world bugs of these WASM runtimes (see 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Based on the collected bugs, we manually construct a taxonomy of 31 bug categories (see 4), indicating the diversity of WASM runtime bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Moreover, we summarize common fix patterns for each bug category (see 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These empirical results provide a high-level categorization that can serve as a guide for developers to resolve common faults and for researchers to develop tools for detecting and fixing common WASM runtime bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Furthermore, we develop a pattern-based bug detection framework based on the knowledge summarized from the bug taxonomy, to test the presence of bugs in WASM runtimes (see 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To evaluate the generalizability of our study, beyond the three analyzed WASM runtimes, we further consider two emerging WASM runtimes (wasm3 and WASMEdge) for bug detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We have successfully identified 53 previously-unknown bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We report these bugs to the developers of corresponding WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' By the time of this writing, 14 bugs have been confirmed by the developers, and 6 of them have been fixed based on our suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To summarize, this paper makes the following contributions: We conduct the first systematic study of bugs in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We summarize common bug categories and their corresponding fix strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Our results can help understand and characterize bugs in WASM runtimes while shedding lights on future WASM related studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We develop a pattern-based bug detection framework based on the knowledge summarized from bug categories we created to automatically detect bugs in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' By applying the detection framework to real-world WASM runtimes, it shows that our proposed framework can effectively detect bugs and provide useful information to facilitate bug diagnosis and fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We will make the scripts, datasets, and bug detector available to the research community for other researchers to replicate and build upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 2 BACKGROUND 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 WASM binaries WASM is a low-level assembly-like language that is designed for efficient execution and compact representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The WASM binary file is compact like Java class files and is saved with the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='wasm suffix [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The WASM specification Manuscript submitted to ACM Characterizing and Detecting WebAssembly Runtime Bugs 3 defines a conceptual stack virtual machine for most WASM instructions to work on, performing numbers’ pop and push and leaving the result on the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' A pretty-printed textual format (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='wat) [22] is also provided for developers, which can be used to learn the syntax, understand the WASM module, test WASM program, optimize applications, debug code, and write WASM programs by hand, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Developers and users can use the wabt [10] tool to translate WASM binaries to WASM textual format or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' High-level language WASM binaries WASM compiler WASM textual format wabt WASM runtime Operating system WASM binaries Hardware Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The execution process of WASM binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2 Execution of WASM binaries As a binary instruction format, WASM is designed as a portable compilation target for high-level programming languages [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As shown in Figure 1, developers can use WASM compilers to translate high-level language programs to WASM binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' There are dozens of compilers available to compile different source language programs to WASM binaries, such as AssemblyScript, Emscripten, Rustc/WASM-Bindgen, etc [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM can be executed at native speed [39] on a wide range of platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The tool for this critical process is a WASM runtime, an intermediate layer between the WASM binaries and the hardware platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' A WASM runtime should consider the structure, operating system, and other differences between various platforms and provide a relatively secure execution environment for the WASM binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As shown in Figure 1, developers can create applications in high-level languages, compile them into WASM binaries [3, 52], and execute WASM binaries in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Alternatively, they could develop simple WASM programs in the textual format, convert them to WASM binaries through wabt [10], and execute the binaries in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3 WASM Runtime Architecture Based on the implementation of well known WASM runtimes [12, 14, 15, 17, 20, 24?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' ], we have summarized the general architecture of WASM runtimes in Figure 2, which can be divided into six major components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Backend compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtimes support executing WASM binaries in the following modes: interpreter mode, Ahead-of-Time compilation mode (AoT), and Just-in-Time compilation mode (JIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtimes support compiling WASM binaries into native code before executing it locally using AoT compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To speed up the execution efficiency, some WASM runtimes use the just-in-time compilation of hot code through JIT compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' JIT compilers and AoT compilers are considered backend compilers in the WebAssembly workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Some WASM runtimes provide interpretive execution on the WASM binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM 4 Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WebAssembly System Interface High-level language API Auxiliary tools Interpreter Backend compiler System calls Hardware Operating system Runtime environment WASM runtime Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The general architecture of a WASM runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Runtime environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The runtime environment supports allocating memory, performing stack operations, reporting execution error messages, and other features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' High-level language API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The WASM runtimes can be embedded in different high-level languages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', C/C++, Java, Python, Rust, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') as a library to allow users to use WASM in any scenarios with various languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WebAssembly system interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtimes provide WASM applications with WebAssembly system interface (WASI) [25] as a modular system interface [12], focusing on security and portability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASI is the bridge between the sandbox environment and operating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASI is an API that provides access to several OS-like features, including file operation and clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Auxiliary tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtimes also provide handy little tools for the users, such as WASM module cache, WASM textual file format validation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 3 CHARACTERIZATION METHODOLOGY We first perform an empirical study to characterize WASM runtime bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Specifically, we seek to investigate: 1) the taxonomy of bugs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', the reasons leading to the bugs, and 2) the fix strategies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', how to address these bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To approach the answer, we collect and analyze the bug reports posted on Github and Stack Overflow, following the traditional empirical methods in the SE community [32, 36, 41, 51, 52, 54, 57, 60, 62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Figure 3 shows the overview of our study methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 Collection of WASM Runtime Bugs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 Selecting WASM Runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As shown in Table 1, we select three most popular WASM runtimes as target, including wasmer [17], wasmtime [20], and wasm-micro-runtime (WAMR) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We believe they are the most representative WASM runtimes for us to characterize real-world WASM runtime bugs across different implementations, as 1) all of them are mature projects (with over 100,000 LOC) that have gained the thousands of stars on GitHub, 2) they have covered different kinds of execution modes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', Interpreter, JIT and AoT), and 3) they have implemented in different languages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', Rust and C/C++).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM Characterizing and Detecting WebAssembly Runtime Bugs 5 Data Collection form GitHub issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Data Collection form SO questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Refined dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' RQ1 : Faults taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' RQ2 : Fix strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Analyze data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Overview of the methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Statistics of our harvested dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Runtime Stars Commits GitHub issues SO posts Total wasmer 12,026 11,332 403 (179) 41 (0) 444 (179) wasmtime 7,360 9,754 167 (94) 52 (0) 219 (94) WAMR 2,720 686 333 (38) 4 (0) 337 (38) Total 903 (311) 97 (0) 1000 (311) ∗The refined numbers are in the parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2 Data Collection from GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Following previous work [32, 51, 52, 57, 62, 63], we extract issues in the official GitHub repositories of the selected WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' GitHub issues contain many bug information, including source code, detailed reports, and contributors’ discussions [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These characteristics make GitHub issues suitable for analyzing bug root causes and summarizing fix strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For details, we use the GitHub search API [5] to extract the related issues on May 14, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' GitHub issues include various topics, including bug reports, feature requests, documentation updates, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Thus, to highlight the purposes of bugs, we take advantage of the bug issue label to identify related issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We collect issues related to wasmer and wasmtime by filtering labels with “bug”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Due to all the issues from WAMR are not labeled, we extract all the issues from WAMR for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Overall, we obtained 403 issues from wasmer, 167 issues from wasmtime, and 333 from WAMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3 Data Collection from SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Initially, we also considered posts from Stack Overflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Each SO question has at least one tag based on its topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We extract the posts related to the selected WASM runtimes on May 14, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As a result, we obtain 41 posts for wasmer, 52 posts for wasmtime, and 4 posts for WAMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Table 1 shows the collected raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4 Refining the Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We perform manual investigation on the collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' First, we filtered out issues and posts with no definite answers, to ensure the accuracy and certainty of bugs and fix strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Second, we exclude installation/build bugs, documentation bugs, and other issues and posts unrelated to WASM binaries’ execution from the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Finally, as shown in Table 1, the total number of WASM runtime bugs is 311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The scale of this dataset is comparable and more extensive than those used in existing bug-related studies [28, 30, 32, 34, 52, 60, 62] that also require manual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' All the 311 issues are from GitHub because all the reports we collected from SO are not related to the WASM binaries execution in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' It is probably because there are few WASM experts on Manuscript submitted to ACM 6 Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' SO since WASM is an emerging language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Therefore, WASM developers tend to report the bugs they encounter to the official WASM runtime repositories to seek immediate help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2 Labelling Bugs and Fix Strategies The refined 311 bug reports are used for distilling features and fix strategies through manual labelling by two authors and an intercessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 Pilot Labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' First, we randomly sample 50% of the posts (𝑁 = 155) from the selected WASM runtimes for pilot labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The first two authors of the paper jointly participate in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' According to the WASM runtime architecture and the root causes, they create the bug categories and fix strategies by analyzing the GitHub issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2 Reliability Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For reliability analysis, the first two authors independently label the remaining 40% issues based on the taxonomy constructed in the prior stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' In detail, they mark each issue with the posted bug, fix strategy categories, and the issues that cannot be classified into the current taxonomies as a new category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To measure the reliability during the independent labelling, we employ the widely used Cohen’s Kappa indicator (𝜅) for bug and fix strategies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='921 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='915, indicating almost perfect agreement [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The agreement levels demonstrate the reliability of our labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The divergence in the labelling process is then discussed and settled after the labeling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For the newly added categories by the first two authors, we discuss them with the intercessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As a result, we add two new categories to the bug taxonomy and three new categories into the fix strategy taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Furthermore, the first two authors independently label the remaining 10% issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' During this process, no more bug taxonomy or fix strategy is added, indicating saturation of the taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' After finishing the whole labelling stage, the Cohen’s Kappa indicator (𝜅) for bug and fix strategies is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='929 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='925, showing almost perfect agreement [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Additionally, the three authors involved in the taxonomy check the final labeling result together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We will detail the bugs and fix patterns in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 4 TAXONOMY OF WASM RUNTIME BUGS We present the hierarchical taxonomy of WASM runtime bugs according to the WASM runtime architecture (see 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As shown in Figure 4, the taxonomy is organized into three-level categories, including a root category (WASM Runtime Bugs), four inner categories linked to different components in a WASM runtime (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', Backend Compilation), and 31 specific leaf categories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', Register allocation error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The backend compilers (JIT compilers and AoT compilers) of the architecture are summarized into one inner bug category, called Backend Compilation (A), which converts WASM binaries into native code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The bugs in the lowest part in a WASM runtime are called WASI Robustness (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The handy little tools in WASM runtimes are called Auxiliary Tools (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Other bugs that occur while running WASM binaries are classified into Runtime Environment (C), including memory allocation, calling host functions, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' It is worth mentioning that bugs that occur while using high-level language API are either divided into Backend Compilation (A) or Runtime Environment (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM users could use the API to compile WASM binaries and take advantage of functionalities in the Runtime Environment, and it is an interface for users to make good use of a WASM runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Moreover, the interpreter part is merged in a leaf category of Backend Compilation (A), as only WAMR provides an interpreter, and only one bug is found in the interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM Characterizing and Detecting WebAssembly Runtime Bugs 7 WASM Runtime Bugs 311 [B] WASI Robustness 54 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1] Incompatible infrastructure version 3 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2] Incorrect compilation 28 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Compilation failure [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4] Register allocation error 15 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5] Incomplete operating system support 9 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6] Incomplete hardware support 7 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7] Unsupported data operation 14 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8] Validation error 7 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9] WASM debugging informatino error [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='10] Others 8 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1] File operation error 14 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2] Import error 3 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Unsupported operation 5 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4] Input and output stream error 7 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5] operating system support error 4 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6] WASI version error 4 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7] Other counterpart error 5 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8] Clock bugs 3 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9] Others 9 [C] Runtime Environment 120 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1] Module instantiation bugs 16 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2] Module import error 8 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Calling host functions 12 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7] Thread safety issue [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8] Stack issue 4 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9] Entry point error 3 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4] Memory issue 19 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='10] Unhandled error 9 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='11] Data type conversion 4 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6] Unsupported features 4 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='12] Others 18 [D] Auxiliary Tools 18 20 [A] Backend Compilation 119 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5] Trap error 9 12 11 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Taxonomy of bug symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The number in the top right corner indicates the number of bugs for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Highlight 1: We construct a taxonomy of 31 leaf bug symptom categories in WASM runtimes, indicating the root causes and the diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 Backend Compilation As the first stage of executing WASM binaries, backend compilation is used to translate WASM binaries into native code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' In general, backend compilers convert WASM binaries into their intermediate representation (IR), allocate registers and optimize the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Note that backend compilers could convert WASM binaries into the IR proposed in other compilation framework infrastructures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', LLVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The whole process needs to support various OSes and CPU architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We observe 119 bugs in this category, accounting for 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3% of all the classified bugs and covering 10 leaf categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Various backend compilers use their own IR as the intermediate step to translate WASM instructions to native code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' During the process of compiling, compilers could generate incorrect IR or incorrect native code during the translation of WASM binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Besides, optimizing the code could also lead to an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These bugs are summarized as Incorrect compilation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' A compiler may raise an exception when generating native code or even fail to generate the native node (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3), which accounts for 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8% bugs in Backend Compilation (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Moreover, some WASM runtimes rely on the existing compilation framework, such as LLVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Thus, Using the incorrect version of infrastructure (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1) could lead to unexpected results, accounting for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5% bugs in Backend Compilation (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Besides converting WASM instructions into native machine instructions, the backend compilers must allocate registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, they may result in the Incorrect register allocation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4), including incorrectly using special registers, loading data from an unexpected register and exhausting registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These bugs account for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6% of bugs in Backend Compilation (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As shown in Example (a) (Figure 5), the backend compiler in wasmtime gets saved and restored in r15 Manuscript submitted to ACM 8 Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' as a CSR (control and status register), which is expected to be used as a pinned register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The allocation of r15 poses a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As another example in Example b) (Figure 6), wasmtime allocates registers for the given wat file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, during lowering SIMD instructions, the allocation of registers shows a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The movdqa instruction moves out of v6, but v6 is never set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This kind of bugs will cause panic during the execution of WASM binaries, and the execution process cannot be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Most WASM runtimes only support JIT or AoT compilation, while WAMR also provides an interpreter to deal with WASM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' There is only one bug in the interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The interpreter could not correctly pass parameters to submodules, leading to an incorrect result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Moreover, this is summarized into Others (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fault description: As shown in the Assembly code converting from wasmtime IR , r15, the pinned register, gets saved and restored as a CSR, making it impossible to use as a pinned register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fault symptom: Register allocation error Assembly code: 0: 55 push rbp 1: 48 89 e5 mov rbp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' rsp 4: 48 83 ec 10 sub rsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 0x10 8: 4c 89 3c 24 mov qword ptr [rsp],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' r15 c: 4c 8b 0f mov r9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' qword ptr [rdi] f: 49 83 c7 01 add r15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 1 13: 4c 8b 3c 24 mov r15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' qword ptr [rsp] 17: 48 83 c4 10 add rsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 0x10 1b: 48 89 ec mov rsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' rbp 1e: 5d pop rbp 1f: c3 ret Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Example (a) - GitHub wasmtime issue #4170 In the compilation process, WASM runtimes run the WASM file across various operating systems (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' They account for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6% of bugs in the current inner category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The backend compilers encounter problems only caused by specific operating systems and lack consideration for their particular circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' With its architecture and instruction set, WASM runtimes also run the WASM files across different CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Some problems are only present in specific CPUs or specific architecture machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These problems are summarized as Incomplete hardware support (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6) which account for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9% bugs in Backend Compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Further, we also observe that a portion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8%) of bugs in Unsupported data operation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example, in the IR used by the backend compiler in wasmtime, the srem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8 and srem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='16 are not supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Besides, wasmtime converts the WASM binaries into cranelift IR before execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' It lacks the design of supporting the data operation in big endianness machines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', GitHub wasmtime issue #3288).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Executing the clif file on s390 hardware shows wrong results only for i16, i32, and i64 types, while i8 passes these tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The s390 architecture is big-endian, while the data operation in wasmtime was taken from the lower bites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Thus, the data operation of i16, i32, and i64 was not supported in the big-endianness machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This kind of bug could pose different execution results on or execution exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This bug can result in inconsistent results or execution exceptions for the same WASM binaries executed on different machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Besides, WASM specification introduced Single Instruction Multiple Data (SIMD) instructions to improve the execution efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Backend compilers may lack the support for data operation related to SIMD instructions, such as the operation of the v128 data type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM Characterizing and Detecting WebAssembly Runtime Bugs 9 Fault description: Wasmtime convert the wat file into Vcode, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The movdqa instruction moves out of v6, which is never set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fault symptom: Register allocation error Wat file: (module (type (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') (func)) (func (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') (type 0) v128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const i32x4 0x00000000 0x00000000 0x00000000 0x00000000 i64x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='extend_low_i32x4_u v128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const i32x4 0x00000000 0x00000000 0x00000000 0x00000000 i64x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='mul i32x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='all_true i64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='load offset=1 align=1 drop unreachable) (func (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') (type 0) nop) (memory (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') 5613 17832)) Vcode: VCode_ShowWithRRU {{ Entry block: 0 Block 0: (original IR block: block0) (instruction range: 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='. 13) Inst 0: movq %rdi, %v0J Inst 1: movq %rsi, %v1J Inst 2: movdqa %v6V, %v7V Inst 3: pxor %v14V, %v14V Inst 4: pcmpeqd %v7V, %v14V Inst 5: ptest %v14V, %v14V Inst 6: setz %v8Jb Inst 7: movq %v8J, %v9J Inst 8: andq $1, %v9J Inst 9: movl %v9Jl, %v10Jl Inst 10: movq 36(%v0J), %v11J Inst 11: movq 1(%v11J,%v10J,1), %v13J Inst 12: ud2 unreachable }} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Example (b) - GitHub wasmtime issue #3337 During the compilation process, the verifier must validate the legalization of IR, WASM instructions, and the temporary files emitted by AoT compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The incorrectness or strictness of validation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8) causes errors in the backend compilation process, accounting for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9% bugs in Backend Compilation (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To make it easier for the users to utilize the WASM runtimes and learn about the code error, the WASM runtime developers should consider the debugging information during the backend compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The WASM debugging information (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9) bugs lead to several consequences, such as failing to provide debugging information or even influencing the compilation of WASM information, accounting for 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7% bugs in Backend Compilation (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM 10 Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Highlight 2: Bugs in Backend Compilation account for 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3% of WASM runtime bugs, covering 10 leaf categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' A large proportion (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5%) of these bugs are thrown with incorrect compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2 WASI Robustness WASI is the fundamental part of a WASM runtime, allowing WASM to run outside the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASI supports WASM with several OS-like features, including files, sockets and the clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Each WASM runtime could implement its specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Bugs in this part account for 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4% of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASI allows the WASM binaries to perform file operations, including making a new directory, writing and reading files, deleting files, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The most common bug related to WASI is File operation error (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1), which accounts for 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example, users failed to rename a file through WASI by applying wasmer in the wasmer issue #2297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Besides, Input and output stream error (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4) and Clock Bugs (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8) are also found in this part, accounting for 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6% of bugs in WASI Robustness individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Different WASM runtimes implement their own WASI, which may lack the support for some operations (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3% of bugs in WASI Robustness are triggered due to unsupported operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' In addition to the basic functionalities, WASI relies on different WASI modules and versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Frontend compilers convert the high-level language into WASM binaries which may include a few WASI modules and different versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM should import different WASI modules (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2) to support specific functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These imports encounter a few bugs, such as module not found, incompatible version, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These bugs account for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6% in WASI Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Due to the updated versions of runtimes, new WASI versions are continuously provided by the runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Using the incompatible WASI version (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6) in WASM runtimes could be unsupported and account for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4% of bugs in WASI Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Moreover, WASI is the bridge between WASM and the OS, which should support different OSes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The diverse in these OSes can result in Operating system support error (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5), accounting for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6% bugs in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Furthermore, all the WASM runtimes should support WASI to interact with the low-level system, and wasmer also provides another application binary interface (ABI) to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Bugs in this part are regarded as Other counterpart error (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7) that account for 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3% of bugs in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Highlight 3: 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4% of bugs are related to WASI implementation, covering 9 symptom categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' In particular, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3% of the bugs in this category are related to the basic functionalities of WASI (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3 Runtime Environment After compiling WASM to native code, WASM runtimes support the WASM with an execution environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The runtime environment supports WASM with module import, trap message tracking, metering computing cost, and other functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Bugs happen in the Runtime Environment (C) account for 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6% of bugs in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We observe a significant proportion (20%) of bugs about module operation in Runtime Environment, including Module instantiation bugs (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1) and Module import error (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Specifically, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3% of bugs in this category are related to WASM module instantiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM module has to be instantiated before execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These bugs are related to the instance allocator, module loading, multiple instantiation error, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' High-level language API makes it easier for WASM Manuscript submitted to ACM Characterizing and Detecting WebAssembly Runtime Bugs 11 developers to utilize WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The API could import modules from the host environment or other WASM modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These bugs are about unknown imports, calling host functions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Functions in WASM could call the host functions defined in high-level language (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3) and account for 10% bugs in Runtime Environment (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This process contains bugs of parameter passing, finding host functions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Memory issue (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4) is a common kind of bugs when executing the WASM binaries, accounting for 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8% bugs in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These bugs are about memory management, including memory allocation, multi-memory support, out-of-memory error, memory release and memory growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' When executing WASM binaries, WASM runtime could encounter bugs in dealing with traps (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5) and lead to an abortion, accounting for a total of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5% of bugs in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These bugs are related to the process of the unreachable instructions in WASM binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Besides, WASM runtimes sometimes do nothing with the errors, and the errors were not carefully reported to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The WASM runtime should generate an exception or return a well-defined error (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' In executing native code generated from WASM, many users encounter Thread safety issue (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7) and Stack issue (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Thread safety issue (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7) refers to the thread safety when executing WASM binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Stack issue (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8) refers to the bugs about the stack, such as match rules for popping when calling WASM functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These bugs account for a total of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3% of bugs in Runtime Environment (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We also observe three bugs about the entry point of a WASM module (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The functions named “_main”, “_start”, “main”, and “start” are regarded as entry points of a WASM module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' A WASM runtime will call the entry point function by default without setting the function name through the command line or high-level language API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, some WASM runtimes require each WASM module to hold an entry point which is too strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These bugs are summarized as Entry point error (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Furthermore, when developers use high-level language API to do some operations of a WASM module, they may encounter data type conversion problems (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='11), accounting for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3% of bugs in the current inner bug category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Besides, the Runtime Environment can not meet all expectations of functionalities from users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Runtime Environment lacks support for some features (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6) that users need, which account for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3% of bugs on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Highlight 4: Most (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6%) bugs occur in the Runtime Environment, covering a broad spectrum of symptoms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', 12 leaf categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Among them, memory issue is the most common, accounting for 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8% of bugs in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4 Auxiliary tools Besides executing WASM files, WASM runtimes provide users with handy little tools related to WASM, including validating the format of WASM files, WASM module cache, Wat and WASM files conversion and package manager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As different WASM runtimes differ significantly in this respect, this category is not classified into leaf categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This category accounts for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8% of all the classified bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example, in wasmer issue 2028, when passing environment variables into wasmer run via the –env flag, the program will fail if the environment variable contains an ‘=,’ which should be allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Moreover, when validating the format of WASM binaries, wasmer uses command wasmer validate to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Although setting the parameter –enable-simd, it incorrectly reports an error when validating a WASM module with SIMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM 12 Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 5 FIX STRATEGIES OF WASM RUNTIME BUGS To figure out how developers fix various types of bugs, we distill their fix strategies in this section for each inner bug category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Due to bugs in categories Auxiliary tools are either too specific or irrelevant to WASM runtime themselves, and they only account for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8% of bugs, we do not study the fix strategies for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We have summarized the general fix strategies for the remaining three inner symptom categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As shown in Figure 7, 9 and 10, the X axis shows each leaf bug category in Figure 4, and the Y axis represents the corresponding fix strategies following with their totally used frequency under the inner category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We elaborate on the summarized fix strategies of their frequent symptoms and demonstrate some examples of bugs and corresponding fixes in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 Fix Strategies for Backend Compilation We summarize eight systematic fix strategies for bugs in Backend Compilation and illustrate the distribution of these strategies on leaf categories in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9 Bug leaf category Fix register allocation(18) Eliminate unreasonable operation(5) Fix compilation rules(44) Add compilation functionality for SIMD instructions(3) Fix data operation(10) Supplement validation rules(9) Using the correct infrastructure version(3) Fix debug information(8) Others(11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Distribution of fix strategies for Backend Compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix compilation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6% of bugs in Backend Compilation can be solved by modifying compilation rules in different backend compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Compilation rules are the guide for translating WASM instructions to native code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example, wasmtime developers modify the inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='isle file to change the rules of emitting native code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Wasmer fixes the emitter file for different CPU architectures to emit native code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This fix strategy covers five bug symptoms and is especially frequently adopted in the Incorrect compilation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2) and Compilation failure (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3) bug categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example, 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4% of Incorrect compilation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2) bugs are fixed after modifying the compilation rules in the backend compilers to support more reasonable translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As for Compilation failure (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3 bugs, the backend compilers may encounter unexpected exceptions and abortion due to the unreasonable compilation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Therefore, developers fix these bugs by changing the compilation rules to meet the actual requirements and support emitting correct native code in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As shown in Example (c) (Figure 8), a developer reports that the i64˙rotr instruction in WASM is incorrectly compiled with LLVM in wasmer when given a rotate amount of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The corresponding fix is to modify the translation process of this instruction in the LLVM backend compiler in the wasmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Compilation rules such as lowering rules guide translating WASM instructions to native code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Even worse than emitting incorrect native code is that the backend compilers fail to compile some instructions or the whole WASM module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example (wasmtime issue #2347), a user reports that the backend compiler in wasmtime (V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='20 and main branch) fails to compile a WASM module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The wasmtime developers fix it by modifying the compilation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' In detail, they do block manipulation in the wasmtime translation of some table-related instructions and explicitly call the ensure_inserted_block().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM Characterizing and Detecting WebAssembly Runtime Bugs 13 Fault description: The i64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='rotr instruction is mis-compiled with LLVM backend compiler in wasmer when given a rotate amount of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fault symptom: Incorrect compilation Fix strategy: Fix compilation rules Wat file code: (func (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') (result i64) i64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const 4 i64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const 0 i64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='rotr) (export "_main" (func 0))) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Example (c) - GitHub wasmer issue #2215 Fix register allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2% of bugs in Backend Compilation, involving three frequent bug categories, can be fixed by changing the register allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As aforementioned in Section 4, while generating native code from WASM instructions, the backend compilers are expected to allocate registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Nevertheless, they may use incorrect registers, load data from an unexpected register, exhaust registers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Various WASM instructions and instruction set architectures (ISA) make it hard for the backend compilers to allocate suitable registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix data operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The fix strategy is used for 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0% of bugs in Backend Compilation, covering a wide range of bug categories, including Incorrect compilation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2), Compilation failure(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3), Incomplete operating system support (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5) and Unsupported data operation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The strategies include fixing data alignment, adding support for i8, i16, fixing byte order, dealing with undefined upper bits, converting data types, returning multi-value data, supporting v128 data type, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Supplement validation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Supplement rules of verifier will tackle the problems in validation, repairing 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1% of bugs in Backend Compilation, and mainly fix the Validation error (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example (wasmer issues #2187), a developer reports that he can not get the memory page in WASM of 65536, although the user sets memory minimum and maximum sizes range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='.65536 inclusive by WASM instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The verifier is expected to block 65537 and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, wasmer only works on 65535 and lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This corresponding fix strategy is to modify its validation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix debug information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This fix strategy repairs 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2% of bugs in Backend Compilation (A), dealing with 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5% bugs in WASM debugging information error (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='WASM debugging information error (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9 may lead to the failure of providing incorrect debug information that misleads the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' By fixing debug information, these bugs could be well settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Eliminate unreasonable operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Some functionalities in backend compilers of WASM runtimes lead to an unexpected consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These functionalities are meaningless and need to be limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example (wasmtime issues #2883), wasmtime users try to use ssub_sat with two I64 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' They use cranelift-object with a triple in the intermediate presentation (IR) in the wasmtime backend compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' It is worth mentioning that ssub_sat is a vector command, but it is used with a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Moreover, the developers eliminate the unreasonable operation, limiting saturating arithmetic instructions: uadd_sat, sadd_sat, usub_sat, and ssub_sat, and applying them only to vector types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This kind of fix strategy fixes 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5% of bugs in Backend Compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM 14 Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Add compilation functionality for SIMD instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Some WASM runtimes do not support the intact func- tionalities to deal with SIMD instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Adding the support will address the bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This strategy fixes 3 bugs in Incorrect compilation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2) or Compilation failure (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3) related to SIMD instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Using the correct version of the infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Since some backend compilers in WASM runtimes rely on existing frameworks such as LLVM, adjusting the LLVM’s version can handle some problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This fix strategy fixes all the bugs in Incompatible infrastructure version (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Highlight 5: We identify eight systematic fix strategies for bugs in Backend Compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The three most common strategies are fixing compilation rules, registering allocation, and data operation, resolving 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6%, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2%, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0% of bugs in this category, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2 Fix Strategies for WASI Robustness As illustrated in Figure 9, we identify seven frequent fix strategies for bugs in WASI Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8 Bug leaf category Supplement features(8) Fix the file operation(16) Fix intput and output stream error(6) Fix WASI import(4) Fix counterpart error(3) Fix clock error(3) Fix WASI version(4) Other(1) 0 2 4 6 8 10 12 14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Distribution of fix strategies for WASI Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix the file operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This strategy fixes 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6% bugs in WASI Robustness, including all the bugs in File operation error (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1) and half of the bugs in Operating system support error (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example (wasmer issue #2297), a user reports that renaming a temporary file through a WASM file fails and prints unable to rename temporary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The developers fix the implementation of WASI to allow this operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Supplement features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtimes are expected to implement the necessary features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, some WASM runtimes do not fulfill this expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' In such cases, WASM runtime developers need to implement the expected functionalities in WASI and thus the Unsupported operation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3) bugs can be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix input and output stream error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' When the required message is not successfully printed, or the necessary information is not correctly imported into WASM, the Input and output stream error (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' In these cases, developers need to fix the input and output streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This fix strategy fixes 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3% bugs in WASI Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix WASI import & Fix the WASI version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The two strategies are mainly used to tackle the Import error (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2) and WASI version error (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6) which occur when using different modules from WASI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The developers fix WASI import to use the suitable WASI module to support specific functionalities, and fix WASI version to make the WASM binaries compatible with the current circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The two strategies all fix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9% of bugs in WASI Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM Characterizing and Detecting WebAssembly Runtime Bugs 15 Fix counterpart error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This fix strategy can resolve the Other counterpart error (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7), accounting for 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7% of bugs in WASI Robustness, which could be regarded as the repair method for all the counterparts ABI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix clock error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The fix strategy only focuses on the Clock bugs (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8) in WASI Robustness and reslove 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7% bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Highlight 6: We distill seven systematic strategies for bugs in WASI Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The most common one is fixing the file operation, which resolves 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6% of bugs in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3 Fix Strategies for Runtime Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We identify eleven frequent fix strategies for bugs in Runtime Environment and Figure 10 shows the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='10C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='11 Bug leaf category Complement unimplemented features(12) Fix memory allocation(13) Fix memory leak(14) Fix thread operation(10) Fix memory release(3) Fix trap issue(8) Fix entry point detecting(3) Fix parameters and return values for host functions(4) Fix error message(12) Repair data operation(9) Fix stack operation(4) Others(10) 0 2 4 6 8 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Distribution of fix strategies for Runtime Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix memory allocation & Fix memory leak & Fix memory release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4% of bugs in Runtime Environment can be resolved by the three fix strategies for memory management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The three strategies mainly fix the Module instantiation bugs (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1) and Memory issues (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' After compiling the WASM binaries into native code, WASM runtimes need to instantiate the current WASM module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Errors about instance allocation and module loading errors could happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Besides, other bugs related to multi-memory support, out-of-memory, and memory growth could occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The developers mainly use these methods to deal with such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix error message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This fix strategy is to modify the error message to make it more reasonable or catch unexpected failures with an error message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This strategy fixes 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8% bugs in Runtime Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example, a user reports that (wasmer issues #868) the WASM runtime gives the wrong message about the error in the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' In another case, the user reports that (wasmer issue #830) the WASM runtime should throw an error instead of the occurrence of the panic when giving a string instead of a digit in the WASM program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Both two cases need to be fixed by modifying the error message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Complement unimplemented features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The fix strategy resolves a wide range of bugs in the Runtime Environ- ment, including Module instantiation bugs (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1), Module import error (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2), Calling host functions (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3), Memory issue (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4), Trap error (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5), Unsupported features (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='6) and Thread safety issue (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7), accounting for 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8% of the total number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix thread operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3% of the bugs in the Thread safety issue (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7) are fixed by this resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example (WAMR issue #1144), a user reports that the wasm_runtime_atomic_wait is not thread-safe and one of subthreads call wasm_runtime_spawn_exec_env return nullptr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The developers use this strategy to fix atomic wait to be thread-safe by a lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM 16 Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix trap issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This strategy is entirely used to fix trap error (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5), accounting for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8% of bugs in Runtime Environ- ment, including fixing trap catch, complementing the missing trap information, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Repair data operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9% of bugs in Runtime Environment are fixed by repairing data operation, which mainly address the data type mismatch between WASM and high-level language API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Besides, it also fixes data alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' All the Data type conversion (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='11) bugs are fixed by repairing data operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix parameters and return values for host functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As suggested in Figure 10, 25% of bugs in Calling host functions (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3) are fixed by modifying the parameters and return values because there is a mismatch or passing error between the parameters and return values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix entry point detecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9% of bugs in Runtime Environment are resolved by fixing the detection of the default entry point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The WASM runtime needs to check the existence of the specially named entry function before execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fix stack operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' All the bugs in Stack issues (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8) are resolved by this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The WASM runtime is expected to fix the order of the items when popped from the stack to match the WASM invocation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Highlight 7: The fix strategies for bugs in Runtime Environment are diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WAMR is the runtime with the best thread support of the three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Developers use WAMR to apply threads more often than wasmer and wasmtime, thus revealing more problems in WAMR of the thread safety bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 6 PATTERN-BASED BUG DETECTOR FOR WASM RUNTIMES Our aforementioned analysis suggests that most bugs have specific patterns and share similarities across different WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Thus, in this section, we seek to develop a pattern-based bug detection framework to identify bugs in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Our key idea is to construct test cases that can trigger various kinds of bugs we summarized in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Specifically, we seek to construct one or more test cases to trigger each of the summarized bug category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Note that, the constructed test cases are either re-constructed from the bug reports or we create the from scratch according to the bug patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Next, we will present the details of our bug-triggering test cases for the three major bug categories: Backend compilation (A), WASI Robustness (B) and Runtime envrionment (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Note that, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Auxiliary Tools is an additional part provided by WASM runtimes, and the tools provided by WASM runtimes in this part vary considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Thus, the category Auxiliary Tools (D) is not considered in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Further, for some leaf categories, it is hard for us to construct test cases, which thus are not covered by our detection framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' In total, our detection framework is constituted of the following 19 bug detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 Bug detectors for Backend Compilation [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2] Incorrect compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' SIMD instructions are the newly introduced features for WASM binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtimes show the bug pattern when compiling specific SIMD instructions, such as using i64x2 and i32x4 to simulate the v128 type and the optimization for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To identify such bugs, we select typical WASM binaries with these instructions to do the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Compilation failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Even worse, WASM runtimes could fail to generate native code for some instructions, especially those with v128 as parameters or the large WASM modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We construct the bug detector to detect the select with v128 as the parameters, a large WASM module such as Ti database with all the backend compilers in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM Characterizing and Detecting WebAssembly Runtime Bugs 17 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4] Register allocation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtimes could load data from an undefined register or get fused with other instructions when compiling the specific instructions, such as i64x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='extend_low_i32x4_u and f64x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='replace_lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To identify such bugs, we select typical WASM binaries with these instructions to do the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We extract the specific OS-related bug-triggering WASM modules for [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5] Incomplete operating system support and unsupported data operation such as alignment of SIMD for [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7] Unsupported data operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We use the max value linear memory to detect the [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8] Validation error and WASM binaries fromm bug reports which easily trigger debugging information to detect the [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9] WASM debugging information error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2 Bug detectors for WASI Robustness [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1] File operation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Different WASM runtimes show similar bug patterns about file operation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These easily bug-triggering file operations include renaming, moving, counting, and mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Thus, based on the shared file operation bug types mentioned in the bug issues, we design the bug detector to detect these bug types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example, we test whether WASM runtimes could rename a file or report error information when the file does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Besides, the detector could test whether WASM runtimes can move a file, count the file number in a directory, or do a mapping operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2] Import error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The most commonly found bug about import is that some WASM runtimes could not support im- porting multiple WASI versions in one WASM module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Thus, we import both wasi_snapshot_preview1 and wasi_unstable, the most used WASI versions, in one WASM module to detect this bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We extract the WASM binaries, including the unsupported pre-opened directories with / and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='/ for WASI, to detect the bug in [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Unsupported operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4] Input and output stream error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To support detecting bugs about standard input and output stream, we use C++ and compile the C++ program into WASM binaries by emscripten[27] to see the rights and types in __wasi_filestat_t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' If the WASM runtime could not successfully print the expected result, it could be a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example, wasmtime print OS error when detecting this kind of bug, which the developer confirms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5] Operating system support error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' There are two OS-specific parts of WASI implementation: clocks and polling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Due to the difference among OSes, the same operation could fail in a specific OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example, the QuickJS engine based on WASM binaries only fails in windows due to the differences between POSIX and windows async APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We extract the QuickJS engine from bug issue to detect this bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3 Bug detectors for Runtime environment [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1] Module instantiation faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtimea provide various high-level language APIs for users to execute WASM binaries embedded in different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' When running WASM binaries in a high-level language, the first step is to load the WASM module from a file or directly load the textual format WASM module in a string variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' And then, instantiate the WASM module, including validating the WASM module, compiling the WASM binaries with the appointed backend compiler, allocating the memory allocation for the table, global, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, WASM runtimes could not support the instantiation for an empty module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Some WASM runtimes will encounter memory leaks when instantiating multiple WASM modules in a short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Thus, we use the bug detector to detect whether WASM runtimes support instantiates an empty WASM module and whether it will lead to memory leaks when instantiating multiple WASM modules in a short period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM 18 Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2] Module import error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We observe that some WASM runtimes omit the step to check the index of imported items, such as skipping to report the error of ‘index out of bounds‘ errors when import_global_index is greater than imports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' globals length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We extract the related WASM binaries from the raw bug report to detect this bug by the bug detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Calling host functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We use the bugs detector for this kind of bug to detect whether WASM runtimes could support importing a self-defined module, not only from ’env.’ Besides, some WASM runtimes show the bug pattern about mis-mapping multiple host functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We use the bug detector to test whether WASM runtimes could successfully run the functions by importing them in the correct order or if the runtime could inspect the mapping by importing them in the wrong order and report the error message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4] Memory issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' By the bug detector, we detect whether WASM runtimes could grow the linear memory dynami- cally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We extract the WASM module from bug issues and modify it to grow the memory using memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='grow instruction and using memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='size instruction to check the linear memory size after the growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5] Trap error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' These bugs are related to the process of the unreachable instructions in WASM modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' By the bug detector, we use a WASM module with unreachable instructions to test whether WASM runtimes could successfully break the execution and report the information in the location where unreachable is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9] Entry point error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtimes are expected to regard the function labeled with ’start’ or ’_start’ as the entry point and execute this function default and allow the WASM module without an entry point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtimes show a similar bug pattern about the entry point: do not run the entry point function or reject the WASM modules without an entry point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We construct the WASM module without or with an entry point to detect this kind of bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='10] Unhandled error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Some WASM runtimes usually encounter panic directly without any operation to avoid it by reporting the error information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The most commonly found are unhandled errors with unsupported operation and invalid access to the data section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We extract typical WASM module examples to detect this bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4 Reliability of the bug detectors As a portion of the bug detectors are curated by ourselves based on the code snippets and bug description provided in the bug reports, we first need to evaluate the reliability of the constructed bug detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Note that, we already have the ground truth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', WASM runtimes (with specific version) have some kinds of bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Thus, we make effort to reconstruct the environment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', OS, WASM runtime version, configuration, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') to replicate the reported bug for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' At last, the bug detector can trigger the reported bugs, which suggest the reliability of our detection framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5 Detecting new bugs As the bug detector we create was constructed based on the knowledge summarized from wasmer, wasmtime, and WAMR, we further apply it to different WASM runtimes, seeking to identify new bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Experimental Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' In this experiment, besides the studied WASM runtimes (wasmer, wasmtime and WAMR,), we further consider two unexplored runtimes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', wasm3 and WASMEdge, to investigate the generalizability of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The bug detector is applied to the following WASM runtimes: wasmer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0, wasmtime 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0, WAMR 05-18-2022, wasm3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0, WASMEdge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 on different execution modes (interpreter, AoT, JIT) and across three different operating systems (macOS 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='15, Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='04, and Windows 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As shown in Table 2, we find 53 new bugs, covering all the tested WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' By the time of this submission, 14 of them have been confirmed by the developers, with 6 already been fixed in the main branch based on our suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM Characterizing and Detecting WebAssembly Runtime Bugs 19 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The experiment result of the bug detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We mark a leaf category on a WASM runtime as ✓if it passes all the execution modes across all the OS platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Otherwise, it is marked with the number of detected bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Leaf category wasmer wasmtime WAMR wasm3 WasmEdge [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2] Incorrect compilation 1 ✓ 3 1 2 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Compilation failure 1 1 1 2 3 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4] Register allocation error ✓ ✓ 1 ✓ ✓ [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5] Incomplete operating system support ✓ ✓ 1 ✓ 1 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7] Unsupported data operation ✓ ✓ 1 ✓ 1 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8] Validation error ✓ ✓ 1 1 ✓ [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9] WASM debugging information error 1 ✓ 1 ✓ 2 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1] File operation error ✓ ✓ 3 4 2 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2] Import error ✓ ✓ ✓ ✓ 1 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Unsupported operation ✓ ✓ ✓ 1 1 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4] Input and output stream error ✓ 1 1 1 1 [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5] Operating system support error ✓ ✓ ✓ ✓ ✓ [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1] Module instantiation faults ✓ ✓ ✓ ✓ 3 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2] Module import error ✓ ✓ ✓ 1 ✓ [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Calling host functions ✓ ✓ ✓ ✓ 1 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='4] Memory issue ✓ ✓ 1 ✓ 1 [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='5] Trap error ✓ ✓ ✓ 1 ✓ [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9] Entry point error ✓ ✓ ✓ ✓ ✓ [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='10] Unhandled error ✓ ✓ ✓ 2 1 Test target: This test case tests whether a Wasm runtime could correctly compile the rotr instruction in Wasm binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Wat file code: (module (func (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') (result i64) i64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const 4 i64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const 0 i64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='rotr) (export "_main" (func 0))) Expected result: 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Case study of [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Incorrect compilation Case Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As shown in Figure 11, it is expected to print the number 4 when testing the rotr instruction for WASM binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, the actual output in WAMR is a random number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Every time executing, it leads to a different output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The developers have confirmed it is a bug and fixed it in the main branch, dealing with the parameter 0 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This bug belongs to Incorrect compilation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' An additional example is shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' It is expected to print the correct directory number 203 when testing WASI in the runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, WasmEdge prints 147 as a result, which is already confirmed as a new bug by the Manuscript submitted to ACM 20 Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Test target: This test case tests the read dir functions for WASI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Wasm file code: The Wasm file is more than 2000 lines and is limited to show here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' It is provided in the artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Expected result: 203 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Case study of [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1] File operation error developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Once the number of files is larger than 147, it will be truncated in WasmEdge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' And the file renaming belongs to [B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1] File operation error fails in wasm3, which is also confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As shown in Figure 13, it is expected to allocate the linear memory to the max value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, the allocation fails in WAMR and wasm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This bug belongs to Validation error (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8) since the max value is not permitted by the validator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The developers in WAMR updated the max memory page value in the interpreter, and the developers from wasm3 updated the max linear memory pages from 32768 to 65535 in the commit fbbacefeaf28e019244bbfa281fc4dea3dbdedc9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Besides, it is expected to print the v128 data type to support the SIMD instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, it prints nothing in WasmEdge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This bug belongs to Unsupported data operation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The developers have confirmed it is a bug and fixed it in the main branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Test target: When allocating WASM linear memory with the max value, memory allocation failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' This is rejected by the validator in the backend compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Wat file code: (module (memory (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') 65536) ) Expected result: Successfully allocate WASM linear memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Case study of [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='8] Validation error Interestingly, we found that the test cases in our detection framework can trigger more than one types of bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For Example, the WASM module in Figure 14 is used to test whether WASM runtimes could successfully compile the div and copysign instructions for float data ([A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Compilation failure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Beyond this, we found that it can identify bugs that belong to [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9] Entry point error in wasm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The WASM module could be successfully compiled in wasm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, ‘_start’ is not considered the entry point in wasm3, although other WASM runtimes do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The developer confirmed it and considered fixing it by checking the return type of ’_start’ and acting according to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Moreover, the WASM module in Figure 15 is used to test whether WASM runtimes could successfully compile the select instruction with two v128 parameters ([A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='3] Compilation failure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' It detects a bug in wasm3 which should be summarized to [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7] Unsupported data operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Because WasmEdge could compile the module, it does not support printing v128 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' The developer confirmed that they only support print i32, i64, f32, and f64, which posed a bug, and they would fix it in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM Characterizing and Detecting WebAssembly Runtime Bugs 21 Test target: This test case tests whether a Wasm runtime could successfully compile the div instruction and copysign instruction for float type and execute the start function by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Wat file code: (module (type (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') (func (result f64))) (func (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') (type 0) (result f64) f64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const 0x0p+0 (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') f64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const 0x0p+0 (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') f64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const 0x0p+0 (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=') f64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='div f64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='copysign) (export "_start" (func 0))) Expected result: 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Case study of [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='9] Entry point error Test target: This test case tests whether a Wasm runtime could successfully compile the select instruction with two v128 parameters and execute the start function by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Wat file code: (module (func (result v128) v128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const i32x4 0x00000009 0x00000000 0x00000000 0x00000000 v128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const i32x4 0x00000007 0x00000000 0x00000000 0x00000000 i32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='const 0 select) (export "func1" (func 0))) Expected result: 79228162514264337593543950336 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Case study of [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='7] Unsupported data operation Highlight 8: Our crafted bug detector can effectively detect bugs in real-world WASM runtimes and provide helpful information to facilitate bug diagnosis and fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Interestingly, we found that the test cases in our detection framework can trigger more than one types of bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' It further suggests that the summarized bugs show similar patterns among different WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Manuscript submitted to ACM 22 Yixuan Zhang, Shangtong Cao and Haoyu Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 7 DISCUSSION 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='1 Implications Given the rapidly increasing popularity of WASM, our study has timely and practical implications for both WASM runtime developers and researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' First, our contribution could help developers dive into and resolve common bugs in WASM runtimes more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Our proposed bug detection framework could effectively detect bugs and provide useful information to facilitate bug fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Second, as an emerging research direction, our study sheds lights on future studies on WASM, including automated testing of WASM runtimes, and developing more advance techniques to fix bugs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='2 Threats to Validity First, our analysis pipeline involves a manual analysis of bugs, which might introduce bias to our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To lower the influence of subjective threat, three authors take part in the analysis of bug and fix strategy analysis, discussing the inconsistent issues until reaching an agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Second, our empirical study only targets the most popular WASM runtimes, while there are many WASM runtimes in the wild, and they may pose other kinds of bugs that we did not cover in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Nevertheless, we believe the selected three projects are representative enough for us to characterize common kinds of runtime bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Third, it is difficult to ensure that our crafted bug detectors are sound and can cover all the bug patterns of WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' To deal with the problem, we perform a reliability evaluation of the bug detector and show that they can indeed trigger the known WASM runtime bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Nevertheless, for some bug reports, we cannot reproduce them to trigger the bugs the authors mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We believe that some advanced techniques like fuzzing and differential testing can be adopted for complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 8 RELATED WORK WebAssembly Runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtime has been used in a wide spectrum of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Ménétrey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' proposed WebAssembly trusted runtime, TWINE [50], to execute unmodified, language-independent applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' They leverage Intel SGX to build the runtime environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Gadepalli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [37] proposed a light-weight WASM runtime, Sledge, for the edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' propose Wasmachine [58], an OS aiming to efficiently and securely execute WebAssembly applications in IoT and Fog devices with constrained resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM runtime is the fundamental part of various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' However, there are no studies about bugs in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' We present the first comprehensive study on characterizing and detecting bugs in WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Other WASM Related Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' WASM is a promising and newly emerged area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' There have been studies on several aspects of WASM, including the WASM execution efficiency [39, 40, 43, 56], WASM compilers [1, 42, 52], WASM binary security [1, 31, 41, 44, 45, 48], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' As WASM runtime is one of the fundamental components, our study provides timely insights to all stakeholders in the ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Empirical Study on bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' There have been a large number of empirical studies focusing on software bugs across a wide range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [32] studied the faults related to the deployment of DL models on mobile devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [62] conducted an empirical study of TensorFlow program bugs, Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [46] provided the comprehensive real-world concurrency bug characteristic study, Di Franco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [35] presented the first comprehensive study of real-world numerical bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Recently, the rapid development of WebAssembly has inspired empirical studies on WebAsssmebly binaries and compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' For example, Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [52] conducted an empirical study of bugs in WebAssembly compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' They investigated 146 bug reports in Emscripten related to the unique challenges WebAssembly Manuscript submitted to ACM Characterizing and Detecting WebAssembly Runtime Bugs 23 compilers encounter compared with traditional compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Moreover, Hilbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' [41] presented a comprehensive empirical study of 8,461 unique WebAssembly binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Following the widely used bug-studying method in the prior studies, we apply these bug characterization methods to the bugs in a different domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=', WebAssembly runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Furthermore, we construct a bug detector for these summarized bugs based on the characterization and find 14 confirmed bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' 9 CONCLUSION This paper has presented the first comprehensive study of bugs and the corresponding fix strategies of WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' By manually analyzing 311 real-world bugs extracted from the most popular WASM runtimes, we have constructed a taxonomy of bug symptoms with 31 categories, and distilled the fix strategies for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' Based on the knowledge extracted, we further develop a pattern-based bug detection framework to automatically detect bugs across WASM runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' By the time of this study, we have identified 53 bugs that have never been reported in the community, and 14 of them have been confirmed by the official developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' REFERENCES [1] Reverse engineering webassembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FLT4oBgHgl3EQfhS9Z/content/2301.12102v1.pdf'} +page_content='pnfsoftware.' metadata={'source': 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0000000000000000000000000000000000000000..8c282bcb84bd384e63db1a1aec420ed7254cf48d --- /dev/null +++ b/oNE3T4oBgHgl3EQfjQoa/content/tmp_files/2301.04586v1.pdf.txt @@ -0,0 +1,2905 @@ +MNRAS 000, 1–15 (2022) +Preprint 12 January 2023 +Compiled using MNRAS LATEX style file v3.0 +AI-assisted reconstruction of cosmic velocity field from redshift-space +spatial distribution of halos +Ziyong Wu1,2, Liang Xiao4,5★, Xu Xiao4, Jie Wang7,8, Xi Kang3,2, Yang Wang6, Xin Wang4,5, +Le Zhang4,5,6†, Xiao-Dong Li4,5‡ +1School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei 230026, China +2Purple Mountain Observatory, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210033, China, +3Institute for Astronomy, The school of Physics, Zhejiang University, Hangzhou 310037, China +5CSST Science Center for the Guangdong–Hong Kong–Macau Greater Bay Area, SYSU, Zhuhai 519082, P. R. China +6Peng Cheng Laboratory, No. 2, Xingke 1st Street, Shenzhen 518000, P. R. China +7National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Beijing 100101, China +8University of Chinese Academy of Sciences, Beijing 100049, China +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +The peculiar velocities of dark matter halos are crucial to study many issues in cosmology and galaxy evolution. In this +study, by using the state-of-the-art deep learning technique, a UNet-based neural network, we propose to reconstruct the peculiar +velocity field from the redshift-space distribution of dark matter halos. Through a point-to-point comparison and examination of +various statistical properties, we demonstrate that, the reconstructed velocity field is in good agreement with the ground truth. The +power spectra of various velocity field components, including velocity magnitude, divergence and vorticity, can be successfully +recovered when 𝑘 ≲ 1.1 ℎ/Mpc (the Nyquist frequency of the simulations) at about 80% accuracy. This approach is very +promising and presents an alternative method to correct the redshift-space distortions using the measured 3D spatial information +of halos. Additionally, for the reconstruction of the momentum field of halos, UNet achieves similar good results. Hence the +applications in various aspects of cosmology are very broad, such as correcting redshift errors and improving measurements in +the structure of the cosmic web, the kinetic Sunyaev-Zel’dovich effect, BAO reconstruction, etc. +Key words: methods: data analysis, numerical; cosmology: large-scale structure of Universe, theory +CONTENTS +1 +Introduction +2 +Method +2.1 +Dataset +2.2 +Input Preprocessing +2.3 +Neural Network Model +2.4 +Loss Function +3 +Results +3.1 +Analysis on Velocity Field +3.2 +RSD Corrections +4 +Conclusion +A Momentum field reconstruction from Unet +★ xiaoliang5@mail2.sysu.edu.cn +† zhangle7@mail.sysu.edu.cn +‡ lixiaod25@mail.sysu.edu.cn +1 INTRODUCTION +The Large Scale Structure (LSS) of the Universe is crucial to our +study of the expansion and structure formation history of the Uni- +verse. In the next decade, the stage IV surveys such as DESI1, EU- +CLID2, LSST3, WFIRST4,CSST,Roman and Subaru will map the +Universe with extraordinary precision on an unprecedented large +volume, deepening the understanding of dark energy, dark matter, +gravity, the Hubble constant, the neutrino mass, and the initial con- +dition of the Universe. +Due to the initial inhomogeneity, the peculiar velocity field of the +universe is generated together with the density field during the pro- +cess of structure formation, and thus contains enormous information +about LSS. Accurate observations or reconstruction of the cosmic +velocity field will greatly help us to quantify and understand the red- +shift spatial distortions (Jackson 1972c; Kaiser 1987), baryon acous- +tic oscillations (Eisenstein et al. 2005, 2007), the Alcock-Paczynski +effect (Alcock & Paczyński 1979b; Li et al. 2014; Li et al. 2015b; Li +1 https://desi.lbl.gov/ +2 http://sci.esa.int/euclid/ +3 http://sci.esa.int/euclid/ +4 https://wfirst.gsfc.nasa.gov/ +© 2022 The Authors +arXiv:2301.04586v1 [astro-ph.CO] 11 Jan 2023 + +2 +Ziyong Wu et al. +et al. 2016; Ramanah et al. 2019a), the cosmic web (Bardeen et al. +1986b; Hahn et al. 2007; Forero-Romero et al. 2009; Hoffman et al. +2012a; Forero-Romero et al. 2014; Fang et al. 2019), the kinematic +Sunyaev-Zeldovich effect (Sunyaev & Zeldovich 1972, 1980), the +integrated Sachs Wolfe effect (Sachs & Wolfe 1967; Rees & Sciama +1968; Crittenden & Turok 1996), etc. +Observationally, however, measuring the peculiar velocity of +galaxies is extremely difficult, mainly as it requires redshift- +independent distance estimates that can only be made by distance +indicators such as type Ia Supernovae (Phillips 1993; Riess et al. +1997; Radburn-Smith et al. 2004; Turnbull et al. 2012; Mathews +et al. 2016), the Tully-Fisher relation (Tully & Fisher 1977; Masters +et al. 2006, 2008) the "fundamental plane" relation (Dressler et al. +1987; Djorgovski & Davis 1987; Springob et al. 2007), etc. There- +fore, great efforts have been devoted to developing alternative meth- +ods of reconstructing the cosmic velocity field from the halo density +field according to theoretical predictions. Here, the difficulty lies in +the complexity arising from the nonlinear evolution of the structure +and the gravitational collapse, and many studies have been made +in this direction (Nusser et al. (1991); Bernardeau (1992); Zaroubi +et al. (1995); Croft & Gaztanaga (1997); Bernardeau et al. (1999); +Kudlicki et al. (2000); Branchini et al. (2002); Mohayaee & Tully +(2005); Lavaux et al. (2008); Bilicki & Chodorowski (2008); Kitaura +et al. (2012); Wang et al. (2012); Jennings & Jennings (2015); Ata +et al. (2017)). +Recent tremendous advances in machine learning algorithms, es- +pecially those based on deep neural networks, provide us with a great +opportunity to extract useful information from complex data. In more +recent years, deep learning-based techniques have been applied to al- +most all areas of cosmology and astrophysics (Mehta et al. 2019; +Jennings et al. 2019; Carleo et al. 2019; Ntampaka et al. 2019), such +as weak gravitational lensing (Schmelzle et al. 2017; Gupta et al. +2018; Springer et al. 2018; Fluri et al. 2019; Jeffrey et al. 2019; +Merten et al. 2019; Peel et al. 2019; Tewes et al. 2019), the Cosmic +Microwave Background (Caldeira et al. 2018; Rodriguez et al. 2018; +Perraudin et al. 2019; Münchmeyer & Smith 2019; Mishra et al. +2019), the LSS including estimating cosmological parameters from +the distribution of matter (Ravanbakhsh et al. 2017a; Lucie-Smith +et al. 2018; Pan et al. 2020; Lazanu 2021), identifying dark matter +halos and reconstruct the initial conditions of the universe using ma- +chine learning (Modi et al. 2018; Berger & Stein 2019; Lucie-Smith +et al. 2019; Ramanah et al. 2019c), mapping rough cosmology to +fine one (He et al. 2019; Li et al. 2021), extracting line intensity +maps (Pfeffer et al. 2019), foreground removal in 21cm intensity +mapping (Makinen et al. 2021), augmenting N-body simulations +with gas (Tröster et al. 2019), a mapping between the 3D galaxy dis- +tribution in hydrodynamic simulations and its underlying dark mat- +ter distribution (Zhang et al. 2019a), modelling small-scale galaxy +formation physics in large cosmological volumes (Ni et al. 2021), +reconstructing the baryon acoustic oscillations (Mao et al. 2020) and +reconstructing the initial linear-regime matter density field (Shallue +& Eisenstein 2022), searching for gravitational waves (Dreissigacker +et al. 2019; Gebhard et al. 2019) and cosmic reionization (La Plante +& Ntampaka 2018; Gillet et al. 2019; Hassan et al. 2019b; Chardin +et al. 2019; Hassan et al. 2019a), as well as supernovae (Lochner et al. +2016; Moss 2018; Ishida et al. 2019; Li et al. 2019a; Muthukrishna +et al. 2019), etc. +For velocity reconstruction, the pioneering work (Wu et al. 2021) +shows that a UNet network can reconstruct the nonlinear velocity +field of dark matter particles with high precision down to a scale +of 2 ℎ−1Mpc. When pushing down to highly non-linear scales of +𝑘 ≲ 1.4 ℎ−1Mpc, they could achieve 90% accuracy in reconstructing +the power spectra of the velocity and momentum fields of the magni- +tude, the divergence and the vorticity components. This demonstrates +that, compared with the traditional perturbation-based theory, deep +learning methods would be more effective and have a great advantage +in reconstructing the cosmic velocity field at nonlinear scales. +More importantly, it is widely believed that the dark matter halos +and subhalos well trace the galaxy distributions, and their clustering +properties approach those of real observations. However, technically, +it is more challenging to reconstruct the velocity field from halos, +since the halos only reside at density peaks and become much more +sparse than simulation particles. +Therefore, in this study, we propose a modified UNet model dedi- +cated to the reconstruction of the velocity field of dark matter halos +(and subhalos). From our simulation tests, this proposed method can +reconstruct the peculiar velocities of each individual halos on high +accuracy. It turns out that both the velocity and real-space density +fields down to the non-linear scales can be well inferred from a +redshift-space measurement alone. Therefore, this study is a major +step toward applying the deep leaning technique to real observational +data, which is extremely important for cosmology. +The layout of this paper is as follows. In Sect. 2, we introduce the +simulation data used in this study and detail the architecture choice +of the neural network and the training procedure, as well as the +validation tests in velocity reconstruction. Results for our network +are presented in Sect. 3, and finally the conclusion and discussion +are present in Sect. 4. +For a comparison to the velocity reconstruction we discuss in this +paper, we present reconstruction results for the corresponding mo- +mentum field (the number-weighted velocity) of halos in Appendix A, +obatined with the same UNet model. +2 METHOD +2.1 Dataset +To train and validate our deep learning framework, the training and +tests data sets are based on the dark matter halos/subhalos of the Big- +MultiDark (BigMD) Planck simulation5, which is the high-resolution +N-body simulation described in Klypin et al. (2016b) and was per- +formed with GADGET-2 (Springel 2005b). The simulation was cre- +ated in a box of 2.5ℎ−1 Gpc on each side, with 38403 dark matter +particles and the mass resolution of 𝑀DM = 2.4 × 1010ℎ−1M⊙. +The initial conditions are generated with Zeldovich approximation +at 𝑧init = 100. The simulation provides 79 redshift snapshots in the +range of 0-8.8. For the analysis, we use the ROCKSTAR (Robust +Overdensity Calculation using K-Space Topologically Adaptive Re- +finement) halo finder (Behroozi et al. 2013b) to identify spherical +dark matter halos/subhalos in the simulation, based on adaptive hi- +erarchical refinement of friends-of-friends groups in six phase-space +dimensions and one time dimension. ROCKSTAR provides halo +mass using spherical overdensities of a virial structure. +The cosmology we assume in this study is the standard flat +ΛCDM, compatible with Planck 2018 results (Aghanim et al. +2020), with the fiducial parameters of {Ω𝑚, Ω𝑏, ℎ, 𝑛𝑠, 𝜎8} += +{0.307, 0.048, 0.677, 0.961, 0.828}. +We construct the redshift-space halo/subhalo catalogue at 𝑧 = 0 +with the number density of 10−3(Mpc/ℎ)3 fixed, to be compatible +with current spectral observations. The redshift-space position s is +5 http://www.cosmosim.org +MNRAS 000, 1–15 (2022) + +3 +related to the real-space position r for a distant observer along the +line of sight by +𝒔 = 𝒓 + +𝒗 · ˆ𝑧 +𝑎𝐻(𝑎) , +(1) +where 𝒗 is the peculiar velocity, 𝑎 is scalar factor and 𝐻 is the Hubble +parameter, and the unit vector ˆ𝑧 denotes the line-of-sight direction. +Based on the catalogue samples, we compute the density field and +velocity field in the mesh cells by assigning the particle mass to a 9003 +mesh using the CIC (Cloud-in-Cell) scheme, with a cell resolution +of (2.78ℎ−1Mpc)3. +2.2 Input Preprocessing +Our framework consists of the prepossessing of the input dataset and +there are some points need to be clarified, as detailed below. +1) In our UNet model, the input is a 6-channel 3D number density +map of halos (and subhalos) in redshift space. For each channel, +the map contains only halos in a certain mass range. To do so, we +sort the halo (and subhalo) sample by mass in descending order and +split it into six mass intervals, with the bin edges: log10(𝑀/𝑀⊙) ∈ +[13.52, 13.12, 12.84, 12.62, 12.43, 12.30], corresponding to binning +the halo mass in percentiles of [5, 15, 30, 50, 75, 100]. The reason +for doing so is that 1) the features of the velocity field may be +significantly different among different masses of halos, which may +be more effective for neural network learning, and 2) in observations +we can have approximately estimated mass of halos. +2) Based on the limitations in size of GPU memory, training time +and model size, we have to divide the large box of side length 2500 +Mpc/ℎ into 8000 smaller boxes, each with side 125 Mpc/ℎ (453 +meshgrid points in CIC). +3) Given that the box division procedure and physically small +boxes lead to loss of large-scale velocity modes, we thus use the +linear perturbation theory to compensate for such loss in the training +data with a set of small box simulations (125 Mpc/ℎ). The velocity +field 𝒗 in the linear regime are directly related to the density field 𝛿 +through +𝒗(𝒌) = 𝑎 𝑓 𝐻 𝑖𝒌 +𝑘2 𝛿(𝒌) , +(2) +where both fields are expressed in Fourier space, and +𝑓 += +𝑑 ln 𝐷/𝑑 ln 𝑎 denotes the growth rate with 𝐷 the growth factor. Using +this linear prediction, the velocity field at large scale can be exactly +calculated in our simulations. +4) To ensure that the training results of the model achieve good +rotational invariance, we use a data augmentation method, in which +the input data to the model for training are randomly rotated by one +of 18 rotational transformations. +5) Instead of learning the 3D velocity vector field directly, we de- +compose the velocity vector into two parts: magnitude and direction. +The predicted velocity field is then reconstructed using these two +parts. +6) Since the dynamic range of the velocity field is very wide, +to improve the accuracy and the convergence speed, the velocity +magnitude in the output isnormalized,wherethenormalization factor +𝑐 is chosen by 𝑐 = 1/200 for 𝑣 ⩾ 60 km/s and 𝑐 = 1/12 otherwise. +Therefore, the output of the velocity magnitude contains two parts, +the large velocity one and the small one, labelled by 𝑣large & 𝑣smaller, +respectively. +7) In addition to the velocity field, we have also trained the model +to account for the momentum field as an output and the results are +summarized in Appendix A. +As our dataset is large enough, in order to test the performance +of our neural network model, we split the dataset consisting of 8000 +subboxes into a training set of 500 subboxes, a validation set of 300 +subboxes to prevent overfitting and the rest are for a test set which is +used for the final test of the performance of the model. Note that the +training and test sets are selected in different regions of the large box +to prevent any potential correlations between them. +2.3 Neural Network Model +Motivated by the neural network model (Wu et al. 2021), we use a +modified UNet neural network architecture for model construction. +The architecture of our neural network and its components are shown +in Fig. 1. The input is the 6-channel number density field of halos, +each channel corresponding to number density field for a certain +mass range of halos. As mentioned in Sect. 2.2, as the velocity +field is decomposed into the two parts: velocity magnitude and the +velocity direction, we build two structurally similar neural networks +to deal with them separately. The network ends with the output +layer of 2+3 fields, three of which correspond to the components +of velocity direction (ˆ𝑣𝑥, ˆ𝑣𝑦, ˆ𝑣𝑧) and two to the velocity magnitude +(𝑣large, 𝑣smaller). A complete reconstruction of the 3D velocity field +is finally achieved by combining all of the output field components. +More specifically, the detail of the UNet network are shown on +the bottom-left panel in Fig.1. The colored plates represent different +operations in the neural network, which are connected from the in- +puts to the outputs by means of arrow lines. The size of the input, +the intermediate and the output fields (number of channels × spatial +pixels) is specified. Also, the size and the number of 3D convolu- +tion kernels ("conv") are also labelled. Moreover, the combination +of padding schemes gives the desired dimensionality after each con- +volution. Note that, 1) the "init" 3D convolutional layers allow for a +sufficiently large receptive field, enabling the network to quickly learn +the large-scale information in the beginning, 2) the "output" convo- +lutional layers increase complexity, followed by the dropout layers to +avoid overfitting and to change the number of channels at the end, +and 3) the batch normalization (BN) layer and the rectified linear unit +(ReLU) activation layer after convolutional layers can speed up the +training convergence and prevents neural networks from overfitting +as well as increasing nonlinear effect. With the trained UNet, the +velocity field of halos is predicted by feeding the number density +field of halos in the redshift space, and the relevant statistics such as +clustering information can be measured straightforwardly. +A crucial ingredient in our model is a three-block structure: an +lower convolution block (red), a upper convolution block (pink), and +a final one (blue). The advantages of these blocks are capable of pass- +ing the initial information to the deep network structure. In parallel, +to avoid the bias in small-box simulations, the linear theory-predicted +velocity field is used as an additional input in the lower block, com- +pensating for the lack of information on large-scale velocities. The +final convolution block ensures that we can correctly learn the ve- +locity field at the center of the number density field and prevent any +spurious signals due to boundary effects. +2.4 Loss Function +The objective of the training our UNet is to minimize a loss func- +tion between prediction, 𝒗, and simulation truth, 𝒗true of each voxel. +Specifically, to account for the contributions from the velocity mag- +nitude (𝑣 ≡ |𝒗|) and the velocity direction (unit vector ˆ𝒗 ≡ 𝒗/𝑣), we +MNRAS 000, 1–15 (2022) + +4 +Ziyong Wu et al. +model +linear velocity +field +6x513 +halo density field +2x453 +3x453 +halo velocity/ +momentum +magnitude field +halo velocity/ +momentum +direction field +conv +36x51 +3 +36x51 +3 +conv +72x25 +3 +72x25 +3 +144x12 +3 +conv +18x27 +3 +3x27 +3 +3x25 +3 +trans +144x12 +3 +conv +conv +conv +init +linear velocity field +72x25 +3 +trans +trans +conv +36x51 +3 +upper convolution +block +conv +36x49 +3 +36x47 +3 +(2+3)x45 +3 +36x45 +3 +conv +conv +output1 +output2 +36x45 +3 +final convolution +block +72x25 +3 +lower +convolution +block +conv 5 +3 +batchnorm +relu +init +conv 3 +3 +batchnorm +relu +conv +ConvTrans 1 +3 +batchnorm +relu +dropout +if 1 +if 2 +ConvTrans 3 +3 +batchnorm +relu +trans +output +U-net +U-net +U-net +6x51 +3 +upper +convolution +block +lower +convolution +block +final +convolution +block +Figure 1. UNet neural network architecture and training scheme used for the velocity (momentum) reconstruction. Starting with a 6-channel 513-voxel input +layer that corresponds to the number density field (a side length of 142 Mpc/ℎ) of halos for the six different mass intervals (over the mass range of 1012–1015𝑀⊙) +in redshift space, our model is consisted of two U-net neural network architecture for reconstruction of velocity magnitude and direction, where one contains +two channels corresponding to the large and small velocity fields (𝑣large, 𝑣small), and the other consists of three channels corresponding to the three velocity +directions (𝑣𝑥, 𝑣𝑦, 𝑣𝑧) (upper left). This U-net architecture essentially consists of the upper, lower and final convolution blocks, together with a compensation +for the linear velocity field (upper right). The dimension of each output field is 453, corresponding to a box volume of 1253 Mpc3ℎ−3. The lower-right part shows +the details of the components given in the three-block structure of the UNet. The layers of "init", "conv", "trans" and "output" are detailed on the lower-left part. +In the final convolution block, a dropout layer between the convolution transform and batchnorm layers is used to enhance the UNet performance and prevent +overfitting, where the dropuout value is chosen as 0.3. +choose the following loss function with two terms, +L = 1 +𝑁 +𝑁 +∑︁ +𝑖=1 +� 2 +5 (𝑣𝑖 − 𝑣true +𝑖 +)2 + 3 +5 (1 − cos 𝜃𝑖) +� +, +(3) +where cos 𝜃𝑖 ≡ ˆ𝒗𝑖 · ˆ𝒗true +𝑖 +, and the index 𝑖 denotes the 𝑖-th voxel. +As observed, the first term is responsible for 𝑣, and corresponds to +the standard and simple mean error (MSE) loss that is essentially +equivalent to the maximum likelihood solution under a Gaussian +assumption with constant variance. The second term naturally mea- +sures the deviation between the reconstructed and the true values of +ˆ𝒗. The coefficients of these two terms can be regarded as normal- +ization factors and are determined by the number of channels, i.e., 2 +for magnitude (𝑣large, 𝑣small), and 3 for direction (ˆ𝑣𝑥, ˆ𝑣𝑦, ˆ𝑣𝑧). Em- +pirically, such loss function is effective, and has proven to be stable +and effective during our training process, providing good results in +the velocity (momentum) reconstruction. We trained our UNet using +the most popular algorithm Adam (Kingma & Ba 2014) for train- +ing deep neural networks, which can iteratively decrease the training +loss by calculating its gradient with respect to model parameters +and performing a small step along the direction with the maximum +decrease. +3 RESULTS +In this section, we test the performance of the trained UNet model +and present our results predicted from 27 test sets, each consisting of +125 small boxes with same volume as that of the training sets (side +length 125 Mpc/ℎ). Therefore, the box volume for each test set we +have performed the analysis on is 6253(ℎ−1Mpc)3. We chose this +because measurements on large boxes would have a better statistical +behavior, reducing the statistical errors. We also find the results for +the other simulation boxes are very similar. To ensure reliable test +results, these simulation boxes was not used for the model training +and refining the model structure/training parameters. +First we shall describe the statistics we will use throughout the +paper. The 2-point correlation function is one of the most commonly +used statistics to characterize a homogeneous density field, +𝜉(𝒓) = ⟨𝛿 (𝒙) 𝛿 (𝒙 + 𝒓)⟩ , +(4) +where 𝛿(𝒙) is the density contrast field, 𝒙 denote for any point, 𝒓 is +a separation vector, and ⟨·⟩ stands for the ensemble mean, computed +with a spatial mean over 𝒙 in practice. +The power spectrum of 𝛿 (𝒙) is just related to 𝜉(𝒓) by the Fourier +transform, i.e., +𝑃(𝒌) = +∫ +𝜉(𝒓)e𝑖𝒌·𝒓d3𝒓 , +(5) +where 𝒌 is the 3D wavevector of the plane wave, with the magnitude +𝑘 ≡ |𝒌| (the wavenumber) related to the wavelength 𝜆 by 𝑘 = 2𝜋/𝜆. +MNRAS 000, 1–15 (2022) + +125Mpc/hrlarge +Mpc/hrsmall +Mpc/h125 Mpc/h125Mpc/hredshift space +60.05 +142Mpc/hredshift space +60.15 +142 Mpc/hredshift space +00.30 +142Mpc/hredshift space +60.50 +oc/hredshiftspace +60.75 +Mpc/hredshift space +61.00 +142Mpc/h5 +Similar to the scalar field 𝛿, we can also define power spectra +for velocity and momentum vector fields of interest. As known, the +velocity field, 𝒗, is completely described by its divergence, 𝜃 ≡ ∇ · 𝒗 +and its vorticity, 𝝎 = ∇ × 𝒗, which, in Fourier space, become purely +radial and transversal velocity modes, respectively, defined by 𝜃(𝒌) = +𝑖𝒌 · 𝒗(𝒌) and 𝝎(𝒌) = 𝑖𝒌 × 𝒗(𝒌). The power spectra of the velocity, +divergence, vorticity and velocity magnitude are given by +⟨𝜃(𝒌)𝜃∗(𝒌′)⟩ =(2𝜋)3𝑃𝜃 (𝑘)𝛿(𝒌 − k′) , +⟨𝜔𝑖(𝒌)𝜔∗𝑗 (𝒌′)⟩ =(2𝜋)3 1 +2 +� +𝛿𝑖 𝑗 − 𝑘𝑖𝑘 𝑗 +𝑘2 +� +𝑃𝜔(𝒌)𝛿(𝒌 − 𝒌′) , +⟨𝒗(𝒌) · 𝒗∗(𝒌′)⟩ =(2𝜋)3𝑃𝑣 (𝒌)𝛿(𝒌 − 𝒌′) , +(6) +where indices 𝑖, 𝑗 denote the components in the Fourier space coor- +dinates. +In the linear perturbation theory, the continuity equation leads to +𝜃 = −H 𝑓 𝛿, where H = 𝑎𝐻 is the conformal Hubble parameter, +𝑎 denotes the cosmic scale factor and 𝑓 is the linear growth rate +defined by 𝑓 = 𝑑 ln 𝐷/𝑑 ln 𝑎, with 𝐷 being the linear density growth +factor. In a ΛCDM model, 𝑓 ≈ Ω0.6 +m +(Peebles 1980), with a good +approximation. +3.1 Analysis on Velocity Field +First we shall describe the metrics that we will use throughout this +section for evaluating the reconstruction accuracy. For an arbitrary +reconstructed field of halos from UNet, denoted by the shorthand +notation 𝑓 , where 𝑓 ∈ {𝜃, 𝝎, 𝒗} for velocity, we use the following +metrics, the so-called transfer function and correlation coefficient, to +compare a reconstructed field ( 𝑓 ) with the true one ( 𝑓 ′): +𝑇𝑓 = +O 𝑓 +O 𝑓 ′ − 1 , +𝐶 𝑓 = +1 +𝑁pix − 1 +∑︁ +𝑖 +( 𝑓𝑖 − ¯𝑓 )( 𝑓 ′ +𝑖 − ¯𝑓 ′) +𝜎𝑓 𝜎𝑓 ′ +, +(7) +where O 𝑓 stands for an arbitrary observable for 𝑓 . The correlation +𝐶 𝑓 is defined between reconstructed ( 𝑓 ) and true fields ( 𝑓 ′) with +the same total number of pixels 𝑁pix. The sample mean and the +standard deviation of field 𝑓 are denoted by ¯𝑓 and 𝜎𝑓 , respectively. +In this study, we make a comparison of the statistical properties of +clustering, through various observables such as the power spectra of +velocity components and the two-point correlation function (2PCF). +Both metrics provide a physical insight for comparison such that the +perfect reconstruction is equivalent to 𝑇𝑓 = 0 and to 𝐶 𝑓 = 1. +3.1.1 Visual inspection and point-wise comparison +As a first validation, we perform a point-wise comparison between +the UNet-predicted halo velocity field to the simulation truth. To do +so, we randomly selected three thin slices in the test sets, each with +volume of 83 × 83 × 28 ℎ−3Mpc3, with spatial resolution of 2.78 +Mpc/ℎ. +Fig. 2 visualizes the number density distribution of dark mat- +ter halos (and subhalos) distribution and velocity field in these +three slices. As seen, there are many massive halos in the range +of 𝑀/𝑀⊙ ∈ [1012, 1015], typically with 60 halos per slice. In the +middle and right panels, we display the true and the predicted velocity +fields, respectively. The colored arrows show the average velocities +at the meshgrid points and are projected onto the image plane. The +length and the direction of the arrow represent the magnitude and the +direction of the projected velocity, respectively. To show clearly, the +projected velocity magnitude is also scaled by the color from purple +to red, reflecting the halo velocities from small to large. As seen, a +Table 1. Summary of the correlation coefficients 𝐶 𝑓 between the recon- +structed and true fields of the velocity 𝒗, the divergence component 𝜃 and the +vorticity 𝝎 via Eq. 7, estimated by averaging over 27 test sets, each with the +box size of 625 Mpc/ℎ. +field +𝒗 +𝜃 +𝝎 +𝐶 𝑓 +0.71 +0.66 +0.65 +high number density region typically leads to a larger velocity field, +since the gravitational collapse and non-linear structure formation +occur intensively there. Moreover, the visualized morphology for the +UNet-predicted and the true velocity fields clearly indicates the effec- +tiveness of our neural network, as they are almost indistinguishable +by eye. Interestingly, although the simulated halo velocity field is +sparse, we can still reliably reconstruct it, especially for the regions +with small velocities. To quantitatively validate such reconstruction, +we show the histogram distributions (in the rightmost panel right +panels) of magnitude and direction of the velocities in these three +slices. We do find that, statistically, the distributions of the recon- +structed velocity magnitude and direction agree well with the true +values. +Specifically, the mean value and its 1𝜎 dispersion (with a Guassian +fit) for the UNet-reconstructed velocity magnitude in each slice are +953.88 ± 731.08 and 1222.17 ± 904.43 km/s, respectively. These +mean values are consistent with the true ones at about higher than +99% accuracy among all these slices. Similarly, we obtain very good +results in the direction reconstruction, with 1 − cos 𝜃 (defined in +Eq. 3) of 0.01 ± 0.07 and 0.01 ± 0.06 for these slices, respectively. +By averaging over the slices, the deviation in the velocity direction, +Δ𝜃 = |𝜃true −𝜃UNet|, achieves Δ𝜃 = 8.1◦ ±19.9◦, implying Δ𝜃 < 30◦ +at 1𝜎 level. +Furthermore, let us focus on the vorticity field halo momentum +fields, ˜𝝎, which generally appears in high-density regions of halos +and is essentially induced by non-linear structure formation. The re- +construction of the vorticity field for two randomly selected slices is +present in Fig. 3. As seen, the vorticity field is tightly concentrated +on high-density regions where the nonlinear processes such as shell +crossing are occurring. Thus its magnitude is tightly coupled to the +local density and decays rapidly at the linear regime (low density) +of structure formation. Also, its direction seems to be distributed +randomly, indicating that these high-density regions have strong a +nonlinear process. Due to the effect of nonlinearity on small scales, +the vorticity fields are theoretically very difficult to reconstruct, es- +pecially through sparse halo samples. Here, however, we show the +advantages of UNet, which can provide the reconstruction in |𝝎| at +very high accuracy, with the deviations in the mean and dispersion +only about |𝝎true−𝝎UNet| = 1.71±3.15 and 5.5±6.90 h· km/s/Mpc +for these two slices, respectively. Also, from the histogram distribu- +tion, the reconstructed directions for the vorticity component indicate +that the reconstruction is unbiased, with the mean and 1𝜎 error of +Δ𝜃 ≈ 20.0◦ ± 40.5◦. +In order to quantitatively compare the reconstructed and real fields, +we compute the coefficients, 𝐶 𝑓 , through Eq. 7 for various velocity +components. The resulting coefficients are summarized in Tab. 1, +estimated by averaging over 27 test sets, each with the box size of side +625 Mpc/ℎ. Our proposed UNet model has excellent performance in +terms of the linear correlation in real-space domain, demonstrating +that the network produces high-fidelity reconstructions in 𝒗, 𝜃 and +𝝎, with 𝐶 𝑓 in the range of [0.71, 0.65]. As seen, the reconstruction +in 𝝎 is almost as good as in the other components, which indicates +MNRAS 000, 1–15 (2022) + +6 +Ziyong Wu et al. +0 +16 +32 +48 +64 +80 +X(Mpc/h) +0 +16 +32 +48 +64 +80 +Y(Mpc/h) +0 +16 +32 +48 +64 +80 + + + + + + +vtruth +0 +16 +32 +48 +64 +80 + + + + + + +vUNet +0 +1000 +2000 +3000 +4000 +5000 +|v| [km/s] +0 +20 +40 +60 +UNet +953.88±731.08 +truth +953.31±700.04 +0.0 +0.1 +0.2 +1 - cos +0 +50 +100 +150 +200 +250 +UNet +0.01±0.07 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +200 +400 +600 +800 +1000 +1200 +1400 +0 +16 +32 +48 +64 +80 +X(Mpc/h) +0 +16 +32 +48 +64 +80 +Y(Mpc/h) +0 +16 +32 +48 +64 +80 + + + + + + +vtruth +0 +16 +32 +48 +64 +80 + + + + + + +vUNet +0 +1000 +2000 +3000 +4000 +5000 +|v| [km/s] +0 +20 +40 +60 +UNet +1222.17±904.43 +truth +1231.59±889.87 +0.0 +0.1 +0.2 +1 - cos +0 +100 +200 +300 +UNet +0.01±0.06 +0 +1 +2 +3 +4 +5 +6 +7 +8 +0 +200 +400 +600 +800 +1000 +1200 +1400 +Figure 2. Point-wise comparison between the UNet-reconstructed velocity field and true one. From top to bottom, we show the results for the three three thin +slices in the test sets, each with volume of 83 × 83 × 28 ℎ−3Mpc3. From left to right, the fields of halo number density, the UNet-reconstructed velocity, the +true velocity and the corresponding histogram are shown, respectively, where all quantities are measured in redshift space. The length and the orientation of +the colored arrows in the velocity fields represent the velocity magnitude and direction. For each slice, the rightmost panels show the statistical histogram +distributions of the velocity samples for magnitude and direction, respectively. UNet is able to reconstruct velocity well via visual inspection and the statistical +analysis on the histogram. +0 +16 +32 +48 +64 +80 +X(Mpc/h) +0 +16 +32 +48 +64 +80 +Y(Mpc/h) +0 +16 +32 +48 +64 +80 + + + + + + +truth +0 +16 +32 +48 +64 +80 + + + + + + +UNet +0 +200 +400 +600 +800 +1000 +| +| [h km/s/Mpc] +0 +50 +100 +150 +UNet +124.86±100.44 +truth +123.15±103.29 +0.0 +0.1 +0.2 +1 - cos +0 +100 +200 +300 +UNet +0.04±0.20 +0 +1 +2 +3 +0 +50 +100 +150 +200 +250 +300 +0 +16 +32 +48 +64 +80 +X(Mpc/h) +0 +16 +32 +48 +64 +80 +Y(Mpc/h) +0 +16 +32 +48 +64 +80 + + + + + + +truth +0 +16 +32 +48 +64 +80 + + + + + + +UNet +0 +200 +400 +600 +800 +1000 +| +| [h km/s/Mpc] +0 +25 +50 +75 +100 +125 +UNet +151.42±134.05 +truth +145.92±127.15 +0.0 +0.1 +0.2 +1 - cos +0 +100 +200 +300 +UNet +0.08±0.28 +0 +1 +2 +3 +4 +5 +6 +7 +8 +0 +50 +100 +150 +200 +250 +300 +Figure 3. Same as in Fig. 2, but for the vorticity field of halo velocities, 𝜔. As seen, the vorticity field, exclusively generated from the nonlinear structure +formation, has a more complex distribution pattern than that of the velocity field. In spite of this, the Unet approach still performs very well in reconstruction. +MNRAS 000, 1–15 (2022) + +7 +0 +1 +2 +3 +4 +-1600 +-1280 +-960 +-640 +-320 +0 +320 +640 +960 +1280 + [h km/s/Mpc] +truth +0 +1 +2 +3 +4 + + + + + +UNet +100 +101 +102 +103 +104 +105 +0 +1 +2 +3 +4 +-1600 +-1280 +-960 +-640 +-320 +0 +320 +640 +960 +1280 + [h km/s/Mpc] +truth +0 +1 +2 +3 +4 + + + + + +UNet +100 +101 +102 +103 +104 +105 +Figure 4. Joint probability distributions of density-divergence, 𝜌( 𝛿, 𝜃) (up- +per), and density-vorticity, 𝜌( 𝛿, |𝝎|) (lower). In each case, the distribution +results are calculated from 5 test sets, each with the box size of 625 Mpc/ℎ. As +seen, the predicted velocity distributions (right) agree well with the simulation +truth (left) for all of the halo number densities of 𝛿 ∈ [0, 5]. +that the UNet model can give a good prediction for the vorticity field +with complicated morphological properties. +To further test the reconstructed velocity field with the ground +truth, in Fig. 4, a visual inspection for the joint probability distri- +butions of density-divergence and density-vorticity are shown. Obvi- +ously, all predictions are in good agreement with the the true values, +even in the very high density regions (𝛿 ≫ 1). Furthermore, we find +that, for a given 𝛿, the reconstructed distributions appear slightly +narrower than the true ones. This is probably because the neural +network would slightly lose some random perturbation information +when learning and compressing the information in the training sets. +3.1.2 Comparison for power spectrum +Here we describe the reconstruction accuracy for the power spec- +tra of velocity components, 𝑃 𝑓 (𝒌). For each component, we have +computed the transfer function (see Eq. 7) in terms of the predicted +power spectrum and the true one. Intuitively, by use of power spec- +trum as the observable, 𝑇𝑓 measures the accuracy of reconstruction +in magnitude as a function of wavevector in Fourier domain. Taking +the directional average, the function 𝑇𝑓 (𝑘) represents the transfer +function with spatial averaging over |𝒌| bins. Also, in general, 𝑇(𝑘) +is not explicitly optimized during the training stage, since the training +minimizes the proposed loss function (see Eq. 3) composed of the +velocity magnitude and direction in real-space domain. +As observed in Fig. 5, there is a constant systematic bias to slightly +underestimate the power spectrum for the velocity field 𝒗 over all +scales, 𝑇𝑓 (𝑘) ≈ −0.2. This underestimate may be due to the fact that +the compensation of the linear theory-predicted velocity field is not +perfect in UNet learning process. For the divergence component 𝜃, +the deviation varies for positive to negative, i.e., 𝑇𝑓 (𝑘) ∈ [0, 0.2] for +𝑘 ≲ 0.25 ℎ/Mpc and 𝑇𝑓 (𝑘) ∈ [−0.2, 0] otherwise. As known, the +vorticity 𝝎 is generated by nonlinear evolution, and so its reconstruc- +tion has always been a challenge. However, we find the reconstructed +vorticity power spectrum successfully match the true one, yielding a +similar deviation level as in 𝒗 and 𝜃, with 𝑇𝑓 (𝑘) ∈ [−0.25, −0.20] +at 𝑘 ≲ 0.4 ℎ/Mpc and 𝑇𝑓 (𝑘) ≈ −0.2 otherwise. This is remarkable +considering that the UNet model performs well from the linear to +deeply nonlinear scales. All these test results highlight the ability of +UNet in learning various velocity components from the halo number +density field, especially on the nonlinear scales. +3.2 RSD Corrections +An important application of the UNet-based velocity reconstruction +is to map a halo distribution from redshift to real space as well +as inferring the distances of individual halos (galaxies). To do so, +redshift-space distortions (RSD) are corrected by moving the ha- +los from redshift to real space according to their peculiar velocities +reconstructed from the halo number density field using the trained +UNet network. By performing a tri-linear interpolation of the recon- +structed velocity field, the velocities at every halo positions can be +obtained with reasonable accuracy. In the following, we will present +the performance of such RSD correction. +3.2.1 Two-point correlation function +In redshift space, anisotropic two-point correlation function (2PCF), +𝜉(𝒓), provides a measurement for halo (galaxy) clustering through +the standard Landy & Szalay (1993a) estimator, +𝜉(𝑟, 𝜇) = 𝐷𝐷(𝑟, 𝜇) − 2𝐷𝑅(𝑟, 𝜇) + 𝑅𝑅(𝑟, 𝜇) +𝑅𝑅(𝑟, 𝜇) +(8) +where 𝐷𝐷, 𝐷𝑅, and 𝑅𝑅 are the normalized galaxy-galaxy, galaxy- +random, and random-random number of pairs with separation (𝑟, 𝜇), +respectively. Here the 3D separation vector between pairs of objects, +𝒓, has been decomposed into (𝑟, 𝜇) coordinates, where 𝑟 is the norm +of the separation vector and 𝜇 is the cosine of the angle between the +line-of-sight and separation vector directions. +It is common to expand 2PCF into Legendre polynomials as +𝜉(𝒓) = +∞ +∑︁ +ℓ=0 +𝜉ℓ (𝑟)𝐿ℓ (𝜇) , +(9) +with +𝜉ℓ (𝑟) = 2ℓ + 1 +2 +∫ 1 +−1 +𝜉(𝑟, 𝜇)𝐿ℓ (𝜇)𝑑𝜇 , +(10) +where 𝐿ℓ is the Legendre polynomial of order ℓ. Throughout this +study, ignoring the more noisy subsequent orders, we only take +into account ℓ = 0, 2 and 4 multipoles, referred to as monopole, +quadrupole, and hexadecapole, respectively, where 𝐿0 = 1, 𝐿2 = +� +3𝜇2 − 1 +� +/2 and 𝐿4 = +� +35𝜇4 − 30𝜇2 + 3 +� +/8. Due to the symmetry +of object pairs, only even multipoles do not vanish. In practice, the +pair counts are linearly binned with width of Δ𝑟 = 1.4 Mpc in 𝑟 and +Δ𝜇 = 0.025 in 𝜇 for the above estimation. +We often measure a 1D 2PCF, 𝜉(𝜇), that projects the 2D correla- +tion 𝜉(𝑟, 𝜇) along the 𝑟 axis, +𝜉1D(𝜇) = +∫ ∞ +0 +𝑑𝑟𝜉(𝑟, 𝜇)𝑑𝑟 . +(11) +Fig. 6 shows the projected 1D 2PCF and the resultant monopole, +quadrupole and hexadecapole of 2PCF. The line and the shaded +area in each panel give the mean and 1𝜎 standard deviation, mea- +sured from 27 test sets. As seen, due to the Kaiser effect (Kaiser +1987), the enhancement of power over all scales is very remarkable. +Meanwhile, in Tab. 2, we summarize the ratios of the redshift-space +measurements with and without UNet correction to the real-space +MNRAS 000, 1–15 (2022) + +8 +Ziyong Wu et al. + + +104 +105 +106 +k3Pvv [km2/s2] +UNet +truth +0.1 +0.2 +0.4 +0.7 +1.0 +k [h Mpc +1] +0.2 +0.1 +0.0 +Tf(k) + + +104 +105 +106 +kP + [km2/s2] +UNet +truth +0.1 +0.2 +0.4 +0.7 +1.0 +k [h Mpc +1] +0.2 +0.0 +0.2 +Tf(k) + + +103 +104 +105 +kP + [km2/s2] +UNet +truth +0.1 +0.2 +0.4 +0.7 +1.0 +k [h Mpc +1] +0.3 +0.2 +0.1 +0.0 +Tf(k) +Figure 5. Comparison of the UNet-predicted power spectrum and the simulation truth. From left to right, we show the the reconstruction for the velocity, the +velocity divergence and the vorticity, respectively. These results are based on the 27 test sets, each with the box of side length 625 Mpc/ℎ. For each case, the +shaded area gives 1𝜎 deviation measured from these test sets. +measurements (the simultion truth), which are shown in the lower +panels in Fig. 6. Overall, the RSD effects are prominent, whereas +they can be highly corrected by UNet with the differences within 1𝜎 +level. To highlight the changes due to RSD (Kasier and fingers-of- +god effects), we show the 1D projected 2PCF, 𝜉1D, with the variable +𝑟 integrated out, where the 2PCF without any correction strongly +deviates from the real-space one (shown in the upper-left panel) with +errors of tens of percent. However, the UNet model can accurately +correct the RSD effects using the reconstructed velocity field, statisti- +cally leading to the correct clustering of halos in almost all directions +with an error of 1–3% (except for 𝜇 = 1 with the relative deviation of +0.14). More importantly, after the UNet correction, the results for 𝜉0 +at ∼ 100 Mpc/ℎ demonstrate that, the baryon acoustic oscillations +(BAO) can be well recovered from redshift space, deriving a very +close BAO peak to the real-space one, with about 12% lower than +the true one. Interestingly, the correction for the quadrupole leads +to good agreement with the true real-space one not only on small +scales, but also on large scales. Even for 𝜉4 with a much smaller +signal-to-noise ratio than the other multipoles, the RSD effects can +also be removed successfully, without any visible artificial effects +such as oscillations and spikes. +The high-quality in the velocity reconstruction can be also appre- +ciated in Fig. 7, displaying the 2D anisotropic correlation function +of redshift-space halos, 𝜉(𝒓), where the separation vector has been +decomposed into line-of-sight and transverse separations such that +𝒓 = (𝑟⊥, 𝑟 ∥). The contours are calculated based on the averaged result +of 𝜉(𝑟⊥, 𝑟 ∥) on the 27 test sets. As observed, without any RSD cor- +rections, the anisotropic pattern is very distinctive. The Kaiser effect +leads to galaxy clusters appearing "squashed" along the line-of-sight +by a coherent infall onto galaxy clusters cancel some of the Hub- +ble flow. Besides, the random velocities attained by galaxies in the +non-linear regime produce the so-called fingers-of-god (FoG) effect, +making structures elongated along the line of sight. As expected, the +measured anisotropic correlation function in redshfit space present a +BAO feature at 𝑟 ≃ 100 Mpc/ℎ, as well as the impacts of the Kasier +and the FoG effects. Compared with the UNet-corrected results, we +find the isotropy of the correlation function is well recovered at all +scales, demonstrating the effectiveness of the UNet approach. Re- +markably, our proposed method not only corrects Kasier effect on +large scales, but also on small scales with 𝑟 ≲ 10 ℎ/Mpc, where the +FoG effect is well removed, indicating that UNet can even accurately +reconstruct the velocity field in the nonlinear regime. +4 CONCLUSION +3D velocity (and momentum) fields constructed by galaxies and clus- +ters are very important in cosmology because they provide more +information than the density field alone, and would help to im- +prove/correct various cosmological measurements. High-fidelity re- +construction may even result in unexpected findings. +Accurate reconstruction is often a challenge for traditional recon- +struction methods, typically relying on many assumptions and ap- +proximations. In this study, we have proposed an alternative scheme, +a deep learning approach based on the UNet neural network to recon- +struct the 3D velocity/momentum fields of halos. We find the UNet is +well-suited for reconstructing such fields directly from the halo (and +subhalos) density field, because the UNet is an elegant architecture +that can effectively capture various features/structures of the fields at +all scales and is very effective in transforming high-dimensional and +structured inputs. Using multiple redshift-space halo number density +fields in different mass ranges, the UNet surprisingly learned how to +transform halo density fields directly into velocity/momentum fields +from the training data. We have performed a detailed validation with +various statistics tests, and find the reconstructed velocity/momentum +fields well agree the ground truth. +Furthermore, using the inferred velocity fields, the RSD effects can +be well corrected by Unet. As an important application, we find that, +the reconstructed velocity field directly provides a recovery of the +real-space positions of individual halos, offering a perfect correction +for the RSD effects down to a highly non-linear scale of 1.13 Mpc/ℎ, +which is the Nyquist frequency (𝑘Ny = 𝜋𝑁/𝐿) of the simulations. +This UNet-based approach is promising for many cosmological ap- +plications in terms of correcting the peculiar velocities. For example, +the reconstruction of cosmic volume-weighted velocity suffers se- +vere sampling artifacts in measurements (Zhang et al. 2015; Yu et al. +2015; Chen et al. 2018). We will further extend our UNet model to +MNRAS 000, 1–15 (2022) + +9 + + + + + +7 +8 +9 +10 +11 +12 +13 +( ) +UNet +redshift space +real space +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.00 +0.25 +0.50 +Tf( ) + + + + +10 +0 +10 +20 +30 +40 +50 +60 +s2 +0(s)[h +2Mpc2] +UNet +redshift space +real space +20 +40 +60 +80 +100 +s (Mpc/h) +0.25 +0.00 +0.25 +Tf(s) + + + + +20 +15 +10 +5 +0 +5 +10 +s2 +2(s)[h +2Mpc2] +UNet +redshift space +real space +20 +40 +60 +80 +100 +s (Mpc/h) +0 +1 +Tf(s) + + + + +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +s2 +4(s)[h +2Mpc2] +UNet +redshift space +real space +5 +10 +15 +20 +25 +30 +35 +s (Mpc/h) +0.25 +0.00 +0.25 +0.50 +Tf(s) +Figure 6. Comparison of the projected 2PCF, 𝜉1D(𝜇) of the halo distribution (top-left) and the multipoles of 2PCF, including the monopole 𝜉0 (top-right), the +quadrupole 𝜉2 (bottom-left) and the hexadecapole 𝜉4 (bottom-right). The line and shaded area give the the mean and 1𝜎 standard deviation measured from +the 27 test sets. The measurements from the halo (and subhalos) positions in real space (gray), the redshift space (pink) and the redshift space corrected by +the UNet-predicted velocity field (blue) are shown, respectively. The lower panel gives the transfer function calculated by the ratio between the redshift-space +measurements after the UNet correction and the real-space measurement. As seen, the real-space and UNet-corrected redshift-space measurements of 2PCFs +are in good agreement through the transfer function, with differences within the 1𝜎 standard deviation. +tackle this long-standing problem and leave such a study for future +work. +As the stage IV galaxy surveys will provide more detailed mea- +surements of the LSS of the Universe than ever before, new com- +puting technologies are being called upon to fully analyze these +high-dimensional, massive amounts of data. Therefor, UNet-based +neural networks promise to be a powerful tool to overcome the prob- +lems that traditional methods are difficult to deal with and to extract +cosmological information in more depth and in a holistic manner. +ACKNOWLEDGEMENTS +This work is supported by the National Key R&D Program of China +(2018YFA0404504, 2018YFA0404601, 2020YFC2201600), the +Ministry of Science and Technology of China (2020SKA0110402, +2020SKA0110401, 2020SKA0110100), National Science Founda- +tion of China (11890691, 11621303, 11653003, 11803094), the +China Manned Space Project with No. CMS-CSST-2021 (A02, +A03, B01), the Major Key Project of PCL, the 111 project No. +B20019, the CAS Interdisciplinary Innovation Team (JCTD-2019- +05), and the Science and Technology Program of Guangzhou, China +(202002030360). We acknowledge the use of Kunlun cluster located +at School of Physics and Astronomy, Sun Yat-Sen University. +DATA AVAILABILITY +The BigMD simulation used in this paper is available in the Cos- +moSim data base (https://www.cosmosim.org/). The datasets gener- +MNRAS 000, 1–15 (2022) + +10 +Ziyong Wu et al. +Table 2. Summary of the resultant ratios between the redshift-space measurements with and without the UNet correction to the simulation truth (measured in +real space) for 1D projected 2PCF, 𝜉1𝐷 (𝜇), and the multipoles of 2PCF, 𝜉ℓ (𝑟) (shown in the lower panels of Fig. 6). The mean and 1𝜎 uncertainty are based +on the 27 test sets, each with the box of side length 625 Mpc/ℎ. +𝜇 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +𝜉 (𝜇)/𝜉true − 1 (UNet correction) +0.14 ± 0.02 +0.03 ± 0.01 +−0.00 ± 0.02 +−0.02 ± 0.02 +−0.02 ± 0.01 +−0.01 ± 0.02 +𝜉 (𝜇)/𝜉true − 1 (redshift space) +0.38 ± 0.03 +0.05 ± 0.02 +0.24 ± 0.02 +0.42 ± 0.02 +0.54 ± 0.02 +0.58 ± 0.03 +𝑟 (Mpc/ℎ) +20 +40 +60 +80 +100 +120 +𝜉0(𝑟)/𝜉true − 1 (UNet correction) +0.03 ± 0.02 +0.14 ± 0.04 +0.35 ± 0.06 +0.19 ± 0.05 +−0.12 ± 0.04 +−0.97 ± 0.04 +𝜉0(𝑟)/𝜉true − 1 (redshift space) +0.33 ± 0.03 +0.37 ± 0.06 +0.41 ± 0.08 +0.62 ± 0.08 +0.30 ± 0.06 +1.18 ± 0.07 +𝑟 (Mpc/ℎ) +20 +40 +60 +80 +100 +120 +𝜉2(𝑟)/𝜉true − 1 (UNet correction) +−0.49 ± 0.09 +−0.29 ± 0.04 +0.05 ± 0.04 +1.13 ± 0.02 +1.23 ± 0.03 +−0.50 ± 0.02 +𝜉2(𝑟)/𝜉true − 1 (redshift space) +6.51 ± 0.17 +10.22 ± 0.06 +8.40 ± 0.05 +6.21 ± 0.05 +4.91 ± 0.04 +−7.47 ± 0.04 +𝑟 (Mpc/ℎ) +5 +10 +15 +20 +25 +30 +𝜉4(𝑟)/𝜉true − 1 (UNet correction) +0.27 ± 0.18 +0.68 ± 0.10 +0.97 ± 0.08 +0.82 ± 0.06 +0.78 ± 0.04 +0.13 ± 0.04 +𝜉4(𝑟)/𝜉true − 1 (redshift space) +3.66 ± 0.60 +5.00 ± 0.13 +5.28 ± 0.09 +4.85 ± 0.06 +4.76 ± 0.06 +3.80 ± 0.05 +-120 +-60 +0 +60 +120 +r [Mpc/h] +-120 +-60 +0 +60 +120 +r//[Mpc/h] +reshift space +-120 +-60 +0 +60 +120 +r [Mpc/h] +real space +-120 +-60 +0 +60 +120 +r [Mpc/h] +UNet reconstruction +0.000 0.008 0.020 0.050 0.160 0.230 0.410 0.800 2.600 8.000 +(r , r//) +Figure 7. Contour of the anisotropic 2PCF 𝜉 (𝑟⊥, 𝑟∥). For comparison, the left and middle panels show the 2PCF of redshift-space halos with and without UNet +correction, and the right one shows the true 2PCF measured from real space. The contours are produced by averaging over the results of 27 test sets in bins of +size 1 Mpc/ℎ. +ated in the current study are available from the corresponding authors +on reasonable request. +REFERENCES +Abbott B. P., et al., 2016, Phys. Rev. Lett., 116, 241103 +Abbott B. P., et al., 2017, Nature, 551, 85 +Ade P. A., et al., 2016, Astronomy & Astrophysics, 594, A13 +Aghanim N., et al., 2020, Astron. 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P., 1986b, ApJ, 302, L1 +et al. T.-G., , in prep +van de Weygaert R., 2014, IAU Symp., 308, 493 +van de Weygaert R., 2016, in van de Weygaert R., Shandarin S., Saar E., +Einasto J., eds, IAU Symposium Vol. 308, The Zeldovich Universe: Gen- +esis and Growth of the Cosmic Web. pp 493–523 (arXiv:1611.01222), +doi:10.1017/S1743921316010504 +APPENDIX A: MOMENTUM FIELD RECONSTRUCTION +FROM UNET +The momentum field of galaxies and clusters of galaxies is also +cosmologically very important (Okumura et al. 2014). In the fol- +lowing, we will use the tilde symbol to denote the momentum field +and its components , i.e., ˜𝒗 for the momentum field, ˜𝜽 and ˜𝝎 for its +divergence and vorticity, respectively. The momentum field can be +defined as ˜𝒗(𝒙) = [1 + 𝛿(𝒙)]𝒗(𝒙), where 𝛿 = 𝑛/¯𝑛 − 1 is the pertur- +bation of number density field, and 𝒗 is the comoving velocity field. +The momentum field thus is the number-weighted velocity. We can +also decompose the momentum field into the divergence and vortic- +ity components, with ˜𝜃(𝒌) = 𝑖𝒌 · ˜𝒗(𝒌) and ˜𝝎(𝒌) = 𝑖𝒌 × ˜𝒗(𝒌), very +similar to the velocity field. The corresponding momentum power +spectra can be defined in the same way (as defined in Eq. 6), e.g., +𝑃 ˜𝜃 and 𝑃 ˜𝜔 for the divergence and vorticity power spectra of the +momentum field, respectively. +The reconstruction results for the momentum field are summarized +as follows. Overall, for the reconstruction of the momentum field, +UNet can achieve even better results. From Tab. A1, the resulting +correlation coefficients are at the level of 0.8, about 10% larger than +the ones shown in Tab. 1 for the velocity field. +Furthermore, let us first compare the joint probability distribu- +tions of density-divergence, and density-vorticity for the momentum +field, which are shown in Fig. A1. We do see that the reconstructed +distributions are pretty consistent with the true ones morphologi- +cally. Additionally, the reconstructions of the momentum field and +its vorticity component for two randomly selected slices are present +in Figs. A2 & A3. As seen, both of these reconstructed fields indeed +MNRAS 000, 1–15 (2022) + +15 +0 +1 +2 +3 +4 +-800 +-640 +-480 +-320 +-160 +0 +160 +320 +480 +640 + [h km/s/Mpc] +truth +0 +1 +2 +3 +4 + + + + + +UNet +100 +101 +102 +103 +104 +105 +0 +1 +2 +3 +4 +-800 +-640 +-480 +-320 +-160 +0 +160 +320 +480 +640 + [h km/s/Mpc] +truth +0 +1 +2 +3 +4 + + + + + +UNet +100 +101 +102 +103 +104 +105 +Figure A1. Same as in Fig. 4, but for the joint probability distributions of +density-divergence (upper), and density-vorticity (lower) for the momentum +field. +correlate strongly with the true ones, providing very high reconstruc- +tion accuracy at the level of 1%. Also, from the histogram distribu- +tions, the deviations on average are about 18◦ for the direction of +the momentum field, and about 23◦ for the direction of the vorticity, +respectively. +Compared with the reconstruction in power spectrum, as observed +in Fig. A4, the transfer functions demonstrate the UNet model +yielding excellent reconstruction at all scales of 𝑘 ≲ 1.1 ℎ/Mpc, +|𝑇(𝑘)| ≲ 0.15 for the momentum field, and |𝑇(𝑘)| ≲ 0.2 for both +momentum divergence and vorticity components. More interestingly, +we can also correct the peculiar velocity of each individual halo from +the UNet-reconstructed momentum field via 𝒗 = ˜𝒗/(1 + 𝛿𝑛), where +we assume the halo number density contrast 𝛿𝑛 is exactly known from +the simulations. The projected 2PCF and the associated multipoles +of 2PCF are illustrated in Fig. A5, and the corresponding transfer +function, the relative deviation between the reconstructed one and +the truth are detailed in Tab. A2. Furthermore, the comparison of +the anisotropic 2PCF between the reconstruction and the true one +are shown in Fig. A6. All of there results obviously demonstrate a +high-fidelity reconstruction of UNet. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–15 (2022) + +16 +Ziyong Wu et al. +0 +16 +32 +48 +64 +80 +X(Mpc/h) +0 +16 +32 +48 +64 +80 +Y(Mpc/h) +0 +16 +32 +48 +64 +80 + + + + + + +vtruth +0 +16 +32 +48 +64 +80 + + + + + + +vUNet +0 +200 +400 +600 +800 +1000 +|v| [km/s] +0 +25 +50 +75 +100 +125 +UNet +271.99±306.72 +truth +272.58±313.11 +0.0 +0.1 +0.2 +1 - cos +0 +50 +100 +150 +200 +250 +UNet +0.05±0.17 +0 +1 +2 +3 +4 +5 +0 +50 +100 +150 +200 +250 +300 +350 +400 +0 +16 +32 +48 +64 +80 +X(Mpc/h) +0 +16 +32 +48 +64 +80 +Y(Mpc/h) +0 +16 +32 +48 +64 +80 + + + + + + +vtruth +0 +16 +32 +48 +64 +80 + + + + + + +vUNet +0 +200 +400 +600 +800 +1000 +|v| [km/s] +0 +20 +40 +60 +80 +UNet +320.56±357.96 +truth +321.46±378.38 +0.0 +0.1 +0.2 +1 - cos +0 +50 +100 +150 +200 +UNet +0.04±0.16 +0 +1 +2 +3 +4 +5 +0 +50 +100 +150 +200 +250 +300 +350 +400 +Figure A2. Same as in Fig. 2, but for the momentum field ˜𝒗. +0 +16 +32 +48 +64 +80 +X(Mpc/h) +0 +16 +32 +48 +64 +80 +Y(Mpc/h) +0 +16 +32 +48 +64 +80 + + + + + + +truth +0 +16 +32 +48 +64 +80 + + + + + + +UNet +0 +200 +400 +600 +800 +1000 +| +| [h km/s/Mpc] +0 +100 +200 +300 +UNet +64.48±66.71 +truth +64.38±65.12 +0.0 +0.1 +0.2 +1 - cos +0 +100 +200 +300 +400 +UNet +0.05±0.23 +0 +1 +2 +3 +4 +5 +0 +20 +40 +60 +80 +100 +120 +140 +0 +16 +32 +48 +64 +80 +X(Mpc/h) +0 +16 +32 +48 +64 +80 +Y(Mpc/h) +0 +16 +32 +48 +64 +80 + + + + + + +truth +0 +16 +32 +48 +64 +80 + + + + + + +UNet +0 +200 +400 +600 +800 +1000 +| +| [h km/s/Mpc] +0 +50 +100 +150 +200 +250 +UNet +77.36±83.25 +truth +78.00±83.34 +0.0 +0.1 +0.2 +1 - cos +0 +100 +200 +300 +400 +UNet +0.08±0.29 +0 +1 +2 +3 +4 +5 +6 +7 +8 +0 +20 +40 +60 +80 +100 +120 +140 +Figure A3. Same as in Fig. 3, but for the vorticity component of the momentum field ˜𝝎. +MNRAS 000, 1–15 (2022) + +17 + + +104 +105 +106 +k3Pvv [km2/s2] +UNet +truth +0.1 +0.2 +0.4 +0.7 +1.0 +k [h Mpc +1] +0.10 +0.05 +0.00 +Tf(k) + + +103 +104 +105 +kP + [km2/s2] +UNet +truth +0.1 +0.2 +0.4 +0.7 +1.0 +k [h Mpc +1] +0.2 +0.0 +0.2 +Tf(k) + + +103 +104 +105 +kP + [km2/s2] +UNet +truth +0.1 +0.2 +0.4 +0.7 +1.0 +k [h Mpc +1] +0.2 +0.1 +0.0 +0.1 +Tf(k) +Figure A4. Same as in Fig. 5, but for the momentum field. + + + + + +7 +8 +9 +10 +11 +12 +13 +( ) +UNet +redshift space +real space +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.00 +0.25 +0.50 +Tf( ) + + + + +10 +0 +10 +20 +30 +40 +50 +60 +s2 +0(s)[h +2Mpc2] +UNet +redshift space +real space +20 +40 +60 +80 +100 +s (Mpc/h) +0.0 +0.2 +Tf(s) + + + + +20 +15 +10 +5 +0 +5 +10 +s2 +2(s)[h +2Mpc2] +UNet +redshift space +real space +20 +40 +60 +80 +100 +s (Mpc/h) +0.50 +0.25 +0.00 +Tf(s) + + + + +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +s2 +4(s)[h +2Mpc2] +UNet +redshift space +real space +5 +10 +15 +20 +25 +30 +35 +s (Mpc/h) +0 +1 +2 +Tf(s) +Figure A5. Same as in 6, but for the 2PCFs reconstructed from the momentum field. +MNRAS 000, 1–15 (2022) + +18 +Ziyong Wu et al. +-120 +-60 +0 +60 +120 +r [Mpc/h] +-120 +-60 +0 +60 +120 +r//[Mpc/h] +reshift space +-120 +-60 +0 +60 +120 +r [Mpc/h] +real space +-120 +-60 +0 +60 +120 +r [Mpc/h] +UNet reconstruction +0.000 0.008 0.020 0.050 0.160 0.230 0.410 0.800 2.600 8.000 +(r , r//) +Figure A6. Same as in Fig. 7, but for the contour of the anisotropic 2PCF reconstructed from the the momentum field. +Table A2. Same as in Tab. 2, but for the momentum field. +𝜇 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +𝜉 (𝜇)/𝜉true − 1 (UNet correction) +0.16 ± 0.02 +−0.01 ± 0.02 +−0.02 ± 0.02 +−0.03 ± 0.02 +−0.01 ± 0.02 +0.0 ± 0.02 +𝜉 (𝜇)/𝜉true − 1 (redshift space) +0.38 ± 0.03 +0.05 ± 0.02 +0.24 ± 0.02 +0.42 ± 0.02 +0.54 ± 0.02 +0.58 ± 0.03 +𝑟 (Mpc/ℎ) +20 +40 +60 +80 +100 +120 +𝜉0(𝑟)/𝜉true − 1 (UNet correction) +−0.01 ± 0.02 +0.04 ± 0.03 +0.18 ± 0.04 +0.07 ± 0.05 +−0.06 ± 0.04 +−0.34 ± 0.05 +𝜉0(𝑟)/𝜉true − 1 (redshift space) +0.33 ± 0.03 +0.37 ± 0.06 +0.41 ± 0.08 +0.62 ± 0.08 +0.30 ± 0.06 +3.81 ± 0.07 +𝑟 (Mpc/ℎ) +20 +40 +60 +80 +100 +120 +𝜉2(𝑟)/𝜉true − 1 (UNet correction) +−0.54 ± 0.07 +−0.50 ± 0.04 +−0.53 ± 0.04 +0.04 ± 0.03 +0.12 ± 0.02 +−0.50 ± 0.02 +𝜉2(𝑟)/𝜉true − 1 (redshift space) +6.51 ± 0.17 +10.22 ± 0.06 +8.40 ± 0.05 +6.21 ± 0.05 +4.91 ± 0.04 +−7.47 ± 0.04 +𝑟 (Mpc/ℎ) +5 +10 +15 +20 +25 +30 +𝜉4(𝑟)/𝜉true − 1 (UNet correction) +1.11 ± 0.31 +1.73 ± 0.09 +2.06 ± 0.08 +1.30 ± 0.06 +0.97 ± 0.04 +0.18 ± 0.04 +𝜉4(𝑟)/𝜉true − 1 (redshift space) +3.66 ± 0.60 +5.00 ± 0.13 +5.28 ± 0.09 +4.85 ± 0.06 +4.76 ± 0.06 +3.80 ± 0.05 +MNRAS 000, 1–15 (2022) + diff --git a/oNE3T4oBgHgl3EQfjQoa/content/tmp_files/load_file.txt b/oNE3T4oBgHgl3EQfjQoa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..709e7dccd0b0a4f418e08db515c25cc8ad3f7877 --- /dev/null +++ b/oNE3T4oBgHgl3EQfjQoa/content/tmp_files/load_file.txt @@ -0,0 +1,2833 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf,len=2832 +page_content='MNRAS 000, 1–15 (2022) Preprint 12 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 AI-assisted reconstruction of cosmic velocity field from redshift-space spatial distribution of halos Ziyong Wu1,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 10 Yuanhua Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Nanjing 210033,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The school of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Hangzhou 310037,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' China 5CSST Science Center for the Guangdong–Hong Kong–Macau Greater Bay Area,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' SYSU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Zhuhai 519082,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' China 6Peng Cheng Laboratory, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2, Xingke 1st Street, Shenzhen 518000, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' China 7National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Beijing 100101, China 8University of Chinese Academy of Sciences, Beijing 100049, China Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The peculiar velocities of dark matter halos are crucial to study many issues in cosmology and galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In this study, by using the state-of-the-art deep learning technique, a UNet-based neural network, we propose to reconstruct the peculiar velocity field from the redshift-space distribution of dark matter halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Through a point-to-point comparison and examination of various statistical properties, we demonstrate that, the reconstructed velocity field is in good agreement with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The power spectra of various velocity field components, including velocity magnitude, divergence and vorticity, can be successfully recovered when 𝑘 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 ℎ/Mpc (the Nyquist frequency of the simulations) at about 80% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' This approach is very promising and presents an alternative method to correct the redshift-space distortions using the measured 3D spatial information of halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Additionally, for the reconstruction of the momentum field of halos, UNet achieves similar good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Hence the applications in various aspects of cosmology are very broad, such as correcting redshift errors and improving measurements in the structure of the cosmic web, the kinetic Sunyaev-Zel’dovich effect, BAO reconstruction, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Key words: methods: data analysis, numerical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' cosmology: large-scale structure of Universe, theory CONTENTS 1 Introduction 2 Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 Dataset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 Input Preprocessing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='3 Neural Network Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 Loss Function 3 Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 Analysis on Velocity Field 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 RSD Corrections 4 Conclusion A Momentum field reconstruction from Unet ★ xiaoliang5@mail2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='cn † zhangle7@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='cn ‡ lixiaod25@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='cn 1 INTRODUCTION The Large Scale Structure (LSS) of the Universe is crucial to our study of the expansion and structure formation history of the Uni- verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In the next decade, the stage IV surveys such as DESI1, EU- CLID2, LSST3, WFIRST4,CSST,Roman and Subaru will map the Universe with extraordinary precision on an unprecedented large volume, deepening the understanding of dark energy, dark matter, gravity, the Hubble constant, the neutrino mass, and the initial con- dition of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Due to the initial inhomogeneity, the peculiar velocity field of the universe is generated together with the density field during the pro- cess of structure formation, and thus contains enormous information about LSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Accurate observations or reconstruction of the cosmic velocity field will greatly help us to quantify and understand the red- shift spatial distortions (Jackson 1972c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Kaiser 1987), baryon acous- tic oscillations (Eisenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2005, 2007), the Alcock-Paczynski effect (Alcock & Paczyński 1979b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2015b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Li 1 https://desi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='gov/ 2 http://sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='int/euclid/ 3 http://sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='int/euclid/ 4 https://wfirst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='gov/ © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04586v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='CO] 11 Jan 2023 2 Ziyong Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Ramanah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019a), the cosmic web (Bardeen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 1986b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Hahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Forero-Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Forero-Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019), the kinematic Sunyaev-Zeldovich effect (Sunyaev & Zeldovich 1972, 1980), the integrated Sachs Wolfe effect (Sachs & Wolfe 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Rees & Sciama 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Crittenden & Turok 1996), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Observationally, however, measuring the peculiar velocity of galaxies is extremely difficult, mainly as it requires redshift- independent distance estimates that can only be made by distance indicators such as type Ia Supernovae (Phillips 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Radburn-Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Turnbull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Mathews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2016), the Tully-Fisher relation (Tully & Fisher 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Masters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2006, 2008) the "fundamental plane" relation (Dressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Djorgovski & Davis 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Springob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2007), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' There- fore, great efforts have been devoted to developing alternative meth- ods of reconstructing the cosmic velocity field from the halo density field according to theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Here, the difficulty lies in the complexity arising from the nonlinear evolution of the structure and the gravitational collapse, and many studies have been made in this direction (Nusser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' (1991);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Bernardeau (1992);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Zaroubi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Croft & Gaztanaga (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Bernardeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Kudlicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Branchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Mohayaee & Tully (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Lavaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Bilicki & Chodorowski (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Kitaura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Jennings & Jennings (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Ata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Recent tremendous advances in machine learning algorithms, es- pecially those based on deep neural networks, provide us with a great opportunity to extract useful information from complex data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In more recent years, deep learning-based techniques have been applied to al- most all areas of cosmology and astrophysics (Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Jennings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Carleo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Ntampaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019), such as weak gravitational lensing (Schmelzle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2018;' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Peel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Tewes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019), the Cosmic Microwave Background (Caldeira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Perraudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Münchmeyer & Smith 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019), the LSS including estimating cosmological parameters from the distribution of matter (Ravanbakhsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Lucie-Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Lazanu 2021), identifying dark matter halos and reconstruct the initial conditions of the universe using ma- chine learning (Modi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Berger & Stein 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Lucie-Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Ramanah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019c), mapping rough cosmology to fine one (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2021), extracting line intensity maps (Pfeffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019), foreground removal in 21cm intensity mapping (Makinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2021), augmenting N-body simulations with gas (Tröster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019), a mapping between the 3D galaxy dis- tribution in hydrodynamic simulations and its underlying dark mat- ter distribution (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019a), modelling small-scale galaxy formation physics in large cosmological volumes (Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2021), reconstructing the baryon acoustic oscillations (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2020) and reconstructing the initial linear-regime matter density field (Shallue & Eisenstein 2022), searching for gravitational waves (Dreissigacker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Gebhard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019) and cosmic reionization (La Plante & Ntampaka 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Gillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Chardin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019a), as well as supernovae (Lochner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Moss 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Ishida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Muthukrishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2019), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' For velocity reconstruction, the pioneering work (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2021) shows that a UNet network can reconstruct the nonlinear velocity field of dark matter particles with high precision down to a scale of 2 ℎ−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' When pushing down to highly non-linear scales of 𝑘 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 ℎ−1Mpc, they could achieve 90% accuracy in reconstructing the power spectra of the velocity and momentum fields of the magni- tude, the divergence and the vorticity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' This demonstrates that, compared with the traditional perturbation-based theory, deep learning methods would be more effective and have a great advantage in reconstructing the cosmic velocity field at nonlinear scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' More importantly, it is widely believed that the dark matter halos and subhalos well trace the galaxy distributions, and their clustering properties approach those of real observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' However, technically, it is more challenging to reconstruct the velocity field from halos, since the halos only reside at density peaks and become much more sparse than simulation particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Therefore, in this study, we propose a modified UNet model dedi- cated to the reconstruction of the velocity field of dark matter halos (and subhalos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' From our simulation tests, this proposed method can reconstruct the peculiar velocities of each individual halos on high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' It turns out that both the velocity and real-space density fields down to the non-linear scales can be well inferred from a redshift-space measurement alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Therefore, this study is a major step toward applying the deep leaning technique to real observational data, which is extremely important for cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The layout of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2, we introduce the simulation data used in this study and detail the architecture choice of the neural network and the training procedure, as well as the validation tests in velocity reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Results for our network are presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3, and finally the conclusion and discussion are present in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' For a comparison to the velocity reconstruction we discuss in this paper, we present reconstruction results for the corresponding mo- mentum field (the number-weighted velocity) of halos in Appendix A, obatined with the same UNet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2 METHOD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 Dataset To train and validate our deep learning framework, the training and tests data sets are based on the dark matter halos/subhalos of the Big- MultiDark (BigMD) Planck simulation5, which is the high-resolution N-body simulation described in Klypin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' (2016b) and was per- formed with GADGET-2 (Springel 2005b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The simulation was cre- ated in a box of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='5ℎ−1 Gpc on each side, with 38403 dark matter particles and the mass resolution of 𝑀DM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 × 1010ℎ−1M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The initial conditions are generated with Zeldovich approximation at 𝑧init = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The simulation provides 79 redshift snapshots in the range of 0-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' For the analysis, we use the ROCKSTAR (Robust Overdensity Calculation using K-Space Topologically Adaptive Re- finement) halo finder (Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2013b) to identify spherical dark matter halos/subhalos in the simulation, based on adaptive hi- erarchical refinement of friends-of-friends groups in six phase-space dimensions and one time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' ROCKSTAR provides halo mass using spherical overdensities of a virial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The cosmology we assume in this study is the standard flat ΛCDM, compatible with Planck 2018 results (Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2020), with the fiducial parameters of {Ω𝑚, Ω𝑏, ℎ, 𝑛𝑠, 𝜎8} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='307, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='048, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='677, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='961, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='828}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We construct the redshift-space halo/subhalo catalogue at 𝑧 = 0 with the number density of 10−3(Mpc/ℎ)3 fixed, to be compatible with current spectral observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The redshift-space position s is 5 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='cosmosim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='org MNRAS 000, 1–15 (2022) 3 related to the real-space position r for a distant observer along the line of sight by 𝒔 = 𝒓 + 𝒗 · ˆ𝑧 𝑎𝐻(𝑎) , (1) where 𝒗 is the peculiar velocity, 𝑎 is scalar factor and 𝐻 is the Hubble parameter, and the unit vector ˆ𝑧 denotes the line-of-sight direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Based on the catalogue samples, we compute the density field and velocity field in the mesh cells by assigning the particle mass to a 9003 mesh using the CIC (Cloud-in-Cell) scheme, with a cell resolution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='78ℎ−1Mpc)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 Input Preprocessing Our framework consists of the prepossessing of the input dataset and there are some points need to be clarified, as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 1) In our UNet model, the input is a 6-channel 3D number density map of halos (and subhalos) in redshift space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' For each channel, the map contains only halos in a certain mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' To do so, we sort the halo (and subhalo) sample by mass in descending order and split it into six mass intervals, with the bin edges: log10(𝑀/𝑀⊙) ∈ [13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='52, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='12, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='84, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='62, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='43, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='30], corresponding to binning the halo mass in percentiles of [5, 15, 30, 50, 75, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The reason for doing so is that 1) the features of the velocity field may be significantly different among different masses of halos, which may be more effective for neural network learning, and 2) in observations we can have approximately estimated mass of halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2) Based on the limitations in size of GPU memory, training time and model size, we have to divide the large box of side length 2500 Mpc/ℎ into 8000 smaller boxes, each with side 125 Mpc/ℎ (453 meshgrid points in CIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3) Given that the box division procedure and physically small boxes lead to loss of large-scale velocity modes, we thus use the linear perturbation theory to compensate for such loss in the training data with a set of small box simulations (125 Mpc/ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The velocity field 𝒗 in the linear regime are directly related to the density field 𝛿 through 𝒗(𝒌) = 𝑎 𝑓 𝐻 𝑖𝒌 𝑘2 𝛿(𝒌) , (2) where both fields are expressed in Fourier space, and 𝑓 = 𝑑 ln 𝐷/𝑑 ln 𝑎 denotes the growth rate with 𝐷 the growth factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Using this linear prediction, the velocity field at large scale can be exactly calculated in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 4) To ensure that the training results of the model achieve good rotational invariance, we use a data augmentation method, in which the input data to the model for training are randomly rotated by one of 18 rotational transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 5) Instead of learning the 3D velocity vector field directly, we de- compose the velocity vector into two parts: magnitude and direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The predicted velocity field is then reconstructed using these two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 6) Since the dynamic range of the velocity field is very wide, to improve the accuracy and the convergence speed, the velocity magnitude in the output isnormalized,wherethenormalization factor 𝑐 is chosen by 𝑐 = 1/200 for 𝑣 ⩾ 60 km/s and 𝑐 = 1/12 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Therefore, the output of the velocity magnitude contains two parts, the large velocity one and the small one, labelled by 𝑣large & 𝑣smaller, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 7) In addition to the velocity field, we have also trained the model to account for the momentum field as an output and the results are summarized in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As our dataset is large enough, in order to test the performance of our neural network model, we split the dataset consisting of 8000 subboxes into a training set of 500 subboxes, a validation set of 300 subboxes to prevent overfitting and the rest are for a test set which is used for the final test of the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Note that the training and test sets are selected in different regions of the large box to prevent any potential correlations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='3 Neural Network Model Motivated by the neural network model (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2021), we use a modified UNet neural network architecture for model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The architecture of our neural network and its components are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The input is the 6-channel number density field of halos, each channel corresponding to number density field for a certain mass range of halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2, as the velocity field is decomposed into the two parts: velocity magnitude and the velocity direction, we build two structurally similar neural networks to deal with them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The network ends with the output layer of 2+3 fields, three of which correspond to the components of velocity direction (ˆ𝑣𝑥, ˆ𝑣𝑦, ˆ𝑣𝑧) and two to the velocity magnitude (𝑣large, 𝑣smaller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' A complete reconstruction of the 3D velocity field is finally achieved by combining all of the output field components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' More specifically, the detail of the UNet network are shown on the bottom-left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The colored plates represent different operations in the neural network, which are connected from the in- puts to the outputs by means of arrow lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The size of the input, the intermediate and the output fields (number of channels × spatial pixels) is specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Also, the size and the number of 3D convolu- tion kernels ("conv") are also labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Moreover, the combination of padding schemes gives the desired dimensionality after each con- volution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Note that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 1) the "init" 3D convolutional layers allow for a sufficiently large receptive field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' enabling the network to quickly learn the large-scale information in the beginning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2) the "output" convo- lutional layers increase complexity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' followed by the dropout layers to avoid overfitting and to change the number of channels at the end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' and 3) the batch normalization (BN) layer and the rectified linear unit (ReLU) activation layer after convolutional layers can speed up the training convergence and prevents neural networks from overfitting as well as increasing nonlinear effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' With the trained UNet, the velocity field of halos is predicted by feeding the number density field of halos in the redshift space, and the relevant statistics such as clustering information can be measured straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' A crucial ingredient in our model is a three-block structure: an lower convolution block (red), a upper convolution block (pink), and a final one (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The advantages of these blocks are capable of pass- ing the initial information to the deep network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In parallel, to avoid the bias in small-box simulations, the linear theory-predicted velocity field is used as an additional input in the lower block, com- pensating for the lack of information on large-scale velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The final convolution block ensures that we can correctly learn the ve- locity field at the center of the number density field and prevent any spurious signals due to boundary effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 Loss Function The objective of the training our UNet is to minimize a loss func- tion between prediction, 𝒗, and simulation truth, 𝒗true of each voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Specifically, to account for the contributions from the velocity mag- nitude (𝑣 ≡ |𝒗|) and the velocity direction (unit vector ˆ𝒗 ≡ 𝒗/𝑣), we MNRAS 000, 1–15 (2022) 4 Ziyong Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='linear velocity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='6x513 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='halo density 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='final ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' UNet neural network architecture and training scheme used for the velocity (momentum) reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Starting with a 6-channel 513-voxel input layer that corresponds to the number density field (a side length of 142 Mpc/ℎ) of halos for the six different mass intervals (over the mass range of 1012–1015𝑀⊙) in redshift space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' our model is consisted of two U-net neural network architecture for reconstruction of velocity magnitude and direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' where one contains two channels corresponding to the large and small velocity fields (𝑣large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 𝑣small),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' and the other consists of three channels corresponding to the three velocity directions (𝑣𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 𝑣𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 𝑣𝑧) (upper left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' This U-net architecture essentially consists of the upper, lower and final convolution blocks, together with a compensation for the linear velocity field (upper right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The dimension of each output field is 453, corresponding to a box volume of 1253 Mpc3ℎ−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The lower-right part shows the details of the components given in the three-block structure of the UNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The layers of "init", "conv", "trans" and "output" are detailed on the lower-left part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In the final convolution block, a dropout layer between the convolution transform and batchnorm layers is used to enhance the UNet performance and prevent overfitting, where the dropuout value is chosen as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' choose the following loss function with two terms, L = 1 𝑁 𝑁 ∑︁ 𝑖=1 � 2 5 (𝑣𝑖 − 𝑣true 𝑖 )2 + 3 5 (1 − cos 𝜃𝑖) � , (3) where cos 𝜃𝑖 ≡ ˆ𝒗𝑖 · ˆ𝒗true 𝑖 , and the index 𝑖 denotes the 𝑖-th voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As observed, the first term is responsible for 𝑣, and corresponds to the standard and simple mean error (MSE) loss that is essentially equivalent to the maximum likelihood solution under a Gaussian assumption with constant variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The second term naturally mea- sures the deviation between the reconstructed and the true values of ˆ𝒗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The coefficients of these two terms can be regarded as normal- ization factors and are determined by the number of channels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', 2 for magnitude (𝑣large, 𝑣small), and 3 for direction (ˆ𝑣𝑥, ˆ𝑣𝑦, ˆ𝑣𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Em- pirically, such loss function is effective, and has proven to be stable and effective during our training process, providing good results in the velocity (momentum) reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We trained our UNet using the most popular algorithm Adam (Kingma & Ba 2014) for train- ing deep neural networks, which can iteratively decrease the training loss by calculating its gradient with respect to model parameters and performing a small step along the direction with the maximum decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3 RESULTS In this section, we test the performance of the trained UNet model and present our results predicted from 27 test sets, each consisting of 125 small boxes with same volume as that of the training sets (side length 125 Mpc/ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Therefore, the box volume for each test set we have performed the analysis on is 6253(ℎ−1Mpc)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We chose this because measurements on large boxes would have a better statistical behavior, reducing the statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We also find the results for the other simulation boxes are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' To ensure reliable test results, these simulation boxes was not used for the model training and refining the model structure/training parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' First we shall describe the statistics we will use throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The 2-point correlation function is one of the most commonly used statistics to characterize a homogeneous density field, 𝜉(𝒓) = ⟨𝛿 (𝒙) 𝛿 (𝒙 + 𝒓)⟩ , (4) where 𝛿(𝒙) is the density contrast field, 𝒙 denote for any point, 𝒓 is a separation vector, and ⟨·⟩ stands for the ensemble mean, computed with a spatial mean over 𝒙 in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The power spectrum of 𝛿 (𝒙) is just related to 𝜉(𝒓) by the Fourier transform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', 𝑃(𝒌) = ∫ 𝜉(𝒓)e𝑖𝒌·𝒓d3𝒓 , (5) where 𝒌 is the 3D wavevector of the plane wave, with the magnitude 𝑘 ≡ |𝒌| (the wavenumber) related to the wavelength 𝜆 by 𝑘 = 2𝜋/𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' MNRAS 000, 1–15 (2022) 125Mpc/hrlarge Mpc/hrsmall Mpc/h125 Mpc/h125Mpc/hredshift space 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05 142Mpc/hredshift space 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='15 142 Mpc/hredshift space 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='30 142Mpc/hredshift space 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='50 oc/hredshiftspace 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='75 Mpc/hredshift space 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 142Mpc/h5 Similar to the scalar field 𝛿, we can also define power spectra for velocity and momentum vector fields of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As known, the velocity field, 𝒗, is completely described by its divergence, 𝜃 ≡ ∇ · 𝒗 and its vorticity, 𝝎 = ∇ × 𝒗, which, in Fourier space, become purely radial and transversal velocity modes, respectively, defined by 𝜃(𝒌) = 𝑖𝒌 · 𝒗(𝒌) and 𝝎(𝒌) = 𝑖𝒌 × 𝒗(𝒌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The power spectra of the velocity, divergence, vorticity and velocity magnitude are given by ⟨𝜃(𝒌)𝜃∗(𝒌′)⟩ =(2𝜋)3𝑃𝜃 (𝑘)𝛿(𝒌 − k′) , ⟨𝜔𝑖(𝒌)𝜔∗𝑗 (𝒌′)⟩ =(2𝜋)3 1 2 � 𝛿𝑖 𝑗 − 𝑘𝑖𝑘 𝑗 𝑘2 � 𝑃𝜔(𝒌)𝛿(𝒌 − 𝒌′) , ⟨𝒗(𝒌) · 𝒗∗(𝒌′)⟩ =(2𝜋)3𝑃𝑣 (𝒌)𝛿(𝒌 − 𝒌′) , (6) where indices 𝑖, 𝑗 denote the components in the Fourier space coor- dinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In the linear perturbation theory, the continuity equation leads to 𝜃 = −H 𝑓 𝛿, where H = 𝑎𝐻 is the conformal Hubble parameter, 𝑎 denotes the cosmic scale factor and 𝑓 is the linear growth rate defined by 𝑓 = 𝑑 ln 𝐷/𝑑 ln 𝑎, with 𝐷 being the linear density growth factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In a ΛCDM model, 𝑓 ≈ Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='6 m (Peebles 1980), with a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 Analysis on Velocity Field First we shall describe the metrics that we will use throughout this section for evaluating the reconstruction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' For an arbitrary reconstructed field of halos from UNet, denoted by the shorthand notation 𝑓 , where 𝑓 ∈ {𝜃, 𝝎, 𝒗} for velocity, we use the following metrics, the so-called transfer function and correlation coefficient, to compare a reconstructed field ( 𝑓 ) with the true one ( 𝑓 ′): 𝑇𝑓 = O 𝑓 O 𝑓 ′ − 1 , 𝐶 𝑓 = 1 𝑁pix − 1 ∑︁ 𝑖 ( 𝑓𝑖 − ¯𝑓 )( 𝑓 ′ 𝑖 − ¯𝑓 ′) 𝜎𝑓 𝜎𝑓 ′ , (7) where O 𝑓 stands for an arbitrary observable for 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The correlation 𝐶 𝑓 is defined between reconstructed ( 𝑓 ) and true fields ( 𝑓 ′) with the same total number of pixels 𝑁pix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The sample mean and the standard deviation of field 𝑓 are denoted by ¯𝑓 and 𝜎𝑓 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In this study, we make a comparison of the statistical properties of clustering, through various observables such as the power spectra of velocity components and the two-point correlation function (2PCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Both metrics provide a physical insight for comparison such that the perfect reconstruction is equivalent to 𝑇𝑓 = 0 and to 𝐶 𝑓 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 Visual inspection and point-wise comparison As a first validation, we perform a point-wise comparison between the UNet-predicted halo velocity field to the simulation truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' To do so, we randomly selected three thin slices in the test sets, each with volume of 83 × 83 × 28 ℎ−3Mpc3, with spatial resolution of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='78 Mpc/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2 visualizes the number density distribution of dark mat- ter halos (and subhalos) distribution and velocity field in these three slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As seen, there are many massive halos in the range of 𝑀/𝑀⊙ ∈ [1012, 1015], typically with 60 halos per slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In the middle and right panels, we display the true and the predicted velocity fields, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The colored arrows show the average velocities at the meshgrid points and are projected onto the image plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The length and the direction of the arrow represent the magnitude and the direction of the projected velocity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' To show clearly, the projected velocity magnitude is also scaled by the color from purple to red, reflecting the halo velocities from small to large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As seen, a Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Summary of the correlation coefficients 𝐶 𝑓 between the recon- structed and true fields of the velocity 𝒗, the divergence component 𝜃 and the vorticity 𝝎 via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 7, estimated by averaging over 27 test sets, each with the box size of 625 Mpc/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' field 𝒗 𝜃 𝝎 𝐶 𝑓 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='65 high number density region typically leads to a larger velocity field, since the gravitational collapse and non-linear structure formation occur intensively there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Moreover, the visualized morphology for the UNet-predicted and the true velocity fields clearly indicates the effec- tiveness of our neural network, as they are almost indistinguishable by eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Interestingly, although the simulated halo velocity field is sparse, we can still reliably reconstruct it, especially for the regions with small velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' To quantitatively validate such reconstruction, we show the histogram distributions (in the rightmost panel right panels) of magnitude and direction of the velocities in these three slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We do find that, statistically, the distributions of the recon- structed velocity magnitude and direction agree well with the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Specifically, the mean value and its 1𝜎 dispersion (with a Guassian fit) for the UNet-reconstructed velocity magnitude in each slice are 953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='88 ± 731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='08 and 1222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='17 ± 904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='43 km/s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' These mean values are consistent with the true ones at about higher than 99% accuracy among all these slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Similarly, we obtain very good results in the direction reconstruction, with 1 − cos 𝜃 (defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='07 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='06 for these slices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' By averaging over the slices, the deviation in the velocity direction, Δ𝜃 = |𝜃true −𝜃UNet|, achieves Δ𝜃 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1◦ ±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='9◦, implying Δ𝜃 < 30◦ at 1𝜎 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Furthermore, let us focus on the vorticity field halo momentum fields, ˜𝝎, which generally appears in high-density regions of halos and is essentially induced by non-linear structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The re- construction of the vorticity field for two randomly selected slices is present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As seen, the vorticity field is tightly concentrated on high-density regions where the nonlinear processes such as shell crossing are occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Thus its magnitude is tightly coupled to the local density and decays rapidly at the linear regime (low density) of structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Also, its direction seems to be distributed randomly, indicating that these high-density regions have strong a nonlinear process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Due to the effect of nonlinearity on small scales, the vorticity fields are theoretically very difficult to reconstruct, es- pecially through sparse halo samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Here, however, we show the advantages of UNet, which can provide the reconstruction in |𝝎| at very high accuracy, with the deviations in the mean and dispersion only about |𝝎true−𝝎UNet| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='71±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='15 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='5±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='90 h· km/s/Mpc for these two slices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Also, from the histogram distribu- tion, the reconstructed directions for the vorticity component indicate that the reconstruction is unbiased, with the mean and 1𝜎 error of Δ𝜃 ≈ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0◦ ± 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In order to quantitatively compare the reconstructed and real fields, we compute the coefficients, 𝐶 𝑓 , through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 7 for various velocity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The resulting coefficients are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 1, estimated by averaging over 27 test sets, each with the box size of side 625 Mpc/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Our proposed UNet model has excellent performance in terms of the linear correlation in real-space domain, demonstrating that the network produces high-fidelity reconstructions in 𝒗, 𝜃 and 𝝎, with 𝐶 𝑓 in the range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='71, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As seen, the reconstruction in 𝝎 is almost as good as in the other components, which indicates MNRAS 000, 1–15 (2022) 6 Ziyong Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 0 16 32 48 64 80 X(Mpc/h) 0 16 32 48 64 80 Y(Mpc/h) 0 16 32 48 64 80 vtruth 0 16 32 48 64 80 vUNet 0 1000 2000 3000 4000 5000 |v| [km/s] 0 20 40 60 UNet 953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='88±731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='08 truth 953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='31±700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 1 - cos 0 50 100 150 200 250 UNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0 200 400 600 800 1000 1200 1400 0 16 32 48 64 80 X(Mpc/h) 0 16 32 48 64 80 Y(Mpc/h) 0 16 32 48 64 80 vtruth 0 16 32 48 64 80 vUNet 0 1000 2000 3000 4000 5000 |v| [km/s] 0 20 40 60 UNet 1222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='17±904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='43 truth 1231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='59±889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 1 - cos 0 100 200 300 UNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='06 0 1 2 3 4 5 6 7 8 0 200 400 600 800 1000 1200 1400 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Point-wise comparison between the UNet-reconstructed velocity field and true one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' From top to bottom, we show the results for the three three thin slices in the test sets, each with volume of 83 × 83 × 28 ℎ−3Mpc3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' From left to right, the fields of halo number density, the UNet-reconstructed velocity, the true velocity and the corresponding histogram are shown, respectively, where all quantities are measured in redshift space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The length and the orientation of the colored arrows in the velocity fields represent the velocity magnitude and direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' For each slice, the rightmost panels show the statistical histogram distributions of the velocity samples for magnitude and direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' UNet is able to reconstruct velocity well via visual inspection and the statistical analysis on the histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 0 16 32 48 64 80 X(Mpc/h) 0 16 32 48 64 80 Y(Mpc/h) 0 16 32 48 64 80 truth 0 16 32 48 64 80 UNet 0 200 400 600 800 1000 | | [h km/s/Mpc] 0 50 100 150 UNet 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='86±100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='44 truth 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='15±103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 1 - cos 0 100 200 300 UNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='20 0 1 2 3 0 50 100 150 200 250 300 0 16 32 48 64 80 X(Mpc/h) 0 16 32 48 64 80 Y(Mpc/h) 0 16 32 48 64 80 truth 0 16 32 48 64 80 UNet 0 200 400 600 800 1000 | | [h km/s/Mpc] 0 25 50 75 100 125 UNet 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='42±134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05 truth 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='92±127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 1 - cos 0 100 200 300 UNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='28 0 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2, but for the vorticity field of halo velocities, 𝜔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As seen, the vorticity field, exclusively generated from the nonlinear structure formation, has a more complex distribution pattern than that of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In spite of this, the Unet approach still performs very well in reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' MNRAS 000, 1–15 (2022) 7 0 1 2 3 4 1600 1280 960 640 320 0 320 640 960 1280 [h km/s/Mpc] truth 0 1 2 3 4 UNet 100 101 102 103 104 105 0 1 2 3 4 1600 1280 960 640 320 0 320 640 960 1280 [h km/s/Mpc] truth 0 1 2 3 4 UNet 100 101 102 103 104 105 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Joint probability distributions of density-divergence, 𝜌( 𝛿, 𝜃) (up- per), and density-vorticity, 𝜌( 𝛿, |𝝎|) (lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In each case, the distribution results are calculated from 5 test sets, each with the box size of 625 Mpc/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As seen, the predicted velocity distributions (right) agree well with the simulation truth (left) for all of the halo number densities of 𝛿 ∈ [0, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' that the UNet model can give a good prediction for the vorticity field with complicated morphological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' To further test the reconstructed velocity field with the ground truth, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 4, a visual inspection for the joint probability distri- butions of density-divergence and density-vorticity are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Obvi- ously, all predictions are in good agreement with the the true values, even in the very high density regions (𝛿 ≫ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Furthermore, we find that, for a given 𝛿, the reconstructed distributions appear slightly narrower than the true ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' This is probably because the neural network would slightly lose some random perturbation information when learning and compressing the information in the training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 Comparison for power spectrum Here we describe the reconstruction accuracy for the power spec- tra of velocity components, 𝑃 𝑓 (𝒌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' For each component, we have computed the transfer function (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 7) in terms of the predicted power spectrum and the true one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Intuitively, by use of power spec- trum as the observable, 𝑇𝑓 measures the accuracy of reconstruction in magnitude as a function of wavevector in Fourier domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Taking the directional average, the function 𝑇𝑓 (𝑘) represents the transfer function with spatial averaging over |𝒌| bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Also, in general, 𝑇(𝑘) is not explicitly optimized during the training stage, since the training minimizes the proposed loss function (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3) composed of the velocity magnitude and direction in real-space domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 5, there is a constant systematic bias to slightly underestimate the power spectrum for the velocity field 𝒗 over all scales, 𝑇𝑓 (𝑘) ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' This underestimate may be due to the fact that the compensation of the linear theory-predicted velocity field is not perfect in UNet learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' For the divergence component 𝜃, the deviation varies for positive to negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', 𝑇𝑓 (𝑘) ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2] for 𝑘 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 ℎ/Mpc and 𝑇𝑓 (𝑘) ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2, 0] otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As known, the vorticity 𝝎 is generated by nonlinear evolution, and so its reconstruc- tion has always been a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' However, we find the reconstructed vorticity power spectrum successfully match the true one, yielding a similar deviation level as in 𝒗 and 𝜃, with 𝑇𝑓 (𝑘) ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='20] at 𝑘 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 ℎ/Mpc and 𝑇𝑓 (𝑘) ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' This is remarkable considering that the UNet model performs well from the linear to deeply nonlinear scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' All these test results highlight the ability of UNet in learning various velocity components from the halo number density field, especially on the nonlinear scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 RSD Corrections An important application of the UNet-based velocity reconstruction is to map a halo distribution from redshift to real space as well as inferring the distances of individual halos (galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' To do so, redshift-space distortions (RSD) are corrected by moving the ha- los from redshift to real space according to their peculiar velocities reconstructed from the halo number density field using the trained UNet network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' By performing a tri-linear interpolation of the recon- structed velocity field, the velocities at every halo positions can be obtained with reasonable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In the following, we will present the performance of such RSD correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 Two-point correlation function In redshift space, anisotropic two-point correlation function (2PCF), 𝜉(𝒓), provides a measurement for halo (galaxy) clustering through the standard Landy & Szalay (1993a) estimator, 𝜉(𝑟, 𝜇) = 𝐷𝐷(𝑟, 𝜇) − 2𝐷𝑅(𝑟, 𝜇) + 𝑅𝑅(𝑟, 𝜇) 𝑅𝑅(𝑟, 𝜇) (8) where 𝐷𝐷, 𝐷𝑅, and 𝑅𝑅 are the normalized galaxy-galaxy, galaxy- random, and random-random number of pairs with separation (𝑟, 𝜇), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Here the 3D separation vector between pairs of objects, 𝒓, has been decomposed into (𝑟, 𝜇) coordinates, where 𝑟 is the norm of the separation vector and 𝜇 is the cosine of the angle between the line-of-sight and separation vector directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' It is common to expand 2PCF into Legendre polynomials as 𝜉(𝒓) = ∞ ∑︁ ℓ=0 𝜉ℓ (𝑟)𝐿ℓ (𝜇) , (9) with 𝜉ℓ (𝑟) = 2ℓ + 1 2 ∫ 1 −1 𝜉(𝑟, 𝜇)𝐿ℓ (𝜇)𝑑𝜇 , (10) where 𝐿ℓ is the Legendre polynomial of order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Throughout this study, ignoring the more noisy subsequent orders, we only take into account ℓ = 0, 2 and 4 multipoles, referred to as monopole, quadrupole, and hexadecapole, respectively, where 𝐿0 = 1, 𝐿2 = � 3𝜇2 − 1 � /2 and 𝐿4 = � 35𝜇4 − 30𝜇2 + 3 � /8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Due to the symmetry of object pairs, only even multipoles do not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In practice, the pair counts are linearly binned with width of Δ𝑟 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 Mpc in 𝑟 and Δ𝜇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='025 in 𝜇 for the above estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We often measure a 1D 2PCF, 𝜉(𝜇), that projects the 2D correla- tion 𝜉(𝑟, 𝜇) along the 𝑟 axis, 𝜉1D(𝜇) = ∫ ∞ 0 𝑑𝑟𝜉(𝑟, 𝜇)𝑑𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' (11) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 6 shows the projected 1D 2PCF and the resultant monopole, quadrupole and hexadecapole of 2PCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The line and the shaded area in each panel give the mean and 1𝜎 standard deviation, mea- sured from 27 test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As seen, due to the Kaiser effect (Kaiser 1987), the enhancement of power over all scales is very remarkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Meanwhile, in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2, we summarize the ratios of the redshift-space measurements with and without UNet correction to the real-space MNRAS 000, 1–15 (2022) 8 Ziyong Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 104 105 106 k3Pvv [km2/s2] UNet truth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 k [h Mpc 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 Tf(k) 104 105 106 kP [km2/s2] UNet truth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 k [h Mpc 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 Tf(k) 103 104 105 kP [km2/s2] UNet truth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 k [h Mpc 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 Tf(k) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Comparison of the UNet-predicted power spectrum and the simulation truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' From left to right, we show the the reconstruction for the velocity, the velocity divergence and the vorticity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' These results are based on the 27 test sets, each with the box of side length 625 Mpc/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' For each case, the shaded area gives 1𝜎 deviation measured from these test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' measurements (the simultion truth), which are shown in the lower panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Overall, the RSD effects are prominent, whereas they can be highly corrected by UNet with the differences within 1𝜎 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' To highlight the changes due to RSD (Kasier and fingers-of- god effects), we show the 1D projected 2PCF, 𝜉1D, with the variable 𝑟 integrated out, where the 2PCF without any correction strongly deviates from the real-space one (shown in the upper-left panel) with errors of tens of percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' However, the UNet model can accurately correct the RSD effects using the reconstructed velocity field, statisti- cally leading to the correct clustering of halos in almost all directions with an error of 1–3% (except for 𝜇 = 1 with the relative deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' More importantly, after the UNet correction, the results for 𝜉0 at ∼ 100 Mpc/ℎ demonstrate that, the baryon acoustic oscillations (BAO) can be well recovered from redshift space, deriving a very close BAO peak to the real-space one, with about 12% lower than the true one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Interestingly, the correction for the quadrupole leads to good agreement with the true real-space one not only on small scales, but also on large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Even for 𝜉4 with a much smaller signal-to-noise ratio than the other multipoles, the RSD effects can also be removed successfully, without any visible artificial effects such as oscillations and spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The high-quality in the velocity reconstruction can be also appre- ciated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 7, displaying the 2D anisotropic correlation function of redshift-space halos, 𝜉(𝒓), where the separation vector has been decomposed into line-of-sight and transverse separations such that 𝒓 = (𝑟⊥, 𝑟 ∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The contours are calculated based on the averaged result of 𝜉(𝑟⊥, 𝑟 ∥) on the 27 test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As observed, without any RSD cor- rections, the anisotropic pattern is very distinctive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The Kaiser effect leads to galaxy clusters appearing "squashed" along the line-of-sight by a coherent infall onto galaxy clusters cancel some of the Hub- ble flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Besides, the random velocities attained by galaxies in the non-linear regime produce the so-called fingers-of-god (FoG) effect, making structures elongated along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As expected, the measured anisotropic correlation function in redshfit space present a BAO feature at 𝑟 ≃ 100 Mpc/ℎ, as well as the impacts of the Kasier and the FoG effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Compared with the UNet-corrected results, we find the isotropy of the correlation function is well recovered at all scales, demonstrating the effectiveness of the UNet approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Re- markably, our proposed method not only corrects Kasier effect on large scales, but also on small scales with 𝑟 ≲ 10 ℎ/Mpc, where the FoG effect is well removed, indicating that UNet can even accurately reconstruct the velocity field in the nonlinear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 4 CONCLUSION 3D velocity (and momentum) fields constructed by galaxies and clus- ters are very important in cosmology because they provide more information than the density field alone, and would help to im- prove/correct various cosmological measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' High-fidelity re- construction may even result in unexpected findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Accurate reconstruction is often a challenge for traditional recon- struction methods, typically relying on many assumptions and ap- proximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In this study, we have proposed an alternative scheme, a deep learning approach based on the UNet neural network to recon- struct the 3D velocity/momentum fields of halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We find the UNet is well-suited for reconstructing such fields directly from the halo (and subhalos) density field, because the UNet is an elegant architecture that can effectively capture various features/structures of the fields at all scales and is very effective in transforming high-dimensional and structured inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Using multiple redshift-space halo number density fields in different mass ranges, the UNet surprisingly learned how to transform halo density fields directly into velocity/momentum fields from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We have performed a detailed validation with various statistics tests, and find the reconstructed velocity/momentum fields well agree the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Furthermore, using the inferred velocity fields, the RSD effects can be well corrected by Unet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As an important application, we find that, the reconstructed velocity field directly provides a recovery of the real-space positions of individual halos, offering a perfect correction for the RSD effects down to a highly non-linear scale of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='13 Mpc/ℎ, which is the Nyquist frequency (𝑘Ny = 𝜋𝑁/𝐿) of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' This UNet-based approach is promising for many cosmological ap- plications in terms of correcting the peculiar velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' For example, the reconstruction of cosmic volume-weighted velocity suffers se- vere sampling artifacts in measurements (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We will further extend our UNet model to MNRAS 000, 1–15 (2022) 9 7 8 9 10 11 12 13 ( ) UNet redshift space real space 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='50 Tf( ) 10 0 10 20 30 40 50 60 s2 0(s)[h 2Mpc2] UNet redshift space real space 20 40 60 80 100 s (Mpc/h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 Tf(s) 20 15 10 5 0 5 10 s2 2(s)[h 2Mpc2] UNet redshift space real space 20 40 60 80 100 s (Mpc/h) 0 1 Tf(s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 s2 4(s)[h 2Mpc2] UNet redshift space real space 5 10 15 20 25 30 35 s (Mpc/h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='50 Tf(s) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Comparison of the projected 2PCF, 𝜉1D(𝜇) of the halo distribution (top-left) and the multipoles of 2PCF, including the monopole 𝜉0 (top-right), the quadrupole 𝜉2 (bottom-left) and the hexadecapole 𝜉4 (bottom-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The line and shaded area give the the mean and 1𝜎 standard deviation measured from the 27 test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The measurements from the halo (and subhalos) positions in real space (gray), the redshift space (pink) and the redshift space corrected by the UNet-predicted velocity field (blue) are shown, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The lower panel gives the transfer function calculated by the ratio between the redshift-space measurements after the UNet correction and the real-space measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As seen, the real-space and UNet-corrected redshift-space measurements of 2PCFs are in good agreement through the transfer function, with differences within the 1𝜎 standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' tackle this long-standing problem and leave such a study for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As the stage IV galaxy surveys will provide more detailed mea- surements of the LSS of the Universe than ever before, new com- puting technologies are being called upon to fully analyze these high-dimensional, massive amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Therefor, UNet-based neural networks promise to be a powerful tool to overcome the prob- lems that traditional methods are difficult to deal with and to extract cosmological information in more depth and in a holistic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work is supported by the National Key R&D Program of China (2018YFA0404504, 2018YFA0404601, 2020YFC2201600), the Ministry of Science and Technology of China (2020SKA0110402, 2020SKA0110401, 2020SKA0110100), National Science Founda- tion of China (11890691, 11621303, 11653003, 11803094), the China Manned Space Project with No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' CMS-CSST-2021 (A02, A03, B01), the Major Key Project of PCL, the 111 project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' B20019, the CAS Interdisciplinary Innovation Team (JCTD-2019- 05), and the Science and Technology Program of Guangzhou, China (202002030360).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We acknowledge the use of Kunlun cluster located at School of Physics and Astronomy, Sun Yat-Sen University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' DATA AVAILABILITY The BigMD simulation used in this paper is available in the Cos- moSim data base (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='cosmosim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='org/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The datasets gener- MNRAS 000, 1–15 (2022) 10 Ziyong Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Summary of the resultant ratios between the redshift-space measurements with and without the UNet correction to the simulation truth (measured in real space) for 1D projected 2PCF, 𝜉1𝐷 (𝜇), and the multipoles of 2PCF, 𝜉ℓ (𝑟) (shown in the lower panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The mean and 1𝜎 uncertainty are based on the 27 test sets, each with the box of side length 625 Mpc/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 𝜇 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 𝜉 (𝜇)/𝜉true − 1 (UNet correction) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 𝜉 (𝜇)/𝜉true − 1 (redshift space) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='03 𝑟 (Mpc/ℎ) 20 40 60 80 100 120 𝜉0(𝑟)/𝜉true − 1 (UNet correction) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04 𝜉0(𝑟)/𝜉true − 1 (redshift space) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='07 𝑟 (Mpc/ℎ) 20 40 60 80 100 120 𝜉2(𝑟)/𝜉true − 1 (UNet correction) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 𝜉2(𝑟)/𝜉true − 1 (redshift space) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='17 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04 𝑟 (Mpc/ℎ) 5 10 15 20 25 30 𝜉4(𝑟)/𝜉true − 1 (UNet correction) 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04 𝜉4(𝑟)/𝜉true − 1 (redshift space) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='60 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='09 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05 120 60 0 60 120 r [Mpc/h] 120 60 0 60 120 r//[Mpc/h] reshift space 120 60 0 60 120 r [Mpc/h] real space 120 60 0 60 120 r [Mpc/h] UNet reconstruction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='410 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='800 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='600 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='000 (r , r//) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Contour of the anisotropic 2PCF 𝜉 (𝑟⊥, 𝑟∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' For comparison, the left and middle panels show the 2PCF of redshift-space halos with and without UNet correction, and the right one shows the true 2PCF measured from real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The contours are produced by averaging over the results of 27 test sets in bins of size 1 Mpc/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' ated in the current study are available from the corresponding authors on reasonable request.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', 2014, IAU Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', 308, 493 van de Weygaert R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', 2016, in van de Weygaert R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', Shandarin S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', Saar E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', Einasto J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', eds, IAU Symposium Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 308, The Zeldovich Universe: Gen- esis and Growth of the Cosmic Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' pp 493–523 (arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='01222), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1017/S1743921316010504 APPENDIX A: MOMENTUM FIELD RECONSTRUCTION FROM UNET The momentum field of galaxies and clusters of galaxies is also cosmologically very important (Okumura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' In the fol- lowing, we will use the tilde symbol to denote the momentum field and its components , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', ˜𝒗 for the momentum field, ˜𝜽 and ˜𝝎 for its divergence and vorticity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The momentum field can be defined as ˜𝒗(𝒙) = [1 + 𝛿(𝒙)]𝒗(𝒙), where 𝛿 = 𝑛/¯𝑛 − 1 is the pertur- bation of number density field, and 𝒗 is the comoving velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The momentum field thus is the number-weighted velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We can also decompose the momentum field into the divergence and vortic- ity components, with ˜𝜃(𝒌) = 𝑖𝒌 · ˜𝒗(𝒌) and ˜𝝎(𝒌) = 𝑖𝒌 × ˜𝒗(𝒌), very similar to the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The corresponding momentum power spectra can be defined in the same way (as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 6), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=', 𝑃 ˜𝜃 and 𝑃 ˜𝜔 for the divergence and vorticity power spectra of the momentum field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The reconstruction results for the momentum field are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Overall, for the reconstruction of the momentum field, UNet can achieve even better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' From Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' A1, the resulting correlation coefficients are at the level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='8, about 10% larger than the ones shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 1 for the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Furthermore, let us first compare the joint probability distribu- tions of density-divergence, and density-vorticity for the momentum field, which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' We do see that the reconstructed distributions are pretty consistent with the true ones morphologi- cally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Additionally, the reconstructions of the momentum field and its vorticity component for two randomly selected slices are present in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' A2 & A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' As seen, both of these reconstructed fields indeed MNRAS 000, 1–15 (2022) 15 0 1 2 3 4 800 640 480 320 160 0 160 320 480 640 [h km/s/Mpc] truth 0 1 2 3 4 UNet 100 101 102 103 104 105 0 1 2 3 4 800 640 480 320 160 0 160 320 480 640 [h km/s/Mpc] truth 0 1 2 3 4 UNet 100 101 102 103 104 105 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 4, but for the joint probability distributions of density-divergence (upper), and density-vorticity (lower) for the momentum field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' correlate strongly with the true ones, providing very high reconstruc- tion accuracy at the level of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Also, from the histogram distribu- tions, the deviations on average are about 18◦ for the direction of the momentum field, and about 23◦ for the direction of the vorticity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Compared with the reconstruction in power spectrum, as observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' A4, the transfer functions demonstrate the UNet model yielding excellent reconstruction at all scales of 𝑘 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 ℎ/Mpc, |𝑇(𝑘)| ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='15 for the momentum field, and |𝑇(𝑘)| ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 for both momentum divergence and vorticity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' More interestingly, we can also correct the peculiar velocity of each individual halo from the UNet-reconstructed momentum field via 𝒗 = ˜𝒗/(1 + 𝛿𝑛), where we assume the halo number density contrast 𝛿𝑛 is exactly known from the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' The projected 2PCF and the associated multipoles of 2PCF are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' A5, and the corresponding transfer function, the relative deviation between the reconstructed one and the truth are detailed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Furthermore, the comparison of the anisotropic 2PCF between the reconstruction and the true one are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' All of there results obviously demonstrate a high-fidelity reconstruction of UNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' MNRAS 000, 1–15 (2022) 16 Ziyong Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 0 16 32 48 64 80 X(Mpc/h) 0 16 32 48 64 80 Y(Mpc/h) 0 16 32 48 64 80 vtruth 0 16 32 48 64 80 vUNet 0 200 400 600 800 1000 |v| [km/s] 0 25 50 75 100 125 UNet 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='99±306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='72 truth 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='58±313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 1 - cos 0 50 100 150 200 250 UNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='17 0 1 2 3 4 5 0 50 100 150 200 250 300 350 400 0 16 32 48 64 80 X(Mpc/h) 0 16 32 48 64 80 Y(Mpc/h) 0 16 32 48 64 80 vtruth 0 16 32 48 64 80 vUNet 0 200 400 600 800 1000 |v| [km/s] 0 20 40 60 80 UNet 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='56±357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='96 truth 321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='46±378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 1 - cos 0 50 100 150 200 UNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='16 0 1 2 3 4 5 0 50 100 150 200 250 300 350 400 Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2, but for the momentum field ˜𝒗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 0 16 32 48 64 80 X(Mpc/h) 0 16 32 48 64 80 Y(Mpc/h) 0 16 32 48 64 80 truth 0 16 32 48 64 80 UNet 0 200 400 600 800 1000 | | [h km/s/Mpc] 0 100 200 300 UNet 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='48±66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='71 truth 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='38±65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 1 - cos 0 100 200 300 400 UNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='23 0 1 2 3 4 5 0 20 40 60 80 100 120 140 0 16 32 48 64 80 X(Mpc/h) 0 16 32 48 64 80 Y(Mpc/h) 0 16 32 48 64 80 truth 0 16 32 48 64 80 UNet 0 200 400 600 800 1000 | | [h km/s/Mpc] 0 50 100 150 200 250 UNet 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='36±83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 truth 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00±83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 1 - cos 0 100 200 300 400 UNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='29 0 1 2 3 4 5 6 7 8 0 20 40 60 80 100 120 140 Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 3, but for the vorticity component of the momentum field ˜𝝎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' MNRAS 000, 1–15 (2022) 17 104 105 106 k3Pvv [km2/s2] UNet truth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 k [h Mpc 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 Tf(k) 103 104 105 kP [km2/s2] UNet truth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 k [h Mpc 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 Tf(k) 103 104 105 kP [km2/s2] UNet truth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 k [h Mpc 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='1 Tf(k) Figure A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 5, but for the momentum field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 7 8 9 10 11 12 13 ( ) UNet redshift space real space 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='50 Tf( ) 10 0 10 20 30 40 50 60 s2 0(s)[h 2Mpc2] UNet redshift space real space 20 40 60 80 100 s (Mpc/h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 Tf(s) 20 15 10 5 0 5 10 s2 2(s)[h 2Mpc2] UNet redshift space real space 20 40 60 80 100 s (Mpc/h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 Tf(s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='25 s2 4(s)[h 2Mpc2] UNet redshift space real space 5 10 15 20 25 30 35 s (Mpc/h) 0 1 2 Tf(s) Figure A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Same as in 6, but for the 2PCFs reconstructed from the momentum field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' MNRAS 000, 1–15 (2022) 18 Ziyong Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 120 60 0 60 120 r [Mpc/h] 120 60 0 60 120 r//[Mpc/h] reshift space 120 60 0 60 120 r [Mpc/h] real space 120 60 0 60 120 r [Mpc/h] UNet reconstruction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='410 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='800 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='600 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='000 (r , r//) Figure A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 7, but for the contour of the anisotropic 2PCF reconstructed from the the momentum field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Table A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' Same as in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 2, but for the momentum field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content=' 𝜇 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 𝜉 (𝜇)/𝜉true − 1 (UNet correction) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 𝜉 (𝜇)/𝜉true − 1 (redshift space) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='03 𝑟 (Mpc/ℎ) 20 40 60 80 100 120 𝜉0(𝑟)/𝜉true − 1 (UNet correction) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} +page_content='04 ± 0.' metadata={'source': 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MNRAS 000, 1–15 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE3T4oBgHgl3EQfjQoa/content/2301.04586v1.pdf'} diff --git a/qdAzT4oBgHgl3EQfOvvj/content/tmp_files/2301.01173v1.pdf.txt b/qdAzT4oBgHgl3EQfOvvj/content/tmp_files/2301.01173v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3bc4399accb43cabe8e230ef492ce0d88cfe05b4 --- /dev/null +++ b/qdAzT4oBgHgl3EQfOvvj/content/tmp_files/2301.01173v1.pdf.txt @@ -0,0 +1,2547 @@ +1 +Message Passing-Based 9-D Cooperative Localization and +Navigation with Embedded Particle Flow +Lukas Wielandner∗†, Erik Leitinger∗, Florian Meyer‡, Klaus Witrisal∗† +∗ Graz University of Technology +† Christian Doppler Laboratory for Location-Aware Electronic Systems +‡ University of California San Diego +Abstract—Cooperative localization (CL) is an important tech- +nology for innovative services such as location-aware commu- +nication networks, modern convenience, and public safety. We +consider wireless networks with mobile agents that aim to localize +themselves by performing pairwise measurements amongst agents +and exchanging their location information. Belief propagation +(BP) is a state-of-the-art Bayesian method for CL. In CL, +particle-based implementations of BP often are employed that can +cope with non-linear measurement models and state dynamics. +However, particle-based BP algorithms are known to suffer from +particle degeneracy in large and dense networks of mobile agents +with high-dimensional states. +This paper derives the messages of BP for CL by means +of particle flow, leading to the development of a distributed +particle-based message-passing algorithm which avoids particle +degeneracy. Our combined particle flow-based BP approach +allows the calculation of highly accurate proposal distributions +for agent states with a minimal number of particles. It out- +performs conventional particle-based BP algorithms in terms of +accuracy and runtime. Furthermore, we compare the proposed +method to a centralized particle flow-based implementation, +known as the exact Daum-Huang filter, and to sigma point BP in +terms of position accuracy, runtime, and memory requirement +versus the network size. We further contrast all methods to the +theoretical performance limit provided by the posterior Cram´er- +Rao lower bound (PCRLB). Based on three different scenarios, +we demonstrate the superiority of the proposed method. +I. INTRODUCTION +Location awareness is crucial for various applications, such +as Internet-of-Things, autonomous navigation, or public safety +[1]–[4]. Cooperative localization (CL) methods aim to estimate +the locations of agents in a wireless sensor network, where +agents can communicate among their neighbors and exchange +information about their position [3], [5]–[9]. This leads to an +improvement of the positioning accuracy as well as an increas- +ing localizability [10] while preventing the use of high-density +anchor deployment as needed for non-CL [5], [6], [11]–[15]. +In fact, the anchor infrastructure can be fully avoided when +using multipath channel information contained in radio-signals +[9]. Due to the increased localizability, CL is more robust than +non-CL since more information in the network can be used. +This increased robustness is especially useful for scenarios +with very uninformative measurement models such as RSS +based localization [13], [15]–[17]. CL algorithms are scalable +and can be implemented in a distributed manner, which makes +This work was supported in part by the Christian Doppler Research +Association; the Austrian Federal Ministry for Digital and Economic Affairs; +and the National Foundation for Research, Technology, and Development. +agent 1 +agent 2 +agents +anchors +particle flow +measurements to agent 1 +measurements to agent 2 +Fig. 1: Visualization of the particle flow (dash-dotted green lines) of two +cooperating agents in the vicinity of three anchors. Each agent has only +connections to two anchors (grey circles) indicated by the multimodal PDF +of the agent positions (color map). +them particularly useful for large-scale networks [18]–[20]. +A further crucial aspect of CL is to track high-dimensional +agent states accurately. This paper proposes a new method for +this purpose where different state-of-the-art algorithms fail as +described in the following. +A. State-of-the-Art +Promising methods for CL are based on the framework of +factor graphs (FGs) and message-passing (MP) calculations, +which can be categorized into mean-field message-passing- +based methods [21], [22] and belief propagation (BP)-based +methods [16], [18], [20], [23]–[26]. In particular, BP-based +methods are known to provide accurate solutions to high- +dimensional Bayesian estimation problems efficiently by exe- +cuting message-passing on a cyclic FG. The sum-product rule +is used to compute approximations (”beliefs”) of the marginal +posterior probability density functions (PDFs) of agent po- +sitions [18], [20]. BP-based methods are very flexible and +have been successfully applied to many diverse applications +as for example radio signal-based simultaneous localization +and mapping (SLAM) [27]–[29], multiobject tracking [30]– +[32], and cooperative multiobject tracking [33]. Their excellent +scalability and distributed nature make BP-based algorithms a +arXiv:2301.01173v1 [eess.SP] 3 Jan 2023 + +2 +powerful tool for CL on large-scale networks [18]–[20]. BP- +based methods are categorized into parametric BP algorithms +[25] and non-parametric BP algorithms [23]. Since the mea- +surement models are usually non-linear and the calculations of +the messages and beliefs cannot be evaluated in closed form, +it is common to use non-parametric BP algorithms, resorting +to conventional bootstrap particle-based implementations [16], +[20]. A common drawback of such methods is the curse of +dimensionality, a known problem of sample-based estimation +in high dimensions, and the presence of informative mea- +surements. The curse of dimensionality can lead to particle +collapse, also known as particle degeneracy [34]. It can often +only be avoided by using an infeasible number of particles +to represent the state accurately. Since the required memory +and the computational demand are proportional to the number +of particles, new strategies need to be developed for online +estimation. A common approach to avoid particle degeneracy +is to design an accurate proposal distribution or to make use +of regularization [20], [35], [36]. For the former, we have +to address the problem of how to design accurate proposal +distributions. Furthermore, regularization has to be treated very +carefully since it can introduce biases if not correctly chosen. +Recently, particle flow (PF) [37]–[42] was suggested for +estimation in nonlinear systems with high-dimensional states +and highly informative likelihood models. It is shown that +the resulting PF particle filter is asymptotically optimal for +nonlinear estimation problems and avoids particle degeneracy +even for a relatively small number of particles. PF particle +filters are successfully applied to multi-sensor localization +[43] and BP-based multi-target tracking [44] with the benefit +that a significantly smaller number of particles are needed +compared to bootstrap particle-based implementations. The +main disadvantage of those methods is that they perform +estimation based on the joint state. This increases the com- +putational complexity excessively. Furthermore, some particle +flow-based algorithms have an inherently large complexity, +which provides an additional scaling by the number of used +particles, for example, the localized EDH (LEDH) filter given +in [43] or the stochastic flow described in [40]. This makes +it unattractive for large networks and does not allow for a +distributed implementation. +B. Contributions and Organization of the Paper +This paper introduces a hybrid particle-based PF-BP +message-passing algorithm for CL of mobile agents with 9-D +states (three-dimensional position, velocity, and acceleration +state vectors) and very informative measurement models. In +this scenario, bootstrap particle-based BP methods that draw +samples from predicted agent beliefs fail since an infeasible +large number of particles is needed to represent the belief +of agents accurately. Our approach avoids particle degeneracy +using invertible PF [43] to compute BP messages. Invertible +PF enables the migration of particles towards regions of high +probability, leading to an accurate approximation of BP mes- +sages with a relatively small number of particles. Therefore, +the proposed algorithm combines the computational efficiency +and scalability of BP methods with the benefits of the PF +method. The proposed algorithm exploits the factorization +structure of the cooperative localization problem. This leads to +an inherent reduction of the number of dimensions per calcula- +tion, which also counteracts the particle degeneration problem +and allows for a distributed implementation. As an example, +Figure 1 shows the particle flow of two cooperating agents, +which are in the vicinity of three anchors. Since each agent +has only connections to two anchors, the PDFs of the agent +positions are multimodal. After considering the cooperative +measurement, the particles flow to the “correct mode” of the +posterior PDF, representing the “true” distribution of the agent +positions. +Numerical simulations demonstrate that the proposed PF-BP +algorithm can significantly outperform a conventional boot- +strap particle-based BP algorithm using sampling-importance- +resampling (abbreviated with SIR-BP) [20], a sigma point BP +(SP-BP) algorithm [25], and a particle-based exact Daum- +Huang (EDH) filter (with a stacked state vector containing all +agent state vectors) [43] in terms of position accuracy. The re- +sults show that the proposed algorithm is Bayes-optimal in that +it reaches the posterior Cram´er-Rao lower bound (PCRLB) [5], +[45], which can also be expressed in the framework of the +equivalent Fisher information matrix [6]–[8]. The proposed +algorithm has much lower memory requirements than the SIR- +BP algorithm since it needs a significantly smaller number +of particles for the same level of position accuracy. The +particle-based EDH filter calculates the matrix inversions and +multiplications for the stacked state vector containing all agent +states. Therefore, the memory requirements are also in favor +of the proposed algorithm for the same number of particles. +This is due to the fact that using PF-BP, the matrix inversions +and multiplications reduce to the dimensions of a subset of +the joint agent state. The key contributions of this paper can +be summarized as follows. +• We develop a distributed particle-based message-passing +method for the CL of dynamic agents that computes BP +messages using PF. +• We compare the proposed PF-BP method to state-of-the- +art CL methods and demonstrate its superiority in terms +of accuracy, runtime, and communication overhead. +• We demonstrate numerically that the proposed PF-BP +method for CL can reach the PCRLB if the agents are +localizable. +• We comprehensively analyze the investigated methods +and highlight their benefits depending on different sce- +narios and applications. +In this work, we do not consider uncertainties beyond Gaussian +noise, like missed detections, clutter/false alarm measure- +ments, and data association uncertainty of measurements [31], +[33], [46], [47]. This paper focuses on dynamic networks. The +behavior of static networks can be analyzed by considering a +single time step of the statistical model. This paper advances +over the preliminary account of our method provided in the +conference publication [48] by (i) also considering the uncer- +tainties of cooperating neighbor agents in the PF-BP belief +update equations, (ii) a detailed description of the proposed +algorithm, (iii) an extension to higher state dimensions, (iv) +a comprehensive comparison to established state-of-the-art +algorithms and to the theoretical performance limit in terms + +3 +of the PCRLB. The remainder of this paper is organized as +follows. Section II introduces the system and measurement +model. We state the problem formulation in Section III. +In Section IV, we provide a review of PF. In Section V, +we describe the message-passing framework and explain the +proposed method. The results of numerical experiments are +reported in Section VI. Section VII concludes the paper. +Notation: Column vectors are denoted by boldface lower- +case letters and matrices in boldface uppercase letters. Random +variables are indicated with sans serif, upright fonts and their +realizations in serif, italic fonts as, for example, x and x and +its respective realization as x and x. We define the PDF of +a continuous random variable as f(x). For a vector x, we +indicate its transpose by xT and the Euclidean norm by ∥x∥. +The mean value of a vector is denoted as x. We will also +use this notation to indicate the sample-based mean value +and the minimum mean-square error (MMSE) estimate. The +cardinality of a set C is defined as |C|. Furthermore, we use +the notation C\{i} to indicate the exclusion of member {i} +from the set C. The notation A ⊗ B denotes the Kronecker +product between matrix A and B, whereas ⊙ indicates the +Hadamard product. diag(·) stands for a diagonal matrix or a +block diagonal matrix with elements on the main diagonal +given by the elements or matrices in brackets, respectively. +Im is an identity matrix of dimensions m. [X]k:l,m:n denotes +a submatrix of X containing k to l rows and m to n columns. +The notation [x]k:l denotes a subvector of x containing k +to l elements. The time step k is indicated by a superscript +(k) whereas the uth message passing iteration with [u]. ∇x +indicates the Nabla operator with respect to x(k). +II. SYSTEM MODEL +We consider a set of agents C and a set of anchors A. +The state of the agents is unknown, whereas the state of the +anchors is exactly known. The number of agents and anchors is +indicated by the cardinality of C and A, respectively. We define +two types of measurements: (i) measurements between agents +and anchors z(k) +i,a at time step k with i ∈ C and a ∈ A(k) +i +where +A(k) +i +⊆ A is the set of anchors that perform measurements to +agent i at time k and (ii) measurements in-between agents +z(k) +i,j with i ∈ C and j ∈ D(k) +i +where D(k) +i +⊆ C\{i} is the set +of agents that cooperate with agent i at time k. The stacked +vector of all measurements for all time steps is written as +z = [z(1:K) +i,l +]i∈C,l∈A(1:K) +i +∪D(1:K) +i +with K being the total number +of time steps. Each anchor has a fixed position which does not +vary with time. The state of the i-th agent at time step k is +denoted as x(k) +i += [p(k)T +i +v(k)T +i +a(k)T +i +]T ∈ R9×1, where p(k) +i +∈ +R3×1, v(k) +i +∈ R3×1, a(k) +i +∈ R3×1 are, respectively, the posi- +tion, velocity, and acceleration vectors. Thus, the number of +dimensions per agent state is ND = 9. We define the joint state +of agent i for all time steps as x(1:K) +i += [x(1)T +i +. . . x(K)T +i +] +T. +The states of the anchors are time-independent and assumed +to be known. We write the state of the a-th anchor as +xa = [pxa pya pza]T ∈ R3×1. The vector x denotes the stacked +vector of all agent and anchor states for all time steps. It +is defined as x = [x(1:K)T +1 +. . . x(1:K)T +|C| +, xT +|C|+1 . . . xT +|C|+|A|]T. +The i-th agent state x(k) +i +is assumed to evolve according to a +constant acceleration model given by +x(k) +i += F x(k−1) +i ++ Gu(k−1) +(1) +with the state transition matrix F ∈ R9×9 and the matrix +G ∈ R9×3 relating the state noise to the state variables. +The state noise vector u(k) ∈ R3×1 is an independent and +identically distributed (iid) sequence of 3-D Gaussian random +vectors with standard deviation σa. The matrices are given as +F = +� +� +1 +∆T +(∆T )2 +2 +0 +1 +∆T +0 +0 +1 +� +� ⊗ I3 +(2) +and +G = +� +� +(∆T )2 +2 +∆T +1 +� +� ⊗ I3. +(3) +Given the motion model, we can define the state transition +probability and define the joint prior PDF for all agent states up +to time K using common statistical independence assumptions +[18], [20] as +f(x(1:K)) = +K +� +k=1 +� +i∈C +f(x(0) +i )f(x(k) +i +|x(k−1) +i +). +(4) +The joint posterior PDF up to time K is given as +f(x(1:K)|z(1:K)) ∝ f(z(1:K)|x(1:K))f(x(1:K)). +(5) +By assuming that measurements between nodes and time steps +are independent of each other [18], [20], we can factorize the +joint likelihood function as +f(z(1:K)|x(1:K)) = +K +� +k=1 +� +i∈C +� +a∈A(k) +i +f(z(k) +i,a |x(k) +i +, xa) +× +� +j∈D(k) +i +f(z(k) +i,j |x(k) +i +, x(k) +j ). +(6) +The joint posterior PDF can now be written in terms of its +factorization by plugging (4) and (6) into (5), which results in +f(x(1:K)|z(1:K)) +∝ +K +� +k=1 +� +i∈C +f(x(0) +i )f(x(k) +i +|x(k−1) +i +) +× +� +a∈A(k) +i +f(z(k) +i,a |x(k) +i +, xa) +� +j∈D(k) +i +f(z(k) +i,j |x(k) +i +, x(k) +j ). +(7) +We use distance measurements to infer the state of the agents. +A measurement between two agents or between an agent and +an anchor with indices i and j, respectively, is given by +z(k) +i,j = h(x(k) +i +, x(k) +j ) + n(k) +i,j +(8) +where h(x(k) +i +, x(k) +j ) = ∥p(k) +j +− p(k) +i +∥. The measurement noise +ni,j is iid across i and j, zero-mean, Gaussian with variance +σ2. + +4 +III. PROBLEM FORMULATION +We aim to estimate mobile agent states x(k) +i +cooperatively. +Our Bayesian approach determines the marginal posterior PDF +f(x(k) +i +|z(1:k)) based on all measurements z(1:k) up to time k. +Estimates of the agent state x(k) +i +are obtained by the minimum +mean-square error (MMSE) estimator [49, Ch. 4] given by +x(k) +i += +� +x(k) +i +f(x(k) +i +|z(1:k))dx(k) +i +. +(9) +Since direct marginalization of the joint posterior in (7) +typically cannot be evaluated in closed form, usually bootstrap +particle-based BP [50], [51] implementations are chosen to +approximate the marginal PDFs. This conventional particle- +based implementation suffers from particle degeneracy [34] +when agent states are high-dimensional, or measurements are +very informative. Particle degeneracy leads to a “wrong” rep- +resentation of agent beliefs that deteriorates the convergence +behavior and performance of the particle-based BP algorithms. +To overcome this issue, we propose a hybrid PF-BP algorithm. +Before the proposed algorithm is introduced, a short review +of the PF method is presented. +IV. REVIEW OF PARTICLE FLOW +In the case of a nonlinear measurement model as in (8), the +posterior PDF f(x|z) ∝ f(z|x)f(x) is often approximated by +a set of weighted samples {wm, xm}M +m=1 with �M +m=1 wm=1 +and the number of samples M. They are calculated based on +the importance sampling principle [36] as +wm ∝ f(z|xm)f(xm) +q(xm|z) +(10) +with the proposal PDF q(x|z), from which the set of particles +{xm}M +m=1 is drawn. The only restriction to the proposal PDF +is that it has to have the same support as the posterior PDF +and heavier-tails [52], i.e., it is less informative. Otherwise, +it can be arbitrary. Importance sampling can provide an arbi- +trarily good approximation of the posterior PDF by choosing +M sufficiently large. Even though importance sampling is +asymptotically optimal, if q(x|z) is correctly chosen, it is often +infeasible to implement due to the large number of particles +required for correct state estimation in high-dimensions. +A. Derivation of the PF Equation +Particle flow is an approach that migrates particles from the +prior PDF to the posterior PDF by solving a partial differential +equation [37], [38], [40], [43], [53]. The particle flow is +described by making use of the homotopy property and the +Fokker-Planck equation (FPE) [54]. The FPE is used to find a +flow of particles that is equivalent to the flow of the probability +density according to the log-homotopy function for the joint +state x(k) at time k. The log-homotopy function is given by +[37], [43] +logf(x(k); λ) = logf(x(k)|x(k−1)) ++ λlogf(z(k)|x(k)) − logZ(λ) +(11) +where λ ∈ [0, 1] is the pseudo time of the flow process, +f(x(k); λ) is the pseudo posterior during the flow process +at time λ, and Z(λ) is the evidence. We want to mention +that Z(λ = 0) = 1. The log-homotopy function describes a +continuous and smooth deformation of the distribution starting +from the prior PDF f(x(k)|x(k−1)), i.e., logf(x(k); 0) = +logf(x(k)|x(k−1)) to finally result in the posterior PDF +logf(x(k); 1) ∝ logf(x(k)|x(k−1)) + logf(z(k)|x(k)). +It is assumed that the flow follows a stochastic differential +equation of the form of [37], [38] +dx(k) = ζ(x(k), λ)dλ + dw. +(12) +A detailed derivation of the flow equations can be found in +Appendix A. +B. Exact Daum-Huang (EDH) Filter +This filter estimates the joint agent state x(k) for each time +step k. We review it since it will be a reference method and +a fundamental cornerstone of our proposed approach. +An analytic solution for ζ(x(k), λ) in (39), given in Ap- +pendix A, can be found for Gaussian distributions [37], result- +ing in the EDH filter [37], [43]. To satisfy these conditions, we +approximate the prior PDF as Gaussian distributed where R(k) +and P (k|k−1) are the measurement noise covariance matrix +and the predicted covariance matrix of the joint state at time +k, respectively. The solution for ζ(x(k), λ), according to the +EDH filter, is given by [53] +ζ(x(k), λ) = A(k) +λ x(k) + c(k) +λ . +(13) +A detailed description of the EHD filter and its implementation +can be found in Appendix B, providing also the solution for +(12). We would like to point out that the EDH in this form +can only be implemented in a centralized manner. +V. MESSAGE PASSING ALGORITHMS AND PROPOSED +METHOD +In a Bayesian framework, we estimate the position of each +agent based on the marginal posterior PDFs. Since a direct +marginalization of the joint posterior (7) is often infeasible, +we perform message passing (MP) by means of the sum- +product-algorithm rules on the factor graph that represents +our statistical model. This so-called “belief propagation (BP)” +yields approximations (“beliefs”) of the marginal posterior +PDFs in an efficient way [50], [51]. It gives the exact marginal +PDFs for a tree-like graph but provides only an approximate +marginalization if the underlying factor graph has cycles [50]. +In this case, the BP message passing becomes iterative, and +there exist different orders in which the messages can be +calculated. We have chosen that in each iteration, the beliefs +of all agents i ∈ C are updated in parallel. In the following +section, we derive the MP scheme based on the factor graph +in Figure 2. In Section V-B, we shortly present the standard +particle-based implementation of BP, whereas in Section V-C, +we state the proposed method based on the same MP scheme. +A. BP Message Passing +Based on the factor graph in Figure 2, we define the +MP scheme to approximate the marginal posterior PDFs. +For a better readability, we use the following shorthand + +5 +notation: In a distributed implementation of BP, the factor +fij +≜ +f(z(k) +i,j |x(k) +i +, x(k) +j ) represents the likelihood function +with respect to the involved agents i and j at time k since only +measurement z(k) +i,j is available at node x(k) +i +. Therefore fij ̸= +fji. In a centralized implementation, both measurements be- +tween agent i and j at time k are available. Therefore the factor +is given as the product of the likelihood of both measurements +as fij ≜ f(z(k) +i,j |x(k) +i +, x(k) +j )f(z(k) +j,i |x(k) +i +, x(k) +j ), which results +in fij = fji. The factor f (k) +i +≜ f(x(k) +i +|x(k−1) +i +) corresponds +to the state transition PDF. At time k = 0 it corresponds to +the prior PDF f(x(0) +i ). The factor fai +≜ +f(zi,a|x(k) +i +, xa) +represents information from an anchor measurement. Since +the factor graph has loops, we use an iterative MP scheme to +approximate the marginal PDF (belief) of agent state i at time +step k. We define the belief at MP iteration u ∈ {1, . . . , U} +as the product of all incoming messages as +b[u](x(k) +i +) = η(x(k) +i +) +� +a∈A(k) +i +ϕa(x(k) +i +) +� +j∈D(k) +i +ν[u−1] +j +(x(k) +i +). +(14) +The messages are defined in the following manner: The +message representing the state transition of agent i is given +as +η(x(k) +i +) = +� +f(x(k) +i +|x(k−1) +i +)b[U](x(k−1) +i +)dx(k−1) +i +(15) +whereas the message from anchor a to agent i is +ϕa(x(k) +i +) = +� +f(zi,a|x(k) +i +, xa)δ(xa − xtrue,a)dxa += f(zi,a|x(k) +i +; xtrue,a) +(16) +where xtrue,a corresponds to the true position of anchor a. +Using the extrinsic information ψ[u−1] +i +(x(k) +j ) from the coop- +erative agent j, the messages of the cooperative part can be +written in the form of +ν[u−1] +j +(x(k) +i +) = +� +f(z(k) +i,j |x(k) +i +, x(k) +j )ψ[u−1] +i +(x(k) +j )dx(k) +j +(17) +for a distributed implementation since only measurement z(k) +i,j +is available at node x(k) +i +. In a centralized manner, it is given +as +ν[u−1] +j +(x(k) +i +) = +� +f(z(k) +i,j |x(k) +i +, x(k) +j ) +× f(z(k) +j,i |x(k) +i +, x(k) +j )ψ[u−1] +i +(x(k) +j )dx(k) +j +(18) +since both measurements between agent i and j at time k are +available. The extrinsic information is given as +ψ[u] +i (x(k) +j ) = η(x(k) +j ) +� +a∈A(k) +j +ϕa(x(k) +j ) +� +l∈D(k) +j +\{i} +ν[u−1] +l +(x(k) +j ) +(19) +where the notation D(k) +j +\{i} indicates that i is excluded +from the set D(k) +j +. It is very common to approximate the +extrinsic information by the corresponding belief, resulting in +ψ[u] +i (x(k) +j ) ≈ b[u](x(k) +j ) [18], [20], [24]. This reduces the com- +putational complexity significantly since it avoids calculating +the extrinsic information, which is different for each cooperat- +ing agent pair. An additional benefit is that it also reduces the +time step: k + 1 +f (k+1) +i +f (k+1) +j +fai +x(k+1) +i +fij +fji +x(k) +j +faj +fmi +fim +flj +fjl +ϕ(k+1) +ai +η(k+1) +i +ψ(k+1)[u] +ij +ν(k+1)[u−1] +ij +ν(k+1)[u−1] +ji +ψ(k+1)[u] +ji +time step: k +f (k) +i +f (k) +j +fai +x(k) +i +fij +fji +x(k) +j +faj +fmi +fim +flj +fjl +ϕ(k) +ai +η(k) +i +ψ(k)[u] +ij +ν(k)[u−1] +ij +ν(k)[u−1] +ji +ψ(k)[u] +ji +i ∈ C\{j} +a ∈ A(k) +i +m ∈ D(k) +i +\{j} +l ∈ D(k) +j +\{i} +a ∈ A(k) +j +b(k)[U] +i +Fig. 2: This figure shows a graphical representation of the system model +in terms of a factor graph at time step k. The notation D(k) +m \{l} means +all members of D(k) +m +except l. We use the short hand notation: b(k)[U] +i +≜ +b[U](x(k) +i +), η(k) +i +≜ η(x(k) +i +), ν(k)[u−1] +ji +≜ ν[u−1] +j +(x(k) +i +), ϕ(k) +ai ≜ ϕa(x(k) +i +) +and ψ(k)[u] +ij +≜ ψ[u] +i +(x(k) +j +). Factors fij change depending on a distributed or +centralized processing scheme. +communication between the agents since exchanging extrinsic +information requires point-to-point communication, whereas +the belief can be broadcast [18], [20], [24]. Throughout the +paper, we use the approximation of extrinsic information. +The agent marginal PDF f(x(0:k) +i +|z(1:k)) is approximated +up to a normalization constant by the belief b[u](x(k) +i +). We +estimate the state of the i-th agent at the end of the MP +iterations according to the MMSE estimator [49] as +¯x(k) +i += +� +xib[U](x(k) +i +)dx(k) +i +. +(20) +B. SIR-BP Algorithm +We represent the belief at MP iteration u with a weighted +set of particles {w(k)[u],m +i +, x(k)[u],m +i +}M +m=1. For further insights, +please refer to [20]. After each iteration u, we use systematic +resampling [36] to approximate the belief of the ith agent state +by a set of equally weighted particles as {1/M, x(k)[u],m +i +}M +m=1, +where M is the number of particles. To avoid particle degener- +acy after resampling, we can use regularization to convolve the +resampled set of particles with a kernel that could be estimated +or predefined [55]. I.e., the m-th particle ´x(k)[u],m +i +is drawn +from a Gaussian distribution with a mean value of x(k)[u],m +i +and a covariance of Σr. +C. PF-BP Algorithm +This approach uses the same BP MP to approximate the +marginal PDF of the state as mentioned in Section V-A. The +only difference is that instead of a point-wise multiplication +of the incoming messages at a variable node, we use particle +flow to determine the product of the messages. We represent +the agent state i at time k by a set of equally weighted particles +{1/M, x(k),m +i +}M +m=1. In the following, we present the particle- +based implementation of PF-BP. +Comparing to Section IV-B and Appendix B, the flow of the +m-th particle, representing the approximate marginal posterior + +6 +PDF of agent i at time step k, pseudo-time step λl and message +passing iteration u is given as +x(k)[u],m +λl,i += x(k)[0],m +λl−1,i ++ ˜ζ(x(k)[0],m +λl−1,i , x(k)[u−1],m +→i +, λl)∆l. (21) +This recursive equation represents the particle-based multipli- +cation of the incoming messages ϕa(x(k) +i +) and ν[u−1] +j +(x(k) +i +) +for a ∈ A(k) +i +and j ∈ D(k) +i +. The message η(x(k) +i +) is obtained +by propagating the particle representation through the motion +model. Therefore, we define the m-th particle, drawn from the +proposal PDF as x(k)[u=0],m +λl=0,i += x(k|k−1),m +i +, being equal to the +predicted particle by the motion model. +The variable x(k)[u] +→i +can be seen as a joint state representing +the beliefs of agents that perform measurements to agent i at +time k, evaluated at MP iteration u. x(k)[u],m +→i +indicates the +m-th particle of the stacked representation of this joint state. +It will be explained in what follows. The particles represented +in (21) at λ = 1 do not exactly match the particles drawn +from the corresponding proposal density. Therefore, we have +to use the invertible flow, as mentioned in [43] and recalculate +the weights of the particles. This is done based on the particle +representation at the end (λ = 1) and the beginning (λ = 0) +of the flow as +w(k)[u],m +i +∝ +f(x(k)[u],m +λ=1,i +|x(k−1),m +i +) +f(x(k)[u=0],m +λ=0,i +|x(k−1),m +i +) +× +� +j∈A(k) +i +∪D(k) +i +f(zi,j|x(k)[u],m +λ=1,i +, x(k)[u−1],m +j +). +(22) +The belief of agent state i at time k and MP iteration u, +given in (14), is represented by the weighted set of parti- +cles {w(k)[u],m +i +, x(k)[u],m +λ=1,i +}M +m=1. Using the weighted particle +representation, we perform systematic resampling to approx- +imate b[u](x(k) +i +) by a set of particles with uniform weights +{1/M, x(k)[u],m +i +}M +m=1 where we again drop the index λ to +indicate the resampled particles. At this point, we want to +mention that the final approximation of the marginal posterior +PDF at MP iteration U is indicated by {1/M, x(k),m +i +}M +m=1, +neglecting the MP index. +We introduce a new variable χ(k)[u] +i +that corresponds to the +resampled set of particles. The covariance matrix of the belief +of agent i is indicated as P (k)[u] +i +. Even though it is possible, +we do not determine P (k)[u] +i +using the particle representation +but based on the UKF update step as described in what follows. +We chose this approach since it was observed that the particle +representation could collapse after resampling. +For each MP iteration u, we let the particles of the +agent state i flow for all λ-steps. In addition, we define +x(k)[u−1] +→i += [χ(k)[u−1] +j +]j∈D(k) +i +, which indicate the states of +agents that perform a measurement to agent i at time k, and +the sample-based mean value of it as x(k)[u−1] +→i +. The states of +the cooperating agents are represented by their beliefs at the +previous iteration [u − 1]. Furthermore we define the stacked +representation of the joint state of agent i at pseudo time +step λl−1 and its cooperative partners at MP iteration u as +β(k)[u] +λl−1,i = [x(k)[0]T +λl−1,i , x(k)[u−1]T +→i +]T and its sample-based mean +value as β +(k)[u] +λl−1,i = [x(k)[0]T +λl−1,i , x(k)[u−1]T +→i +]T. With that, we can +write the drift of each particle m as +ζ(x(k)[0],m +λl−1,i , x(k)[u−1],m +→i +, λl) = Aiβ(k)[u],m +λl−1,i ++ ci +(23) +with +Ai +≜ +A(x(k)[0] +λl−1,i, x(k)[u−1] +→i +, λl) +and +ci +≜ +c(x(k)[0] +λl−1,i, ˆx(k)[u−1] +→i +, λl). +For +the +flow +update +in +(21), +˜ζ(·) consists of the first ND elements of ζ(·) in (23). This +corresponds to the drift of the marginal distribution of agent +state i, since the dimension of x(k) +i +is ND. The flow of the +mean value of the agent state is similar to (21) where we +replace the particle representation of the agent state with the +mean values as in (43). +With that in mind, we can define Ai and ci as +Ai = − 1 +2 +˜PiH(k)T +i +(λlH(k) +i +˜PiH(k)T +i ++ R(k) +i +)−1H(k) +i +(24) +ci =(IND(|D(k) +i +|+1)+2λlAi) +� +(IND(|D(k) +i +|+1)+λlAi) +× ˜PiH(k)T +i +(R(k) +i +)−1(zi−νi) + Aiβ +(k)[u] +λ=0,i +� +(25) +with +νi = [h(x(k)[0] +λl−1,i, ϑ +(k) +q )]q∈A(k) +i +∪D(k) +i +− H(k) +i +β +(k)[u] +λl−1,i +(26) +where +νi +corresponds +to +the +model +mis- +match +due +to +the +linearization +and +ϑ +(k) += +[xtrueT,Ai(1), . . . , xT +true,Ai(|Ai|), x(k)[u−1]T +→i +]T, +zi += +[zi,j]j∈A(k) +i +∪D(k) +i , and x(k)[u] +λl,i += +(1/M) �M +m=1 x(k)[u],m +λl,i +. +In what follows, we define all other involved vectors and +matrices. +The observation matrix H(k) +i +has the dimensions (|A(k) +i +| + +|D(k) +i +|) × ND(1 + |D(k) +i +|), which is equivalent to the number +of measurements of agent i times the sum of the dimensions +of all involved states. H(k) +i +consists of the ND-dimensional +elements +[H(k) +i +]˜o,ND ˜p−ND+1:ND ˜p = ∂h(x(k) +p , x(k) +o ) +∂x(k) +p +����x(k) +p +=ˆβ ˜ +p +(27) +for p ∈ {i} ∪ Di, which is a sorted set with index ˜p, +representing the index of the cooperative partner, and the +sorted set o ∈ A(k) +i +∪ D(k) +i +, with index ˜o, determining the +index of the o-th measurement. The derivative is evaluated at +ˆβ ˜p = [β +(k)[u] +λl−1,i]ND ˜p−ND+1:ND ˜p. +The first three elements in (27) correspond to the derivative +with respect to the position coordinates. The following three +elements correspond to the derivative with respect to the +velocity coordinates, and the last three elements correspond +to the derivative with respect to the acceleration coordinates. +The elements containing the derivative with respect to velocity +and acceleration are zero since the observation model depends +only on the position. +The measurement noise covariance matrix R(k) +i +has the +dimensions (|A(k) +i +|+|D(k) +i +|)×(|A(k) +i +|+|D(k) +i +|) with σ2 at the +main diagonal and zeros elsewhere. We also define the block- +diagonal covariance matrix of the involved states at time k +as +˜Pi = diag +� +P (k|k−1) +i +, . . . , P (k)[u−1] +m +, . . . +� +(28) + +7 +Algorithm 1 Proposed PF-BP Algorithm +1: for i = 1 : |C| do +2: +initialize Gaussian prior distribution with mean value +x(0) +i +and covariance matrix P (0) +i +. +3: +draw particles {1/M, x(0),m +i +}M +m=1 from prior +distribution +4: end for +5: for k =1:K do +6: +for i = 1 : |C| do +7: +predict particles and covariance matrix according +to (1) and (29). +8: +determine sample-based mean value x(k)[0] +λ=0,i +9: +end for +10: +for u = 1 : U do +11: +for i = 1 : |C| do +12: +calculate flow according to (21) +(using (23)–(28)) for all λ-steps +13: +resample particles according to (22) to get +{1/M, x(k)[u],m +i +}M +m=1 +14: +determine sample-based mean value x(k)[u] +i +15: +calculate P (k)[u] +i +according to (30) at x(k)[u] +i +16: +optional: regularization of resampled particles +and P (k)[u] +i +according to (33) +17: +end for +18: +end for +19: +for i = 1 : |C| do +20: +determine MMSE estimate according to +sample-based mean value x(k)[U] +i +21: +end for +22: end for +where P (k|k−1) +i +is the predicted covariance matrix of agent +state i and P (k)[u−1] +m +are the covariance matrices of the states +of all other connected agents m ∈ D(k) +i +determined at flow +time λ = 1 of the previous MP iteration u − 1. Similarly +to [43], these covariance matrices are calculated, respectively, +using a UKF covariance matrix prediction and update, i.e., +P (k|k−1) +i += F P (k−1)[U] +i +F T + Q +(29) +P (k)[u] +i += P (k|k−1) +i +− ˜ +K[u] ˜Pzz ˜ +K[u]T +(30) +where ˜ +K[u] again represents the Kalman gain at MP iteration +u since it depends on the beliefs of the involved agent states, +and ˜Pzz is the measurement covariance matrix. As discussed +above, we perform systematic resampling at the end of each +MP iteration resulting in {1/M, x(k)[u],m +i +}M +m=1. Note that the +covariance matrices P (k)[u] +i +are calculated at sample-based +mean value x(k)[u] +i +. In addition to the particles, we represent +the marginal posterior PDF of agent i at time k and MP +iteration u, with a mean value and a covariance matrix. +At MP iteration U, we determine the MMSE estimate of +each agent state according to the sample-based mean value +of each agent state. We use an exponentially spaced λ as +suggested in [38], which results in a more accurate position +estimate in our simulations compared to a linear spacing with +the same number of steps. A summary of the particle-based +implementation of PF-BP is provided in Algorithm 1. +0 +10 +20 +0 +20 +0 +10 +20 +x in m +y in m +z in m +1 +2 +3 +4 +5 +anchor measurements +Fig. 3: A realization of the trajectories for 20 agents. Anchors are given in +black. The initial positions of the agents are marked with red diamonds, and +the trajectory is given in red. The colored scatter points indicate how many +connections an agent has to anchors along its trajectory. The communication +range is rmax = 18 m. Agents have at least one connection to an anchor at +every time step. +VI. EVALUATION OF ALGORITHMS +In this section, we evaluate the proposed algorithm based on +dynamic networks for various network sizes and connectivi- +ties. We use a constant acceleration motion model in 9D (three- +dimensional position, velocity, and acceleration state vectors) +given in (1). We compare the performance to a bootstrap +particle-based BP algorithm (termed SIR-BP) described in +Section V-B, a SP-BP algorithm [25], and to a fully joint +particle-based EDH filter [43]. Furthermore, we show the +theoretical performance limit w.r.t. the PCRLB [5], [45]. We +determine the performance in terms of the root-mean-square +error (RMSE) of the MMSE estimates of position (RMSEp), +velocity (RMSEv) and acceleration (RMSEa), the cumulative +frequency (CF) of the position error, and the runtime per time +step. In addition, we show the probability of outage of the +position error versus a position error threshold. The outage is +defined as position errors above the position error threshold. +The uncertainty of the measurement model is σ = 0.1 m. In +the following simulations, we use 9 anchors and two different +numbers of agents defined as Nagent ∈ {5, 20}. The true +agent positions are uniformly drawn for each realization in +a volume of 20 m × 20 m × 20 m. The true velocity of +each agent is initialized with a unit vector in the direction +of the center of the scenario, while the true acceleration is +initialized with zero. The agent trajectories are generated in +3D based on a constant acceleration model given in (1) with +∆T = 0.1 s and the standard deviation of u(k) is σa = +0.15 m/s2. The prior distribution for position (except for the + +8 +SIR-BP algorithm), velocity and acceleration of each agent +state xi is initialized with a Gaussian distribution with a mean +value of x(0) +i += [p(0)T +i +v(0)T +i +a(0)T +i +]T, which will be defined later +on, and a covariance matrix according to +P (0) +i += diag([(σ2 +p)T, ∆T 2(σ2 +ainit)T, (σ2 +ainit)T]) +(31) +where σ2 +p = [σ2 +px, σ2 +py, σ2 +pz]T. We define the prior stan- +dard deviation of the position to be identical in all di- +mensions and set it to 20 m. For σ2 +ainit, we also define it +to be identical in all dimensions. It is given as σ2 +ainit += +[(10σa)2, (10σa)2, (10σa)2]T. The mean values v(0) +i +and a(0) +i , +corresponding to velocity and acceleration respectively, are +drawn from the zero-mean Gaussian distribution defined by the +covariance matrix in (31). The mean value p(0) +i , corresponding +to the position, is drawn uniformly in the support volume. For +the SIR-BP algorithm, the particles representing the position +are drawn uniformly in the support volume. In contrast, for the +EDH filter and the PF-BP algorithm, the particles are drawn +from the Gaussian prior distribution. One realization of the +dynamic scenario with 20 agents and a communication range +of rmax = 18 m is given in Figure 3. This figure also shows the +anchors’ placement at the corners of the support volume and +the placement of a single anchor in the center. In addition, we +indicate in color how many anchor measurements an agent has +at each point of its trajectory. The setup is chosen such that +each agent lies within the communication range of at least one +anchor at each time step. For an agent to be fully localizable +based on anchor measurements, one needs measurements from +four different anchors where the positions of the anchors do +not lay on a plane. As we see in Figure 3, agents would not +be localizable without cooperative measurements for most of +the trajectories. +We simulate 200 trajectories of the agents for K = 40 time- +steps. We use 20 λ-steps and 200 particles for the PF-based +algorithms and 100 000 particles for the SIR-BP algorithm. +As an additional benchmark, we use 1 000 000 particles for +the SIR-BP algorithm indicated as SIR-BPMil. We fix the +number of MP-iterations to 2. More iterations would be more +time-consuming, and the benefit regarding the convergence +behavior of the BP-based algorithms would be negligible. +Further insights regarding this topic is provided later on in +this section. Since it is common to use regularization to +avoid particle degeneracy [55], we investigate the impact of +regularization on all presented methods. For that purpose, we +regularize velocity and acceleration with σrvel = 0.15 m/s and +σracc = 0.15 m/s2 for all investigate algorithms. This is done +as follows: We define a Gaussian kernel with a covariance +matrix +Σr = diag([0, 0, 0, σ2 +rvel, σ2 +rvel, σ2 +rvel, σ2 +racc, σ2 +racc, σ2 +racc]). +(32) +For the UKF update and SP-BP, we add this covariance to the +estimated covariance of each marginal state. Using for example +(30), it would result in +P (k)[u] +i += P (k|k−1) +i +− ˜ +K[u] ˜Pzz ˜ +K[u]T + Σr. +(33) +For the particle-based methods, we draw for each particle after +resampling x(k),m +i +a new particle ´x(k),m +i +, which is distributed +according to a Gaussian distribution with mean value x(k),m +i +TABLE I: Runtime per time step for the results with 5 agents with respect +to a joint and a distributed (distr.) processing. For the distributed processing, +the results are given in runtime per agent. +rmax +SP-BP +EDH +PF-BP +SIR-BP +SIR-BPMil +joint +18 +3 ms +10 ms +50 ms +0.44 s +4.4 s +∞ +4 ms +20 ms +60 ms +0.51 s +5.1 s +distr. +18 +0.6 ms +- +10 ms +0.09 s +0.9 s +∞ +0.8 ms +- +12 ms +0.10 s +1.2 s +and covariance Σr. Results with regularization are indicated +with dashed or dotted lines in the following figures and with +“reg” in the legends. +1) Scenario I: We evaluate a scenario with 5 agents for +different communication ranges rmax. For rmax = 18 m, agents +have at least one connection to an anchor, which is a similar +scenario as given in Figure 3. The results for that setting are +given in Figure 4a-4d where we show the CF of the overall +trajectory and the RMSE of position, velocity, and acceleration +for each time step. We see clearly that the EDH filter and the +proposed PF-BP algorithm outperform the SP-BP algorithm +and the SIR-BP algorithm significantly in terms of accuracy +without regularization. Table I shows the runtime per time-step +for each algorithm with respect to a joint and a distributed +processing. For a distributed processing, the runtime is given +per time-step and agent. For a small number of agents and the +chosen numbers of particles, the SP-BP algorithm outperforms +all other methods in terms of runtime. +At the first few time-steps, some of the marginal posterior +PDFs of the agent states are still multimodal, which can be +well represented by the particles of the SIR-BP algorithm. +Hence, the SIR-BP algorithm converges much faster to the +“correct mode” of the posterior PDF leading to a much lower +position error at the beginning of the agent trajectories (see +Figure 4b). However, after a few steps, we can observe that +the SIR-BP algorithm diverges in almost every simulation run +since the chosen number of particles (100 000) is still too +small to sufficiently represent the 9-D agent state vectors. With +regularization, the SIR-BP algorithm achieves a much better +performance. However, we can still observe a significant bias +in the RMSE, indicating that the chosen number of particles +is still too low. With 1 000 000 particles and regularization, +the SIR-BP algorithm reaches almost PCRLB level; however, +with the cost of a significant increase of runtime (see Table I) +making it not applicable for real-time applications and systems +with memory restrictions. The small bias, that occurs, can be +avoided using even more particles (not shown). The SP-BP +algorithm also benefits from the regularization since it leads to +faster convergence of the MMSE estimate over time towards +the PCRLB. However, the achievable accuracy is still very +low compared to the PCRLB. Furthermore, it was observed +that the posterior covariance matrices provided by SP-BP are +significantly overconfident (not shown). For both PF-based +methods, regularization has only a slight impact. +For a fully connected agent network (highly informative +measurement models), we see clearly in Figure 4e-4h the +superiority of both PF-based methods. The proposed PF- +BP algorithm reaches the theoretical performance limit much +faster compared to the other methods. The EDH filter reaches + +9 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +position error in m +CF +(a) rmax = 18 m +0 +10 +20 +30 +40 +0 +0.5 +1 +1.5 +2 +time steps +RMSEp in m +(b) rmax = 18 m +0 +10 +20 +30 +40 +2 +4 +time steps +RMSEv in m/s +(c) rmax = 18 m +0 +10 +20 +30 +40 +0 +2 +4 +6 +8 +10 +time steps +RMSEa in m/s2 +(d) rmax = 18 m +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +position error in m +CF +(e) rmax = ∞ +0 +10 +20 +30 +40 +0 +0.2 +0.4 +time steps +RMSEp in m +(f) rmax = ∞ +0 +10 +20 +30 +40 +0 +1 +2 +3 +time steps +RMSEv in m/s +(g) rmax = ∞ +0 +10 +20 +30 +40 +0 +2 +4 +6 +time steps +RMSEa in m/s2 +(h) rmax = ∞ +SIR-BP +SIR-BP reg +SIR-BPMil reg +SP-BP +SP-BP reg +EDH +EDH reg +PF-BP +PF-BP reg +Fig. 4: Influence of the communication range rmax on the performance in terms of accuracy for 5 agents and σ = 0.1 m for 200 simulation runs. We show +the CF of the position error over the whole trajectory as well as the RMSE of the agent states at each time step, where we look separately at the position, +velocity, and acceleration. The theoretical performance limit is given in terms of the PCRLB. Regularization is indicated by reg. +0 +10 +20 +30 +40 +0 +1 +2 +time steps +RMSE in m +1 +2 +3 +0.5 +1 +1.5 +2 +2.5 +time steps +RMSE in m +MP: 2 / λ-steps: 10 +MP: 2 / λ-steps: 20 +MP: 2 / λ-steps: 30 +MP: 4 / λ-steps: 10 +MP: 4 / λ-steps: 20 +MP: 4 / λ-steps: 30 +MP: 6 / λ-steps: 10 +MP: 6 / λ-steps: 20 +MP: 6 / λ-steps: 30 +PCRLB +Fig. 5: Convergence behaviour of PF-BP with respect to message passing +iterations and pseudo-time-steps. The results are averaged over 200 simulation +runs. The setting corresponding to the green line is used for all other +simulations. +the PCRLBs after a few time-steps. The SP-BP algorithm +needs significantly more time-steps until converging towards +the PCRLBs. Using 100 000 particles, the SIR-BP algorithm +obviously diverges with and without regularization in every +simulation run. Even with 1 000 000 particles, the SIR-BP al- +gorithm only converges if regularization is activated. Figure 4f +shows that in this case, the SIR-BP algorithm also reaches +the position PCRLB; however, due to the regularization, the +velocity and acceleration RMSEs are biased. As a consequence +of the large runtime and huge memory requirements, we do +not present results with even more particles. +Both PF-based methods reach the PCRLBs without the +need for regularization. Figure 4g shows that regularizing +the PF-based methods only induces error biases to all states +and is counterproductive for highly informative measurement +models. Figure 4g also indicates that the SP-BP and SIR-BP +algorithms benefit from the regularization since their estimates +of velocity and acceleration need more time-steps to converge +or even diverge without regularization. We conclude that +regularization should be treated cautiously, as it has a sensitive +effect on error biases. +The runtimes of the investigated algorithms for both agent +network are reported in Table I. They were determined based +on a centralized and a distributed processing. The results +indicate that even though PF-BP has a higher computation +time compared to the EDH filter if processed centralized, the +per-agent computations for a distributed processing are lower +or of similar computation time. +In addition, we investigated the convergence behaviour of +our proposed method with respect to rmax = 18 m. Figure 5 +depicts the convergence over time-steps of the trajectory +towards the PCRLB with regard to different MP iterations and +different numbers of λ-steps. It can be observed that a larger +number of λ-steps is always more beneficial than more MP +iterations. Therefore, we fixed the number of MP iterations +to 2 and the number of λ-steps to 20 for all simulations as +mentioned in the beginning of this section. The result with +this set of parameters is indicated in green. +Furthermore, we show in Figure 6 the probability of outage + +10 +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +threshold τ in m +Pout(ϵ > τ) +(a) k = 1, rmax = 18 m +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +threshold τ in m +Pout(ϵ > τ) +(b) k = 20, rmax = 18 m +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +threshold τ in m +Pout(ϵ > τ) +(c) k = 40, rmax = 18 m +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +threshold τ in m +Pout(ϵ > τ) +(d) k = 1, rmax = ∞ +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +threshold τ in m +Pout(ϵ > τ) +(e) k = 20, rmax = ∞ +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +threshold τ in m +Pout(ϵ > τ) +(f) k = 40, rmax = ∞ +SIR-BP +SIR-BP reg +SIR-BPMil reg +SP-BP +SP-BP reg +EDH +EDH reg +PF-BP +PF-BP reg +Fig. 6: Probability of outage of the position error for the investigated algorithms for the scenario with five agents. The first row shows the probability of an +outage for a communication range of rmax=18 m, whereas the second row presents the probability of an outage for the fully connected case. It is evaluated +at certain time-steps k. Regularization is indicated by reg. +TABLE II: Runtime per time step for the results with 20 agents with respect +to a joint and a distributed (distr.) processing. For the distributed processing, +the results are given in runtime per agent. +SP-BP +EDH +PF-BP +SIR-BP +SIR-BPMil +joint +0.07 s +0.25 s +0.9 s +3.6 s +40 s +distr. +0.004 s +- +0.05 s +0.18 s +2 s +Pout(ϵ > τ) of the position error ϵ, where τ is the position +threshold in meters. We evaluate it at three time-steps k ∈ +{1, 20, 40}. At k = 1, we can see the benefits of the different +algorithms. Figure 6a shows for rmax = 18 m at k = 1, that the +SIR-BP algorithm with 1 000 000 provides the most accurate +results, followed by SIR-BP with 100 000. This is because not +every agent is localizable in the first step, and as mentioned +above, SIR-BP can represent any PDF if enough particles are +available. In Figure 6d, there are no multimodalities in the +position state due to the fully connected scenario. Therefore +the unimodal approximation of the PF-BP algorithm is suffi- +cient to represent the agent state correctly. Hence, it achieves +higher accuracy than the SIR-BP with 1 000 000. For k = 20, +all particle-based methods have a similar performance except +the SIR-BP algorithm without regularization. The estimates +of SP-BP are still biased in Figure 6b, whereas they are close +to the optimum result in Figure 6e. At the last step, we see +that if converged, all algorithms perform approximately the +same, which is equivalent to the results in Figure 4f where all +investigated methods reach the PCRLB at the last time step. +2) Scenario II: In Figure 7, we show the results for 20 +agents and a communication range of rmax = 18 m. The +results look similar to those given in Figure 4 but with +two major differences. At first, we can observe that none of +the investigated methods reach the PCRLB with the defined +parametrization. However, PF-BP has the smallest bias. Fur- +thermore, we see that the estimates of the PF-based methods +at k += 1 differ significantly. Since the joint state now +has 180 dimensions compared to the 45 dimensions of the +scenario with five agents, the EDH filter has many more +problems representing the state correctly. The PF-BP algorithm +determines the marginal posterior PDFs of the agents and +calculates the flow only based on a subset of the joint state, +i.e., the state of agent i and all other agents connected to it. +Therefore the state dimension is much smaller, which also +reduces the effect of particle degeneracy. This leads, with the +same parameter setting, to a similar result to the one with five +agents in Figure 4b. The discrepancy to the SIR-BP algorithm +at k = 1 shows that the PF-BP algorithm can not resolve +multimodalities. We can observe that all investigated methods +benefit from the regularization for this scenario and the specific +parameter setting. The RMSE of the PF-BP algorithm has a +constant bias without regularization in Figure 7b. This could +be resolved with more particles, which increases the runtime. +The same is true for the EDH filter. We can also see that the +PF-based methods are the only ones that can reach the PCRLB +within the time of the trajectory with a reasonable calculation +time. The runtimes per time step are summarized in Table II +for a joint and a distributed processing. We see that the SIR- +BP algorithm has a long runtime and is, therefore, unsuitable +for real-time applications. The PF-BP algorithm also has a +larger runtime than the EDH filter but only if processed jointly, +hence making it suitable for real-time applications. SP-BP +outperforms all other methods in terms of runtime but does +not converge at all to the theoretical limit of the estimation + +11 +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +position error in m +CF +(a) CF of the overall trajectory +0 +10 +20 +30 +40 +0 +1 +2 +3 +time steps +RMSEp in m +(b) Position RMSE +0 +10 +20 +30 +40 +2 +4 +6 +time steps +RMSEv in m/s +(c) Velocity RMSE +0 +10 +20 +30 +40 +0 +2 +4 +6 +8 +10 +time steps +RMSEa in m/s2 +(d) Acceleration RMSE +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +threshold τ in m +Pout(ϵ > τ) +(e) k = 1 +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +threshold τ in m +Pout(ϵ > τ) +(f) k = 20 +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +threshold τ in m +Pout(ϵ > τ) +(g) k = 40 +SIR-BP +SIR-BP reg +SIR-BPMil reg +SP-BP +SP-BP reg +EDH +EDH reg +PF-BP +PF-BP reg +Fig. 7: Visualization of the performance of the investigated algorithms in terms of accuracy for the scenario with 20 agents and rmax = 18 m. The first row +shows the CF of the position error over the whole trajectory as well as the RMSE of the agent states at each time step, where we look separately at the +position, velocity, and acceleration. The second row depicts the probability of outage of the position error at certain time-steps k. The results are given for +σ = 0.1 m for 200 simulation runs. Regularization is indicated by reg. +performance. +Note that for highly informative prior distributions of the +agent states at time k = 1, the PF-based methods would still +have higher accuracy than the SIR-BP and SP-BP algorithms. +However, specifically for the SP-BP algorithm, the difference +is significantly smaller. +In what follows, we summarize the advantages and dis- +advantages of the comparison methods and the proposed +algorithm. +• The SIR-BP algorithm requires many particles to repre- +sent the posterior PDFs of the 9-D agent states correctly. +Therefore, the algorithm has a long runtime and requires +significant memory. However, the SIR-BP algorithm has +the potential to correctly represent the posterior PDFs of +the agent states asymptotically in the number of particles. +It can therefore capture multimodalities in the posterior +PDFs. +• The SP-BP algorithm has low computational demand +and, therefore, a low run time. However, it shows slow +convergence toward smaller RMSEs for high dimensional +agent states over time. +• The particle-based EDH filter is suitable for small agent +networks since it provides PCRLB-level position accu- +racy and has a low runtime. However, for larger networks, +the convergence of the MMSE estimates over time is +relatively slow, i.e., it needs many time-steps to reach +PCRLB-level. Due to the joint state representation, it also +does not scale well in the number of agents. +• The proposed PF-BP algorithm provides high position ac- +curacy at the PCRLB level and exhibits low running time +per time step for distributed processing. It also converges +quickly over time and scales well in the number of agents +due to the possibility of a distributed implementation. +Regarding the communication overhead, we can draw the +following conclusions: SP-BP and PF-BP use a Gaussian +approximation, which means that Gaussian distributions repre- +sent the agent states. Therefore, each agent has to transmit only +the mean value and the covariance corresponding to its belief +instead of all particles, as is the case for SIR-BP. For PF-BP, +each agent has to sample locally from that Gaussian distribu- +tion to perform the particle flow process in the measurement +update step. The EDH cannot be implemented in a distributed +manner, leading to the case where a central computation unit +has to collect all measurements and perform the computation. +To make the advantages of the proposed method even +clearer, the runtimes of the investigated algorithms were deter- +mined for centralized and distributed processing. The results +indicate that even though PF-BP has a higher computation +time compared to the EDH filter if processed centralized, the +per-agent computations for a distributed processing are lower +or of similar computation time. +VII. CONCLUSION +We have proposed a Bayesian method based on belief prop- +agation (BP) and particle flow for cooperative localization and +navigation. Our method is particularly suitable for scenarios +with high-dimensional agent states and informative nonlinear +measurement models. To avoid particle degeneracy in such +scenarios, invertible PF is used to compute BP messages. As +a result, the proposed PF-BP algorithm can reach position + +12 +accuracy at PCRLB level in a cooperative localization sce- +nario with 9-D agent states and range measurements. Our +numerical results demonstrate a reduced computational de- +mand and memory requirement compared to the conventional +SIR-BP algorithm and a particle-based EDH filter applied +to cooperative localization. In addition, the communication +overhead is reduced significantly with respect to SIR-BP and +is comparable to SP-BP, which relies on a similar Gaus- +sian representation. We performed simulations with different +numbers of agents and communication ranges, demonstrating +the superior estimation performance of the proposed PF-BP +approach compared to state-of-the-art reference methods. We +highlight the benefits and disadvantages of each investigated +method in various scenarios. +Possible future work is to extent the measurement model +beyond Gaussian noise, like missed detections, clutter/false +alarm measurements, and data association uncertainty of mea- +surements [31], [33], [46], [47], or to cooperative radio signal- +based SLAM algorithm with highly informative measurement +models [28], [29], [56]. +APPENDIX A +DERIVATION OF THE PF EQUATION +The drift term ζ(x(k), λ) can be determined using the FPE, +which is given as +∂f(x(k); λ) +∂λ += − ∇T +x(f(x(k); λ)ζ(x(k), λ)) ++ 1 +2∇T +x(f(x(k); λ)Q(x(k), λ))∇x +(34) +where Q(x(k), λ) corresponds to the diffusion term. The +solutions of (34) for ζ(x(k), λ) can be categorized into zero- +diffusion, i.e., Q(x(k), λ) = 0 [37], [43] and nonzero-diffusion +[38], [40]. The following two useful relations are used in the +further derivation of the method: +1) Using the chain rule of the divergence, the fist term in (34) +can be rewritten as +∇T +x(f(x(k); λ)ζ(x(k), λ)) +=f(x(k); λ)∇T +xζ(x(k), λ)+(∇T +xf(x(k); λ))ζ(x(k), λ). +(35) +2) Using (11), the left side of the FPE, namely the partial +derivative with respect to λ, can be rewritten as +∂f(x(k); λ) +∂λ += f(x(k)|x(k−1)) +�∂f(z(k)|x(k))λ +∂λ +� +Z(λ)−1 ++ f(x(k)|x(k−1)) f(z(k)|x(k))λ +�∂Z(λ)−1 +∂λ +� += f(x(k)|x(k−1)) f(z(k)|x(k))λ +× logf(z(k)|x(k)) Z(λ)−1 − f(x(k)|x(k−1)) +× f(z(k)|x(k))λZ(λ)−2 +�∂Z(λ) +∂λ +� += f(x(k); λ) +� +logf(z(k)|x(k)) − Z(λ)−1 ∂Z(λ) +∂λ +� += f(x(k); λ) +� +logf(z(k)|x(k)) − ∂logZ(λ) +∂λ +� +. +(36) +By assuming zero-diffusion, (34) simplifies to +∂f(x(k); λ) +∂λ += −∇T +x(f(x(k); λ)ζ(x(k), λ)). +(37) +Neglecting the derivative of the evidence Z(λ) with respect to +λ [37], and substituting (36) and (35) into (37), we get +logf(z(k)|x(k)) = − [f(x(k); λ)−1∇T +xf(x(k); λ)]ζ(x(k), λ) +− ∇T +xζ(x(k), λ) +(38) +resulting in +∇T +xζ(x(k), λ) = − logf(z(k)|x(k)) +− (∇xlogf(x(k); λ))Tζ(x(k), λ). +(39) +APPENDIX B +IMPLEMENTATION OF THE EDH FILTER +Given (12) and (13), we will describe here the state repre- +sentation, matrices and vectors for the implementation of the +EDH. Regarding (13), A(k) +λ +and c(k) +λ +are given as +A(k) +λ += − 1 +2P (k|k−1)H(k)T +× (λH(k)P (k|k−1)H(k)T+R(k))−1H(k) +(40) +c(k) +λ +=(IND|C|+2λA) +× [(IND|C|+λA)P (k|k−1)H(k)T(R(k))−1 +× (z(k)+ν(k))+Ax(k) +λ=0] +(41) +where ν(k) = h(x(k) +λ )−H(k)x(k) +λ +and H(k) = ∂h(x) +∂x +���x=x(k) +λ , +h(x) represents a shorthand notation to indicate all measure- +ment hypotheses for all connected agents and anchors, and, +x(k) +λ +represents the mean value of the state at pseudo time +λ and time step k [43]. For λ = 0, x(k) +λ=0 corresponds to +the mean value of the proposal PDF. Due to the Gaussian +assumption, the proposal PDF is fully described by the mean +value x(k) +λ=0 ≜ x(k|k−1) and the covariance matrix P (k|k−1) +of the predicted agent state x(k|k−1). The predicted mean and +the predicted covariance matrix can either be determined by +the set of particles, i.e., x(k) +λ=0 = (1/M) �M +m=1 x(k),m +λ=0 +and +P (k|k−1) = (1/M) �M +m=1(x(k),m +λ=0 +− x(k) +λ=0)(x(k),m +λ=0 +− x(k) +λ=0)T +or by means of the Kalman-filter prediction equation as it will +be described later on in this section. +The particle representation {1/M, x(k),m +λl +}M +m=1 of the joint +state at pseudo-time-step λl with l ∈ {1, . . . , Nλ}, where +Nλ is the maximum number of pseudo-time-steps, as well +as the mean value of the particle representation can now be +determined as +x(k),m +λl += x(k),m +λl−1 + ζ(x(k),m +λl−1 , λl)∆l +(42) +x(k) +λl = x(k) +λl−1 + ζ(x(k) +λl−1, λl)∆l +(43) +with ∆l = λl − λl−1 being the step size of the flow process +between two consecutive pseudo time steps. This corresponds +to the solution of (12). +To evaluate the proposal distribution corresponding to the +particles (42) at the end of the flow (λ = 1), we make use of + +13 +the invertible flow principle introduced in [43]. Following that +principle, the weights of the particles are recalculated based on +the particle representation at the end (λ = 1) and the beginning +(λ = 0) of the flow, i.e., +w(k),m ∝ f(x(k),m +λ=1 |x(k),m +λ=0 ) f(z(k)|x(k),m +λ=1 ) +f(x(k),m +λ=0 ) +. +(44) +Here, x(k),m +λ=0 +is a particle sampled from the proposal PDF, +represented by a Gaussian distribution. The posterior PDF of +the joint agent state x(k) is then represented by the set of +weighted particles {w(k),m, x(k),m +λ=1 }M +m=1. As final operation, +we perform systematic resampling of the joint state resulting +in the posterior PDF of the joint agent state at time k given +by {1/M, x(k),m}M +m=1 [36] where we drop the index λ. +Similar to [43] we calculate the posterior covariance matrix +P (k) based on an unscented-Kalman-filter (UKF) update step +[25], [57] at the sample-based mean value of the particle repre- +sentation of the posterior PDF x(k) +λ=1 = (1/M) �M +m=1 x(k),m +λ=1 +and the predicted covariance P (k|k−1). The predicted covari- +ance matrix is given by +P (k|k−1) = ˜F P (k−1) ˜F T + W +(45) +where +˜F = I|C| ⊗ F +(46) +W = I|C| ⊗ Q +(47) +Q = G(I3 ⊙ σ2 +a)GT +(48) +The update step is given as +P (k) = P (k|k−1) − KPzzKT +(49) +with K being the Kalman gain defined in [25], [57] and the +measurement covariance matrix Pzz. More details on the UKF +filter can be found in [25], [57]. +It is possible to reduce the computation time of the EDH +filter by comparing the rank of R(k) and P (k|k−1). 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Van Der Merwe, “The unscented Kalman filter for non- +linear estimation,” in Proceedings of the IEEE 2000 Adaptive Systems +for Signal Processing, Communications, and Control Symposium (Cat. +No.00EX373), Oct. 2000, pp. 153–158. + diff --git a/qdAzT4oBgHgl3EQfOvvj/content/tmp_files/load_file.txt b/qdAzT4oBgHgl3EQfOvvj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..52cb364cbc1281edb3b305fb6699308e1d72a413 --- /dev/null +++ b/qdAzT4oBgHgl3EQfOvvj/content/tmp_files/load_file.txt @@ -0,0 +1,1213 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf,len=1212 +page_content='1 Message Passing-Based 9-D Cooperative Localization and Navigation with Embedded Particle Flow Lukas Wielandner∗†, Erik Leitinger∗, Florian Meyer‡, Klaus Witrisal∗† ∗ Graz University of Technology † Christian Doppler Laboratory for Location-Aware Electronic Systems ‡ University of California San Diego Abstract—Cooperative localization (CL) is an important tech- nology for innovative services such as location-aware commu- nication networks, modern convenience, and public safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We consider wireless networks with mobile agents that aim to localize themselves by performing pairwise measurements amongst agents and exchanging their location information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Belief propagation (BP) is a state-of-the-art Bayesian method for CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In CL, particle-based implementations of BP often are employed that can cope with non-linear measurement models and state dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' However, particle-based BP algorithms are known to suffer from particle degeneracy in large and dense networks of mobile agents with high-dimensional states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This paper derives the messages of BP for CL by means of particle flow, leading to the development of a distributed particle-based message-passing algorithm which avoids particle degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Our combined particle flow-based BP approach allows the calculation of highly accurate proposal distributions for agent states with a minimal number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It out- performs conventional particle-based BP algorithms in terms of accuracy and runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Furthermore, we compare the proposed method to a centralized particle flow-based implementation, known as the exact Daum-Huang filter, and to sigma point BP in terms of position accuracy, runtime, and memory requirement versus the network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We further contrast all methods to the theoretical performance limit provided by the posterior Cram´er- Rao lower bound (PCRLB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Based on three different scenarios, we demonstrate the superiority of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' INTRODUCTION Location awareness is crucial for various applications, such as Internet-of-Things, autonomous navigation, or public safety [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Cooperative localization (CL) methods aim to estimate the locations of agents in a wireless sensor network, where agents can communicate among their neighbors and exchange information about their position [3], [5]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This leads to an improvement of the positioning accuracy as well as an increas- ing localizability [10] while preventing the use of high-density anchor deployment as needed for non-CL [5], [6], [11]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In fact, the anchor infrastructure can be fully avoided when using multipath channel information contained in radio-signals [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Due to the increased localizability, CL is more robust than non-CL since more information in the network can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This increased robustness is especially useful for scenarios with very uninformative measurement models such as RSS based localization [13], [15]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' CL algorithms are scalable and can be implemented in a distributed manner, which makes This work was supported in part by the Christian Doppler Research Association;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' the Austrian Federal Ministry for Digital and Economic Affairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' and the National Foundation for Research, Technology, and Development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' agent 1 agent 2 agents anchors particle flow measurements to agent 1 measurements to agent 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 1: Visualization of the particle flow (dash-dotted green lines) of two cooperating agents in the vicinity of three anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Each agent has only connections to two anchors (grey circles) indicated by the multimodal PDF of the agent positions (color map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' them particularly useful for large-scale networks [18]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' A further crucial aspect of CL is to track high-dimensional agent states accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This paper proposes a new method for this purpose where different state-of-the-art algorithms fail as described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' State-of-the-Art Promising methods for CL are based on the framework of factor graphs (FGs) and message-passing (MP) calculations, which can be categorized into mean-field message-passing- based methods [21], [22] and belief propagation (BP)-based methods [16], [18], [20], [23]–[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In particular, BP-based methods are known to provide accurate solutions to high- dimensional Bayesian estimation problems efficiently by exe- cuting message-passing on a cyclic FG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The sum-product rule is used to compute approximations (”beliefs”) of the marginal posterior probability density functions (PDFs) of agent po- sitions [18], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' BP-based methods are very flexible and have been successfully applied to many diverse applications as for example radio signal-based simultaneous localization and mapping (SLAM) [27]–[29], multiobject tracking [30]– [32], and cooperative multiobject tracking [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Their excellent scalability and distributed nature make BP-based algorithms a arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='01173v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='SP] 3 Jan 2023 2 powerful tool for CL on large-scale networks [18]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' BP- based methods are categorized into parametric BP algorithms [25] and non-parametric BP algorithms [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Since the mea- surement models are usually non-linear and the calculations of the messages and beliefs cannot be evaluated in closed form, it is common to use non-parametric BP algorithms, resorting to conventional bootstrap particle-based implementations [16], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' A common drawback of such methods is the curse of dimensionality, a known problem of sample-based estimation in high dimensions, and the presence of informative mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The curse of dimensionality can lead to particle collapse, also known as particle degeneracy [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It can often only be avoided by using an infeasible number of particles to represent the state accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Since the required memory and the computational demand are proportional to the number of particles, new strategies need to be developed for online estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' A common approach to avoid particle degeneracy is to design an accurate proposal distribution or to make use of regularization [20], [35], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For the former, we have to address the problem of how to design accurate proposal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Furthermore, regularization has to be treated very carefully since it can introduce biases if not correctly chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Recently, particle flow (PF) [37]–[42] was suggested for estimation in nonlinear systems with high-dimensional states and highly informative likelihood models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It is shown that the resulting PF particle filter is asymptotically optimal for nonlinear estimation problems and avoids particle degeneracy even for a relatively small number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' PF particle filters are successfully applied to multi-sensor localization [43] and BP-based multi-target tracking [44] with the benefit that a significantly smaller number of particles are needed compared to bootstrap particle-based implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The main disadvantage of those methods is that they perform estimation based on the joint state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This increases the com- putational complexity excessively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Furthermore, some particle flow-based algorithms have an inherently large complexity, which provides an additional scaling by the number of used particles, for example, the localized EDH (LEDH) filter given in [43] or the stochastic flow described in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This makes it unattractive for large networks and does not allow for a distributed implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Contributions and Organization of the Paper This paper introduces a hybrid particle-based PF-BP message-passing algorithm for CL of mobile agents with 9-D states (three-dimensional position, velocity, and acceleration state vectors) and very informative measurement models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In this scenario, bootstrap particle-based BP methods that draw samples from predicted agent beliefs fail since an infeasible large number of particles is needed to represent the belief of agents accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Our approach avoids particle degeneracy using invertible PF [43] to compute BP messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Invertible PF enables the migration of particles towards regions of high probability, leading to an accurate approximation of BP mes- sages with a relatively small number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Therefore, the proposed algorithm combines the computational efficiency and scalability of BP methods with the benefits of the PF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The proposed algorithm exploits the factorization structure of the cooperative localization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This leads to an inherent reduction of the number of dimensions per calcula- tion, which also counteracts the particle degeneration problem and allows for a distributed implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' As an example, Figure 1 shows the particle flow of two cooperating agents, which are in the vicinity of three anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Since each agent has only connections to two anchors, the PDFs of the agent positions are multimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' After considering the cooperative measurement, the particles flow to the “correct mode” of the posterior PDF, representing the “true” distribution of the agent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Numerical simulations demonstrate that the proposed PF-BP algorithm can significantly outperform a conventional boot- strap particle-based BP algorithm using sampling-importance- resampling (abbreviated with SIR-BP) [20], a sigma point BP (SP-BP) algorithm [25], and a particle-based exact Daum- Huang (EDH) filter (with a stacked state vector containing all agent state vectors) [43] in terms of position accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The re- sults show that the proposed algorithm is Bayes-optimal in that it reaches the posterior Cram´er-Rao lower bound (PCRLB) [5], [45], which can also be expressed in the framework of the equivalent Fisher information matrix [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The proposed algorithm has much lower memory requirements than the SIR- BP algorithm since it needs a significantly smaller number of particles for the same level of position accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The particle-based EDH filter calculates the matrix inversions and multiplications for the stacked state vector containing all agent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Therefore, the memory requirements are also in favor of the proposed algorithm for the same number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This is due to the fact that using PF-BP, the matrix inversions and multiplications reduce to the dimensions of a subset of the joint agent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The key contributions of this paper can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We develop a distributed particle-based message-passing method for the CL of dynamic agents that computes BP messages using PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We compare the proposed PF-BP method to state-of-the- art CL methods and demonstrate its superiority in terms of accuracy, runtime, and communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We demonstrate numerically that the proposed PF-BP method for CL can reach the PCRLB if the agents are localizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We comprehensively analyze the investigated methods and highlight their benefits depending on different sce- narios and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In this work, we do not consider uncertainties beyond Gaussian noise, like missed detections, clutter/false alarm measure- ments, and data association uncertainty of measurements [31], [33], [46], [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This paper focuses on dynamic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The behavior of static networks can be analyzed by considering a single time step of the statistical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This paper advances over the preliminary account of our method provided in the conference publication [48] by (i) also considering the uncer- tainties of cooperating neighbor agents in the PF-BP belief update equations, (ii) a detailed description of the proposed algorithm, (iii) an extension to higher state dimensions, (iv) a comprehensive comparison to established state-of-the-art algorithms and to the theoretical performance limit in terms 3 of the PCRLB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Section II introduces the system and measurement model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We state the problem formulation in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In Section IV, we provide a review of PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In Section V, we describe the message-passing framework and explain the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The results of numerical experiments are reported in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Section VII concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Notation: Column vectors are denoted by boldface lower- case letters and matrices in boldface uppercase letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Random variables are indicated with sans serif, upright fonts and their realizations in serif, italic fonts as, for example, x and x and its respective realization as x and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We define the PDF of a continuous random variable as f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For a vector x, we indicate its transpose by xT and the Euclidean norm by ∥x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The mean value of a vector is denoted as x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We will also use this notation to indicate the sample-based mean value and the minimum mean-square error (MMSE) estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The cardinality of a set C is defined as |C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Furthermore, we use the notation C\\{i} to indicate the exclusion of member {i} from the set C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The notation A ⊗ B denotes the Kronecker product between matrix A and B, whereas ⊙ indicates the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' diag(·) stands for a diagonal matrix or a block diagonal matrix with elements on the main diagonal given by the elements or matrices in brackets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Im is an identity matrix of dimensions m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' [X]k:l,m:n denotes a submatrix of X containing k to l rows and m to n columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The notation [x]k:l denotes a subvector of x containing k to l elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The time step k is indicated by a superscript (k) whereas the uth message passing iteration with [u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' ∇x indicates the Nabla operator with respect to x(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' SYSTEM MODEL We consider a set of agents C and a set of anchors A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The state of the agents is unknown, whereas the state of the anchors is exactly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The number of agents and anchors is indicated by the cardinality of C and A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We define two types of measurements: (i) measurements between agents and anchors z(k) i,a at time step k with i ∈ C and a ∈ A(k) i where A(k) i ⊆ A is the set of anchors that perform measurements to agent i at time k and (ii) measurements in-between agents z(k) i,j with i ∈ C and j ∈ D(k) i where D(k) i ⊆ C\\{i} is the set of agents that cooperate with agent i at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The stacked vector of all measurements for all time steps is written as z = [z(1:K) i,l ]i∈C,l∈A(1:K) i ∪D(1:K) i with K being the total number of time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Each anchor has a fixed position which does not vary with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The state of the i-th agent at time step k is denoted as x(k) i = [p(k)T i v(k)T i a(k)T i ]T ∈ R9×1, where p(k) i ∈ R3×1, v(k) i ∈ R3×1, a(k) i ∈ R3×1 are, respectively, the posi- tion, velocity, and acceleration vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Thus, the number of dimensions per agent state is ND = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We define the joint state of agent i for all time steps as x(1:K) i = [x(1)T i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' x(K)T i ] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The states of the anchors are time-independent and assumed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We write the state of the a-th anchor as xa = [pxa pya pza]T ∈ R3×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The vector x denotes the stacked vector of all agent and anchor states for all time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It is defined as x = [x(1:K)T 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' x(1:K)T |C| , xT |C|+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' xT |C|+|A|]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The i-th agent state x(k) i is assumed to evolve according to a constant acceleration model given by x(k) i = F x(k−1) i + Gu(k−1) (1) with the state transition matrix F ∈ R9×9 and the matrix G ∈ R9×3 relating the state noise to the state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The state noise vector u(k) ∈ R3×1 is an independent and identically distributed (iid) sequence of 3-D Gaussian random vectors with standard deviation σa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The matrices are given as F = � � 1 ∆T (∆T )2 2 0 1 ∆T 0 0 1 � � ⊗ I3 (2) and G = � � (∆T )2 2 ∆T 1 � � ⊗ I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (3) Given the motion model, we can define the state transition probability and define the joint prior PDF for all agent states up to time K using common statistical independence assumptions [18], [20] as f(x(1:K)) = K � k=1 � i∈C f(x(0) i )f(x(k) i |x(k−1) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (4) The joint posterior PDF up to time K is given as f(x(1:K)|z(1:K)) ∝ f(z(1:K)|x(1:K))f(x(1:K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (5) By assuming that measurements between nodes and time steps are independent of each other [18], [20], we can factorize the joint likelihood function as f(z(1:K)|x(1:K)) = K � k=1 � i∈C � a∈A(k) i f(z(k) i,a |x(k) i , xa) × � j∈D(k) i f(z(k) i,j |x(k) i , x(k) j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (6) The joint posterior PDF can now be written in terms of its factorization by plugging (4) and (6) into (5), which results in f(x(1:K)|z(1:K)) ∝ K � k=1 � i∈C f(x(0) i )f(x(k) i |x(k−1) i ) × � a∈A(k) i f(z(k) i,a |x(k) i , xa) � j∈D(k) i f(z(k) i,j |x(k) i , x(k) j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (7) We use distance measurements to infer the state of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' A measurement between two agents or between an agent and an anchor with indices i and j, respectively, is given by z(k) i,j = h(x(k) i , x(k) j ) + n(k) i,j (8) where h(x(k) i , x(k) j ) = ∥p(k) j − p(k) i ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The measurement noise ni,j is iid across i and j, zero-mean, Gaussian with variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' PROBLEM FORMULATION We aim to estimate mobile agent states x(k) i cooperatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Our Bayesian approach determines the marginal posterior PDF f(x(k) i |z(1:k)) based on all measurements z(1:k) up to time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Estimates of the agent state x(k) i are obtained by the minimum mean-square error (MMSE) estimator [49, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 4] given by x(k) i = � x(k) i f(x(k) i |z(1:k))dx(k) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (9) Since direct marginalization of the joint posterior in (7) typically cannot be evaluated in closed form, usually bootstrap particle-based BP [50], [51] implementations are chosen to approximate the marginal PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This conventional particle- based implementation suffers from particle degeneracy [34] when agent states are high-dimensional, or measurements are very informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Particle degeneracy leads to a “wrong” rep- resentation of agent beliefs that deteriorates the convergence behavior and performance of the particle-based BP algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' To overcome this issue, we propose a hybrid PF-BP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Before the proposed algorithm is introduced, a short review of the PF method is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' REVIEW OF PARTICLE FLOW In the case of a nonlinear measurement model as in (8), the posterior PDF f(x|z) ∝ f(z|x)f(x) is often approximated by a set of weighted samples {wm, xm}M m=1 with �M m=1 wm=1 and the number of samples M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' They are calculated based on the importance sampling principle [36] as wm ∝ f(z|xm)f(xm) q(xm|z) (10) with the proposal PDF q(x|z), from which the set of particles {xm}M m=1 is drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The only restriction to the proposal PDF is that it has to have the same support as the posterior PDF and heavier-tails [52], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=', it is less informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Otherwise, it can be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Importance sampling can provide an arbi- trarily good approximation of the posterior PDF by choosing M sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Even though importance sampling is asymptotically optimal, if q(x|z) is correctly chosen, it is often infeasible to implement due to the large number of particles required for correct state estimation in high-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Derivation of the PF Equation Particle flow is an approach that migrates particles from the prior PDF to the posterior PDF by solving a partial differential equation [37], [38], [40], [43], [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The particle flow is described by making use of the homotopy property and the Fokker-Planck equation (FPE) [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The FPE is used to find a flow of particles that is equivalent to the flow of the probability density according to the log-homotopy function for the joint state x(k) at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The log-homotopy function is given by [37], [43] logf(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ) = logf(x(k)|x(k−1)) + λlogf(z(k)|x(k)) − logZ(λ) (11) where λ ∈ [0, 1] is the pseudo time of the flow process, f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ) is the pseudo posterior during the flow process at time λ, and Z(λ) is the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We want to mention that Z(λ = 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The log-homotopy function describes a continuous and smooth deformation of the distribution starting from the prior PDF f(x(k)|x(k−1)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=', logf(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 0) = logf(x(k)|x(k−1)) to finally result in the posterior PDF logf(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 1) ∝ logf(x(k)|x(k−1)) + logf(z(k)|x(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It is assumed that the flow follows a stochastic differential equation of the form of [37], [38] dx(k) = ζ(x(k), λ)dλ + dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (12) A detailed derivation of the flow equations can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Exact Daum-Huang (EDH) Filter This filter estimates the joint agent state x(k) for each time step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We review it since it will be a reference method and a fundamental cornerstone of our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' An analytic solution for ζ(x(k), λ) in (39), given in Ap- pendix A, can be found for Gaussian distributions [37], result- ing in the EDH filter [37], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' To satisfy these conditions, we approximate the prior PDF as Gaussian distributed where R(k) and P (k|k−1) are the measurement noise covariance matrix and the predicted covariance matrix of the joint state at time k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The solution for ζ(x(k), λ), according to the EDH filter, is given by [53] ζ(x(k), λ) = A(k) λ x(k) + c(k) λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (13) A detailed description of the EHD filter and its implementation can be found in Appendix B, providing also the solution for (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We would like to point out that the EDH in this form can only be implemented in a centralized manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' MESSAGE PASSING ALGORITHMS AND PROPOSED METHOD In a Bayesian framework, we estimate the position of each agent based on the marginal posterior PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Since a direct marginalization of the joint posterior (7) is often infeasible, we perform message passing (MP) by means of the sum- product-algorithm rules on the factor graph that represents our statistical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This so-called “belief propagation (BP)” yields approximations (“beliefs”) of the marginal posterior PDFs in an efficient way [50], [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It gives the exact marginal PDFs for a tree-like graph but provides only an approximate marginalization if the underlying factor graph has cycles [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In this case, the BP message passing becomes iterative, and there exist different orders in which the messages can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We have chosen that in each iteration, the beliefs of all agents i ∈ C are updated in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In the following section, we derive the MP scheme based on the factor graph in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In Section V-B, we shortly present the standard particle-based implementation of BP, whereas in Section V-C, we state the proposed method based on the same MP scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' BP Message Passing Based on the factor graph in Figure 2, we define the MP scheme to approximate the marginal posterior PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For a better readability, we use the following shorthand 5 notation: In a distributed implementation of BP, the factor fij ≜ f(z(k) i,j |x(k) i , x(k) j ) represents the likelihood function with respect to the involved agents i and j at time k since only measurement z(k) i,j is available at node x(k) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Therefore fij ̸= fji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In a centralized implementation, both measurements be- tween agent i and j at time k are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Therefore the factor is given as the product of the likelihood of both measurements as fij ≜ f(z(k) i,j |x(k) i , x(k) j )f(z(k) j,i |x(k) i , x(k) j ), which results in fij = fji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The factor f (k) i ≜ f(x(k) i |x(k−1) i ) corresponds to the state transition PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' At time k = 0 it corresponds to the prior PDF f(x(0) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The factor fai ≜ f(zi,a|x(k) i , xa) represents information from an anchor measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Since the factor graph has loops, we use an iterative MP scheme to approximate the marginal PDF (belief) of agent state i at time step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We define the belief at MP iteration u ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' , U} as the product of all incoming messages as b[u](x(k) i ) = η(x(k) i ) � a∈A(k) i ϕa(x(k) i ) � j∈D(k) i ν[u−1] j (x(k) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (14) The messages are defined in the following manner: The message representing the state transition of agent i is given as η(x(k) i ) = � f(x(k) i |x(k−1) i )b[U](x(k−1) i )dx(k−1) i (15) whereas the message from anchor a to agent i is ϕa(x(k) i ) = � f(zi,a|x(k) i , xa)δ(xa − xtrue,a)dxa = f(zi,a|x(k) i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' xtrue,a) (16) where xtrue,a corresponds to the true position of anchor a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Using the extrinsic information ψ[u−1] i (x(k) j ) from the coop- erative agent j, the messages of the cooperative part can be written in the form of ν[u−1] j (x(k) i ) = � f(z(k) i,j |x(k) i , x(k) j )ψ[u−1] i (x(k) j )dx(k) j (17) for a distributed implementation since only measurement z(k) i,j is available at node x(k) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In a centralized manner, it is given as ν[u−1] j (x(k) i ) = � f(z(k) i,j |x(k) i , x(k) j ) × f(z(k) j,i |x(k) i , x(k) j )ψ[u−1] i (x(k) j )dx(k) j (18) since both measurements between agent i and j at time k are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The extrinsic information is given as ψ[u] i (x(k) j ) = η(x(k) j ) � a∈A(k) j ϕa(x(k) j ) � l∈D(k) j \\{i} ν[u−1] l (x(k) j ) (19) where the notation D(k) j \\{i} indicates that i is excluded from the set D(k) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It is very common to approximate the extrinsic information by the corresponding belief, resulting in ψ[u] i (x(k) j ) ≈ b[u](x(k) j ) [18], [20], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This reduces the com- putational complexity significantly since it avoids calculating the extrinsic information, which is different for each cooperat- ing agent pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' An additional benefit is that it also reduces the time step: k + 1 f (k+1) i f (k+1) j fai x(k+1) i fij fji x(k) j faj fmi fim flj fjl ϕ(k+1) ai η(k+1) i ψ(k+1)[u] ij ν(k+1)[u−1] ij ν(k+1)[u−1] ji ψ(k+1)[u] ji time step: k f (k) i f (k) j fai x(k) i fij fji x(k) j faj fmi fim flj fjl ϕ(k) ai η(k) i ψ(k)[u] ij ν(k)[u−1] ij ν(k)[u−1] ji ψ(k)[u] ji i ∈ C\\{j} a ∈ A(k) i m ∈ D(k) i \\{j} l ∈ D(k) j \\{i} a ∈ A(k) j b(k)[U] i Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 2: This figure shows a graphical representation of the system model in terms of a factor graph at time step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The notation D(k) m \\{l} means all members of D(k) m except l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We use the short hand notation: b(k)[U] i ≜ b[U](x(k) i ), η(k) i ≜ η(x(k) i ), ν(k)[u−1] ji ≜ ν[u−1] j (x(k) i ), ϕ(k) ai ≜ ϕa(x(k) i ) and ψ(k)[u] ij ≜ ψ[u] i (x(k) j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Factors fij change depending on a distributed or centralized processing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' communication between the agents since exchanging extrinsic information requires point-to-point communication, whereas the belief can be broadcast [18], [20], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Throughout the paper, we use the approximation of extrinsic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The agent marginal PDF f(x(0:k) i |z(1:k)) is approximated up to a normalization constant by the belief b[u](x(k) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We estimate the state of the i-th agent at the end of the MP iterations according to the MMSE estimator [49] as ¯x(k) i = � xib[U](x(k) i )dx(k) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (20) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' SIR-BP Algorithm We represent the belief at MP iteration u with a weighted set of particles {w(k)[u],m i , x(k)[u],m i }M m=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For further insights, please refer to [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' After each iteration u, we use systematic resampling [36] to approximate the belief of the ith agent state by a set of equally weighted particles as {1/M, x(k)[u],m i }M m=1, where M is the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' To avoid particle degener- acy after resampling, we can use regularization to convolve the resampled set of particles with a kernel that could be estimated or predefined [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=', the m-th particle ´x(k)[u],m i is drawn from a Gaussian distribution with a mean value of x(k)[u],m i and a covariance of Σr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' PF-BP Algorithm This approach uses the same BP MP to approximate the marginal PDF of the state as mentioned in Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The only difference is that instead of a point-wise multiplication of the incoming messages at a variable node, we use particle flow to determine the product of the messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We represent the agent state i at time k by a set of equally weighted particles {1/M, x(k),m i }M m=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In the following, we present the particle- based implementation of PF-BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Comparing to Section IV-B and Appendix B, the flow of the m-th particle, representing the approximate marginal posterior 6 PDF of agent i at time step k, pseudo-time step λl and message passing iteration u is given as x(k)[u],m λl,i = x(k)[0],m λl−1,i + ˜ζ(x(k)[0],m λl−1,i , x(k)[u−1],m →i , λl)∆l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (21) This recursive equation represents the particle-based multipli- cation of the incoming messages ϕa(x(k) i ) and ν[u−1] j (x(k) i ) for a ∈ A(k) i and j ∈ D(k) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The message η(x(k) i ) is obtained by propagating the particle representation through the motion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Therefore, we define the m-th particle, drawn from the proposal PDF as x(k)[u=0],m λl=0,i = x(k|k−1),m i , being equal to the predicted particle by the motion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The variable x(k)[u] →i can be seen as a joint state representing the beliefs of agents that perform measurements to agent i at time k, evaluated at MP iteration u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' x(k)[u],m →i indicates the m-th particle of the stacked representation of this joint state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It will be explained in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The particles represented in (21) at λ = 1 do not exactly match the particles drawn from the corresponding proposal density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Therefore, we have to use the invertible flow, as mentioned in [43] and recalculate the weights of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This is done based on the particle representation at the end (λ = 1) and the beginning (λ = 0) of the flow as w(k)[u],m i ∝ f(x(k)[u],m λ=1,i |x(k−1),m i ) f(x(k)[u=0],m λ=0,i |x(k−1),m i ) × � j∈A(k) i ∪D(k) i f(zi,j|x(k)[u],m λ=1,i , x(k)[u−1],m j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (22) The belief of agent state i at time k and MP iteration u, given in (14), is represented by the weighted set of parti- cles {w(k)[u],m i , x(k)[u],m λ=1,i }M m=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Using the weighted particle representation, we perform systematic resampling to approx- imate b[u](x(k) i ) by a set of particles with uniform weights {1/M, x(k)[u],m i }M m=1 where we again drop the index λ to indicate the resampled particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' At this point, we want to mention that the final approximation of the marginal posterior PDF at MP iteration U is indicated by {1/M, x(k),m i }M m=1, neglecting the MP index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We introduce a new variable χ(k)[u] i that corresponds to the resampled set of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The covariance matrix of the belief of agent i is indicated as P (k)[u] i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Even though it is possible, we do not determine P (k)[u] i using the particle representation but based on the UKF update step as described in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We chose this approach since it was observed that the particle representation could collapse after resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For each MP iteration u, we let the particles of the agent state i flow for all λ-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In addition, we define x(k)[u−1] →i = [χ(k)[u−1] j ]j∈D(k) i , which indicate the states of agents that perform a measurement to agent i at time k, and the sample-based mean value of it as x(k)[u−1] →i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The states of the cooperating agents are represented by their beliefs at the previous iteration [u − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Furthermore we define the stacked representation of the joint state of agent i at pseudo time step λl−1 and its cooperative partners at MP iteration u as β(k)[u] λl−1,i = [x(k)[0]T λl−1,i , x(k)[u−1]T →i ]T and its sample-based mean value as β (k)[u] λl−1,i = [x(k)[0]T λl−1,i , x(k)[u−1]T →i ]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' With that, we can write the drift of each particle m as ζ(x(k)[0],m λl−1,i , x(k)[u−1],m →i , λl) = Aiβ(k)[u],m λl−1,i + ci (23) with Ai ≜ A(x(k)[0] λl−1,i, x(k)[u−1] →i , λl) and ci ≜ c(x(k)[0] λl−1,i, ˆx(k)[u−1] →i , λl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For the flow update in (21), ˜ζ(·) consists of the first ND elements of ζ(·) in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This corresponds to the drift of the marginal distribution of agent state i, since the dimension of x(k) i is ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The flow of the mean value of the agent state is similar to (21) where we replace the particle representation of the agent state with the mean values as in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' With that in mind, we can define Ai and ci as Ai = − 1 2 ˜PiH(k)T i (λlH(k) i ˜PiH(k)T i + R(k) i )−1H(k) i (24) ci =(IND(|D(k) i |+1)+2λlAi) � (IND(|D(k) i |+1)+λlAi) × ˜PiH(k)T i (R(k) i )−1(zi−νi) + Aiβ (k)[u] λ=0,i � (25) with νi = [h(x(k)[0] λl−1,i, ϑ (k) q )]q∈A(k) i ∪D(k) i − H(k) i β (k)[u] λl−1,i (26) where νi corresponds to the model mis- match due to the linearization and ϑ (k) = [xtrueT,Ai(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' , xT true,Ai(|Ai|), x(k)[u−1]T →i ]T, zi = [zi,j]j∈A(k) i ∪D(k) i , and x(k)[u] λl,i = (1/M) �M m=1 x(k)[u],m λl,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In what follows, we define all other involved vectors and matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The observation matrix H(k) i has the dimensions (|A(k) i | + |D(k) i |) × ND(1 + |D(k) i |), which is equivalent to the number of measurements of agent i times the sum of the dimensions of all involved states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' H(k) i consists of the ND-dimensional elements [H(k) i ]˜o,ND ˜p−ND+1:ND ˜p = ∂h(x(k) p , x(k) o ) ∂x(k) p ����x(k) p =ˆβ ˜ p (27) for p ∈ {i} ∪ Di, which is a sorted set with index ˜p, representing the index of the cooperative partner, and the sorted set o ∈ A(k) i ∪ D(k) i , with index ˜o, determining the index of the o-th measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The derivative is evaluated at ˆβ ˜p = [β (k)[u] λl−1,i]ND ˜p−ND+1:ND ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The first three elements in (27) correspond to the derivative with respect to the position coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The following three elements correspond to the derivative with respect to the velocity coordinates, and the last three elements correspond to the derivative with respect to the acceleration coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The elements containing the derivative with respect to velocity and acceleration are zero since the observation model depends only on the position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The measurement noise covariance matrix R(k) i has the dimensions (|A(k) i |+|D(k) i |)×(|A(k) i |+|D(k) i |) with σ2 at the main diagonal and zeros elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We also define the block- diagonal covariance matrix of the involved states at time k as ˜Pi = diag � P (k|k−1) i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' , P (k)[u−1] m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' � (28) 7 Algorithm 1 Proposed PF-BP Algorithm 1: for i = 1 : |C| do 2: initialize Gaussian prior distribution with mean value x(0) i and covariance matrix P (0) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 3: draw particles {1/M, x(0),m i }M m=1 from prior distribution 4: end for 5: for k =1:K do 6: for i = 1 : |C| do 7: predict particles and covariance matrix according to (1) and (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 8: determine sample-based mean value x(k)[0] λ=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='i 9: end for 10: for u = 1 : U do 11: for i = 1 : |C| do 12: calculate flow according to (21) (using (23)–(28)) for all λ-steps 13: resample particles according to (22) to get {1/M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' x(k)[u],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='}M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='determine sample-based mean value x(k)[u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='15: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='calculate P (k)[u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='according to (30) at x(k)[u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='16: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='optional: regularization of resampled particles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='and P (k)[u] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='according to (33) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='17: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='18: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='19: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='for i = 1 : |C| do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='20: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='determine MMSE estimate according to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='sample-based mean value x(k)[U] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='21: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='22: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='where P (k|k−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='is the predicted covariance matrix of agent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='state i and P (k)[u−1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='are the covariance matrices of the states ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='of all other connected agents m ∈ D(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='determined at flow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='time λ = 1 of the previous MP iteration u − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Similarly to [43], these covariance matrices are calculated, respectively, using a UKF covariance matrix prediction and update, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=', P (k|k−1) i = F P (k−1)[U] i F T + Q (29) P (k)[u] i = P (k|k−1) i − ˜ K[u] ˜Pzz ˜ K[u]T (30) where ˜ K[u] again represents the Kalman gain at MP iteration u since it depends on the beliefs of the involved agent states, and ˜Pzz is the measurement covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' As discussed above, we perform systematic resampling at the end of each MP iteration resulting in {1/M, x(k)[u],m i }M m=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Note that the covariance matrices P (k)[u] i are calculated at sample-based mean value x(k)[u] i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In addition to the particles, we represent the marginal posterior PDF of agent i at time k and MP iteration u, with a mean value and a covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' At MP iteration U, we determine the MMSE estimate of each agent state according to the sample-based mean value of each agent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We use an exponentially spaced λ as suggested in [38], which results in a more accurate position estimate in our simulations compared to a linear spacing with the same number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' A summary of the particle-based implementation of PF-BP is provided in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 0 10 20 0 20 0 10 20 x in m y in m z in m 1 2 3 4 5 anchor measurements Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 3: A realization of the trajectories for 20 agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Anchors are given in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The initial positions of the agents are marked with red diamonds, and the trajectory is given in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The colored scatter points indicate how many connections an agent has to anchors along its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The communication range is rmax = 18 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Agents have at least one connection to an anchor at every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' EVALUATION OF ALGORITHMS In this section, we evaluate the proposed algorithm based on dynamic networks for various network sizes and connectivi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We use a constant acceleration motion model in 9D (three- dimensional position, velocity, and acceleration state vectors) given in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We compare the performance to a bootstrap particle-based BP algorithm (termed SIR-BP) described in Section V-B, a SP-BP algorithm [25], and to a fully joint particle-based EDH filter [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Furthermore, we show the theoretical performance limit w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' the PCRLB [5], [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We determine the performance in terms of the root-mean-square error (RMSE) of the MMSE estimates of position (RMSEp), velocity (RMSEv) and acceleration (RMSEa), the cumulative frequency (CF) of the position error, and the runtime per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In addition, we show the probability of outage of the position error versus a position error threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The outage is defined as position errors above the position error threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The uncertainty of the measurement model is σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In the following simulations, we use 9 anchors and two different numbers of agents defined as Nagent ∈ {5, 20}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The true agent positions are uniformly drawn for each realization in a volume of 20 m × 20 m × 20 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The true velocity of each agent is initialized with a unit vector in the direction of the center of the scenario, while the true acceleration is initialized with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The agent trajectories are generated in 3D based on a constant acceleration model given in (1) with ∆T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='1 s and the standard deviation of u(k) is σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='15 m/s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The prior distribution for position (except for the 8 SIR-BP algorithm), velocity and acceleration of each agent state xi is initialized with a Gaussian distribution with a mean value of x(0) i = [p(0)T i v(0)T i a(0)T i ]T, which will be defined later on, and a covariance matrix according to P (0) i = diag([(σ2 p)T, ∆T 2(σ2 ainit)T, (σ2 ainit)T]) (31) where σ2 p = [σ2 px, σ2 py, σ2 pz]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We define the prior stan- dard deviation of the position to be identical in all di- mensions and set it to 20 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For σ2 ainit, we also define it to be identical in all dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It is given as σ2 ainit = [(10σa)2, (10σa)2, (10σa)2]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The mean values v(0) i and a(0) i , corresponding to velocity and acceleration respectively, are drawn from the zero-mean Gaussian distribution defined by the covariance matrix in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The mean value p(0) i , corresponding to the position, is drawn uniformly in the support volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For the SIR-BP algorithm, the particles representing the position are drawn uniformly in the support volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In contrast, for the EDH filter and the PF-BP algorithm, the particles are drawn from the Gaussian prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' One realization of the dynamic scenario with 20 agents and a communication range of rmax = 18 m is given in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This figure also shows the anchors’ placement at the corners of the support volume and the placement of a single anchor in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In addition, we indicate in color how many anchor measurements an agent has at each point of its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The setup is chosen such that each agent lies within the communication range of at least one anchor at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For an agent to be fully localizable based on anchor measurements, one needs measurements from four different anchors where the positions of the anchors do not lay on a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' As we see in Figure 3, agents would not be localizable without cooperative measurements for most of the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We simulate 200 trajectories of the agents for K = 40 time- steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We use 20 λ-steps and 200 particles for the PF-based algorithms and 100 000 particles for the SIR-BP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' As an additional benchmark, we use 1 000 000 particles for the SIR-BP algorithm indicated as SIR-BPMil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We fix the number of MP-iterations to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' More iterations would be more time-consuming, and the benefit regarding the convergence behavior of the BP-based algorithms would be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Further insights regarding this topic is provided later on in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Since it is common to use regularization to avoid particle degeneracy [55], we investigate the impact of regularization on all presented methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For that purpose, we regularize velocity and acceleration with σrvel = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='15 m/s and σracc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='15 m/s2 for all investigate algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This is done as follows: We define a Gaussian kernel with a covariance matrix Σr = diag([0, 0, 0, σ2 rvel, σ2 rvel, σ2 rvel, σ2 racc, σ2 racc, σ2 racc]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (32) For the UKF update and SP-BP, we add this covariance to the estimated covariance of each marginal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Using for example (30), it would result in P (k)[u] i = P (k|k−1) i − ˜ K[u] ˜Pzz ˜ K[u]T + Σr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (33) For the particle-based methods, we draw for each particle after resampling x(k),m i a new particle ´x(k),m i , which is distributed according to a Gaussian distribution with mean value x(k),m i TABLE I: Runtime per time step for the results with 5 agents with respect to a joint and a distributed (distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=') processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For the distributed processing, the results are given in runtime per agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' rmax SP-BP EDH PF-BP SIR-BP SIR-BPMil joint 18 3 ms 10 ms 50 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='44 s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 s ∞ 4 ms 20 ms 60 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='51 s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='1 s distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 ms 10 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='09 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='9 s ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 ms 12 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='10 s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 s and covariance Σr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Results with regularization are indicated with dashed or dotted lines in the following figures and with “reg” in the legends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 1) Scenario I: We evaluate a scenario with 5 agents for different communication ranges rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For rmax = 18 m, agents have at least one connection to an anchor, which is a similar scenario as given in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The results for that setting are given in Figure 4a-4d where we show the CF of the overall trajectory and the RMSE of position, velocity, and acceleration for each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We see clearly that the EDH filter and the proposed PF-BP algorithm outperform the SP-BP algorithm and the SIR-BP algorithm significantly in terms of accuracy without regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Table I shows the runtime per time-step for each algorithm with respect to a joint and a distributed processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For a distributed processing, the runtime is given per time-step and agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For a small number of agents and the chosen numbers of particles, the SP-BP algorithm outperforms all other methods in terms of runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' At the first few time-steps, some of the marginal posterior PDFs of the agent states are still multimodal, which can be well represented by the particles of the SIR-BP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Hence, the SIR-BP algorithm converges much faster to the “correct mode” of the posterior PDF leading to a much lower position error at the beginning of the agent trajectories (see Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' However, after a few steps, we can observe that the SIR-BP algorithm diverges in almost every simulation run since the chosen number of particles (100 000) is still too small to sufficiently represent the 9-D agent state vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' With regularization, the SIR-BP algorithm achieves a much better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' However, we can still observe a significant bias in the RMSE, indicating that the chosen number of particles is still too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' With 1 000 000 particles and regularization, the SIR-BP algorithm reaches almost PCRLB level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' however, with the cost of a significant increase of runtime (see Table I) making it not applicable for real-time applications and systems with memory restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The small bias, that occurs, can be avoided using even more particles (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The SP-BP algorithm also benefits from the regularization since it leads to faster convergence of the MMSE estimate over time towards the PCRLB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' However, the achievable accuracy is still very low compared to the PCRLB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Furthermore, it was observed that the posterior covariance matrices provided by SP-BP are significantly overconfident (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For both PF-based methods, regularization has only a slight impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For a fully connected agent network (highly informative measurement models), we see clearly in Figure 4e-4h the superiority of both PF-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The proposed PF- BP algorithm reaches the theoretical performance limit much faster compared to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The EDH filter reaches 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 position error in m CF (a) rmax = 18 m 0 10 20 30 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 2 time steps RMSEp in m (b) rmax = 18 m 0 10 20 30 40 2 4 time steps RMSEv in m/s (c) rmax = 18 m 0 10 20 30 40 0 2 4 6 8 10 time steps RMSEa in m/s2 (d) rmax = 18 m 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 position error in m CF (e) rmax = ∞ 0 10 20 30 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 time steps RMSEp in m (f) rmax = ∞ 0 10 20 30 40 0 1 2 3 time steps RMSEv in m/s (g) rmax = ∞ 0 10 20 30 40 0 2 4 6 time steps RMSEa in m/s2 (h) rmax = ∞ SIR-BP SIR-BP reg SIR-BPMil reg SP-BP SP-BP reg EDH EDH reg PF-BP PF-BP reg Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 4: Influence of the communication range rmax on the performance in terms of accuracy for 5 agents and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='1 m for 200 simulation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We show the CF of the position error over the whole trajectory as well as the RMSE of the agent states at each time step, where we look separately at the position, velocity, and acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The theoretical performance limit is given in terms of the PCRLB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Regularization is indicated by reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 0 10 20 30 40 0 1 2 time steps RMSE in m 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 time steps RMSE in m MP: 2 / λ-steps: 10 MP: 2 / λ-steps: 20 MP: 2 / λ-steps: 30 MP: 4 / λ-steps: 10 MP: 4 / λ-steps: 20 MP: 4 / λ-steps: 30 MP: 6 / λ-steps: 10 MP: 6 / λ-steps: 20 MP: 6 / λ-steps: 30 PCRLB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 5: Convergence behaviour of PF-BP with respect to message passing iterations and pseudo-time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The results are averaged over 200 simulation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The setting corresponding to the green line is used for all other simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' the PCRLBs after a few time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The SP-BP algorithm needs significantly more time-steps until converging towards the PCRLBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Using 100 000 particles, the SIR-BP algorithm obviously diverges with and without regularization in every simulation run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Even with 1 000 000 particles, the SIR-BP al- gorithm only converges if regularization is activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Figure 4f shows that in this case, the SIR-BP algorithm also reaches the position PCRLB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' however, due to the regularization, the velocity and acceleration RMSEs are biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' As a consequence of the large runtime and huge memory requirements, we do not present results with even more particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Both PF-based methods reach the PCRLBs without the need for regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Figure 4g shows that regularizing the PF-based methods only induces error biases to all states and is counterproductive for highly informative measurement models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Figure 4g also indicates that the SP-BP and SIR-BP algorithms benefit from the regularization since their estimates of velocity and acceleration need more time-steps to converge or even diverge without regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We conclude that regularization should be treated cautiously, as it has a sensitive effect on error biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The runtimes of the investigated algorithms for both agent network are reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' They were determined based on a centralized and a distributed processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The results indicate that even though PF-BP has a higher computation time compared to the EDH filter if processed centralized, the per-agent computations for a distributed processing are lower or of similar computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In addition, we investigated the convergence behaviour of our proposed method with respect to rmax = 18 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Figure 5 depicts the convergence over time-steps of the trajectory towards the PCRLB with regard to different MP iterations and different numbers of λ-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It can be observed that a larger number of λ-steps is always more beneficial than more MP iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Therefore, we fixed the number of MP iterations to 2 and the number of λ-steps to 20 for all simulations as mentioned in the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The result with this set of parameters is indicated in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Furthermore, we show in Figure 6 the probability of outage 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 threshold τ in m Pout(ϵ > τ) (a) k = 1, rmax = 18 m 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 threshold τ in m Pout(ϵ > τ) (b) k = 20, rmax = 18 m 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 threshold τ in m Pout(ϵ > τ) (c) k = 40, rmax = 18 m 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 threshold τ in m Pout(ϵ > τ) (d) k = 1, rmax = ∞ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 threshold τ in m Pout(ϵ > τ) (e) k = 20, rmax = ∞ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 threshold τ in m Pout(ϵ > τ) (f) k = 40, rmax = ∞ SIR-BP SIR-BP reg SIR-BPMil reg SP-BP SP-BP reg EDH EDH reg PF-BP PF-BP reg Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 6: Probability of outage of the position error for the investigated algorithms for the scenario with five agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The first row shows the probability of an outage for a communication range of rmax=18 m, whereas the second row presents the probability of an outage for the fully connected case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It is evaluated at certain time-steps k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Regularization is indicated by reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' TABLE II: Runtime per time step for the results with 20 agents with respect to a joint and a distributed (distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=') processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For the distributed processing, the results are given in runtime per agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' SP-BP EDH PF-BP SIR-BP SIR-BPMil joint 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='07 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='25 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='9 s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 s 40 s distr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='004 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='05 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='18 s 2 s Pout(ϵ > τ) of the position error ϵ, where τ is the position threshold in meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We evaluate it at three time-steps k ∈ {1, 20, 40}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' At k = 1, we can see the benefits of the different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Figure 6a shows for rmax = 18 m at k = 1, that the SIR-BP algorithm with 1 000 000 provides the most accurate results, followed by SIR-BP with 100 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This is because not every agent is localizable in the first step, and as mentioned above, SIR-BP can represent any PDF if enough particles are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In Figure 6d, there are no multimodalities in the position state due to the fully connected scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Therefore the unimodal approximation of the PF-BP algorithm is suffi- cient to represent the agent state correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Hence, it achieves higher accuracy than the SIR-BP with 1 000 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For k = 20, all particle-based methods have a similar performance except the SIR-BP algorithm without regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The estimates of SP-BP are still biased in Figure 6b, whereas they are close to the optimum result in Figure 6e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' At the last step, we see that if converged, all algorithms perform approximately the same, which is equivalent to the results in Figure 4f where all investigated methods reach the PCRLB at the last time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 2) Scenario II: In Figure 7, we show the results for 20 agents and a communication range of rmax = 18 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The results look similar to those given in Figure 4 but with two major differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' At first, we can observe that none of the investigated methods reach the PCRLB with the defined parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' However, PF-BP has the smallest bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Fur- thermore, we see that the estimates of the PF-based methods at k = 1 differ significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Since the joint state now has 180 dimensions compared to the 45 dimensions of the scenario with five agents, the EDH filter has many more problems representing the state correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The PF-BP algorithm determines the marginal posterior PDFs of the agents and calculates the flow only based on a subset of the joint state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=', the state of agent i and all other agents connected to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Therefore the state dimension is much smaller, which also reduces the effect of particle degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This leads, with the same parameter setting, to a similar result to the one with five agents in Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The discrepancy to the SIR-BP algorithm at k = 1 shows that the PF-BP algorithm can not resolve multimodalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We can observe that all investigated methods benefit from the regularization for this scenario and the specific parameter setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The RMSE of the PF-BP algorithm has a constant bias without regularization in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This could be resolved with more particles, which increases the runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The same is true for the EDH filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We can also see that the PF-based methods are the only ones that can reach the PCRLB within the time of the trajectory with a reasonable calculation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The runtimes per time step are summarized in Table II for a joint and a distributed processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We see that the SIR- BP algorithm has a long runtime and is, therefore, unsuitable for real-time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The PF-BP algorithm also has a larger runtime than the EDH filter but only if processed jointly, hence making it suitable for real-time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' SP-BP outperforms all other methods in terms of runtime but does not converge at all to the theoretical limit of the estimation 11 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 position error in m CF (a) CF of the overall trajectory 0 10 20 30 40 0 1 2 3 time steps RMSEp in m (b) Position RMSE 0 10 20 30 40 2 4 6 time steps RMSEv in m/s (c) Velocity RMSE 0 10 20 30 40 0 2 4 6 8 10 time steps RMSEa in m/s2 (d) Acceleration RMSE 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 threshold τ in m Pout(ϵ > τ) (e) k = 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 threshold τ in m Pout(ϵ > τ) (f) k = 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='8 1 threshold τ in m Pout(ϵ > τ) (g) k = 40 SIR-BP SIR-BP reg SIR-BPMil reg SP-BP SP-BP reg EDH EDH reg PF-BP PF-BP reg Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 7: Visualization of the performance of the investigated algorithms in terms of accuracy for the scenario with 20 agents and rmax = 18 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The first row shows the CF of the position error over the whole trajectory as well as the RMSE of the agent states at each time step, where we look separately at the position, velocity, and acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The second row depicts the probability of outage of the position error at certain time-steps k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The results are given for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='1 m for 200 simulation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Regularization is indicated by reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Note that for highly informative prior distributions of the agent states at time k = 1, the PF-based methods would still have higher accuracy than the SIR-BP and SP-BP algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' However, specifically for the SP-BP algorithm, the difference is significantly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In what follows, we summarize the advantages and dis- advantages of the comparison methods and the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The SIR-BP algorithm requires many particles to repre- sent the posterior PDFs of the 9-D agent states correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Therefore, the algorithm has a long runtime and requires significant memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' However, the SIR-BP algorithm has the potential to correctly represent the posterior PDFs of the agent states asymptotically in the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It can therefore capture multimodalities in the posterior PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The SP-BP algorithm has low computational demand and, therefore, a low run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' However, it shows slow convergence toward smaller RMSEs for high dimensional agent states over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The particle-based EDH filter is suitable for small agent networks since it provides PCRLB-level position accu- racy and has a low runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' However, for larger networks, the convergence of the MMSE estimates over time is relatively slow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=', it needs many time-steps to reach PCRLB-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Due to the joint state representation, it also does not scale well in the number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The proposed PF-BP algorithm provides high position ac- curacy at the PCRLB level and exhibits low running time per time step for distributed processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It also converges quickly over time and scales well in the number of agents due to the possibility of a distributed implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Regarding the communication overhead, we can draw the following conclusions: SP-BP and PF-BP use a Gaussian approximation, which means that Gaussian distributions repre- sent the agent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Therefore, each agent has to transmit only the mean value and the covariance corresponding to its belief instead of all particles, as is the case for SIR-BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For PF-BP, each agent has to sample locally from that Gaussian distribu- tion to perform the particle flow process in the measurement update step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The EDH cannot be implemented in a distributed manner, leading to the case where a central computation unit has to collect all measurements and perform the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' To make the advantages of the proposed method even clearer, the runtimes of the investigated algorithms were deter- mined for centralized and distributed processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The results indicate that even though PF-BP has a higher computation time compared to the EDH filter if processed centralized, the per-agent computations for a distributed processing are lower or of similar computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' CONCLUSION We have proposed a Bayesian method based on belief prop- agation (BP) and particle flow for cooperative localization and navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Our method is particularly suitable for scenarios with high-dimensional agent states and informative nonlinear measurement models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' To avoid particle degeneracy in such scenarios, invertible PF is used to compute BP messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' As a result, the proposed PF-BP algorithm can reach position 12 accuracy at PCRLB level in a cooperative localization sce- nario with 9-D agent states and range measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Our numerical results demonstrate a reduced computational de- mand and memory requirement compared to the conventional SIR-BP algorithm and a particle-based EDH filter applied to cooperative localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' In addition, the communication overhead is reduced significantly with respect to SIR-BP and is comparable to SP-BP, which relies on a similar Gaus- sian representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We performed simulations with different numbers of agents and communication ranges, demonstrating the superior estimation performance of the proposed PF-BP approach compared to state-of-the-art reference methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' We highlight the benefits and disadvantages of each investigated method in various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Possible future work is to extent the measurement model beyond Gaussian noise, like missed detections, clutter/false alarm measurements, and data association uncertainty of mea- surements [31], [33], [46], [47], or to cooperative radio signal- based SLAM algorithm with highly informative measurement models [28], [29], [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' APPENDIX A DERIVATION OF THE PF EQUATION The drift term ζ(x(k), λ) can be determined using the FPE, which is given as ∂f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ) ∂λ = − ∇T x(f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ)ζ(x(k), λ)) + 1 2∇T x(f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ)Q(x(k), λ))∇x (34) where Q(x(k), λ) corresponds to the diffusion term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The solutions of (34) for ζ(x(k), λ) can be categorized into zero- diffusion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=', Q(x(k), λ) = 0 [37], [43] and nonzero-diffusion [38], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The following two useful relations are used in the further derivation of the method: 1) Using the chain rule of the divergence, the fist term in (34) can be rewritten as ∇T x(f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ)ζ(x(k), λ)) =f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ)∇T xζ(x(k), λ)+(∇T xf(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ))ζ(x(k), λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (35) 2) Using (11), the left side of the FPE, namely the partial derivative with respect to λ, can be rewritten as ∂f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ) ∂λ = f(x(k)|x(k−1)) �∂f(z(k)|x(k))λ ∂λ � Z(λ)−1 + f(x(k)|x(k−1)) f(z(k)|x(k))λ �∂Z(λ)−1 ∂λ � = f(x(k)|x(k−1)) f(z(k)|x(k))λ × logf(z(k)|x(k)) Z(λ)−1 − f(x(k)|x(k−1)) × f(z(k)|x(k))λZ(λ)−2 �∂Z(λ) ∂λ � = f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ) � logf(z(k)|x(k)) − Z(λ)−1 ∂Z(λ) ∂λ � = f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ) � logf(z(k)|x(k)) − ∂logZ(λ) ∂λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (36) By assuming zero-diffusion, (34) simplifies to ∂f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ) ∂λ = −∇T x(f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ)ζ(x(k), λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (37) Neglecting the derivative of the evidence Z(λ) with respect to λ [37], and substituting (36) and (35) into (37), we get logf(z(k)|x(k)) = − [f(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ)−1∇T xf(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ)]ζ(x(k), λ) − ∇T xζ(x(k), λ) (38) resulting in ∇T xζ(x(k), λ) = − logf(z(k)|x(k)) − (∇xlogf(x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' λ))Tζ(x(k), λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (39) APPENDIX B IMPLEMENTATION OF THE EDH FILTER Given (12) and (13), we will describe here the state repre- sentation, matrices and vectors for the implementation of the EDH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Regarding (13),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' A(k) λ and c(k) λ are given as A(k) λ = − 1 2P (k|k−1)H(k)T × (λH(k)P (k|k−1)H(k)T+R(k))−1H(k) (40) c(k) λ =(IND|C|+2λA) × [(IND|C|+λA)P (k|k−1)H(k)T(R(k))−1 × (z(k)+ν(k))+Ax(k) λ=0] (41) where ν(k) = h(x(k) λ )−H(k)x(k) λ and H(k) = ∂h(x) ∂x ���x=x(k) λ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' h(x) represents a shorthand notation to indicate all measure- ment hypotheses for all connected agents and anchors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' x(k) λ represents the mean value of the state at pseudo time λ and time step k [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' For λ = 0, x(k) λ=0 corresponds to the mean value of the proposal PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Due to the Gaussian assumption, the proposal PDF is fully described by the mean value x(k) λ=0 ≜ x(k|k−1) and the covariance matrix P (k|k−1) of the predicted agent state x(k|k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The predicted mean and the predicted covariance matrix can either be determined by the set of particles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=', x(k) λ=0 = (1/M) �M m=1 x(k),m λ=0 and P (k|k−1) = (1/M) �M m=1(x(k),m λ=0 − x(k) λ=0)(x(k),m λ=0 − x(k) λ=0)T or by means of the Kalman-filter prediction equation as it will be described later on in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The particle representation {1/M, x(k),m λl }M m=1 of the joint state at pseudo-time-step λl with l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' , Nλ}, where Nλ is the maximum number of pseudo-time-steps, as well as the mean value of the particle representation can now be determined as x(k),m λl = x(k),m λl−1 + ζ(x(k),m λl−1 , λl)∆l (42) x(k) λl = x(k) λl−1 + ζ(x(k) λl−1, λl)∆l (43) with ∆l = λl − λl−1 being the step size of the flow process between two consecutive pseudo time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' This corresponds to the solution of (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' To evaluate the proposal distribution corresponding to the particles (42) at the end of the flow (λ = 1), we make use of 13 the invertible flow principle introduced in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Following that principle, the weights of the particles are recalculated based on the particle representation at the end (λ = 1) and the beginning (λ = 0) of the flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=', w(k),m ∝ f(x(k),m λ=1 |x(k),m λ=0 ) f(z(k)|x(k),m λ=1 ) f(x(k),m λ=0 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' (44) Here, x(k),m λ=0 is a particle sampled from the proposal PDF, represented by a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The posterior PDF of the joint agent state x(k) is then represented by the set of weighted particles {w(k),m, x(k),m λ=1 }M m=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' As final operation, we perform systematic resampling of the joint state resulting in the posterior PDF of the joint agent state at time k given by {1/M, x(k),m}M m=1 [36] where we drop the index λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Similar to [43] we calculate the posterior covariance matrix P (k) based on an unscented-Kalman-filter (UKF) update step [25], [57] at the sample-based mean value of the particle repre- sentation of the posterior PDF x(k) λ=1 = (1/M) �M m=1 x(k),m λ=1 and the predicted covariance P (k|k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' The predicted covari- ance matrix is given by P (k|k−1) = ˜F P (k−1) ˜F T + W (45) where ˜F = I|C| ⊗ F (46) W = I|C| ⊗ Q (47) Q = G(I3 ⊙ σ2 a)GT (48) The update step is given as P (k) = P (k|k−1) − KPzzKT (49) with K being the Kalman gain defined in [25], [57] and the measurement covariance matrix Pzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' More details on the UKF filter can be found in [25], [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' It is possible to reduce the computation time of the EDH filter by comparing the rank of R(k) and P (k|k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' If the rank of R(k) is larger than the rank of P (k|k−1), (40) is reformulated using the Woodbury matrix identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' REFERENCES [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' Xu, W.' metadata={'source': 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+page_content=' 2000, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} +page_content=' 153–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfOvvj/content/2301.01173v1.pdf'} diff --git a/t9AyT4oBgHgl3EQf0fl1/content/tmp_files/2301.00719v1.pdf.txt b/t9AyT4oBgHgl3EQf0fl1/content/tmp_files/2301.00719v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..10fa06ceb802d62fc43c02f2740f27e494f03dbb --- /dev/null +++ b/t9AyT4oBgHgl3EQf0fl1/content/tmp_files/2301.00719v1.pdf.txt @@ -0,0 +1,1951 @@ +Detection of Groups with +Biased Representation in Ranking +Yuval Moskovitch +Ben Gurion University of the Negev +yuvalmos@bgu.ac.il +Jinyang Li +University of Michigan +jinyli@umich.edu +H. V. Jagadish +University of Michigan +jag@umich.edu +Abstract—Real-life tools for decision-making in many critical +domains are based on ranking results. With the increasing +awareness of algorithmic fairness, recent works have presented +measures for fairness in ranking. Many of those definitions +consider the representation of different “protected groups”, in +the top-k ranked items, for any reasonable k. Given the protected +groups, confirming algorithmic fairness is a simple task. However, +the groups’ definitions may be unknown in advance. +In this paper, we study the problem of detecting groups with +biased representation in the top-k ranked items, eliminating +the need to pre-define protected groups. The number of such +groups possible can be exponential, making the problem hard. +We propose efficient search algorithms for two different fair- +ness measures: global representation bounds, and proportional +representation. Then we propose a method to explain the bias +in the representations of groups utilizing the notion of Shapley +values. We conclude with an experimental study, showing the +scalability of our approach and demonstrating the usefulness of +the proposed algorithms. +I. INTRODUCTION +Ranking is a commonly used operation in a wide range of +application domains, for example, in presenting results on a +web search engine [8], establishing credit scores [7], school +admission [29] and hiring [17]. While convenient and useful, +these tools can be biased. As a result, they may affect decision- +making in undesirable ways and can even impact human +life [2], [6], [11]. This problem has drawn much attention +from the research community, and a line of recent works has +focused on measuring and mitigating bias and unfairness in +ranking [3], [10], [17], [22], [34], [36], [39]. +The notion of algorithmic fairness was studied extensively +for a broad class of models [25], [30]. Fairness measures +typically refer to a given “protected group” in the data, which +is defined based on the values of some sensitive attributes +(e.g., gender, race, age, or combinations thereof), usually based +on societal history of discrimination. Analyzing the fairness +measure of a system with respect to the given group is a simple +task. However, “non-standard” protected groups cannot always +be specified in advance, and such groups may be overlooked +when examining the performance of a system. +For example, a model developed to assign grades to students +(in place of exams that were canceled due to the COVID- +19 pandemic) was shown to be biased against high-achieving +students from poor school districts1. For instance, students +1https://www.nytimes.com/2020/09/08/opinion/ +international-baccalaureate-algorithm-grades.html +from low-income families were predicted to fail the Spanish +exam, even when they were native Spanish speakers. In this +case, the model was discriminating against Hispanic students +from poor school districts. A primary source of bias was the +use of historical exam results of each school to predict student +performance. However, using the school (identified by school +ID) to define the protected group is not an intuitive choice, +and so may not have been considered. Moreover, even if we +consider the group of Hispanic students as a protected group, +we may not find any fairness issues, since this subgroup of +students is only a small fraction of all Hispanic students. +In this paper we study the problem of detecting groups that +are treated unfairly by a ranking algorithm. In other words, we +want to let the data speak to (potential) unfairness, without +requiring a human modeler to identify protected attributes +ahead of time. Following fairness definitions presented in the +literature on fairness in ranking (see e.g., [10], [30], [36]), we +consider group representation in the top-k ranked items for +any k in a reasonable range as a measure of fairness. +Recent works have studied the problem of automatically +detecting “problematic” or biased subgroups in the data with- +out the need to specify the protected attributes a priori [9], +[12], [21], [23], [28]. However, these works considered only +classification models. In [27] the authors of [28] extend their +framework to consider ranking as well. In contrast to our work, +which builds on fairness measures for ranking from the litera- +ture, they use the notion of divergence to measure performance +differences among data subgroups. This difference leads to +differences in the result sets returned by each method (see +Section VI-D for more details). +We next outline our main contributions. +Problem formulation: We formally define the problem +of detecting groups with biased representation in the top-k +ranked items for a given ranking algorithm R, a dataset D, +and a range of possible k’s. Groups are defined using value +assignment to a set of attributes we denote as patterns (see +Section II). To provide concise and meaningful results we use +a threshold on the returned groups’ size and report only groups +that are not subsumed by any other group in the result set +(referred to as the most general patterns, see Section III). We +start with the fundamental definition of [10], which uses upper +and lower bounds to restrict the number of tuples in the top-k +from different groups in the data. The goal is to report groups +such that their representation (i.e., number of tuples) in the +arXiv:2301.00719v1 [cs.LG] 30 Dec 2022 + +top-k does not lie within the given bounds for a given range +of possible k’s. We refer to this problem as the global bounds +representation bias problem. We then consider the prominent +class of fairness measures utilizing proportional representation +(see, e.g., [36]). Intuitively, the representation of each group +in the top-k should be proportional to its size in the data. +Using this notion we define the proportional representation +bias problem. We show that no polynomial algorithm exists +to solve either problem. +Detection of groups with biased representation: +We +present algorithms for the problem of finding the set of all +substantial groups (in terms of their size in the data, and their +subsumption in other groups) with biased representation in the +top-k ranked items. We first present a simple baseline solution +that utilizes the notion of pattern graph presented in [5]. We +show how to traverse the graph in a top-down fashion in order +to find groups with biased representation. This search is then +applied repeatedly for each k in the given range. Bearing in +mind the complexity of the problem, we focus on optimizing +the search. Our optimized solutions rely on the fact that the set +of top-k and top-(k + 1) tuples differ by a single tuple. As a +result, the search spaces for succeeding k values are typically +very similar. The optimized solutions utilize this observation +to avoid parts of the search tree. +Result analysis: Given a group with biased representation +in the top-k ranked items, an analyst may wish to understand +the cause of bias. To this end, we propose a method that +harnesses the notion of Shapley values to identify attributes +that significantly affect the ranking of the detected group. +To analyze the difference between the detected group and +top-k ranked tuples, we visualize the value distribution of +such attributes. Shapley values have been used to provide +similar explanations for regression and classification models. +Our novelty is in developing a corresponding method for the +ranking problem. +Experimental study: We complement our algorithmic +development with an extensive experimental study. We eval- +uate the performance and properties of the algorithms, i.e., +the scalability and parameter setting effect. We examine the +effect of the number of attributes, groups’ size threshold, +and range of k, using three real-world datasets. Our results +show the applicability of our solution in practice, despite the +theoretical complexity of the problems, and the usefulness of +the optimized algorithms compared to the baseline solution. +We then experimentally demonstrate the usefulness of our +approach for results analysis. Finally, we compare the result +of our algorithms to the results of the method proposed in [27] +through a case study. +Paper organization: The rest of the paper is organized +as follows. We present the necessary preliminaries for our +problem definition in Section II. Then in Section III we +formally define the problems of detecting groups with bi- +ased representation in the data and prove their hardness. +Our solutions is presented in Section IV. In Section V we +introduce a method for analyzing and explaining the results +of our algorithms. We describe our experimental evaluation in +Section VI, overview related work in Section VII and conclude +in Section VIII. +II. PRELIMINARIES +We next provide necessary background on the notion of +patterns to represent data groups and the concept of fairness +in ranking. We will use the following example as our running +example to demonstrate the ideas presented in the paper. +Example 2.1: The Student Performance Data Set [13] con- +tains information from two Portuguese secondary schools in +the Alentejo region of Portugal, Gabriel Pereira (GP) and +Mousinho da Silveira (MS). The data was collected during +the 2005-2006 school year and it contains the performance +of 1044 students in the Math and the Portuguese language +exams, along with demographic, social, and school-related +information. Figure 1 depicts a sample from the data with the +attributes: gender, school, address (urban or rural), and failures +(number of past class failures). The grade attribute depicts the +students’ grades on a scale of 0 − 20. Consider an excellence +student program committee that wishes to select students for +a scholarship based on their academic achievements. To this +end, they use a ranking algorithm R to rank students by +their grades. In the case of similar grades, students with +fewer failures are ranked higher. The scholars’ list is publicly +announced, and should be diverse and inclusive. +A. Data Groups +We assume the data is represented using a single relational +database, and that the relation’s attribute values used for +group definitions are categorical. To include attribute values +drawn from a continuous domain in the group definition, +we render them categorical by bucketizing them into ranges: +very commonly done in practice to present aggregate results. +We use the notion of patterns, value assignment to a set of +attributes, to define groups in the data [5]. +Definition 2.2 (Patterns): Let D be a database with cate- +gorical attributes A = {A1, . . . , An} and let Dom(Ai) be the +active domain of Ai for i ∈ [1..n]. A pattern p over D is the +set of {Ai1 = a1, . . . , Aik = ak} where {Ai1, . . . , Aik} ⊆ A +and aj ∈ Dom(Aij) for each Aij in p. We use Attr(p) to +denote the set of attributes in p. +We say that a tuple t ∈ D satisfies a pattern p if t.Ai = ai +for each Ai ∈ Attr(p). The size sD(p) of a pattern p is then +the number of tuples in D that satisfy p. Given a ranking +algorithm R we use Rk(D) to denote the top-k ranked items in +D. Finally, we use sRk(D)(p) to denote the size of p in Rk(D). +Example 2.3: Consider the dataset given in Figure 1. p = +{School = GP}, is an example of a pattern. Tuples 3, 4, 7, +8, 12, 13, 15 and 16 satisfy p and thus sD(p) = 8. For the +ranking algorithm R whose results are depicted in the Rank +column, we have sR5(D)(p) = 1, since only one tuple in the +top-5 ranked items satisfies p. +B. Fairness measure +The problem of fairness in ranking was studied in a line +of works (see [30] for a survey). A fundamental definition, + +# +Gender +School +Address +Failures +Grade +Rank +1 +F +MS +R +1 +11 +8 +2 +M +MS +R +1 +15 +3 +3 +M +GP +U +1 +8 +10 +4 +M +GP +U +2 +4 +16 +5 +M +MS +R +0 +19 +2 +6 +F +MS +U +1 +4 +15 +7 +F +GP +R +1 +7 +11 +8 +M +GP +R +1 +6 +13 +9 +F +MS +R +0 +14 +4 +10 +F +MS +R +2 +7 +12 +11 +M +MS +R +2 +13 +6 +12 +F +GP +U +0 +20 +1 +13 +F +GP +U +2 +12 +7 +14 +M +MS +U +1 +13 +5 +15 +F +GP +U +1 +5 +14 +16 +M +GP +U +0 +9 +9 +Fig. 1: Students’ data. The Rank column depicts their ranking +based on the grade and number of past failures. The top-5 +ranked students are highlighted +presented in [10], uses constraints over the representations +of different groups in the top-k ranked items. They use an +upper bound Ukl and a lower bound Lkl over the number of +items with the property l (i.e., a protected group) in the top-k +positions of the ranking. Then a ranking algorithm is fair by +the definition of [10], if the number of selected items from the +protected group in the top-k lies within the given boundaries. +Example 2.4: Consider again the dataset given in Figure 1 +and the ranker whose result is presented in the Rank column. +Consider a lower bound of 2 over the number of students from +each school for k = 5 (i.e., L5,school=MS = L5,school=GP = +2). In this case, among the top-5 students, only one is from +the GP school, thus the ranker does not satisfy the constraints. +Another prominent class of fairness measure in ranking +considers the proportional representation of different groups +in the top-k ranked items (see, e.g [36]). Intuitively, these +definitions can be seen as variants of the definition of [10] +such that for each group g, and each k, the bounds on the +number of occurrences of items from g in the top-k ranked +items are defined with respect to the size of g in the dataset. +Example 2.5: Continuing Example 2.4, the total number of +students from each school (MS and GP) is 8. The total dataset +size is 16, thus a proportionate representation of each school +in the top-5 items should be roughly 5 · 8 +16 ≈ 2. +III. PROBLEM DEFINITION +Our goal is to detect groups with biased representation +in the top-k ranked items for a given ranking algorithm R, +dataset D, and a range of k’s. We define our problem by +harnessing fairness measures for ranking algorithms from the +literature. In particular, we present two problem definitions +utilizing prominent fairness measures, both considering the +representation of different groups in the top-k ranked items +for different values of k. Intuitively, accounting for a range +of k’s ensures that the ranking is fair for any position in the +ranking. For instance, if the top-10 items consist of 5 students +with an urban address and 5 students with a rural address, but +the students with urban addresses are in the 1-5 positions of the +ranking, the output may seem “fair” if we are only interested +in selecting the top-10 students, but if the positions within the +top-10 are also important (e.g., position in the ranking affects +an award amount), then clearly this ranking is unfair. +The first definition simply utilizes the definition of [10]. The +fairness definition of [10] restricts the count of different groups +(i.e., patterns) in the top-k using upper and lower bounds. +According to this definition, the result is biased either when +the size of a pattern l exceeds the upper bound Ukl or falls +below the lower bound Lkl among the top-k tuples for some +k. We eliminate the requirement to define l in advance and use +Uk and Lk to denote the upper and lower bound respectively, +on every pattern in the top-k ranked tuples of a given ranking +algorithm. +We say that a group has a biased representation in the output +of a ranking algorithm R, if its size in the top-k ranked items +by R does not lie within the given bounds for any k in a given +range of possible k’s. Intuitively, we wish to avoid reporting +“very specific” descriptions of groups and provide the user +with a concise set of properties that characterize meaningful +groups (in terms of their size) that have biased representation. +To this end, we present the notion of most general patterns. +Given the bounds over the group’s representation in the top- +k ranked items, we say that a pattern p is the most general +pattern with biased representation, if p is used to represent +a group with inadequate representation by the given bounds, +and ∀p′ ⊊ p, the count of p′ in Rk(D) lies within the given +boundaries. We are now ready to formally define our problem. +Problem 3.1 (Global Bounds Representation Bias): Given +a database D, a ranking algorithm R, a size threshold τs2, a +range [kmin, kmax], and lower bounds Lk and upper bounds +Uk for each kmin ⩽ k ⩽ kmax, find for each kmin ⩽ k ⩽ +kmax, all most general patterns p with size ⩾ τs such that +sRk(D)(p) < Lk or sRk(D)(p) > Uk. +Note that the ranking algorithm is treated as a black box, +making the problem to be model agnostic. Following the line +of work on proportional representation, we consider another +problem definition by refining the above definition to account +for the groups’ sizes in the dataset. Intuitively, the number of +items from each group in the top-k ranked items should be +proportionate to the group’s representation in the data. +Problem 3.2 (Proportional Representation Bias): Given a +database D, a ranking algorithm R, a size threshold τs and +a range [kmin, kmax], find for each kmin ⩽ k ⩽ kmax, all +most general patterns p with size ⩾ τs such that sRk(D)(p) < +α · sD(p) k +|D| or sRk(D)(p) > β · sD(p) k +|D| for α < β ∈ R. +Proportional representation gives the user an intuitive +bounds definition. However, the definition of global represen- +tation allows the user to actively control the bounds over the +representation of different groups in the top-k ranked items, +even if their representation in the overall data is low/high. For +instance, consider a ranking algorithm for job applicants in +fields that are dominated by men (e.g., tech companies). If +2We use an absolute value as the size threshold. Equivalently it may be +defined as a fraction of the dataset size. + +A1 +A2 +· · · +An−1 +An +Rank +t1 +1 +0 +· · · +0 +0 +1 +t2 +0 +1 +· · · +0 +0 +2 +... +... +... +... +... +... +... +tn +0 +0 +· · · +0 +1 +n +tn+1 +0 +0 +· · · +0 +0 +n + 1 +Fig. 2: Dataset D in the proof of Theorem 3.3 +the company wishes to increase the representation of women +they hire but their application number is low and only the +top-k ranked applicants are invited for an interview, by using +proportional representation, their number in the top-k, and as +a result, the number of women invited to an interview, would +be low as well. The fundamental definition of [10] allows the +user to define bounds over the representation of the protected +groups in the data for different values of k. Following this +definition, we defined the global bounds representation prob- +lem, which assumes no prior information regarding the identity +of the protected groups and aims to identify all groups with +biased representation. Note that our goal is to report only the +most general patterns (groups), providing a concise description +of these groups. +While there are typically far fewer most general patterns +with biased representation than the set of all patterns with +biased representation, in the worst case, their numbers can be +exponential. This is true even if we consider only the lower +bounds (e.g., if Uk = |D|). +Theorem 3.3: Given a dataset D and a ranking algorithm R, +no polynomial time algorithm can guarantee the enumeration +of the set of all most general patterns with biased representa- +tion in the result of R on D. +Proof 3.4: We prove the theorem by construction. Consider +a dataset D with n (assuming n ≥ 2) binary attributes +{A1 . . . , An} and n + 1 tuples t1, . . . , tn, tn+1 as shown in +Figure 2. I.e., ∀i ∈ [1, n], ti[Ai] = 1, and ∀j ̸= i, ti[Aj] = 0. +All the attributes of tn+1 are zero. Let R be a ranking +algorithm such that the top-k tuples in D are the tuples +t1, . . . , tk in Figure 2. Let kmin = kmax = n, Lk = n +2 +1 for +Problem 3.1 (global representation bounds), and α = n+3 +n+4 for +Problem 3.2 (proportional representation), for some n ≥ 2. +Consider a pattern p with m ≤ n attributes Ai with the +value assignment Ai = 0. Let I be the set of indices of +those attributes. Among t1, · · · , tn, the size of p is n − m, +since ∀i ∈ I, ti[i] = 1, ti does not satisfy p, and all the other +tuples satisfy p. Let m = n +2 , thus the size of p in the top-k +ranked items is sRk(D)(p) = n +2 < Lk = n +2 + 1 in the case of +global representation bounds. For proportional representation, +we have sD(p) = n +2 + 1 since tn+1 also satisfies p. And we +get sRk(D)(p) = n +2 < α · sD(p) · +k +|D| = n+3 +n+4( n +2 + 1) +n +n+1 = +n +2 +(n+3)(n+2) +(n+4)(n+1). To show p is a most general pattern to report, +we examine the parents of p, p′ which has m − 1 attribute +with the value assignment 0. For global representation bounds, +we have sRk(D)(p′) = n − m + 1 = +n +2 + 1 = Lk. For +proportional representation, we have sRk(D)(p′) = n +2 + 1 > +α · sD(p′) · +k +|D| = n+3 +n+4( n +2 + 2) +n +n+1 = n+2 +2 +(n+3)n +(n+1)(n+2). As a +result, every pattern with n +2 attribute assigned to 0 should be in +the result set. The number of such patterns is +� n +n/2 +� +> +√ +2 +n, +which is exponential. Therefore, any algorithm enumerating +these patterns is exponential. +Upper bounds: The notion of the most general patterns is +motivated by the utility of the information they provide. For +example, in the case of global representation bounds, if the +number of females in the top-k is less than the lower bound, +then clearly the number of black females is below the bound. +Unlike the most general patterns for the lower bound, in the +case of the upper bound, the most specific patterns are more +informative. For instance, if the number of black females is +above the upper bound, then so is the number of blacks and +the number of females. A plausible problem definition may +account for the most specific substantial patterns. Analogously +to the definition of the most general patterns, a pattern p is +a most specific substantial pattern if the size of p is above a +given threshold τs and for every pattern p′ such that p ⊊ p′, +the size of p′ is below the threshold τs. The goal is then to +find the most general patterns that do not satisfy the lower +bound and the most specific substantial patterns that exceed +the upper bound. For ease of presentation, in the rest of the +paper we will focus on the solution for Problems 3.1 and 3.2 +considering only the lower bounds. We note that our solutions +can be adjusted to support such problem definition (and other +definitions such as most general for upper bound, and the most +specific for lower bound). +While Theorem 3.3 indicates that the number of most +general groups with biased representation may be exponential, +our experimental evaluation shows that, in practice, their +number is significantly lower. In 97.58% of the times, the +number of the reported groups was less than 100. We note that +presenting a large number of results may be overwhelming to +the user. A user-friendly interface would organize the output +by k value and rank the groups by their overall size in the data +or by the bias in their representation (the difference between +the required representation and the actual representation) +IV. DETECTING GROUPS WITH BIASED REPRESENTATION +We next present our algorithms for detecting groups with +biased representation as defined in Problems 3.1 and 3.2. We +start with a simple solution that can be used to detect groups +based on both of the problem definitions. We then present two +optimized algorithms, designed to optimize the search for each +of the problems. +A. Iterative Top-Down Search (Baseline Solution) +The first, simple solution, utilizes the algorithm presented +in [5] to traverse the set of possible patterns, starting with the +most general ones, and compute the representation of each +group in the top-k ranked items in the data (for each k in +the given range). This is done using the notion of pattern +graph [5]. Briefly, the nodes in the graph are the set of all +possible patterns, and there is an edge between a pair of +patterns p and p′ if p ⊂ p′ and p′ can be obtained from p by + +{} +{G=F} +{S=GP} +{S=MS} +{G=M} +. . . +{G=F, S=GP} +{G=F, S=MS} +{G=M, S=GP} +{G=M, S=MS} +Fig. 3: Part of the pattern graph for the running example. Edges +of the search tree are marked with solid lines. +adding a single attribute value pair. In this case, we say that p +(p′) is a parent (child) of p′ (p). As shown in [5], the pattern +graph can be traversed in a top-down fashion, while visiting +each pattern at most once. Namely, traversing a spanning tree +of the pattern graph, which we denote as the search tree, and +formally define as follows. +Definition 4.1: Let D be a dataset with categorical attributes +A = {A1, . . . , An}. We assume attributes are ordered, and for +a given set of attributes S ⊆ A we use idx(S) to denote the +maximal index value of all attributes in S. Let p be a node +in the pattern graph of D. The children of p in the search +tree T are p′ such that p′ is a child of p in the pattern graph, +and idx(Attr(p′) \ Attr(p)) > idx(Attr(p)). Namely, p′ is +obtained from p by adding a single attribute value pair such +that the index of the newly added attribute is larger than the +maximal index value of attributes in p. +Example 4.2: A part of the pattern graph for the dataset of +the running example is shown in Figure 3, where G and S are +used as shorthands for Gender and School respectively. The +pattern {G=F, S=GP} is a child node of the patterns {G=F} +and {S=GP} in the pattern graph, however, in the search tree +it is only a child of {G=F}. This is because idx({G}) = +1 < 2 = idx({G, S}) \ {G}) whereas idx({S}) = 2 > 1 = +idx({G, S} \ {S}). +Algorithm 1: Top-down search +input : A dataset D, a ranking algorithm R, a size threshold τs, k +and lower bound Lk +output: Res = {p1, . . . , pn} where ∀pi ∈ Res sD(pi) ≥ τs and +pi is a most general pattern with sRk(D)(p) < Lk +/* For proportional bounds α is given as input +and Lk = α · sD(p) k +|D| +*/ +1 Res ← ∅ +2 S ← {generateChildren({})} +3 while S is not empty do +4 +p = S.pop() +5 +if patternSize (p, D) > τs then +6 +top-k c ← patternSize (p, Rk(D)) +/* For proportional bounds use +top-k_c < α · sD(p) k +|D| +*/ +7 +if top-k c < Lk +then +8 +update (Res, p) +9 +else +10 +S.push(generateChildren(p)) +11 return Res +Algorithm 1 detects patterns with adequate size in the data +(namely above a given threshold τs) and low representation +(less than Lk), in the top-k ranked items for a given dataset D, +a ranking algorithm R and a given k. It traverses the search +tree of the pattern graph (top-down), using a queue S and +maintains the set of identified patterns with sRk(D) < Lk. +First it initializes the result set Res to ∅ (line 1) and sets S +to contain the children of the most general (empty) pattern +(line 2). While the queue S is not empty (lines 3 – 10), the +algorithm extracts the first pattern in the queue p (line 4), +and computes its size in D. If it is greater than τs (line 5), +the size of p in Rk(D), sRk(D), is computed (line 6). If +sRk(D) is bellow the lower bound Lk (line 7), Res is updated +using the procedure update (line 8), that checks whether any +ancestor of p in the pattern graph is already in Res (this is +possible since the algorithm traverses the search tree and not +the patterns graph). Otherwise (line 9), sRk(D) exceeds the +lower bound Lk, and the children of p are added to the queue +using the procedure generateChildren (line 10), which +generates the children of a node as defined in Definition 4.1. +Finally, Res is returned (line 11). +ITERTD algorithm (baseline): Given a dataset D and a +ranking algorithm R, a size threshold τs, a range [kmin, kmax] +and lower bounds Lk for each kmin +⩽ k +⩽ kmax, a +simple solution for detecting groups with biased represen- +tation based on the global representation bounds definition +utilizes Algorithm 1, to apply a top-down search for each +kmin ≤ k ≤ kmax. Then, in each iteration report the patterns +with low representation in the top-k ranked items. Similarly, +Algorithm 1 can be used for the case of proportional repre- +sentation, with some slight modifications (shown as comments +in Algorithm 1). The objective is to report the patterns p +with adequate size but insufficient representation in the top- +k tuples Rk(D), where the representation in Rk(D) should +be proportional to the representation in D. In this case, the +bounds Lk are not given as input, instead, a bound for each +pattern is computed based on its size in the data and a value +α. Note that the pattern’s size is computed in line 5, and given +α we can replace the condition in line 7 with the condition +top-k c < α · sD(p) k +|D|. +We next propose more efficient algorithms for the problems. +B. Global Representation Bounds +The key observation is that when the lower bound remains +the same for k and k + 13, the search spaces for patterns with +biased representation in the ranking result of R for k and k+1 +are typically very similar. This is because the set of top-k and +top-(k + 1) tuples differ by a single tuple. Namely, increasing +k implies only local modifications to the search space. Let Tk +be the search tree generated to find the set of unfairly treated +patterns in the top-k tuples, and R(D)[k + 1], the (k + 1) +element in the result of ranking D using R. We can bound the +number of nodes in Tk whose size is affected by increasing k. +Proposition 4.3: R(D)[k + 1] can satisfy at most +|Tk| +2 +patterns (nodes) in Tk, where |Tk| denotes the number of nodes +in Tk. +3We assume Lk ⩽ Lk+1∀k +kmin ⩽ k < kmax. This is a reasonable +assumption since as k increases, so is the number of tuples in the top-k, thus +it is only logical the bounds increase as well. + +Proof 4.4: (Sketch) The basic idea of the proof is that when- +ever a pattern p is generated during the search, at least one +pattern p′ with the same set of attributes (Attr(p) = Attr(p′)) +that differs from p in the value assignment of a single attribute +is generated as well. This is true under the assumption that +every attribute has at least 2 values. For instance, the patterns +{Gender = M, School = MS} and {Gender = M, School = GP} +are both children of the pattern {Gender = M} and are gen- +erated when the procedure generateChilder({Gender = +M}) is invoked. Let Ai be the attribute such that Ai ∈ Attr(p) +and {Ai = ai} ⊆ p while {Ai = a′ +i} ⊆ p′. R(D)[k + 1] +can satisfy at most one of the patterns, as the value of Ai +in R(D)[k + 1] is either ai or a′ +i (or possibly other value, if +|Dom(Ai)| > 2). Since this is true for every node generated +during the search, R(D)[k + 1] can satisfy at most +|(Tk)| +2 +patterns (nodes) in Tk. +In that case, by starting the search for k + 1 from the +endpoint of the search for k, we significantly reduce the search +space (and as a result, the runtime, see Section VI-B). +Algorithm 2: GLOBALBOUNDS. Detecting groups +with biased representation based on global bounds +input : A dataset D, a ranking algorithm R, a size threshold τs, a +range [kmin, kmax] and lower bounds Lk for each +kmin ⩽ k ⩽ kmax +output: Res s.t. for each kmin ⩽ k ⩽ kmax +Res[k] = {p1, . . . , pn} where ∀pi ∈ Res[k] sD(pi) ≥ τs +and pi is a most general pattern with sRk(D)(p) < Lk +1 Res ← ∅ +2 Res[kmin], DRes ← +TopDownSearch(D, R, τs, kmin, Lkmin) +3 for k = kmin + 1 to kmax do +4 +if Lk−1 < Lk then +5 +Res[k], DRes ← +TopDownSearch(D, Rk(D), τs, Lk) +6 +else +7 +Res[k] ← Res[k − 1] +8 +foreach +b ∈ {p ∈ Res[k − 1] | R(D)[k] satisfies p} ∪ DRes do +9 +Res[k], DRes ← +searchFromNode (b, Res[k], DRes) +10 return Res +GLOBALBOUNDS algorithm: Our optimized algorithm +for the problem for detecting groups with global represen- +tation bias in the top-k, GLOBALBOUNDS, is depicted in +Algorithm 2. GLOBALBOUNDS starts by initializing the result +set map Res (line 1). It then performs a top-down search +for the case where k = kmin (line 2). This is done using +the procedure TopDownSearch, similar to the algorithm +depicted in Algorithm 1, with a minor addition. It maintains +a set DRes of patterns p reached during the search with +size below the lower bound in top-k, that are not part of +the result set since it already contains an ancestor of p. +TopDownSearch returns both, the result set of the search +Res, and the set DRes. When k increases (and Lk is kept +intact), the algorithm will utilize this set to initiate a local +search in the pattern graph. +Next, the algorithm preforms the search for each k from +k = kmin + 1 through k = kmax (lines 3–9). For each k, +if the bound increases, TopDownSearch is used to perform +a new top-down search. Otherwise, The algorithm considers +only patterns from DRes and patterns from Res[k − 1] that +the newly inserted tuple R(D)[k] satisfies (line 8). This is +because only their sizes in the top-k are affected by the new +tuple (at most half of the tree, based on Proposition 4.3). +For each such pattern, the algorithm applies the procedure +searchFromNode (line 9) to resume the search in the +relevant parts of the graph. This search updates Res[k] and +DRes. Finally, Res is returned (line 10). +Proposition 4.5: GLOBALBOUNDS returns the set of all +most general patterns p with bias representation using global +bounds in the top-k for each k in the given range. +The proof is by induction on k with a base case of k = +kmin. Details are omitted due to space constraints. +Example 4.6: Consider again D and R from the running +example. Assume we are given the size threshold τs += +4, kmin = 4, kmax = 5, and the lower bounds L4 = L5 = 2. +At the end of the top-down search for k = 4, the result +set Res[4] contains (among others) the patterns {Address = +U} and {Failures = 1}, that appears only once in the top-4 +tuples (namely, below the lower bound). DRes contains, for +instance, the patterns {Gender = F, Address = U}, {Gender = +M, Address = U}, {Gender = F, Failures = 1} and {Address += R, Failures = 1}. These patterns were generated during the +top-down search and have ancestors in Res[4] ({Address = +U} and {Failures = 1}). Next, the algorithm turns to compute +patterns with biased representation for k = 5. The new tuple +in the top-5 is tuple 14. It matches the patterns {Address = +U} and {Failures = 1} in Res[4]. Thus the algorithm performs +the search starting from those nodes. Their sizes in the top- +5 exceed the lower bound. In this search, these two patterns +are extracted from the result set and the pattern {Address = +U, Failures = 1} is added. From the set DRes, the patterns +{Gender = F, Address = U}, {Gender = M, Address = U}, +{Gender = F, Failures = 1} and {Address = R, Failures = 1} +are added to the result set Res[5], as their sizes in the top- +5 tuples are still below the threshold L5 and their respective +ancestors are removed from the result set. +C. Proportional Representation +We next consider the problem of detecting groups with +biased proportional representation as depicted in Problem 3.2. +The inputs are a dataset D, a ranking algorithm R, a range +[kmin, kmax], a size threshold τs and α ∈ R. The objective +is to report the patterns p with adequate size in D, but +insufficient representation in the top-k tuples Rk(D), where +the representation in Rk(D) should be proportional to the +representation of p in D. +First, note that the optimized solution presented for the case +of global representation bounds depicted in Section IV-B is not +applicable in this case. Recall that GLOBALBOUNDS (Algo- +rithm 2) aims at reducing the search space by avoiding search- +ing areas in the pattern graph that were not changed between + +consecutive iterations. When the bound remains unchanged +(Lk = Lk+1), patterns that the (k + 1) tuple in the ranking +does not satisfy, are not affected, and can be eliminated from +the search. This is not the case for proportional representation, +as the bound for each pattern depends on k as well. +Recall that the goal is to find patterns p such that +sRk(D)(p) < α·sD(p)· +k +|D|, and note that α and sD(p) do not +change during the computations. Thus, the inequality holds +depending on the value of k. Given R, D, α, a pattern p and +a value k such that sRk(D)(p) ⩾ α · sD(p) · +k +|D|, we denote by +˜k the minimal value for k such that the inequality does not +hold when fixing the value of sRk(D)(p). Namely, the minimal +value such that sRk(D)(p) < α · sD(p) · +˜k +|D|. +Example 4.7: Let α = 0.9. For D and R from our running +example, p ={Gender = F} satisfies the inequality for k = 4 +since sRk(D)(p) = 2 > 1.8 = 0.9 · 8 · +4 +16. ˜k = 5 in this case +since 0.9 · 8 · 5 +16 = 2.25. +Intuitively, if k is increased up to ˜k but the number of +tuples satisfying p remains the same, then the representation +of p is biased in the ranking result. If there is no ancestor +of p in the result set, then p should be added to it. To this +end, our optimized algorithm, PROPBOUNDS, computes for +each pattern in the search tree its corresponding ˜k value. It +maintains a set K which indicates patterns that potentially, if +they do not satisfy the ˜k element in the ranking, should be +added to the result set. K contains patterns in a branch of +the search tree whose ˜k values are monotonically decreasing. +A pattern p in K with the corresponding ˜k value should +be extracted from K and added to the result set when the +computation reaches k = ˜k if sRk(D)(p) = sRk−1(D)(p) (i.e., +p does not satisfy R(D)[k]). +Algorithm 3: PROPBOUNDS. Detecting groups with +biased proportional representation +input : A dataset D, a ranking algorithm R, a size +threshold τs, a range [kmin, kmax] and α ∈ R +output: Res s.t. for each kmin ⩽ k ⩽ kmax +Res[k] = {p1, . . . , pn} where ∀pi ∈ Res[k] +sD(pi) ≥ τs and pi is a most general pattern with +sRk(D)(p) < α · sD(p) k +|D| +1 Res ← ∅ +2 Res[kmin], K, DRes ← +TopDownSearch(D, R, τs, kmin, α) +3 for k = kmin + 1 to kmax do +4 +Res[k] ← Res[k − 1] +5 +Res[k], K, DRes ← +selectiveTD (D, R, τs, k, α, R(D)[k], K, DRes) +6 +foreach p ∈ DRes � +K[p′]=k{p′} such that R(D)[k] +doesn’t satisfy p do +7 +update (Res, p) +8 return Res +PROPBOUNDS algorithm: PROPBOUNDS (Algorithm 3) +operates as follows. Similarly to GLOBALBOUNDS, it starts +by initializing the result set map Res (line 1) and applying +a top-down search for kmin (line 2), as depicted in proce- +dure TopDownSearch (with the required modification for +proportional bounds), but in addition to sets Res, DRes, it +also maintains the set K. Then, the algorithm iterates over +the values of k from kmin + 1 to kmax (lines 3 – 7). In +each iteration, it first initializes Res[k] with the result from +the previous iteration Res[k − 1] (line 4). Then the algorithm +applies a (partial) search from the root using the procedure +selectiveTD (line 5). This search ignores the areas in +the tree that are not affected by R(D)[k]. The result of the +procedure is used to update the result set, K, and DRes. The +algorithm then iterates over the patterns in DRes (patterns +reached during the search that has an ancestor in the result) +and all patterns in K with ˜k = k that are not affected by +R(D)[k], to determine the changes to the result set (lines 6 – +7). Finally, Res is returned (line 8). +Proposition 4.8: PROPBOUNDS returns the set of all most +general patterns p with bias proportional representation in the +top-k for each k in the given range. +Similarly to Proposition 4.5, the proof is by induction on k +with a base case of k = kmin. Due to space constraints, the +details are omitted. +Example 4.9: Consider again D and R from the running +example. Assume we are given the size threshold τs += +5, kmin = 4, kmax = 5, and α = 0.9. At the end of the +top-down search for k = 4, the result set Res[4] consists of +the patterns {School = GP}, {Address = U} and {Failures = +1}. For each one of them, sD(p) = 8, and thus the bound on +sRk(D)(p) is α·sD(p)· +k +|D| = 0.9·8· 4 +16 = 1.8, but sRk(D)(p) +is only 1. The set K consists of {Gender = M} and {Gender += F}, both with ˜k of 5, and {School = MS} and {Address = +R} with ˜k = 7. Note that the pattern {School = MS, Address += R} was generated in the first top-down search, but was not +added to K since its ˜k value is 9, higher than its parent in the +search tree {School = MS}. +When k is increased to 5, the algorithm reexamines only +the patterns {Gender = M}, {School = MS}, {Address = U} +and {Failures = 1} that are affected by the R(D)[5] (tuple +14). The patterns {Address = U} and {Failures = 1} remain +in the result set for k = 5 even-though their size in the top-5 +is larger, since the bound for k = 5 increases as well. Finally, +the pattern {Gender = F} is added to Res[5] based on the +information from K (it is stored in K with ˜k = 5). +V. RESULT ANALYSIS +With the results of our algorithm in hand, an analyst may +wish to understand the cause of the bias in the representation +of the detected groups. We propose a method to provide +such explanations utilizing the notion of Shapley values [33]. +Shapley values is a concept adopted from game theory to +explain the effect of different attributes on the output of +a model for a given input. The use of Shapley values has +recently gained popularity in the field of interpretability and +explainability of ML models [24], [35]. Given a regression +model (or a classifier with probabilities) M, Shapley values +are used to evaluate the contribution of each attribute on the +output of M for a given input t. This is done by computing the + +weighted marginal contribution of each attribute value using +all possible subsets of attribute values. +Intuitively, an explanation for the bias may be the values +that affected the ranking of tuples in the given group. To this +end, we propose a method for explanations that consists of two +parts: the first identifies the attributes with the highest effect +on the ranking of tuples in the given group (using Shapley +values), and then we compare the values distribution of these +attributes in the top-k and the biased represented group. In +order to adopt the use of Shapley values, we need to tackle +two challenges. The first is to adjust our problem’s setting, +where we are given a ranking algorithm R (as a black box) +rather than a regression model. Second, Shapley values are +used to explain the contribution of the attribute values of +a single tuple, whereas we are interested in explaining the +(inadequate) representation of a group of tuples (in the top-k). +To address the first challenge we compute a regression +model MR that simulates the process of R and can be used +to approximate the effect of attribute values of a given tuple +t on t’s ranking by computing the Shapley values of MR(t). +To this end we define DR = {(t, R(D)[t]) | t ∈ D}, where +R(D)[t] is the ranking of t in R(D), and use it to train a +regression model MR. Then, given a pattern p such that p +was returned by one of our algorithms for detecting groups +with biased representation for a given k, to explain the result, +we compute the Shapley values (st +1, . . . , st +m) for each tuple t +such that t satisfies p, namely, for each tuple in the detected +group. We then aggregate the results into a single Shapley +value vector (s1, . . . , sm) for the pattern p such that +si = +� +t s.t. t satisfies p st +i +sD(p) +To show the differences between the pattern p and the top- +k patterns, we visualize the value distribution of attributes +with large Shapley values of tuples that satisfy the pattern +p compared to their distribution among the tuples in the top- +k. In Section VI-C we show that using our method we are +able to disclose the attributes that were used for ranking (and +thus affect the representation of groups in the top-k) when the +ranking model is given as a black box. Moreover, we show +that the value distribution in the attribute identified as most +significant in the ranking is different for groups detected by +our algorithms than for the top-k tuples, which indicates the +identified attributes values explain the results. +VI. EXPERIMENTS +We experimentally examine the proposed solutions using +three real-life datasets. We start with our setup and then present +a quantitative experimental study whose goal is to assess +the scalability of our algorithms. In particular, we examine +our algorithms’ performance for each fairness definition as a +function of the number of attributes, pattern’s size threshold, +and range of k. We then demonstrate our proposed method for +the analysis of the results presented in Section V. We conclude +with a comparison to the framework presented in [27], showing +the differences in the results between our algorithms and the +algorithm of [27] through a case study. +A. Experiment Setup +a) Datasets: We used three real datasets with different +numbers of tuples and attributes as follows. +• The COMPAS Dataset4 was collected and published by +ProPublica as part of their investigation into racial bias +in criminal risk assessment software. It contains the de- +mographics, recidivism scores produced by the COMPAS +software, and criminal offense information for 6,889 indi- +viduals. We used up to 16 attributes eliminating attributes +such as names, ids, dates, etc. +• Student Performance Dataset (Student dataset)5 shows +the performance of students in secondary education of two +Portuguese schools as described in Example 2.1. We consid- +ered in the experiment the data fragment with information +regarding the Math exam (395 tuples and 33 attributes). +• German Credit Dataset6 with financial and demographic +information about 1,000 loan applicants with 20 attributes. +It was originally used in the context of classification, where +each application is classified as a good or bad credit risk. +Compared algorithms: We evaluate the performance, in +terms of the running time of our proposed algorithms. +• IterTD (baseline). The simple solution for detection of +groups with biased representation, which iteratively applies +a top-down search as depicted in Section IV-A. +• GLOBALBOUNDS (Algorithm 2). The algorithm for de- +tecting groups with biased representation based on global +bounds as described in Section IV-B. +• PROPBOUNDS (Algorithm 3). The algorithm for detecting +groups with biased representation based on proportional +representation bounds as described in Section IV-C. +Parameters setting: For space constraints, unless stated +otherwise, we report the result for the following set of default +parameters: τs = 50, kmin = 10, kmax = 49, and the lower +bounds are 10 for 10 ≤ k < 20, 20 for 20 ≤ k < 30, 30 for +30 ≤ k < 40 and 40 for 40 ≤ k < 50 for global bounds, and +α = 0.8 for proportional representation. The reported results +reflect the algorithm’s performance under expected and typical +usage scenarios. Following the goals of fairness in ranking, +we set gradually increasing bounds on group representation in +the top-k ranked items. Since we aim at reporting the detected +groups to the user, we set the parameters such that the number +of reported groups in most cases is between 1 to 100. The +number of attributes was set to be the maximal number the +baseline solution could handle. and continuous attributes, e.g., +age, were bucketized equally into 3 − 4 bins, based on their +domain and values. We note that the selection of bucketization +affects the patterns graph size and may also affects the possible +group definitions and their representation in the top-k, which +4https://www.propublica.org/datastore/dataset/ +compas-recidivism-risk-score-data-and-analysis +5https://archive.ics.uci.edu/ml/datasets/student+performance +6https://archive.ics.uci.edu/ml/datasets/Statlog+(German+Credit+Data) + +could also affect the running time. However, in this work, we +assume that the attribute values used for group definitions are +categorical (i.e., the bucketization is given). We experimentally +evaluated the algorithms using different parameter settings and +observed similar trends. +Ranking Algorithms: The Student dataset was ranked +based on the value of the attribute G3 showing the stu- +dent’s math final grades. For the COMPAS dataset, we per- +formed similar ranking method as in [4]: We normalized +attribute values c days from compas, juv other count, +days b screening arrest, +start, +end, +age, +and +pri- +ors count as scoring attributes. Values are normalized as +(val − min)/(max − min). Higher values correspond to +higher scores, except for age. Tuples are ranked descendingly +according to their scores. For the German Credit dataset, we +used the ranking presented in [36] based on creditworthiness. +All experiments were performed on a macOS machine with +a 2.8 GHz Quad-Core Intel Core i7 CPU and 8GB memory. +The algorithms were implemented using Python3.7. +B. Experimental results +Both GLOBALBOUNDS and PROPBOUNDS run much faster +than the baseline, particularly as the number of attributes +increases and the baseline becomes exponentially more ex- +pensive. Details below. +Number of attributes: The first set of experiments aims +to study the effect of the number of attributes on the running +time. To this end, we varied the number of attributes in the +datasets from 3 to |A| where A is the set of all attributes +in the dataset. The number of attributes (along with their +cardinality) determines the number of possible patterns, and +as a result, the size of the search space. Thus, as the number +of attributes increases, we expect to see a steep growth in +the running time. The results (using a 10-minute timeout) are +presented in Figures 4–5. Indeed, in all cases we observed a +rapid increase in the running time, while GLOBALBOUNDS +and PROPBOUNDS outperform ITERTD. +Size threshold: In the next set of experiments, we as- +sessed the effect of the size threshold τs on the running time. +To this end, we varied the size threshold from 10 to 100 +while using the default values for the rest of the parameters. +The results are presented in Figures 6 and 7. We observed +a decrease in the running times of the algorithms. This is +because the number of patterns satisfying the size threshold +decreases as the threshold increases, and as a result, the search +space is decreased as well. In all cases, GLOBALBOUNDS and +PROPBOUNDS outperform ITERTD. +Range of k: We examine the scalability with respect to +the range of k considered by the algorithm. We varied the +range from 40 (40) to 990 (340) by setting kmin to be 10, and +increasing kmax from 50 (50) to 1000 (350) for COMPAS +(Student and German Credit) dataset and observed the effect +on the running time. We set different maximum range of k +due to different size of the datasets (6889 for COMPAS, 395 +for Student and 1000 for the German Credit). The results +are presented in Figure 8 and 9. In all cases the optimized +algorithms outperform ITERTD, which illustrates the efficient +reduction in the search space. +Recall that GLOBALBOUNDS and PROPBOUNDS optimize +the search space compared to ITERTD by utilizing the search +result of the iteration for k in order to compute the result set +for k + 1. Thus, as the range of k increase, we expect to see +a greater improvement in the performance of the optimized +algorithm compared to the baseline solution. This trend is +shown in Figure 8 and 9. To further demonstrate the useful- +ness of the approach, we compared the number of patterns +examined during the search for each one of the algorithms. +The observed gain was up to 39.35% in the COMPAS dataset, +56.87% in the student dataset and 29.27% in the credit card +dataset for detecting groups with biased representation using +global bounds, and 39.60%, 20.49% and 56.83% respectively +for proportional representation. +C. Result Analysis using Shapley values +We next demonstrate the usefulness of our proposed method +for results analysis using Shapley values presented in Sec- +tion V. The goal of the experiment is twofold. First, we +show that our Shapley values based method for evaluating the +effects of attributes on the ranking can indeed reveal useful +information on the actual attributes used for ranking when the +ranking algorithm is given as a black box. Then we show +that the value distribution for those attributes can be used to +explain the representation bias, by comparing the distributions +for the values in the top-k with those in the detected groups +and focusing on the differences. +We trained a regression model using the ranked data for each +dataset and examined the Shapley values for groups detected +by our algorithms. We present the results for the patterns +(groups) p1 = {mother’s education = primary education (4th +grade)} in the Student dataset, p2 = {age = younger than 35} +in COMPAS and p3 = {status of existing account = (0 ⩽ +· · · < 200) DM7} from the German Credit dataset, which +were detected by the GLOBALBOUNDS algorithm for k = 49 +and Lk = 40. We observed similar results for other groups +detected by the algorithms and other parameters. Figures 10a, +10b and 10c show the resulting aggregated Shapley values for +each group, as explained in Section V. We show the Shapley +values for the six attributes with the larges values for each +group, as the rest had significantly lower values (lower than +5.79%, 3.31% and 9.54% of largest aggregated Shapley values +for p1, p2 and p3 respectively). +For the group of students whose mother’s education level +is primary education, which was detected by our algorithm +as a group with biased representation in the Student dataset, +the final grade has the largest aggregated Shapley value on +the ranking (Figure 10a). This result agrees with the fact that +the value of the final grade is indeed used for ranking by the +ranking algorithm (and it is in fact the only attribute used). +7A Debit Memo (DM) on a company’s bank statement refers to a deduction +by the bank from the company’s bank account. In other words, a bank debit +memo reduces the bank account balance similar to a check drawn on the bank +account. + +(a) COMPAS dataset +(b) Students dataset +(c) German Credit +Fig. 4: Running time as a function of number of attributes - Ranking with global bounds +(a) COMPAS dataset +(b) Student dataset +(c) German Credit +Fig. 5: Running time as a function of number of attributes - Ranking with proportional representation +(a) COMPAS dataset +(b) Students dataset +(c) German Credit +Fig. 6: Running time as a function of the size threshold τs - Ranking with global bounds +(a) COMPAS dataset +(b) Students dataset +(c) German Credit +Fig. 7: Running time as a function of the size threshold τs - Ranking with proportional representation +Other than the final grade, the first and second period grades +have a notable aggregated Shapley (although significantly +lower aggregated Shapley value than the final grade). This +is due to the high correlation between those attributes and +the final grade [13]. We also noticed the mother’s education +attribute in the result. This may indicate some correlation +between the mother’s education and the final grade. However, +we also note that the aggregation of the Shapley values for +other attributes, e.g., father’s education, show no clear pattern: +some values have a positive effect and some negative, and +different tuples in the group have different values. In contrast, +all the tuples in the group have the same value (primary +education) for mother’s education attribute (since it is used to +define the group). Therefore the Shapley values of the attribute +for the different tuples in the groups are similar. This may also +increase the aggregated value compared to other attributes. +We observed this phenomenon, where the attributes used to +define the detected group are slightly higher, for other groups +detected by our algorithms also. +For the COMPAS dataset, tuples are ranked by a combined +score based on seven attributes: days from compas, the number +of other juvenile convictions, days before screening arrest, +start date, end date, age, and the number of priors crimes +committed. In Figure 10b, showing the aggregated Shapley +values of people younger than 35, six out of the above seven +attributes are the six attributes with the largest Shapley values. +In this case, the attribute end date and the number of priors +crimes committed are identified as the most significant factor +affecting the detected group. +For the German Credit dataset, tuples are ranked according +to their ranking in [36], however the actual ranking algorithm +is unknown. Namely, we do not have the ground truth and + +40 +S +GlobalBounds +Execution time +IterTD +20 +0 +2 +4 +6 +8 +10 +12 +14 +16 +Number of attributes250 +S +PropBounds +Execution time +IterTD +200 +150 +100 +10 +20 +30 +4050 +¥60　70 +80 +90 +100 +Size threshold400 +PropBounds +Execution time +IterTD +200 +10 +20 +30 +4050 +60 +70 +80 +90 +100 +Size thresholdS150 +PropBounds +time +IterTD +100 +Execution +50 +10 +20 +30 +40　50 +60 +70 +80 +90 +100 +Size thresholdS +GlobalBounds +Execution time ( +IterTD +100 +5 +10 +15 +20 +25 +30 +33 +Number of attributesS +GlobalBounds +Execution time ( +IterTD +2 +U +5 +10 +15 +20 +Number of attributess +400 +PropBounds +Execution time +IterTD +200 +0 +2 +4 +6 +8 +10 +12 +14 +Number of attributes400 +S +PropBounds +Execution time +IterTD +200 +0 +2 +4 +6 +8 +10 +12 +14 +Number of attributes200 +S +PropBounds +Execution time +IterTD +100 +5 +10 +15 +20 +Number of attributesS +40 +30 +GlobalBounds +20 +IterTD +10 +20 +30 +40　50 +60 +70 +80 +90 +100 +Size thresholdS +400 +GlobalBounds +Execution time +IterTD +200 +10 +20 +30 + 4050 +¥60　70 +80 +90 +100 +Size thresholdGlobalBounds +Execution time +IterTD +2 +10 +20 +30 +4050 + 60 +70 +80 +90 +100 +Size threshold(a) COMPAS dataset +(b) Students dataset +(c) German Credit +Fig. 8: Running time as a function of the range of k- Ranking with global bounds +(a) COMPAS dataset +(b) Students dataset +(c) German Credit +Fig. 9: Running time as a function of the range of k - Ranking with proportional representation +(a) Aggregated Shapley value of group +p1 ={mother’s education = primary ed- +ucation} in the Student datase +(b) Aggregated Shapley value of group +p2 +={age = younger than 35} in the +COMPAS dataset +(c) Aggregated Shapley value of group +p3 ={status of existing account = (0 ⩽ +· · · < 200DM)} in the German Credit +dataset +(d) Value distribution of the final grade +attribute in the Student dataset +(e) Value distribution of the end date at- +tribute in the COMPAS dataset +(f) Value distribution of residence length +in the German Credit dataset, +Fig. 10: Result analysis using Shapley values +cannot verify the attribute detected as significant for explain- +ing the bias in the representation of the detected group in +the top-k are actually used by the ranking algorithm. The +attributes residence length, duration in month, credit amount, +and installment rate have the largest Shapley values as shown +in Figure 10c. All of these attributes represents reasonable +features to decide one’s credit worthiness. +The Shapley value represents the effect of different at- +tributes on the ranking of groups. To analyze the differences +between the detected groups and top-k tuples (with respect +to these attributes), we visualize the value distribution of +attributes with the largest Shapley values in Figures 10d, 10e, +and 10f. Since the number of tuples in the top-k and the +detected group differ, the y-axis represents the proportion of +tuples (rather than their count) with the values shown on the +x-axis (the set of possible values for the attribute). +For all three datasets, we observed vast differences in the +distributions of the values of the attribute with the largest +Shapley value between the tuples in the top-k and the tuples in +the group detected with biased representation. For the students +dataset (Figure 10d), the final grades of tuples in the top-k all +fall in the range of 15 − 20, while most tuples in the detected +group have a final grade lower than 15. In the COMPAS +dataset (Figure 10e), the value of the end date for all top-k +tuples is 0 while only half of the tuples in the detected group +have the same value, and almost 30% of them have the value +of 2. Similar results were observed in the value distribution of +the residence length attribute as shown in Figure 10f. +D. Comparison with Existing Solution +The problem of identifying subgroups in the data that +behave differently compared to the overall dataset was studied +in [27]. Different from our problem definition, which relies +on fairness measures for ranking to define groups with biased + +S +GlobalBounds +Execution time +150 +IterTD +100 +50 +200 +400 +600 +800 +1000 +Range of k200 +S +Execution time +GlobalBounds +150 +IterTD +100 +100 +200 +300 +350 +Range of kS +GlobalBounds +Execution time +7.5 +IterTD +5.0 +2.5 +100 +200 +300 +350 +Range of k600 +S +PropBounds +Execution time +IterTD +400 +200 +200 +400 +600 +800 +1000 +Range of kS +PropBounds +Execution time ( +400 +IterTD +200 +100 +200 +300 +350 +Range of kS +PropBounds +Execution time +400 +IterTD +200 +100 +200 +300 +350 +Range of kfinal grade +second period grade +Attribute +mother's education +first period grade +number of past class failures +father's education +other positive Shapley values +other negative Shapley values +0. 10 20 30end date +# of prior crimes committed +Attribute +start date +# of other juvenile convictions +days before screening arrest +age +other positive Shapley values +other negative Shapley values +-100 +0residence length +duration in month +Attribute +credit amount +installment rate +employment length +existing credits at this bank +other positive Shapley values +other negative Shapley values +-25 +0 +250.3 +{mother's education = primary +education (4th grade)) +0.2 +top-k +0.1 +0 +0 +46810 12 14 16 18 20 +Value of final grade1.0 +[age = younger +Proportion +than 35) +0.5 +top-k +0.0 +0 +1 +2 +Value of end date1.0 +[ status of existing account += (0 <= ... < 200 DM)) +0.5 +top-k +0.0 +0 +1 +2 +3 +Value of residence lengthrepresentation in the top-k ranked items, the work of [27] uses +the notion of divergence to measure performance differences +among data subgroups. Each data item in the data t ∈ D is +associated with an outcome o(t) where the outcome function is +defined based on the ranking of t by the ranking algorithm. The +outcome of a group o(G), is then the average of the outcome +of every item t ∈ G. The divergence of a subgroup G in +the data D is the difference between the outcome of G and +the entire data, i.e., o(G) − o(D). Given a threshold s on the +subgroup size, the solution of [27] computes the divergence +of all subgroups with sizes larger than s. +To better demonstrate the differences between the defini- +tions and the resulting groups identified by each algorithm, +we conducted an experiment using the Student dataset. We +used the default size threshold of τs = 50 (support in [27] of +0.13, i.e., 13% of the data). Since [27] does not consider a +range of k’s, we fixed kmin = kmax = 10 (namely, compare +the results when k = 10). To allow for easy comparison, we +used only the first 4 attributes of the data: school, sex, age, +and address. We used the outcome function o(t) that assigns +the value 1 for tuples t in the top-k, and 0 for the rest (as +presented in [27]). Finally, for our algorithms, we used the +default parameters of lower bound 10 for ranking with global +bounds and α = 0.8 for proportional representation. +PROPBOUNDS +outputs +2 +patterns: +{sex=F} +and +{address=R}, +both +returned +by +GLOBALBOUNDS +as +well. Additionally, GLOBALBOUNDS returned the patterns +{school=GP}, {sex=M} and {address=U}, which had less +than 10 instances in the top-10 ranked items (9, 7, and 9 +respectively), but considering their overall size (349, 208 and +307 respectively), their representation in the top-k is adequate +and thus are not returned by PROPBOUNDS. The algorithm +of [27] returned 28 groups including the groups detected by +our solution. Since the number of reported groups may be +extremely large, the algorithm of [27] ranks the groups by +their divergence. The 5 patterns with the highest divergence +contain 3 − 5 attributes, with the value assignment sex=M, +i.e., they are descendents of the pattern {sex=M} (in the +pattern graph) returned by GLOBALBOUNDS. The pattern +{sex=M} was ranked at 17 according to its divergence value. +The key difference between our algorithms and the solution +of [27] lies in the definitions of the groups they aim to identify. +The two solutions deal with a similar problem, however, our +solution prefers concise groups (most general pattern) while +the solution of [27] is designed to identify all groups with +sufficient representation in the overall data and high divergence +(a measure of “unfairness”). As a result, the output of [27] is +typically larger and contains subgroups that are consumed by +each other. Finally, the work of [27] considers a single k while +we consider a range of k’s, aligning with fairness definitions +in the literature, making the solution fair for any position in +the top-k. +VII. RELATED WORK +The notion of fairness in ranking algorithms was studied in +a line of works, introducing different fairness definitions [10], +[20], [30], [34], [36], [38]. These definitions typically focus +on top-k positions, as those are usually the most important +positions. In this paper, we consider two such definitions: the +fundamental definition of [10], which measures fairness by +bounding the representation of different groups in the data, +and a refined definition that considers proportional groups +representation. These definitions, as customary in the context +of algorithmic fairness, refer to some given protected groups. +We harness these definitions to define the problem of detecting +groups with biased representation, eliminating the need to +pre-define protected groups. The problem of generating fair +ranking results was studied in [4], [38]. These works consider a +wide range of definitions for fairness in ranking, which rely on +the notion of protected groups. This line of work is orthogonal +to the problem we defined in this paper, and our proposed +method can be used to identify such protected groups, when +they are unknown in advance. +Recent works have studied the problem of automatically +detecting “problematic” or biased subgroups in the data, +without the need to specify the protected attributes a priori, in +the context of classification [9], [12], [27], [28]. In [28], the +authors introduced the notion of divergence to measure the +difference in the behavior of a classifier on data subgroups. +The goal is then to report subgroups with sizes above a +given threshold and high divergence. In [27] they extend +their framework to ranking, where they consider the average +outcome value, which is defined based on the ranking of the +instances in each group, as a measure of the group’s outcome. +In contrast to [27], our problem definitions rely on groups’ +representation in the top-k ranked items as fairness measures +for ranking. We demonstrate the differences in the resulting +groups identified by each definition in Secetion VI-D. The +interactive system MithraCoverage [21] investigates popula- +tion bias in intersectional groups. The notion of coverage is +introduced to identify intersectional subgroups with inadequate +representation in the dataset. Differently, in our work, we only +report patterns with adequate representation in the data, but +inadequate representation in the output of a ranking algorithm. +The use of Shapley values to provide explanations for +ML models was studied in a line of works (see e.g., [24], +[35]). In these works, the Shapley values are computed for +an individual input instance to a classification or regression +model. The Shapley value of a feature is then interpreted as +the contribution of the feature to the output of the given input. +Differently, we are aiming at providing explanations for the +representation of a group of tuples in the output of a ranking +algorithm. In [28] the authors presented a method to measure +the contribution of items to divergence of groups utilizing +Shapley values. However, it considers only the contribution +of attributes that are used to define the group. In contrast, our +solution considers all attributes as possible explanations. This +requires an additional aggregation step in the computation of +the Shapley values. As demonstrated in VI-C, the explanation +is typically buried in the values of attributes used for ranking. +Moreover, our adjustment of Shapley values to explain a +ranking algorithm is novel. + +Our baseline solution, utilizing a top-town search presented +in Section IV-A is built on the algorithm presented in [5] +(for a simpler problem), which in turn shares similar ideas +to the Apriori algorithm [1], the Set-Enumeration Tree for +enumerating sets in a best-first fashion [32], discovering +functional dependencies (FDs) [19], [26] and frequent item- +sets and association rule mining [1], [37]. Similarly to [5], the +key difference from our work lies in the structure of the graph +traversed in the solution: the pattern graph (in our case) com- +pared to the powerset lattice in the other works. Conditional +functional dependencies (CFDs) [15] extend the notion of FDs +by considering patterns to describe dependencies that hold +only on subgroups in the data. Similar to the top-down search +applied by the baseline solution, algorithms for discovering +CFDs [14], [16], [18], [31] also utilize the notion of pattern +and lattice of patterns. However, the difference in the end +goal (discovering CFDs versus identifying groups with biased +representation) leads to differences in the pruning techniques +in the baseline solution. We then present two novel optimized +algorithms designed for each one of the problems we defined. +These algorithms reduce the search space as explained in +Section IV and significantly outperform the baseline solution +as shown in the experimental evaluation. +VIII. CONCLUSION +In this paper, we have studied the problem of detecting +groups with biased representation in the result of a ranking +algorithm. We build on fairness measures previously defined +in the literature, considering the representation of protected +groups in the top-k ranked items, for any reasonable range +of k. Our problem definitions eliminate the need to pre-define +the protected groups. We consider two variants of the problem. +The first is based on global bounds over the representation of +different groups in the top-k ranked items, and the second +restricts the representation of each group in the top-k, based +on its overall representation in the data. +We theoretically analyse the complexity of the problem, +showing that in the worst case, the number of groups can be +exponential in the number of the dataset attributes. We present +a baseline algorithm that can handle both definitions and two +optimized algorithms designed to improve the performance for +each fairness measure. Furthermore, we present a method to +explain the output of our algorithms. There are many intriguing +directions for future research, including the extension of the +framework to support other fairness measures and further +investigation of the automatic suggestion for thresholds. + +REFERENCES +[1] Rakesh Agrawal and Ramakrishnan Srikant. Fast algorithms for mining +association rules in large databases. In VLDB’94, Proceedings of 20th +International Conference on Very Large Data Bases, September 12-15, +1994, Santiago de Chile, Chile, pages 487–499. Morgan Kaufmann, +1994. +[2] Kristen M. Altenburger, Rajlakshmi De, Kaylyn Frazier, Nikolai Avte- +niev, and Jim Hamilton. Are there gender differences in professional +self-promotion? an empirical case study of linkedin profiles among +recent MBA graduates. In Proceedings of the Eleventh International +Conference on Web and Social Media, ICWSM 2017, Montr´eal, Qu´ebec, +Canada, May 15-18, 2017, pages 460–463. AAAI Press, 2017. +[3] Abolfazl Asudeh, H. V. 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ACM, 2017. + diff --git a/t9AyT4oBgHgl3EQf0fl1/content/tmp_files/load_file.txt b/t9AyT4oBgHgl3EQf0fl1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c89958a7b89d9297fcbc36145781fffadc4816cb --- /dev/null +++ b/t9AyT4oBgHgl3EQf0fl1/content/tmp_files/load_file.txt @@ -0,0 +1,1050 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf,len=1049 +page_content='Detection of Groups with Biased Representation in Ranking Yuval Moskovitch Ben Gurion University of the Negev yuvalmos@bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='il Jinyang Li University of Michigan jinyli@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='edu H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Jagadish University of Michigan jag@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='edu Abstract—Real-life tools for decision-making in many critical domains are based on ranking results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' With the increasing awareness of algorithmic fairness, recent works have presented measures for fairness in ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Many of those definitions consider the representation of different “protected groups”, in the top-k ranked items, for any reasonable k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Given the protected groups, confirming algorithmic fairness is a simple task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' However, the groups’ definitions may be unknown in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In this paper, we study the problem of detecting groups with biased representation in the top-k ranked items, eliminating the need to pre-define protected groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The number of such groups possible can be exponential, making the problem hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We propose efficient search algorithms for two different fair- ness measures: global representation bounds, and proportional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Then we propose a method to explain the bias in the representations of groups utilizing the notion of Shapley values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We conclude with an experimental study, showing the scalability of our approach and demonstrating the usefulness of the proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' INTRODUCTION Ranking is a commonly used operation in a wide range of application domains, for example, in presenting results on a web search engine [8], establishing credit scores [7], school admission [29] and hiring [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' While convenient and useful, these tools can be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' As a result, they may affect decision- making in undesirable ways and can even impact human life [2], [6], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This problem has drawn much attention from the research community, and a line of recent works has focused on measuring and mitigating bias and unfairness in ranking [3], [10], [17], [22], [34], [36], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The notion of algorithmic fairness was studied extensively for a broad class of models [25], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Fairness measures typically refer to a given “protected group” in the data, which is defined based on the values of some sensitive attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', gender, race, age, or combinations thereof), usually based on societal history of discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Analyzing the fairness measure of a system with respect to the given group is a simple task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' However, “non-standard” protected groups cannot always be specified in advance, and such groups may be overlooked when examining the performance of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For example, a model developed to assign grades to students (in place of exams that were canceled due to the COVID- 19 pandemic) was shown to be biased against high-achieving students from poor school districts1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For instance, students 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='nytimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='com/2020/09/08/opinion/ international-baccalaureate-algorithm-grades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='html from low-income families were predicted to fail the Spanish exam, even when they were native Spanish speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In this case, the model was discriminating against Hispanic students from poor school districts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' A primary source of bias was the use of historical exam results of each school to predict student performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' However, using the school (identified by school ID) to define the protected group is not an intuitive choice, and so may not have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Moreover, even if we consider the group of Hispanic students as a protected group, we may not find any fairness issues, since this subgroup of students is only a small fraction of all Hispanic students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In this paper we study the problem of detecting groups that are treated unfairly by a ranking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In other words, we want to let the data speak to (potential) unfairness, without requiring a human modeler to identify protected attributes ahead of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Following fairness definitions presented in the literature on fairness in ranking (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', [10], [30], [36]), we consider group representation in the top-k ranked items for any k in a reasonable range as a measure of fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Recent works have studied the problem of automatically detecting “problematic” or biased subgroups in the data with- out the need to specify the protected attributes a priori [9], [12], [21], [23], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' However, these works considered only classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In [27] the authors of [28] extend their framework to consider ranking as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In contrast to our work, which builds on fairness measures for ranking from the litera- ture, they use the notion of divergence to measure performance differences among data subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This difference leads to differences in the result sets returned by each method (see Section VI-D for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We next outline our main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Problem formulation: We formally define the problem of detecting groups with biased representation in the top-k ranked items for a given ranking algorithm R, a dataset D, and a range of possible k’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Groups are defined using value assignment to a set of attributes we denote as patterns (see Section II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To provide concise and meaningful results we use a threshold on the returned groups’ size and report only groups that are not subsumed by any other group in the result set (referred to as the most general patterns, see Section III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We start with the fundamental definition of [10], which uses upper and lower bounds to restrict the number of tuples in the top-k from different groups in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The goal is to report groups such that their representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', number of tuples) in the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='00719v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='LG] 30 Dec 2022 top-k does not lie within the given bounds for a given range of possible k’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We refer to this problem as the global bounds representation bias problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We then consider the prominent class of fairness measures utilizing proportional representation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Intuitively, the representation of each group in the top-k should be proportional to its size in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Using this notion we define the proportional representation bias problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We show that no polynomial algorithm exists to solve either problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Detection of groups with biased representation: We present algorithms for the problem of finding the set of all substantial groups (in terms of their size in the data, and their subsumption in other groups) with biased representation in the top-k ranked items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We first present a simple baseline solution that utilizes the notion of pattern graph presented in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We show how to traverse the graph in a top-down fashion in order to find groups with biased representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This search is then applied repeatedly for each k in the given range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Bearing in mind the complexity of the problem, we focus on optimizing the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Our optimized solutions rely on the fact that the set of top-k and top-(k + 1) tuples differ by a single tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' As a result, the search spaces for succeeding k values are typically very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The optimized solutions utilize this observation to avoid parts of the search tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Result analysis: Given a group with biased representation in the top-k ranked items, an analyst may wish to understand the cause of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To this end, we propose a method that harnesses the notion of Shapley values to identify attributes that significantly affect the ranking of the detected group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To analyze the difference between the detected group and top-k ranked tuples, we visualize the value distribution of such attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Shapley values have been used to provide similar explanations for regression and classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Our novelty is in developing a corresponding method for the ranking problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Experimental study: We complement our algorithmic development with an extensive experimental study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We eval- uate the performance and properties of the algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', the scalability and parameter setting effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We examine the effect of the number of attributes, groups’ size threshold, and range of k, using three real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Our results show the applicability of our solution in practice, despite the theoretical complexity of the problems, and the usefulness of the optimized algorithms compared to the baseline solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We then experimentally demonstrate the usefulness of our approach for results analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Finally, we compare the result of our algorithms to the results of the method proposed in [27] through a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Paper organization: The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We present the necessary preliminaries for our problem definition in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Then in Section III we formally define the problems of detecting groups with bi- ased representation in the data and prove their hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Our solutions is presented in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In Section V we introduce a method for analyzing and explaining the results of our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We describe our experimental evaluation in Section VI, overview related work in Section VII and conclude in Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' PRELIMINARIES We next provide necessary background on the notion of patterns to represent data groups and the concept of fairness in ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We will use the following example as our running example to demonstrate the ideas presented in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='1: The Student Performance Data Set [13] con- tains information from two Portuguese secondary schools in the Alentejo region of Portugal, Gabriel Pereira (GP) and Mousinho da Silveira (MS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The data was collected during the 2005-2006 school year and it contains the performance of 1044 students in the Math and the Portuguese language exams, along with demographic, social, and school-related information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Figure 1 depicts a sample from the data with the attributes: gender, school, address (urban or rural), and failures (number of past class failures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The grade attribute depicts the students’ grades on a scale of 0 − 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Consider an excellence student program committee that wishes to select students for a scholarship based on their academic achievements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To this end, they use a ranking algorithm R to rank students by their grades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In the case of similar grades, students with fewer failures are ranked higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The scholars’ list is publicly announced, and should be diverse and inclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Data Groups We assume the data is represented using a single relational database, and that the relation’s attribute values used for group definitions are categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To include attribute values drawn from a continuous domain in the group definition, we render them categorical by bucketizing them into ranges: very commonly done in practice to present aggregate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We use the notion of patterns, value assignment to a set of attributes, to define groups in the data [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='2 (Patterns): Let D be a database with cate- gorical attributes A = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' , An} and let Dom(Ai) be the active domain of Ai for i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='.n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' A pattern p over D is the set of {Ai1 = a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' , Aik = ak} where {Ai1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' , Aik} ⊆ A and aj ∈ Dom(Aij) for each Aij in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We use Attr(p) to denote the set of attributes in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We say that a tuple t ∈ D satisfies a pattern p if t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='Ai = ai for each Ai ∈ Attr(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The size sD(p) of a pattern p is then the number of tuples in D that satisfy p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Given a ranking algorithm R we use Rk(D) to denote the top-k ranked items in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Finally, we use sRk(D)(p) to denote the size of p in Rk(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='3: Consider the dataset given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' p = {School = GP}, is an example of a pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Tuples 3, 4, 7, 8, 12, 13, 15 and 16 satisfy p and thus sD(p) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For the ranking algorithm R whose results are depicted in the Rank column, we have sR5(D)(p) = 1, since only one tuple in the top-5 ranked items satisfies p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Fairness measure The problem of fairness in ranking was studied in a line of works (see [30] for a survey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' A fundamental definition, # Gender School Address Failures Grade Rank 1 F MS R 1 11 8 2 M MS R 1 15 3 3 M GP U 1 8 10 4 M GP U 2 4 16 5 M MS R 0 19 2 6 F MS U 1 4 15 7 F GP R 1 7 11 8 M GP R 1 6 13 9 F MS R 0 14 4 10 F MS R 2 7 12 11 M MS R 2 13 6 12 F GP U 0 20 1 13 F GP U 2 12 7 14 M MS U 1 13 5 15 F GP U 1 5 14 16 M GP U 0 9 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 1: Students’ data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The Rank column depicts their ranking based on the grade and number of past failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The top-5 ranked students are highlighted presented in [10], uses constraints over the representations of different groups in the top-k ranked items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' They use an upper bound Ukl and a lower bound Lkl over the number of items with the property l (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', a protected group) in the top-k positions of the ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Then a ranking algorithm is fair by the definition of [10], if the number of selected items from the protected group in the top-k lies within the given boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='4: Consider again the dataset given in Figure 1 and the ranker whose result is presented in the Rank column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Consider a lower bound of 2 over the number of students from each school for k = 5 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', L5,school=MS = L5,school=GP = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In this case, among the top-5 students, only one is from the GP school, thus the ranker does not satisfy the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Another prominent class of fairness measure in ranking considers the proportional representation of different groups in the top-k ranked items (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='g [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Intuitively, these definitions can be seen as variants of the definition of [10] such that for each group g, and each k, the bounds on the number of occurrences of items from g in the top-k ranked items are defined with respect to the size of g in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='5: Continuing Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='4, the total number of students from each school (MS and GP) is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The total dataset size is 16, thus a proportionate representation of each school in the top-5 items should be roughly 5 · 8 16 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' PROBLEM DEFINITION Our goal is to detect groups with biased representation in the top-k ranked items for a given ranking algorithm R, dataset D, and a range of k’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We define our problem by harnessing fairness measures for ranking algorithms from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In particular, we present two problem definitions utilizing prominent fairness measures, both considering the representation of different groups in the top-k ranked items for different values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Intuitively, accounting for a range of k’s ensures that the ranking is fair for any position in the ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For instance, if the top-10 items consist of 5 students with an urban address and 5 students with a rural address, but the students with urban addresses are in the 1-5 positions of the ranking, the output may seem “fair” if we are only interested in selecting the top-10 students, but if the positions within the top-10 are also important (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', position in the ranking affects an award amount), then clearly this ranking is unfair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The first definition simply utilizes the definition of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The fairness definition of [10] restricts the count of different groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', patterns) in the top-k using upper and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' According to this definition, the result is biased either when the size of a pattern l exceeds the upper bound Ukl or falls below the lower bound Lkl among the top-k tuples for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We eliminate the requirement to define l in advance and use Uk and Lk to denote the upper and lower bound respectively, on every pattern in the top-k ranked tuples of a given ranking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We say that a group has a biased representation in the output of a ranking algorithm R, if its size in the top-k ranked items by R does not lie within the given bounds for any k in a given range of possible k’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Intuitively, we wish to avoid reporting “very specific” descriptions of groups and provide the user with a concise set of properties that characterize meaningful groups (in terms of their size) that have biased representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To this end, we present the notion of most general patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Given the bounds over the group’s representation in the top- k ranked items, we say that a pattern p is the most general pattern with biased representation, if p is used to represent a group with inadequate representation by the given bounds, and ∀p′ ⊊ p, the count of p′ in Rk(D) lies within the given boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We are now ready to formally define our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='1 (Global Bounds Representation Bias): Given a database D, a ranking algorithm R, a size threshold τs2, a range [kmin, kmax], and lower bounds Lk and upper bounds Uk for each kmin ⩽ k ⩽ kmax, find for each kmin ⩽ k ⩽ kmax, all most general patterns p with size ⩾ τs such that sRk(D)(p) < Lk or sRk(D)(p) > Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Note that the ranking algorithm is treated as a black box, making the problem to be model agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Following the line of work on proportional representation, we consider another problem definition by refining the above definition to account for the groups’ sizes in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Intuitively, the number of items from each group in the top-k ranked items should be proportionate to the group’s representation in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='2 (Proportional Representation Bias): Given a database D, a ranking algorithm R, a size threshold τs and a range [kmin, kmax], find for each kmin ⩽ k ⩽ kmax, all most general patterns p with size ⩾ τs such that sRk(D)(p) < α · sD(p) k |D| or sRk(D)(p) > β · sD(p) k |D| for α < β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Proportional representation gives the user an intuitive bounds definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' However, the definition of global represen- tation allows the user to actively control the bounds over the representation of different groups in the top-k ranked items, even if their representation in the overall data is low/high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For instance, consider a ranking algorithm for job applicants in fields that are dominated by men (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', tech companies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' If 2We use an absolute value as the size threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Equivalently it may be defined as a fraction of the dataset size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' A1 A2 · · An−1 An Rank t1 1 0 · · 0 0 1 t2 0 1 · · 0 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' tn 0 0 · · 0 1 n tn+1 0 0 · · 0 0 n + 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 2: Dataset D in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='3 the company wishes to increase the representation of women they hire but their application number is low and only the top-k ranked applicants are invited for an interview, by using proportional representation, their number in the top-k, and as a result, the number of women invited to an interview, would be low as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The fundamental definition of [10] allows the user to define bounds over the representation of the protected groups in the data for different values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Following this definition, we defined the global bounds representation prob- lem, which assumes no prior information regarding the identity of the protected groups and aims to identify all groups with biased representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Note that our goal is to report only the most general patterns (groups), providing a concise description of these groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' While there are typically far fewer most general patterns with biased representation than the set of all patterns with biased representation, in the worst case, their numbers can be exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is true even if we consider only the lower bounds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', if Uk = |D|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='3: Given a dataset D and a ranking algorithm R, no polynomial time algorithm can guarantee the enumeration of the set of all most general patterns with biased representa- tion in the result of R on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Proof 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='4: We prove the theorem by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Consider a dataset D with n (assuming n ≥ 2) binary attributes {A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' , An} and n + 1 tuples t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' , tn, tn+1 as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', ∀i ∈ [1, n], ti[Ai] = 1, and ∀j ̸= i, ti[Aj] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' All the attributes of tn+1 are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Let R be a ranking algorithm such that the top-k tuples in D are the tuples t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' , tk in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Let kmin = kmax = n, Lk = n 2 +1 for Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='1 (global representation bounds), and α = n+3 n+4 for Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='2 (proportional representation), for some n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Consider a pattern p with m ≤ n attributes Ai with the value assignment Ai = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Let I be the set of indices of those attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Among t1, · · · , tn, the size of p is n − m, since ∀i ∈ I, ti[i] = 1, ti does not satisfy p, and all the other tuples satisfy p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Let m = n 2 , thus the size of p in the top-k ranked items is sRk(D)(p) = n 2 < Lk = n 2 + 1 in the case of global representation bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For proportional representation, we have sD(p) = n 2 + 1 since tn+1 also satisfies p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' And we get sRk(D)(p) = n 2 < α · sD(p) · k |D| = n+3 n+4( n 2 + 1) n n+1 = n 2 (n+3)(n+2) (n+4)(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To show p is a most general pattern to report, we examine the parents of p, p′ which has m − 1 attribute with the value assignment 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For global representation bounds, we have sRk(D)(p′) = n − m + 1 = n 2 + 1 = Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For proportional representation, we have sRk(D)(p′) = n 2 + 1 > α · sD(p′) · k |D| = n+3 n+4( n 2 + 2) n n+1 = n+2 2 (n+3)n (n+1)(n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' As a result, every pattern with n 2 attribute assigned to 0 should be in the result set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The number of such patterns is � n n/2 � > √ 2 n, which is exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Therefore, any algorithm enumerating these patterns is exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Upper bounds: The notion of the most general patterns is motivated by the utility of the information they provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For example, in the case of global representation bounds, if the number of females in the top-k is less than the lower bound, then clearly the number of black females is below the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Unlike the most general patterns for the lower bound, in the case of the upper bound, the most specific patterns are more informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For instance, if the number of black females is above the upper bound, then so is the number of blacks and the number of females.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' A plausible problem definition may account for the most specific substantial patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Analogously to the definition of the most general patterns, a pattern p is a most specific substantial pattern if the size of p is above a given threshold τs and for every pattern p′ such that p ⊊ p′, the size of p′ is below the threshold τs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The goal is then to find the most general patterns that do not satisfy the lower bound and the most specific substantial patterns that exceed the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For ease of presentation, in the rest of the paper we will focus on the solution for Problems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='2 considering only the lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We note that our solutions can be adjusted to support such problem definition (and other definitions such as most general for upper bound, and the most specific for lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' While Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='3 indicates that the number of most general groups with biased representation may be exponential, our experimental evaluation shows that, in practice, their number is significantly lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='58% of the times, the number of the reported groups was less than 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We note that presenting a large number of results may be overwhelming to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' A user-friendly interface would organize the output by k value and rank the groups by their overall size in the data or by the bias in their representation (the difference between the required representation and the actual representation) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' DETECTING GROUPS WITH BIASED REPRESENTATION We next present our algorithms for detecting groups with biased representation as defined in Problems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We start with a simple solution that can be used to detect groups based on both of the problem definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We then present two optimized algorithms, designed to optimize the search for each of the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Iterative Top-Down Search (Baseline Solution) The first, simple solution, utilizes the algorithm presented in [5] to traverse the set of possible patterns, starting with the most general ones, and compute the representation of each group in the top-k ranked items in the data (for each k in the given range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is done using the notion of pattern graph [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Briefly, the nodes in the graph are the set of all possible patterns, and there is an edge between a pair of patterns p and p′ if p ⊂ p′ and p′ can be obtained from p by {} {G=F} {S=GP} {S=MS} {G=M} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' {G=F, S=GP} {G=F, S=MS} {G=M, S=GP} {G=M, S=MS} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 3: Part of the pattern graph for the running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Edges of the search tree are marked with solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' adding a single attribute value pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In this case, we say that p (p′) is a parent (child) of p′ (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' As shown in [5], the pattern graph can be traversed in a top-down fashion, while visiting each pattern at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Namely, traversing a spanning tree of the pattern graph, which we denote as the search tree, and formally define as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='1: Let D be a dataset with categorical attributes A = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' , An}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We assume attributes are ordered, and for a given set of attributes S ⊆ A we use idx(S) to denote the maximal index value of all attributes in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Let p be a node in the pattern graph of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The children of p in the search tree T are p′ such that p′ is a child of p in the pattern graph, and idx(Attr(p′) \\ Attr(p)) > idx(Attr(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Namely, p′ is obtained from p by adding a single attribute value pair such that the index of the newly added attribute is larger than the maximal index value of attributes in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='2: A part of the pattern graph for the dataset of the running example is shown in Figure 3, where G and S are used as shorthands for Gender and School respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The pattern {G=F, S=GP} is a child node of the patterns {G=F} and {S=GP} in the pattern graph, however, in the search tree it is only a child of {G=F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is because idx({G}) = 1 < 2 = idx({G, S}) \\ {G}) whereas idx({S}) = 2 > 1 = idx({G, S} \\ {S}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Algorithm 1: Top-down search input : A dataset D, a ranking algorithm R, a size threshold τs, k and lower bound Lk output: Res = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' , pn} where ∀pi ∈ Res sD(pi) ≥ τs and pi is a most general pattern with sRk(D)(p) < Lk /* For proportional bounds α is given as input and Lk = α · sD(p) k |D| / 1 Res ← ∅ 2 S ← {generateChildren({})} 3 while S is not empty do 4 p = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='pop() 5 if patternSize (p, D) > τs then 6 top-k c ← patternSize (p, Rk(D)) /* For proportional bounds use top-k_c < α · sD(p) k |D| / 7 if top-k c < Lk then 8 update (Res, p) 9 else 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='push(generateChildren(p)) 11 return Res Algorithm 1 detects patterns with adequate size in the data (namely above a given threshold τs) and low representation (less than Lk), in the top-k ranked items for a given dataset D, a ranking algorithm R and a given k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' It traverses the search tree of the pattern graph (top-down), using a queue S and maintains the set of identified patterns with sRk(D) < Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' First it initializes the result set Res to ∅ (line 1) and sets S to contain the children of the most general (empty) pattern (line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' While the queue S is not empty (lines 3 – 10), the algorithm extracts the first pattern in the queue p (line 4), and computes its size in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' If it is greater than τs (line 5), the size of p in Rk(D), sRk(D), is computed (line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' If sRk(D) is bellow the lower bound Lk (line 7), Res is updated using the procedure update (line 8), that checks whether any ancestor of p in the pattern graph is already in Res (this is possible since the algorithm traverses the search tree and not the patterns graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Otherwise (line 9), sRk(D) exceeds the lower bound Lk, and the children of p are added to the queue using the procedure generateChildren (line 10), which generates the children of a node as defined in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Finally, Res is returned (line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' ITERTD algorithm (baseline): Given a dataset D and a ranking algorithm R, a size threshold τs, a range [kmin, kmax] and lower bounds Lk for each kmin ⩽ k ⩽ kmax, a simple solution for detecting groups with biased represen- tation based on the global representation bounds definition utilizes Algorithm 1, to apply a top-down search for each kmin ≤ k ≤ kmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Then, in each iteration report the patterns with low representation in the top-k ranked items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Similarly, Algorithm 1 can be used for the case of proportional repre- sentation, with some slight modifications (shown as comments in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The objective is to report the patterns p with adequate size but insufficient representation in the top- k tuples Rk(D), where the representation in Rk(D) should be proportional to the representation in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In this case, the bounds Lk are not given as input, instead, a bound for each pattern is computed based on its size in the data and a value α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Note that the pattern’s size is computed in line 5, and given α we can replace the condition in line 7 with the condition top-k c < α · sD(p) k |D|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We next propose more efficient algorithms for the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Global Representation Bounds The key observation is that when the lower bound remains the same for k and k + 13, the search spaces for patterns with biased representation in the ranking result of R for k and k+1 are typically very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is because the set of top-k and top-(k + 1) tuples differ by a single tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Namely, increasing k implies only local modifications to the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Let Tk be the search tree generated to find the set of unfairly treated patterns in the top-k tuples, and R(D)[k + 1], the (k + 1) element in the result of ranking D using R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We can bound the number of nodes in Tk whose size is affected by increasing k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='3: R(D)[k + 1] can satisfy at most |Tk| 2 patterns (nodes) in Tk, where |Tk| denotes the number of nodes in Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 3We assume Lk ⩽ Lk+1∀k kmin ⩽ k < kmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is a reasonable assumption since as k increases, so is the number of tuples in the top-k, thus it is only logical the bounds increase as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Proof 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='4: (Sketch) The basic idea of the proof is that when- ever a pattern p is generated during the search, at least one pattern p′ with the same set of attributes (Attr(p) = Attr(p′)) that differs from p in the value assignment of a single attribute is generated as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is true under the assumption that every attribute has at least 2 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For instance, the patterns {Gender = M, School = MS} and {Gender = M, School = GP} are both children of the pattern {Gender = M} and are gen- erated when the procedure generateChilder({Gender = M}) is invoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Let Ai be the attribute such that Ai ∈ Attr(p) and {Ai = ai} ⊆ p while {Ai = a′ i} ⊆ p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' R(D)[k + 1] can satisfy at most one of the patterns, as the value of Ai in R(D)[k + 1] is either ai or a′ i (or possibly other value, if |Dom(Ai)| > 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Since this is true for every node generated during the search, R(D)[k + 1] can satisfy at most |(Tk)| 2 patterns (nodes) in Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In that case, by starting the search for k + 1 from the endpoint of the search for k, we significantly reduce the search space (and as a result, the runtime, see Section VI-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Algorithm 2: GLOBALBOUNDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Detecting groups with biased representation based on global bounds input : A dataset D, a ranking algorithm R, a size threshold τs, a range [kmin, kmax] and lower bounds Lk for each kmin ⩽ k ⩽ kmax output: Res s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' for each kmin ⩽ k ⩽ kmax Res[k] = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' pn} where ∀pi ∈ Res[k] sD(pi) ≥ τs and pi is a most general pattern with sRk(D)(p) < Lk 1 Res ← ∅ 2 Res[kmin],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' DRes ← TopDownSearch(D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' τs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' kmin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Lkmin) 3 for k = kmin + 1 to kmax do 4 if Lk−1 < Lk then 5 Res[k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' DRes ← TopDownSearch(D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Rk(D),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' τs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Lk) 6 else 7 Res[k] ← Res[k − 1] 8 foreach b ∈ {p ∈ Res[k − 1] | R(D)[k] satisfies p} ∪ DRes do 9 Res[k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' DRes ← searchFromNode (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Res[k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' DRes) 10 return Res GLOBALBOUNDS algorithm: Our optimized algorithm for the problem for detecting groups with global represen- tation bias in the top-k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' GLOBALBOUNDS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' is depicted in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' GLOBALBOUNDS starts by initializing the result set map Res (line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' It then performs a top-down search for the case where k = kmin (line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is done using the procedure TopDownSearch, similar to the algorithm depicted in Algorithm 1, with a minor addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' It maintains a set DRes of patterns p reached during the search with size below the lower bound in top-k, that are not part of the result set since it already contains an ancestor of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' TopDownSearch returns both, the result set of the search Res, and the set DRes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' When k increases (and Lk is kept intact), the algorithm will utilize this set to initiate a local search in the pattern graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Next, the algorithm preforms the search for each k from k = kmin + 1 through k = kmax (lines 3–9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For each k, if the bound increases, TopDownSearch is used to perform a new top-down search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Otherwise, The algorithm considers only patterns from DRes and patterns from Res[k − 1] that the newly inserted tuple R(D)[k] satisfies (line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is because only their sizes in the top-k are affected by the new tuple (at most half of the tree, based on Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For each such pattern, the algorithm applies the procedure searchFromNode (line 9) to resume the search in the relevant parts of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This search updates Res[k] and DRes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Finally, Res is returned (line 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='5: GLOBALBOUNDS returns the set of all most general patterns p with bias representation using global bounds in the top-k for each k in the given range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The proof is by induction on k with a base case of k = kmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Details are omitted due to space constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='6: Consider again D and R from the running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Assume we are given the size threshold τs = 4, kmin = 4, kmax = 5, and the lower bounds L4 = L5 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' At the end of the top-down search for k = 4, the result set Res[4] contains (among others) the patterns {Address = U} and {Failures = 1}, that appears only once in the top-4 tuples (namely, below the lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' DRes contains, for instance, the patterns {Gender = F, Address = U}, {Gender = M, Address = U}, {Gender = F, Failures = 1} and {Address = R, Failures = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' These patterns were generated during the top-down search and have ancestors in Res[4] ({Address = U} and {Failures = 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Next, the algorithm turns to compute patterns with biased representation for k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The new tuple in the top-5 is tuple 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' It matches the patterns {Address = U} and {Failures = 1} in Res[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Thus the algorithm performs the search starting from those nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Their sizes in the top- 5 exceed the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In this search, these two patterns are extracted from the result set and the pattern {Address = U, Failures = 1} is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' From the set DRes, the patterns {Gender = F, Address = U}, {Gender = M, Address = U}, {Gender = F, Failures = 1} and {Address = R, Failures = 1} are added to the result set Res[5], as their sizes in the top- 5 tuples are still below the threshold L5 and their respective ancestors are removed from the result set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Proportional Representation We next consider the problem of detecting groups with biased proportional representation as depicted in Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The inputs are a dataset D, a ranking algorithm R, a range [kmin, kmax], a size threshold τs and α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The objective is to report the patterns p with adequate size in D, but insufficient representation in the top-k tuples Rk(D), where the representation in Rk(D) should be proportional to the representation of p in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' First, note that the optimized solution presented for the case of global representation bounds depicted in Section IV-B is not applicable in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Recall that GLOBALBOUNDS (Algo- rithm 2) aims at reducing the search space by avoiding search- ing areas in the pattern graph that were not changed between consecutive iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' When the bound remains unchanged (Lk = Lk+1), patterns that the (k + 1) tuple in the ranking does not satisfy, are not affected, and can be eliminated from the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is not the case for proportional representation, as the bound for each pattern depends on k as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Recall that the goal is to find patterns p such that sRk(D)(p) < α·sD(p)· k |D|, and note that α and sD(p) do not change during the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Thus, the inequality holds depending on the value of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Given R, D, α, a pattern p and a value k such that sRk(D)(p) ⩾ α · sD(p) · k |D|, we denote by ˜k the minimal value for k such that the inequality does not hold when fixing the value of sRk(D)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Namely, the minimal value such that sRk(D)(p) < α · sD(p) · ˜k |D|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='7: Let α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For D and R from our running example, p ={Gender = F} satisfies the inequality for k = 4 since sRk(D)(p) = 2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='9 · 8 · 4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' ˜k = 5 in this case since 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='9 · 8 · 5 16 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Intuitively, if k is increased up to ˜k but the number of tuples satisfying p remains the same, then the representation of p is biased in the ranking result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' If there is no ancestor of p in the result set, then p should be added to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To this end, our optimized algorithm, PROPBOUNDS, computes for each pattern in the search tree its corresponding ˜k value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' It maintains a set K which indicates patterns that potentially, if they do not satisfy the ˜k element in the ranking, should be added to the result set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' K contains patterns in a branch of the search tree whose ˜k values are monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' A pattern p in K with the corresponding ˜k value should be extracted from K and added to the result set when the computation reaches k = ˜k if sRk(D)(p) = sRk−1(D)(p) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', p does not satisfy R(D)[k]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Algorithm 3: PROPBOUNDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Detecting groups with biased proportional representation input : A dataset D, a ranking algorithm R, a size threshold τs, a range [kmin, kmax] and α ∈ R output: Res s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' for each kmin ⩽ k ⩽ kmax Res[k] = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' , pn} where ∀pi ∈ Res[k] sD(pi) ≥ τs and pi is a most general pattern with sRk(D)(p) < α · sD(p) k |D| 1 Res ← ∅ 2 Res[kmin], K, DRes ← TopDownSearch(D, R, τs, kmin, α) 3 for k = kmin + 1 to kmax do 4 Res[k] ← Res[k − 1] 5 Res[k], K, DRes ← selectiveTD (D, R, τs, k, α, R(D)[k], K, DRes) 6 foreach p ∈ DRes � K[p′]=k{p′} such that R(D)[k] doesn’t satisfy p do 7 update (Res, p) 8 return Res PROPBOUNDS algorithm: PROPBOUNDS (Algorithm 3) operates as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Similarly to GLOBALBOUNDS, it starts by initializing the result set map Res (line 1) and applying a top-down search for kmin (line 2), as depicted in proce- dure TopDownSearch (with the required modification for proportional bounds), but in addition to sets Res, DRes, it also maintains the set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Then, the algorithm iterates over the values of k from kmin + 1 to kmax (lines 3 – 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In each iteration, it first initializes Res[k] with the result from the previous iteration Res[k − 1] (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Then the algorithm applies a (partial) search from the root using the procedure selectiveTD (line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This search ignores the areas in the tree that are not affected by R(D)[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The result of the procedure is used to update the result set, K, and DRes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The algorithm then iterates over the patterns in DRes (patterns reached during the search that has an ancestor in the result) and all patterns in K with ˜k = k that are not affected by R(D)[k], to determine the changes to the result set (lines 6 – 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Finally, Res is returned (line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='8: PROPBOUNDS returns the set of all most general patterns p with bias proportional representation in the top-k for each k in the given range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Similarly to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='5, the proof is by induction on k with a base case of k = kmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Due to space constraints, the details are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='9: Consider again D and R from the running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Assume we are given the size threshold τs = 5, kmin = 4, kmax = 5, and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' At the end of the top-down search for k = 4, the result set Res[4] consists of the patterns {School = GP}, {Address = U} and {Failures = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For each one of them, sD(p) = 8, and thus the bound on sRk(D)(p) is α·sD(p)· k |D| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='9·8· 4 16 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='8, but sRk(D)(p) is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The set K consists of {Gender = M} and {Gender = F}, both with ˜k of 5, and {School = MS} and {Address = R} with ˜k = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Note that the pattern {School = MS, Address = R} was generated in the first top-down search, but was not added to K since its ˜k value is 9, higher than its parent in the search tree {School = MS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' When k is increased to 5, the algorithm reexamines only the patterns {Gender = M}, {School = MS}, {Address = U} and {Failures = 1} that are affected by the R(D)[5] (tuple 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The patterns {Address = U} and {Failures = 1} remain in the result set for k = 5 even-though their size in the top-5 is larger, since the bound for k = 5 increases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Finally, the pattern {Gender = F} is added to Res[5] based on the information from K (it is stored in K with ˜k = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' RESULT ANALYSIS With the results of our algorithm in hand, an analyst may wish to understand the cause of the bias in the representation of the detected groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We propose a method to provide such explanations utilizing the notion of Shapley values [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Shapley values is a concept adopted from game theory to explain the effect of different attributes on the output of a model for a given input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The use of Shapley values has recently gained popularity in the field of interpretability and explainability of ML models [24], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Given a regression model (or a classifier with probabilities) M, Shapley values are used to evaluate the contribution of each attribute on the output of M for a given input t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is done by computing the weighted marginal contribution of each attribute value using all possible subsets of attribute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Intuitively, an explanation for the bias may be the values that affected the ranking of tuples in the given group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To this end, we propose a method for explanations that consists of two parts: the first identifies the attributes with the highest effect on the ranking of tuples in the given group (using Shapley values), and then we compare the values distribution of these attributes in the top-k and the biased represented group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In order to adopt the use of Shapley values, we need to tackle two challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The first is to adjust our problem’s setting, where we are given a ranking algorithm R (as a black box) rather than a regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Second, Shapley values are used to explain the contribution of the attribute values of a single tuple, whereas we are interested in explaining the (inadequate) representation of a group of tuples (in the top-k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To address the first challenge we compute a regression model MR that simulates the process of R and can be used to approximate the effect of attribute values of a given tuple t on t’s ranking by computing the Shapley values of MR(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To this end we define DR = {(t, R(D)[t]) | t ∈ D}, where R(D)[t] is the ranking of t in R(D), and use it to train a regression model MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Then, given a pattern p such that p was returned by one of our algorithms for detecting groups with biased representation for a given k, to explain the result, we compute the Shapley values (st 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' , st m) for each tuple t such that t satisfies p, namely, for each tuple in the detected group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We then aggregate the results into a single Shapley value vector (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' , sm) for the pattern p such that si = � t s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' t satisfies p st i sD(p) To show the differences between the pattern p and the top- k patterns, we visualize the value distribution of attributes with large Shapley values of tuples that satisfy the pattern p compared to their distribution among the tuples in the top- k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In Section VI-C we show that using our method we are able to disclose the attributes that were used for ranking (and thus affect the representation of groups in the top-k) when the ranking model is given as a black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Moreover, we show that the value distribution in the attribute identified as most significant in the ranking is different for groups detected by our algorithms than for the top-k tuples, which indicates the identified attributes values explain the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' EXPERIMENTS We experimentally examine the proposed solutions using three real-life datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We start with our setup and then present a quantitative experimental study whose goal is to assess the scalability of our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In particular, we examine our algorithms’ performance for each fairness definition as a function of the number of attributes, pattern’s size threshold, and range of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We then demonstrate our proposed method for the analysis of the results presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We conclude with a comparison to the framework presented in [27], showing the differences in the results between our algorithms and the algorithm of [27] through a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Experiment Setup a) Datasets: We used three real datasets with different numbers of tuples and attributes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The COMPAS Dataset4 was collected and published by ProPublica as part of their investigation into racial bias in criminal risk assessment software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' It contains the de- mographics, recidivism scores produced by the COMPAS software, and criminal offense information for 6,889 indi- viduals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We used up to 16 attributes eliminating attributes such as names, ids, dates, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Student Performance Dataset (Student dataset)5 shows the performance of students in secondary education of two Portuguese schools as described in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We consid- ered in the experiment the data fragment with information regarding the Math exam (395 tuples and 33 attributes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' German Credit Dataset6 with financial and demographic information about 1,000 loan applicants with 20 attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' It was originally used in the context of classification, where each application is classified as a good or bad credit risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Compared algorithms: We evaluate the performance, in terms of the running time of our proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' IterTD (baseline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The simple solution for detection of groups with biased representation, which iteratively applies a top-down search as depicted in Section IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' GLOBALBOUNDS (Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The algorithm for de- tecting groups with biased representation based on global bounds as described in Section IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' PROPBOUNDS (Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The algorithm for detecting groups with biased representation based on proportional representation bounds as described in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Parameters setting: For space constraints, unless stated otherwise, we report the result for the following set of default parameters: τs = 50, kmin = 10, kmax = 49, and the lower bounds are 10 for 10 ≤ k < 20, 20 for 20 ≤ k < 30, 30 for 30 ≤ k < 40 and 40 for 40 ≤ k < 50 for global bounds, and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='8 for proportional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The reported results reflect the algorithm’s performance under expected and typical usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Following the goals of fairness in ranking, we set gradually increasing bounds on group representation in the top-k ranked items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Since we aim at reporting the detected groups to the user, we set the parameters such that the number of reported groups in most cases is between 1 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The number of attributes was set to be the maximal number the baseline solution could handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' and continuous attributes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', age, were bucketized equally into 3 − 4 bins, based on their domain and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We note that the selection of bucketization affects the patterns graph size and may also affects the possible group definitions and their representation in the top-k, which 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='propublica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='org/datastore/dataset/ compas-recidivism-risk-score-data-and-analysis 5https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='edu/ml/datasets/student+performance 6https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='edu/ml/datasets/Statlog+(German+Credit+Data) could also affect the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' However, in this work, we assume that the attribute values used for group definitions are categorical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', the bucketization is given).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We experimentally evaluated the algorithms using different parameter settings and observed similar trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Ranking Algorithms: The Student dataset was ranked based on the value of the attribute G3 showing the stu- dent’s math final grades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For the COMPAS dataset, we per- formed similar ranking method as in [4]: We normalized attribute values c days from compas, juv other count, days b screening arrest, start, end, age, and pri- ors count as scoring attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Values are normalized as (val − min)/(max − min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Higher values correspond to higher scores, except for age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Tuples are ranked descendingly according to their scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For the German Credit dataset, we used the ranking presented in [36] based on creditworthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' All experiments were performed on a macOS machine with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='8 GHz Quad-Core Intel Core i7 CPU and 8GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The algorithms were implemented using Python3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Experimental results Both GLOBALBOUNDS and PROPBOUNDS run much faster than the baseline, particularly as the number of attributes increases and the baseline becomes exponentially more ex- pensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Details below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Number of attributes: The first set of experiments aims to study the effect of the number of attributes on the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To this end, we varied the number of attributes in the datasets from 3 to |A| where A is the set of all attributes in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The number of attributes (along with their cardinality) determines the number of possible patterns, and as a result, the size of the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Thus, as the number of attributes increases, we expect to see a steep growth in the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The results (using a 10-minute timeout) are presented in Figures 4–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Indeed, in all cases we observed a rapid increase in the running time, while GLOBALBOUNDS and PROPBOUNDS outperform ITERTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Size threshold: In the next set of experiments, we as- sessed the effect of the size threshold τs on the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To this end, we varied the size threshold from 10 to 100 while using the default values for the rest of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The results are presented in Figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We observed a decrease in the running times of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is because the number of patterns satisfying the size threshold decreases as the threshold increases, and as a result, the search space is decreased as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In all cases, GLOBALBOUNDS and PROPBOUNDS outperform ITERTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Range of k: We examine the scalability with respect to the range of k considered by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We varied the range from 40 (40) to 990 (340) by setting kmin to be 10, and increasing kmax from 50 (50) to 1000 (350) for COMPAS (Student and German Credit) dataset and observed the effect on the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We set different maximum range of k due to different size of the datasets (6889 for COMPAS, 395 for Student and 1000 for the German Credit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The results are presented in Figure 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In all cases the optimized algorithms outperform ITERTD, which illustrates the efficient reduction in the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Recall that GLOBALBOUNDS and PROPBOUNDS optimize the search space compared to ITERTD by utilizing the search result of the iteration for k in order to compute the result set for k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Thus, as the range of k increase, we expect to see a greater improvement in the performance of the optimized algorithm compared to the baseline solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This trend is shown in Figure 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To further demonstrate the useful- ness of the approach, we compared the number of patterns examined during the search for each one of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The observed gain was up to 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='35% in the COMPAS dataset, 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='87% in the student dataset and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='27% in the credit card dataset for detecting groups with biased representation using global bounds, and 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='60%, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='49% and 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='83% respectively for proportional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Result Analysis using Shapley values We next demonstrate the usefulness of our proposed method for results analysis using Shapley values presented in Sec- tion V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The goal of the experiment is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' First, we show that our Shapley values based method for evaluating the effects of attributes on the ranking can indeed reveal useful information on the actual attributes used for ranking when the ranking algorithm is given as a black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Then we show that the value distribution for those attributes can be used to explain the representation bias, by comparing the distributions for the values in the top-k with those in the detected groups and focusing on the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We trained a regression model using the ranked data for each dataset and examined the Shapley values for groups detected by our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We present the results for the patterns (groups) p1 = {mother’s education = primary education (4th grade)} in the Student dataset, p2 = {age = younger than 35} in COMPAS and p3 = {status of existing account = (0 ⩽ · · < 200) DM7} from the German Credit dataset, which were detected by the GLOBALBOUNDS algorithm for k = 49 and Lk = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We observed similar results for other groups detected by the algorithms and other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Figures 10a, 10b and 10c show the resulting aggregated Shapley values for each group, as explained in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We show the Shapley values for the six attributes with the larges values for each group, as the rest had significantly lower values (lower than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='79%, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='31% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='54% of largest aggregated Shapley values for p1, p2 and p3 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For the group of students whose mother’s education level is primary education, which was detected by our algorithm as a group with biased representation in the Student dataset, the final grade has the largest aggregated Shapley value on the ranking (Figure 10a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This result agrees with the fact that the value of the final grade is indeed used for ranking by the ranking algorithm (and it is in fact the only attribute used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 7A Debit Memo (DM) on a company’s bank statement refers to a deduction by the bank from the company’s bank account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In other words, a bank debit memo reduces the bank account balance similar to a check drawn on the bank account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' (a) COMPAS dataset (b) Students dataset (c) German Credit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 4: Running time as a function of number of attributes - Ranking with global bounds (a) COMPAS dataset (b) Student dataset (c) German Credit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 5: Running time as a function of number of attributes - Ranking with proportional representation (a) COMPAS dataset (b) Students dataset (c) German Credit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 6: Running time as a function of the size threshold τs - Ranking with global bounds (a) COMPAS dataset (b) Students dataset (c) German Credit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 7: Running time as a function of the size threshold τs - Ranking with proportional representation Other than the final grade, the first and second period grades have a notable aggregated Shapley (although significantly lower aggregated Shapley value than the final grade).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This is due to the high correlation between those attributes and the final grade [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We also noticed the mother’s education attribute in the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This may indicate some correlation between the mother’s education and the final grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' However, we also note that the aggregation of the Shapley values for other attributes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', father’s education, show no clear pattern: some values have a positive effect and some negative, and different tuples in the group have different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In contrast, all the tuples in the group have the same value (primary education) for mother’s education attribute (since it is used to define the group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Therefore the Shapley values of the attribute for the different tuples in the groups are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This may also increase the aggregated value compared to other attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We observed this phenomenon, where the attributes used to define the detected group are slightly higher, for other groups detected by our algorithms also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For the COMPAS dataset, tuples are ranked by a combined score based on seven attributes: days from compas, the number of other juvenile convictions, days before screening arrest, start date, end date, age, and the number of priors crimes committed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In Figure 10b, showing the aggregated Shapley values of people younger than 35, six out of the above seven attributes are the six attributes with the largest Shapley values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In this case, the attribute end date and the number of priors crimes committed are identified as the most significant factor affecting the detected group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For the German Credit dataset, tuples are ranked according to their ranking in [36], however the actual ranking algorithm is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' we do not have the ground truth and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='GlobalBounds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='Execution time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='IterTD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='2 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='PropBounds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='Execution time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='IterTD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='Number of attributesS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='GlobalBounds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='IterTD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='40 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='Size thresholdS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='GlobalBounds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='Execution time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='IterTD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='4050 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='¥60 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='Size thresholdGlobalBounds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='Execution time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='IterTD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='4050 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='Size threshold(a) COMPAS dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='(b) Students dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='(c) German Credit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 8: Running time as a function of the range of k- Ranking with global bounds (a) COMPAS dataset (b) Students dataset (c) German Credit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 9: Running time as a function of the range of k - Ranking with proportional representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='(a) Aggregated Shapley value of group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='p1 ={mother’s education = primary ed- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='ucation} in the Student datase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='(b) Aggregated Shapley value of group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='={age = younger than 35} in the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='COMPAS dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='(c) Aggregated Shapley value of group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='p3 ={status of existing account = (0 ⩽ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='· · < 200DM)} in the German Credit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='(d) Value distribution of the final grade ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='attribute in the Student dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='(e) Value distribution of the end date at- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='tribute in the COMPAS dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='(f) Value distribution of residence length ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='in the German Credit dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 10: Result analysis using Shapley values cannot verify the attribute detected as significant for explain- ing the bias in the representation of the detected group in the top-k are actually used by the ranking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The attributes residence length, duration in month, credit amount, and installment rate have the largest Shapley values as shown in Figure 10c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' All of these attributes represents reasonable features to decide one’s credit worthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The Shapley value represents the effect of different at- tributes on the ranking of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To analyze the differences between the detected groups and top-k tuples (with respect to these attributes), we visualize the value distribution of attributes with the largest Shapley values in Figures 10d, 10e, and 10f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Since the number of tuples in the top-k and the detected group differ, the y-axis represents the proportion of tuples (rather than their count) with the values shown on the x-axis (the set of possible values for the attribute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For all three datasets, we observed vast differences in the distributions of the values of the attribute with the largest Shapley value between the tuples in the top-k and the tuples in the group detected with biased representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' For the students dataset (Figure 10d), the final grades of tuples in the top-k all fall in the range of 15 − 20, while most tuples in the detected group have a final grade lower than 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In the COMPAS dataset (Figure 10e), the value of the end date for all top-k tuples is 0 while only half of the tuples in the detected group have the same value, and almost 30% of them have the value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Similar results were observed in the value distribution of the residence length attribute as shown in Figure 10f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Comparison with Existing Solution The problem of identifying subgroups in the data that behave differently compared to the overall dataset was studied in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Different from our problem definition, which relies on fairness measures for ranking to define groups with biased S GlobalBounds Execution time 150 IterTD 100 50 200 400 600 800 1000 Range of k200 S Execution time GlobalBounds 150 IterTD 100 100 200 300 350 Range of kS GlobalBounds Execution time 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='5 IterTD 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content="5 100 200 300 350 Range of k600 S PropBounds Execution time IterTD 400 200 200 400 600 800 1000 Range of kS PropBounds Execution time ( 400 IterTD 200 100 200 300 350 Range of kS PropBounds Execution time 400 IterTD 200 100 200 300 350 Range of kfinal grade second period grade Attribute mother's education first period grade number of past class failures father's education other positive Shapley values other negative Shapley values 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' 10 20 30end date # of prior crimes committed Attribute start date # of other juvenile convictions days before screening arrest age other positive Shapley values other negative Shapley values 100 0residence length duration in month Attribute credit amount installment rate employment length existing credits at this bank other positive Shapley values other negative Shapley values 25 0 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content="3 {mother's education = primary education (4th grade)) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='2 top-k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='1 0 0 46810 12 14 16 18 20 Value of final grade1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='0 [age = younger Proportion than 35) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='5 top-k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='0 0 1 2 Value of end date1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='0 [ status of existing account = (0 <= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' < 200 DM)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='5 top-k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='0 0 1 2 3 Value of residence lengthrepresentation in the top-k ranked items, the work of [27] uses the notion of divergence to measure performance differences among data subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Each data item in the data t ∈ D is associated with an outcome o(t) where the outcome function is defined based on the ranking of t by the ranking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The outcome of a group o(G), is then the average of the outcome of every item t ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The divergence of a subgroup G in the data D is the difference between the outcome of G and the entire data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', o(G) − o(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Given a threshold s on the subgroup size, the solution of [27] computes the divergence of all subgroups with sizes larger than s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To better demonstrate the differences between the defini- tions and the resulting groups identified by each algorithm, we conducted an experiment using the Student dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We used the default size threshold of τs = 50 (support in [27] of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='13, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', 13% of the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Since [27] does not consider a range of k’s, we fixed kmin = kmax = 10 (namely, compare the results when k = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' To allow for easy comparison, we used only the first 4 attributes of the data: school, sex, age, and address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We used the outcome function o(t) that assigns the value 1 for tuples t in the top-k, and 0 for the rest (as presented in [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Finally, for our algorithms, we used the default parameters of lower bound 10 for ranking with global bounds and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='8 for proportional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' PROPBOUNDS outputs 2 patterns: {sex=F} and {address=R}, both returned by GLOBALBOUNDS as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Additionally, GLOBALBOUNDS returned the patterns {school=GP}, {sex=M} and {address=U}, which had less than 10 instances in the top-10 ranked items (9, 7, and 9 respectively), but considering their overall size (349, 208 and 307 respectively), their representation in the top-k is adequate and thus are not returned by PROPBOUNDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The algorithm of [27] returned 28 groups including the groups detected by our solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Since the number of reported groups may be extremely large, the algorithm of [27] ranks the groups by their divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The 5 patterns with the highest divergence contain 3 − 5 attributes, with the value assignment sex=M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', they are descendents of the pattern {sex=M} (in the pattern graph) returned by GLOBALBOUNDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The pattern {sex=M} was ranked at 17 according to its divergence value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The key difference between our algorithms and the solution of [27] lies in the definitions of the groups they aim to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The two solutions deal with a similar problem, however, our solution prefers concise groups (most general pattern) while the solution of [27] is designed to identify all groups with sufficient representation in the overall data and high divergence (a measure of “unfairness”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' As a result, the output of [27] is typically larger and contains subgroups that are consumed by each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Finally, the work of [27] considers a single k while we consider a range of k’s, aligning with fairness definitions in the literature, making the solution fair for any position in the top-k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' RELATED WORK The notion of fairness in ranking algorithms was studied in a line of works, introducing different fairness definitions [10], [20], [30], [34], [36], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' These definitions typically focus on top-k positions, as those are usually the most important positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In this paper, we consider two such definitions: the fundamental definition of [10], which measures fairness by bounding the representation of different groups in the data, and a refined definition that considers proportional groups representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' These definitions, as customary in the context of algorithmic fairness, refer to some given protected groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We harness these definitions to define the problem of detecting groups with biased representation, eliminating the need to pre-define protected groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The problem of generating fair ranking results was studied in [4], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' These works consider a wide range of definitions for fairness in ranking, which rely on the notion of protected groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This line of work is orthogonal to the problem we defined in this paper, and our proposed method can be used to identify such protected groups, when they are unknown in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Recent works have studied the problem of automatically detecting “problematic” or biased subgroups in the data, without the need to specify the protected attributes a priori, in the context of classification [9], [12], [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In [28], the authors introduced the notion of divergence to measure the difference in the behavior of a classifier on data subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The goal is then to report subgroups with sizes above a given threshold and high divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In [27] they extend their framework to ranking, where they consider the average outcome value, which is defined based on the ranking of the instances in each group, as a measure of the group’s outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In contrast to [27], our problem definitions rely on groups’ representation in the top-k ranked items as fairness measures for ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We demonstrate the differences in the resulting groups identified by each definition in Secetion VI-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The interactive system MithraCoverage [21] investigates popula- tion bias in intersectional groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The notion of coverage is introduced to identify intersectional subgroups with inadequate representation in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Differently, in our work, we only report patterns with adequate representation in the data, but inadequate representation in the output of a ranking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The use of Shapley values to provide explanations for ML models was studied in a line of works (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=', [24], [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In these works, the Shapley values are computed for an individual input instance to a classification or regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The Shapley value of a feature is then interpreted as the contribution of the feature to the output of the given input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Differently, we are aiming at providing explanations for the representation of a group of tuples in the output of a ranking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In [28] the authors presented a method to measure the contribution of items to divergence of groups utilizing Shapley values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' However, it considers only the contribution of attributes that are used to define the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' In contrast, our solution considers all attributes as possible explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' This requires an additional aggregation step in the computation of the Shapley values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' As demonstrated in VI-C, the explanation is typically buried in the values of attributes used for ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Moreover, our adjustment of Shapley values to explain a ranking algorithm is novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Our baseline solution, utilizing a top-town search presented in Section IV-A is built on the algorithm presented in [5] (for a simpler problem), which in turn shares similar ideas to the Apriori algorithm [1], the Set-Enumeration Tree for enumerating sets in a best-first fashion [32], discovering functional dependencies (FDs) [19], [26] and frequent item- sets and association rule mining [1], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Similarly to [5], the key difference from our work lies in the structure of the graph traversed in the solution: the pattern graph (in our case) com- pared to the powerset lattice in the other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Conditional functional dependencies (CFDs) [15] extend the notion of FDs by considering patterns to describe dependencies that hold only on subgroups in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Similar to the top-down search applied by the baseline solution, algorithms for discovering CFDs [14], [16], [18], [31] also utilize the notion of pattern and lattice of patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' However, the difference in the end goal (discovering CFDs versus identifying groups with biased representation) leads to differences in the pruning techniques in the baseline solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We then present two novel optimized algorithms designed for each one of the problems we defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' These algorithms reduce the search space as explained in Section IV and significantly outperform the baseline solution as shown in the experimental evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' CONCLUSION In this paper, we have studied the problem of detecting groups with biased representation in the result of a ranking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We build on fairness measures previously defined in the literature, considering the representation of protected groups in the top-k ranked items, for any reasonable range of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Our problem definitions eliminate the need to pre-define the protected groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We consider two variants of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' The first is based on global bounds over the representation of different groups in the top-k ranked items, and the second restricts the representation of each group in the top-k, based on its overall representation in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We theoretically analyse the complexity of the problem, showing that in the worst case, the number of groups can be exponential in the number of the dataset attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' We present a baseline algorithm that can handle both definitions and two optimized algorithms designed to improve the performance for each fairness measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' Furthermore, we present a method to explain the output of our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' There are many intriguing directions for future research, including the extension of the framework to support other fairness measures and further investigation of the automatic suggestion for thresholds.' metadata={'source': 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Informa- tion and Knowledge Management, CIKM 2017, Singapore, November 06 - 10, 2017, pages 1569–1578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} +page_content=' ACM, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQf0fl1/content/2301.00719v1.pdf'} diff --git a/ttE0T4oBgHgl3EQfsAEn/content/tmp_files/2301.02572v1.pdf.txt b/ttE0T4oBgHgl3EQfsAEn/content/tmp_files/2301.02572v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3419360f818d18ef1d9f2e11f933239874ea529e --- /dev/null +++ b/ttE0T4oBgHgl3EQfsAEn/content/tmp_files/2301.02572v1.pdf.txt @@ -0,0 +1,1362 @@ +Springer Nature 2021 LATEX template +Formation of Magnetic Switchbacks +Observed by Parker Solar Probe +Gabor Toth1*, Marco Velli2 and Bart van der Holst1 +1*Department of Climate and Space Sciences and Engineering, +University of Michigan, 2455 Hayward, Ann Arbor, 48109, MI, +USA. +2Department of Earth, Planetary and Space Sciences, University +of California at Los Angeles, 603 Charles E. Young Drive, East, +Los Angeles, 90095, CA, USA. +*Corresponding author(s). E-mail(s): gtoth@umich.edu; +Contributing authors: mvelli@ucla.edu; bartvand@umich.edu; +Keywords: Parker data used, Solar wind, Magnetohydrodynamics, Magnetic +switchbacks, Alfv´en waves +Magnetic switchbacks are rapid high amplitude reversals of the radial magnetic +field in the solar wind that do not involve a heliospheric current sheet crossing. +First seen sporadically in the seventies in Mariner and Helios data, switchbacks +were later observed by the Ulysses spacecraft beyond 1 au and have been +recently identified as a typical component of solar wind fluctuations in the +inner heliosphere by the Parker Solar Probe spacecraft. Here we provide a +simple yet predictive theory for the formation of these magnetic reversals: the +switchbacks are produced by the shear of circularly polarized Alfv´en waves +by a transversely varying radial wave propagation velocity. We provide an +analytic expression for the magnetic field variation, establish the necessary and +sufficient conditions and show that the mechanism works in a realistic solar +wind scenario. +The solar wind, to a good approximation, can be described with the +equations of ideal magnetohydrodynamics (MHD). Parker Solar Probe obser- +vations of switchbacks show a tight correlation of magnetic and velocity +1 +arXiv:2301.02572v1 [astro-ph.SR] 6 Jan 2023 + +Springer Nature 2021 LATEX template +2 +Formation of Magnetic Switchbacks +BR [nT] +0 +5 +10 +15 +20 +Hours from 00UT Nov 5 2018 +-100 +-50 +0 +50 +Ur [km/s] + + + + + + +0 +100 +200 +300 +400 + -Br [nT] + + + + + + +0 +100 +200 +300 +400 + Density [cm-3] + + + + + + +0 +100 +200 +300 +400 +500 +600 +700 + V=Ur+VA [km/s] + + + + + + +0 +100 +200 +300 +400 +500 +600 +700 + MA + + + + + + +0 +2 +4 +6 +8 +10 +Ur [km/s] +0 5 10 15 20 +Time [hour] +0 +100 +200 +300 +400 + Br [nT] +0 5 10 15 20 +Time [hour] +0 +100 +200 +300 +400 + Density [cm-3] +0 5 10 15 20 +Time [hour] +0 +100 +200 +300 +400 +500 +600 +700 + V=Ur+VA [km/s] +0 5 10 15 20 +Time [hour] +0 +100 +200 +300 +400 +500 +600 +700 + MA +0 5 10 15 20 +Time [hour] +0 +2 +4 +6 +8 +10 +BR [nT] +5 +10 +15 +20 +Hours from 00UT June 1 2022 +-200 +0 +200 +400 +600 +800 +BR-800nT (solid), dBT (dots), dBN (dash) +19.70 +19.71 +19.72 +19.73 +19.74 +19.75 +Hours from 00UT June 1 2022 +-150 +-100 +-50 +0 +50 +100 +150 + +19.70 +19.71 +19.72 +19.73 +19.74 +19.75 +Hours from 00UT June 1 2022 +-150 +-100 +-50 +0 +50 +100 +150 + +19.70 +19.71 +19.72 +19.73 +19.74 +19.75 +Hours from 00UT June 1 2022 +-150 +-100 +-50 +0 +50 +100 +150 +14.3 +14.4 +14.5 +14.6 +14.7 +14.8 +Hours from 00UT Nov 5 2018 +-100 +-50 +0 +50 +100 +dBN + + + + + + +-100 +-50 +0 +50 +100 +dBT + + + + + + +-100 +-50 +0 +50 +100 +dBR + + + + + + +-1.0 +-0.5 +0.0 +0.5 +1.0 +r(dBR,dBN) + + + + + + +-1.0 +-0.5 +0.0 +0.5 +1.0 +r(dBR,dBT) +19.2 +19.3 +19.4 +19.5 +19.6 +19.7 +19.8 +Hours from 00UT June 1 2022 +-400 +-200 +0 +200 +400 +dBN + + + + + + + +-400 +-200 +0 +200 +400 +dBT + + + + + + + +-400 +-200 +0 +200 +400 +4*dBR + + + + + + + +-1.0 +-0.5 +0.0 +0.5 +1.0 +r(dBR,dBN) + + + + + + + +-1.0 +-0.5 +0.0 +0.5 +1.0 +r(dBR,dBT) +Fig. 1 +Parker Solar Probe observations of switchbacks. The figures in the top half are from +November 5 2018 during the first encounter at about 35 Rs from the Sun, and the rest are +from May 30 to June 1 2022 during the 12th encounter between 30 Rs and 13.3 Rs distance. +The two panels showing the radial field Br with 0.22 s time cadence contain switchbacks +where the black line is inside the blue areas. The panels in the middle show hourly averages +of radial velocity ur, magnitude of Br, number density, radial wave speed v =ur +vA and the +Alfv´enic Mach number MA =ur/vA. For the 12th encounter the 24h period prior 19:00UT +May 31 is shown as there is no public plasma data after that. The wave speed varies along +the PSP orbit in both cases. The figures on the right show the high cadence observations +of the three components of the magnetic field for half hour periods. The background is +removed by subtracting a 7.4 minute sliding average. During the first encounter all three +components vary with similar amplitudes. For the second encounter dBR is multiplied by 4 to +make its variation similar to the perpendicular oscillations. The dBR−dBT and dBR−dBN +correlation coefficients r(dBR,dBT) and r(dBR,dBN) calculated over a two-minute sliding +window show strong correlations of dBR with the tangential components. The light red +rectangles highlight where dBR is highly correlated with dBT or dBN. The yellow rectangles +highlight anti-correlation, where the signs are opposite. The green rectangles indicate times +when Br is approximately constant. The bottom left panel shows one of these times. The +solid line is Br−800 nT ≈0. The dotted and dashed lines show dBT and dBN varying with +similar amplitudes consistent with roughly circularly polarized Alfv´en waves. + +Springer Nature 2021 LATEX template +Formation of Magnetic Switchbacks +3 +perturbations that are characteristic of Alfv´en waves [1]. Alfv´en waves are typ- +ically thought of as transverse oscillations around a constant guide field Br, +which, in case of the solar wind, points approximately in the radial direction +within Mercury’s orbit. The magnitudes of transverse velocity and magnetic +perturbations, u⊥ and B⊥, are related as u⊥ = B⊥/√µ0ρ, where ρ is the +mass density of the solar wind and µ0 is the magnetic permeability of vac- +uum. Circularly polarized Alfv´en waves are in fact exact solutions of the MHD +equations even when their amplitude B⊥ is large. The most puzzling property +of the observed switchbacks is that the presumed guide field Br changes sign +with frequent large amplitude oscillations. +This suggests that PSP observes spherically polarized Alfv´en waves [2]. For +these waves both the magnetic field vector B and velocity vector u oscillate in +arbitrary directions and u = ±B/√µ0ρ in the coordinate frame moving with +the wave, where the sign determines if the wave propagates parallel or anti- +parallel with the magnetic field direction. This is fully consistent with PSP +measurements [1]. +There is, however, a requirement for nonlinear spherical Alfv´en waves to +be an exact solution: the magnetic pressure pB = B2/(2µ0) must be constant. +This is actually not trivial, because a non-constant divergence-free magnetic +field typically has a spatially varying amplitude with the exception of circu- +larly polarized Alfv´en waves [3]. Spherically polarized Alfv´en waves are only +approximately stationary, but they can travel large distances in the solar wind +without significant dissipation. There have been several ideas put forward +how switchbacks form, including magnetic reconnection [4], Kelvin-Helmholtz +instability [5, 6], compressible turbulence [7, 8], and radial velocity shears and +jets [9, 10], but none of these provide a fully self-consistent explanation for all +observed properties. +We propose a new explanation for the formation of switchbacks and provide +supporting observational, theoretical and numerical evidence. The switch- +backs are produced by circularly polarized Alfv´en waves distorted +and twisted by a transverse shear of the radial wave speed. The radial +speed of an outward traveling Alfv´en wave is v=ur+vA, where vA = |Br|/√ρµ0 +and ur are the Alfv´en and solar wind speeds in the radial direction, respec- +tively. The wave velocity can vary for three reasons: variation of ur, Br, or ρ. +Our numerical tests confirm that any of these can produce switchbacks. +Let us consider a sinusoidally sheared radial wave velocity profile v(y) = +v0 + v1 sin(2πy/λy), where λy is the wavelength in the y direction that is per- +pendicular to the radial direction and z completes the coordinate system. The +wave velocity shear impacts an initially circularly polarized sinusoidal Alfv´en +wave with radial wave length λr. The magnetic field lines of the wave oscillate +within a width w = (B⊥/Br)λr/π. A long wave-length velocity perturbation, +λy ≫ w, will shear the circularly polarized waves and rotate the field in the +r–y plane across several waves. The left side panels of Figure 2 show numerical +simulation results for this case. When λy ∼ w, a much more complex solution +emerges as shown in the right panel. Finally, for λy ≪ w the velocity shear + +Springer Nature 2021 LATEX template +4 +Formation of Magnetic Switchbacks + + + + + + + + + +-1.0 +-0.5 +0.0 +0.5 + +-Br & (Br,By) + + + + +-11 +-10 +-9 +-8 +Y + + + + +-11 +-10 +-9 +-8 +Y + + + + +-11 +-10 +-9 +-8 +Y + + + + +-11 +-10 +-9 +-8 +Y + + + + + + + + +0.0 +0.5 +1.0 + +-By + + + + + + + + + + + + + + + + + + + +-1.0 +-0.5 +0.0 +0.5 +1.0 + +-Bz + + + + + + + + + + + + + + + + + + +-1.0 +-0.5 +0.0 +0.5 + +Ur & (Ur,Uy) +-1 +0 +1 +r +-11 +-10 +-9 +-8 +Y +-1 +0 +1 +r +-11 +-10 +-9 +-8 +Y +-1 +0 +1 +r +-11 +-10 +-9 +-8 +Y +-1 +0 +1 +r +-11 +-10 +-9 +-8 +Y + + + + + + + + + +0.0 +0.5 +1.0 +1.5 + +Uy +-1 +0 +1 +r + + + + + + + + + + + + + + + +-1.0 +-0.5 +0.0 +0.5 +1.0 + +Uz +-1 +0 +1 +r + + + + + + + + + + + + +-1.0 +-0.5 +0.0 + +-Br & (Br,By) + + + + +-4 +-2 +0 +2 +4 +Y + + + + +-4 +-2 +0 +2 +4 +Y + + + + +-4 +-2 +0 +2 +4 +Y + + + + +-4 +-2 +0 +2 +4 +Y + + + + + + + + + +-1.0 +-0.5 +0.0 +0.5 +1.0 + +-By + + + + + + + + + + + + + + + + + + + +-1.0 +-0.5 +0.0 +0.5 +1.0 + +-Bz + + + + + + + + + + + + + + + + + +-1.0 +-0.5 +0.0 + +Ur-Ur(0) & (Ur,Uy) +-1 +0 +1 +r +-4 +-2 +0 +2 +4 +Y +-1 +0 +1 +r +-4 +-2 +0 +2 +4 +Y +-1 +0 +1 +r +-4 +-2 +0 +2 +4 +Y +-1 +0 +1 +r +-4 +-2 +0 +2 +4 +Y + + + + + + + + + +-1.0 +-0.5 +0.0 +0.5 +1.0 + +Uy +-1 +0 +1 +r + + + + + + + + + + + + + + + +-1.0 +-0.5 +0.0 +0.5 +1.0 + +Uz +-1 +0 +1 +r + + + + + + +-Br and Ur +-10 +0 +10 +y +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 + +-10 +0 +10 +y +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-Br and Ur +-4 +-2 +0 +2 +4 +y +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 + +-4 +-2 +0 +2 +4 +y +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +Fig. 2 Numerical solutions of sheared circularly polarized Alfv´en waves in a double peri- +odic box. Colors show components of the magnetic field and velocity. White lines are field +lines and streamlines. Top left: The long wavelength λy/λr = 10 shear produces a highly +distorted Alfv´en wave. Only a part of the domain is shown where the shear is maximal. +Parameters: Br = 1, p = 1 and v1 = 0.5 (by perturbing Br) and time t = 20. Top right: +The comparable wavelength shear λy/λr = 1.25 results in a complex spherically polarized +Alfv´en wave solution. Parameters: Br = 0.5, p = 1, v1 = 0.5 (by perturbing ur) and t = 6.6. +Bottom: cuts along r = 0. Solid lines show the −Br magnetic field component, while the +dotted lines are the ur velocity components coordinate system moving with the wave. +can bend the transverse field lines as found by [11], but this only works if the +magnetic field is weak, which is not the case near PSP. +The long wavelength case can be regarded locally as a constant shear of the +radial wave velocity, dv/dy = const, which can be studied analytically. Let us +consider a circularly polarized wave with Br = const, By = B⊥ cos r and Bz = +B⊥ sin r, so the wavelength is λr = 2π. After time t, the field lines at a distance +y from the center of the wave will be pushed from position r to r′ = r + sy, +where s = t(dv/dy) is the shear at time t. To a first order approximation, the +shear will simply shift By and Bz in the radial direction: B′ +y(r, y) = By(r−sy) +and B′ +z(r, y) = Bz(r − sy) as illustrated in Figure 3. On the other hand, the +originally constant Br will change to B′ +r(r, y) = B′ +r(r −sy) = Br +sBy(r −sy) +that varies proportionally to By. A switchback occurs when B′ +r changes sign. +This happens when the shear exceeds the ratio of the radial and transverse +field magnitudes in the original circularly polarized wave: s > Br/B⊥. The +observations shown in Figure 1 support this assertion too: there are several +switchbacks during encounter 1 when the average Br ≈ −63 nT with oscillation +amplitudes dBR ≈ 32 nT, and the average B⊥ ≈ +� +dBT2 + dBN2 ≈ 49 nT +suggesting s ≈ 0.65, which is comparable to Br/B⊥ ≈ 1.3. During encounter + +Springer Nature 2021 LATEX template +Formation of Magnetic Switchbacks +5 +Fig. 3 Illustration how switchbacks form. The top part of the four panels show the evolution +of three magnetic field lines (solid black curves with arrows) projected to the r − y plane. +The dashed lines follow r = 3π + sy indicating the amount of shear s = 1.5t that grows +linearly with time t. At time t = 0 the original circularly polarized Alfv´en wave is shown with +wavelength λr = 2π and width w = (B⊥/Br)λr/π = 0.8 At this time Br is uniform and +the magnitude of the perpendicular field is B⊥ = 0.4Br. The wave velocity v = 1.5y shown +by the red arrows changes linearly with y. At time t = 1 the field lines are mildly distorted. +By time t = 2 the field line folds over and Br changes sign creating a switchback around the +positions where the dashed line intersects the field lines. At t = 3 there is substantial radial +field reversal. The bottom part of the panels show Br(t) = Br(t = 0) + sBy (solid lines) +and the magnetic pressure pB = (B2 +⊥ + B2 +r)/(2µ0) (dashed lines) at the four time instances. +Switchbacks occur when the line enters the blue regions. The gray arrows show the gradient +of the magnetic pressure that compresses the plasma and the magnetic field. The red lines +show numerical simulation results for comparable conditions. The switchbacks are narrow +peaks between flatter background field regions. +12, Br varies from 400 nT to 800 nT and the average B⊥ ≈ 210 nT is about +four times larger than dBR implying s ≈ 0.25 ≪ Br/B⊥ ≈ 2 to 4, so there +are only a few switchbacks. The observations also show strong correlations +between the oscillations of Br and the perpendicular components at most +times. This confirms that the oscillations are the radial and perpendicular +components of a sheared oscillation. The direction of the shear determines if +dBR is proportional to dBT or dBN (or some linear combination of them), and + +t=0, =0 +t=1, s=1.5 +2.5 +2.0 +2.0 +1.5 +1.5 +1.0 +1.0 +0.5 +0.5 +100 +0.0 +-0.5 +-0.5 +4 +B, +3 +3 +pB + PB +2 +2 +0 +0 +5 +10 +15 +0 +5 +10 +15 +t=2, S=3 +t=3, s=4.5 +2.5 +2.5 +2s +2s +2.0 +2.0 +1.5 +1.5 +1.0 +1.0 +0.5 +0.5 +0.0 +0.0 +-0.5 +-0.5 +4 +4 +R +R +3 +2 +0 +0 +5 +10 +15 +0 +5 +10 +15Springer Nature 2021 LATEX template +6 +Formation of Magnetic Switchbacks +the sign of s determines if there is a positive correlation or an anti-correlation. +On the other hand, the ratio of amplitudes is fairly constant for each encounter +suggesting that the average shear is a function of radial distance from the Sun, +or in other words, it is increasing in time as the wave propagates outward. +The first order approximation satisfies the divergence-free property, but the +magnetic pressure p′ +B = (B2 + 2sBrB′ +y + s2B′2 +y )/(2µ0) is no longer constant. +The magnetic pressure gradient will compress the plasma and modify B′ +x and +B′ +y while maintaining the B′ +r(r − sy) = Br + sB′ +y(r − sy) relationship so that +the field remains divergence free. The plasma will move towards the small +magnetic pressure region where Br is small and the switchbacks form. This +explains why the observed switchbacks are narrow peaks while the regions with +normal Br direction are wide and flat (see Figure 1). +An additional requirement for a switchback to occur is that the shear +velocity has sufficient energy to distort the original wave. A simple estimate +is that the average energy density of the shear motion is comparable to, or +larger than, the magnetic energy density of the transverse magnetic field: +ρv2 +1 > C(B⊥)2/µ0, where C ≈ 0.2 based on numerical experiments. +Finally, the turning needs to happen fast enough while the wave is traveling +outward in the solar wind. It takes the waves t = D/¯v to reach the spacecraft, +where D is the distance from the location where the circularly polarized Alfv´en +waves start to get sheared, and ¯v is a proper average of the radial wave velocity. +For a switchback to occur, s = (D/¯v)(dv/dy) > Br/B⊥ is required. +The shearing does not continue indefinitely. Eventually the energy related +to the shear is exhausted and the perturbed waves will keep propagating with +minimal evolution. The hourly averaged PSP plasma data during the first +encounter (see Figure 1) suggests that this is indeed happening. Ur, Br, and +√µ0ρ vary 5%, 29% and 6%, respectively, which would result in ≈ 30% vari- +ation in the wave speed v if these were independent of each other. But the +observed wave speed only varies 5.5%, which means that the velocity, mag- +netic field and density variations contributing to the wave speed cancel each +other out. This cancellation is caused by the distortion of the field reducing +the energy of the shear as the system tries to find an approximate equilib- +rium solution with a constant wave speed. If the energy density of the shear +exceeds the magnetic energy density by orders of magnitudes, then the shear +will eventually break down due to non-ideal MHD processes, such as magnetic +reconnection or turbulent cascade to kinetic scales. +The basic dynamics of shearing a circularly polarized Alfv´en wave can +be captured in a two-dimensional (2D) MHD simulation with three vector +components for velocity and magnetic field. The simulation domain is a double +periodic rectangle. The r direction corresponds to the radial direction in the +solar wind. The frame of reference is chosen such that the initial circularly +polarized wave, without the perturbation of the wave speed, is at rest. The +setup is normalized by setting the units of distance, time and mass, so that +λr = 4, B⊥ = u⊥ = 1, and µ0¯ρ = 1 where the ¯ρ is the unperturbed density. +The initial magnetic and velocity fields are By = −uy = cos(2πr/λr) and Bz = + +Springer Nature 2021 LATEX template +Formation of Magnetic Switchbacks +7 +−uz = − sin(2πr/λr), which correspond to the Alfv´en wave propagating in the ++R direction relative to the plasma. There are only four free dimensionless +parameters: the relative strength of the unperturbed guide field ¯Br/B⊥ (which +also determines ¯ur = −¯vA = − ¯Br to make the wave standing), the plasma +beta ¯β = p/¯pB that defines the pressure p, and the two parameters, v1/u⊥ +and λy/λr, for the wave velocity perturbation v1 sin(2πy/λy). We can perturb +either ur, Br or ρ to change the wave speed. The size of the domain in the r +direction is λr, while in the y direction a multiple of λy. +The simulations are performed with the BATS-R-US code [12, 13] on a fine +grid (cell size ∆r = ∆y = 0.04 = λr/100) with a fifth order accurate scheme +[14]. The left panels of Figure 2 show the solution for the long wavelength +case, with the perturbation applied to Br, in the part of the domain where the +shear is near maximal. The result is a distorted wave, similar to the analytic +description, with large switchbacks (left bottom panel) that look remarkably +similar to the observations in Figure 1. The right panels show the solution for +a case when the wave length of the perturbation λy is comparable to w. The +solution shows complex structures that do not resemble a circularly polarized +wave, still the Alfv´enic relations, −By ≈ uy, −Bz ≈ uz and −Br ≈ ur − +v1 sin(2πy/λy), hold (subtracting the initial perturbation from ur removes the +background variation). In this case the y = 0 cuts show more complicated +switchback structures. +Finally, we show that the mechanism also works in the radially expand- +ing solar wind. We use physical units for easier comparison with observations. +The 2D computational domain is a spherical wedge extending from r = 25 Rs +to 40 Rs and the azimuthal angle goes from −5◦ to 5◦. The 2D computa- +tional grid consists of 4, 000 × 1, 600 cells. The boundaries are periodic in the +azimuthal direction and outflow condition is applied at r = 40 Rs. The circu- +larly polarized Alfv´en waves enter at r = 25 Rs with amplitude B⊥ = 80 nT +and wavelength λr = 0.1 Rs. The number density, the radial velocity and the +temperature are 800 cm−3, 300 km/s, and 350,000 K, respectively. The radial +field is Br = −120 + 64.8 sin(2πy/λy) nT and λy = 2.18 Rs, which is half of +the width of the domain at the inflow boundary. +Figure 4 shows the solution at t = 10 hours, which is enough for the +solar wind to propagate from 25 Rs to 40 Rs with 300 km/s speed. The figures +shows that switchbacks develop with their characteristic asymmetric shapes +and the Alfv´enic relationship between magnetic and velocity fields are satisfied. +This simulation was set up to demonstrate the formation of switchbacks in +an idealized solar wind. The real solar wind is 3-dimensional with a spectrum +of Alfv´en waves that become turbulent due to the spherical expansion [15]. +According to previous theoretical and numerical studies [7, 8] the turbulence +will preserve the spherically polarized Alfv´en waves and further enhance their +amplitudes. +This paper focused on explaining the puzzling observations by PSP, but +the interaction of wave velocity shear with circularly polarized Alfv´en waves +can play an important role in the physics of the solar wind. The interaction + +Springer Nature 2021 LATEX template +8 +Formation of Magnetic Switchbacks +Fig. 4 +Formation and propagation of spherically polarized Alfv´en waves in the spherically +expanding solar wind. The top panel shows the three components of the magnetic field and +its magnitude in part of the 2D computational domain. The circularly polarized Alfv´en +waves enter through the left boundary at R = 25 Rs. The incoming radial field is perturbed +along the Y direction, which causes a shear in the Alfv´en wave speed and the development +of spherical polarization. Switchbacks with Br > 0 form at r > 27.5 Rs. The total magnetic +field (top right) is relatively smooth. The black curve indicates a possible PSP trajectory +at r ≈ 29 Rs. The bottom left panel shows the magnetic field and velocity components +as well as the magnetic pressure pB, the density, the thermal pressure p, and the total +pressure p + pB along the trajectory. All quantities are comparable to PSP observations +during the first encounter. The gradients of the total pressure are small, but not zero. +Density variations are also substantial. The bottom right panel compares the magnetic +(solid lines) and velocity (dotted lines) perturbations around a switchback. The magnetic +field components are converted to Alfv´en velocity components: VA = B/√ρµ0. For the +radial components the background variation is removed with a smoothing over 100 grid cells +(0.28 Rs). All components satisfy the Alfv´enic relationship to a high accuracy similar to PSP +observations [1]. +can create mode conversion from Alfv´en turbulence to compressive turbulence +heating and accelerating the solar wind [16]. +Acknowledgments. +G. T´oth and B. van der Holst are supported by +NSF grant PHY-2027555 and NASA grant 80NSSC22K0892. PSP data +was obtained through NASA CDAWeb. Simulations were performed on +the Pleiades supercomputer at NASA Ames. BATSRUS is open source at + +BR[nT] +BT [nT] +BN [nT] +B [nT] +200 +2 +100 +100 +0 +150 +50 +50 +1 +[Rs] +-50 +0 +Jo +0E +100 +Y +-10 +-1 +-50 +-50 +50 +-150 +-2 +-106 +-106 +252627 28 +30 +25 +26 +2728 +29 +¥30 +25 +26 +2728 +32930 +25 +26 +2728 +29 +30 +R [Rs] +R [Rs] +R [Rs] +R [Rs] +BR [nT] +UR [km/s] +dVAR & dUr [km/s] +100 E +340 +80E +20 +320 +300 +60 +280 +40 +20 +BT [nT] +UT [km/s] +60 E +-20 日 +40 +20 +0 +20 +-20 +VA,r & Ur [km/s] +.40 +-40 +100 E +BN [nT] +UN [km/s] +60 +50 E +40 +20 +0 +-20 +-40 +-50 +PB [nPa] +Density [cm3] +-100 E +1000 +VA.n & Un [km/s] +800 +100 F +600 +400 +50F +p+PB [nPa] +p [nPa] +50 +-100E +-2 +-1 +1 +2 +-2 +-1 +1 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0 +Y [Rs] +Y [Rs] +Y [Rs]Springer Nature 2021 LATEX template +Formation of Magnetic Switchbacks +9 +http://github.com/MSTEM-QUDA. We thank Prof. Tamas Gombosi at the +University of Michigan for excellent comments and suggestions. +References +[1] Kasper, J.C., Bale, S.D., Belcher, J.W., Berthomier, M., Case, A.W., +Chandran, B.D.G., Curtis, D.W., Gallagher, D., Gary, S.P., Golub, L., +Halekas, J.S., Ho, G.C., T. S. Horbury, Q.H., Huang, J., Klein, K.G., +Korreck, K.E., D. E. Larson, R.L., Maruca, B., Lavraud, B., Louarn, +P., Maksimovic, M., Martinovic, M., McGinnis, D., Pogorelov, N.V., +J. D. Richardson, R.M.S., Steinberg, J.T., Stevens, M.L., A. Szabo, M.V., +Whittlesey, P.L., Wright, K.H., G. P. Zank, R.J.M., McComas, D.J., Jr, +R.L.M., Pulupa, M., Raouafi, N.E., Schwadron, N.A.: Alfv´enic veloc- +ity spikes and rotational flows in the near-sun solar wind. 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Lett. 937, 39 (2022). https: +//doi.org/10.3847/2041-8213/ac913d + diff --git a/ttE0T4oBgHgl3EQfsAEn/content/tmp_files/load_file.txt b/ttE0T4oBgHgl3EQfsAEn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c32bf0cd13580092f7c118fe6a05ede3341239fd --- /dev/null +++ b/ttE0T4oBgHgl3EQfsAEn/content/tmp_files/load_file.txt @@ -0,0 +1,735 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf,len=734 +page_content='Springer Nature 2021 LATEX template Formation of Magnetic Switchbacks Observed by Parker Solar Probe Gabor Toth1*, Marco Velli2 and Bart van der Holst1 1*Department of Climate and Space Sciences and Engineering, University of Michigan, 2455 Hayward, Ann Arbor, 48109, MI, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' 2Department of Earth, Planetary and Space Sciences, University of California at Los Angeles, 603 Charles E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Young Drive, East, Los Angeles, 90095, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' *Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' E-mail(s): gtoth@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Contributing authors: mvelli@ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' bartvand@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Keywords: Parker data used, Solar wind, Magnetohydrodynamics, Magnetic switchbacks, Alfv´en waves Magnetic switchbacks are rapid high amplitude reversals of the radial magnetic field in the solar wind that do not involve a heliospheric current sheet crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' First seen sporadically in the seventies in Mariner and Helios data, switchbacks were later observed by the Ulysses spacecraft beyond 1 au and have been recently identified as a typical component of solar wind fluctuations in the inner heliosphere by the Parker Solar Probe spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Here we provide a simple yet predictive theory for the formation of these magnetic reversals: the switchbacks are produced by the shear of circularly polarized Alfv´en waves by a transversely varying radial wave propagation velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' We provide an analytic expression for the magnetic field variation, establish the necessary and sufficient conditions and show that the mechanism works in a realistic solar wind scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The solar wind, to a good approximation, can be described with the equations of ideal magnetohydrodynamics (MHD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Parker Solar Probe obser- vations of switchbacks show a tight correlation of magnetic and velocity 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='02572v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='SR] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='6 Jan 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='Springer Nature 2021 LATEX template 2 Formation of Magnetic Switchbacks BR [nT] 0 5 10 15 20 Hours from 00UT Nov 5 2018 -100 -50 0 50 Ur [km/s] ' 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The figures in the top half are from November 5 2018 during the first encounter at about 35 Rs from the Sun, and the rest are from May 30 to June 1 2022 during the 12th encounter between 30 Rs and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='3 Rs distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The two panels showing the radial field Br with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='22 s time cadence contain switchbacks where the black line is inside the blue areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The panels in the middle show hourly averages of radial velocity ur, magnitude of Br, number density, radial wave speed v =ur +vA and the Alfv´enic Mach number MA =ur/vA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' For the 12th encounter the 24h period prior 19:00UT May 31 is shown as there is no public plasma data after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The wave speed varies along the PSP orbit in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The figures on the right show the high cadence observations of the three components of the magnetic field for half hour periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The background is removed by subtracting a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='4 minute sliding average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' During the first encounter all three components vary with similar amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' For the second encounter dBR is multiplied by 4 to make its variation similar to the perpendicular oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The dBR−dBT and dBR−dBN correlation coefficients r(dBR,dBT) and r(dBR,dBN) calculated over a two minute sliding window show strong correlations of dBR with the tangential components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The light red rectangles highlight where dBR is highly correlated with dBT or dBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The yellow rectangles highlight anti correlation, where the signs are opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The green rectangles indicate times when Br is approximately constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The bottom left panel shows one of these times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The solid line is Br−800 nT ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The dotted and dashed lines show dBT and dBN varying with similar amplitudes consistent with roughly circularly polarized Alfv´en waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Formation of Magnetic Switchbacks 3 perturbations that are characteristic of Alfv´en waves [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Alfv´en waves are typ- ically thought of as transverse oscillations around a constant guide field Br, which, in case of the solar wind, points approximately in the radial direction within Mercury’s orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The magnitudes of transverse velocity and magnetic perturbations, u⊥ and B⊥, are related as u⊥ = B⊥/√µ0ρ, where ρ is the mass density of the solar wind and µ0 is the magnetic permeability of vac- uum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Circularly polarized Alfv´en waves are in fact exact solutions of the MHD equations even when their amplitude B⊥ is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The most puzzling property of the observed switchbacks is that the presumed guide field Br changes sign with frequent large amplitude oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' This suggests that PSP observes spherically polarized Alfv´en waves [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' For these waves both the magnetic field vector B and velocity vector u oscillate in arbitrary directions and u = ±B/√µ0ρ in the coordinate frame moving with the wave, where the sign determines if the wave propagates parallel or anti- parallel with the magnetic field direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' This is fully consistent with PSP measurements [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' There is, however, a requirement for nonlinear spherical Alfv´en waves to be an exact solution: the magnetic pressure pB = B2/(2µ0) must be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' This is actually not trivial, because a non-constant divergence-free magnetic field typically has a spatially varying amplitude with the exception of circu- larly polarized Alfv´en waves [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Spherically polarized Alfv´en waves are only approximately stationary, but they can travel large distances in the solar wind without significant dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' There have been several ideas put forward how switchbacks form, including magnetic reconnection [4], Kelvin-Helmholtz instability [5, 6], compressible turbulence [7, 8], and radial velocity shears and jets [9, 10], but none of these provide a fully self-consistent explanation for all observed properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' We propose a new explanation for the formation of switchbacks and provide supporting observational, theoretical and numerical evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The switch- backs are produced by circularly polarized Alfv´en waves distorted and twisted by a transverse shear of the radial wave speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The radial speed of an outward traveling Alfv´en wave is v=ur+vA, where vA = |Br|/√ρµ0 and ur are the Alfv´en and solar wind speeds in the radial direction, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The wave velocity can vary for three reasons: variation of ur, Br, or ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Our numerical tests confirm that any of these can produce switchbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Let us consider a sinusoidally sheared radial wave velocity profile v(y) = v0 + v1 sin(2πy/λy), where λy is the wavelength in the y direction that is per- pendicular to the radial direction and z completes the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The wave velocity shear impacts an initially circularly polarized sinusoidal Alfv´en wave with radial wave length λr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The magnetic field lines of the wave oscillate within a width w = (B⊥/Br)λr/π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' A long wave-length velocity perturbation, λy ≫ w, will shear the circularly polarized waves and rotate the field in the r–y plane across several waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The left side panels of Figure 2 show numerical simulation results for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' When λy ∼ w, a much more complex solution emerges as shown in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Finally, for λy ≪ w the velocity shear Springer Nature 2021 LATEX template 4 Formation of Magnetic Switchbacks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 Br & (Br,By) 11 10 9 8 Y 11 10 9 8 Y 11 10 9 8 Y 11 10 9 8 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 Ur Ur(0) & (Ur,Uy) 1 0 1 r 4 2 0 2 4 Y 1 0 1 r 4 2 0 2 4 Y 1 0 1 r 4 2 0 2 4 Y 1 0 1 r 4 2 0 2 4 Y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 Uy 1 0 1 r 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 Br and Ur 4 2 0 2 4 y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' 2 Numerical solutions of sheared circularly polarized Alfv´en waves in a double peri odic box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Colors show components of the magnetic field and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' White lines are field lines and streamlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Top left: The long wavelength λy/λr = 10 shear produces a highly distorted Alfv´en wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Only a part of the domain is shown where the shear is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Parameters: Br = 1, p = 1 and v1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 (by perturbing Br) and time t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Top right: The comparable wavelength shear λy/λr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='25 results in a complex spherically polarized Alfv´en wave solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Parameters: Br = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5, p = 1, v1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 (by perturbing ur) and t = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Bottom: cuts along r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Solid lines show the −Br magnetic field component, while the dotted lines are the ur velocity components coordinate system moving with the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' can bend the transverse field lines as found by [11], but this only works if the magnetic field is weak, which is not the case near PSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The long wavelength case can be regarded locally as a constant shear of the radial wave velocity, dv/dy = const, which can be studied analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Let us consider a circularly polarized wave with Br = const, By = B⊥ cos r and Bz = B⊥ sin r, so the wavelength is λr = 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' After time t, the field lines at a distance y from the center of the wave will be pushed from position r to r′ = r + sy, where s = t(dv/dy) is the shear at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' To a first order approximation, the shear will simply shift By and Bz in the radial direction: B′ y(r, y) = By(r−sy) and B′ z(r, y) = Bz(r − sy) as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' On the other hand, the originally constant Br will change to B′ r(r, y) = B′ r(r −sy) = Br +sBy(r −sy) that varies proportionally to By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' A switchback occurs when B′ r changes sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' This happens when the shear exceeds the ratio of the radial and transverse field magnitudes in the original circularly polarized wave: s > Br/B⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The observations shown in Figure 1 support this assertion too: there are several switchbacks during encounter 1 when the average Br ≈ −63 nT with oscillation amplitudes dBR ≈ 32 nT, and the average B⊥ ≈ � dBT2 + dBN2 ≈ 49 nT suggesting s ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='65, which is comparable to Br/B⊥ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' During encounter Springer Nature 2021 LATEX template Formation of Magnetic Switchbacks 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' 3 Illustration how switchbacks form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The top part of the four panels show the evolution of three magnetic field lines (solid black curves with arrows) projected to the r − y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The dashed lines follow r = 3π + sy indicating the amount of shear s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5t that grows linearly with time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' At time t = 0 the original circularly polarized Alfv´en wave is shown with wavelength λr = 2π and width w = (B⊥/Br)λr/π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='8 At this time Br is uniform and the magnitude of the perpendicular field is B⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='4Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The wave velocity v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5y shown by the red arrows changes linearly with y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' At time t = 1 the field lines are mildly distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' By time t = 2 the field line folds over and Br changes sign creating a switchback around the positions where the dashed line intersects the field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' At t = 3 there is substantial radial field reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The bottom part of the panels show Br(t) = Br(t = 0) + sBy (solid lines) and the magnetic pressure pB = (B2 ⊥ + B2 r)/(2µ0) (dashed lines) at the four time instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Switchbacks occur when the line enters the blue regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The gray arrows show the gradient of the magnetic pressure that compresses the plasma and the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The red lines show numerical simulation results for comparable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The switchbacks are narrow peaks between flatter background field regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' 12, Br varies from 400 nT to 800 nT and the average B⊥ ≈ 210 nT is about four times larger than dBR implying s ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='25 ≪ Br/B⊥ ≈ 2 to 4, so there are only a few switchbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The observations also show strong correlations between the oscillations of Br and the perpendicular components at most times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' This confirms that the oscillations are the radial and perpendicular components of a sheared oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The direction of the shear determines if dBR is proportional to dBT or dBN (or some linear combination of them), and t=0, =0 t=1, s=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 4 B, 3 3 pB PB 2 2 0 0 5 10 15 0 5 10 15 t=2, S=3 t=3, s=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 2s 2s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 4 4 R R 3 2 0 0 5 10 15 0 5 10 15Springer Nature 2021 LATEX template 6 Formation of Magnetic Switchbacks the sign of s determines if there is a positive correlation or an anti-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' On the other hand, the ratio of amplitudes is fairly constant for each encounter suggesting that the average shear is a function of radial distance from the Sun, or in other words, it is increasing in time as the wave propagates outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The first order approximation satisfies the divergence-free property, but the magnetic pressure p′ B = (B2 + 2sBrB′ y + s2B′2 y )/(2µ0) is no longer constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The magnetic pressure gradient will compress the plasma and modify B′ x and B′ y while maintaining the B′ r(r − sy) = Br + sB′ y(r − sy) relationship so that the field remains divergence free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The plasma will move towards the small magnetic pressure region where Br is small and the switchbacks form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' This explains why the observed switchbacks are narrow peaks while the regions with normal Br direction are wide and flat (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' An additional requirement for a switchback to occur is that the shear velocity has sufficient energy to distort the original wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' A simple estimate is that the average energy density of the shear motion is comparable to, or larger than, the magnetic energy density of the transverse magnetic field: ρv2 1 > C(B⊥)2/µ0, where C ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='2 based on numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Finally, the turning needs to happen fast enough while the wave is traveling outward in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' It takes the waves t = D/¯v to reach the spacecraft, where D is the distance from the location where the circularly polarized Alfv´en waves start to get sheared, and ¯v is a proper average of the radial wave velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' For a switchback to occur, s = (D/¯v)(dv/dy) > Br/B⊥ is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The shearing does not continue indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Eventually the energy related to the shear is exhausted and the perturbed waves will keep propagating with minimal evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The hourly averaged PSP plasma data during the first encounter (see Figure 1) suggests that this is indeed happening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Ur, Br, and √µ0ρ vary 5%, 29% and 6%, respectively, which would result in ≈ 30% vari- ation in the wave speed v if these were independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' But the observed wave speed only varies 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5%, which means that the velocity, mag- netic field and density variations contributing to the wave speed cancel each other out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' This cancellation is caused by the distortion of the field reducing the energy of the shear as the system tries to find an approximate equilib- rium solution with a constant wave speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' If the energy density of the shear exceeds the magnetic energy density by orders of magnitudes, then the shear will eventually break down due to non-ideal MHD processes, such as magnetic reconnection or turbulent cascade to kinetic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The basic dynamics of shearing a circularly polarized Alfv´en wave can be captured in a two-dimensional (2D) MHD simulation with three vector components for velocity and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The simulation domain is a double periodic rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The r direction corresponds to the radial direction in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The frame of reference is chosen such that the initial circularly polarized wave, without the perturbation of the wave speed, is at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The setup is normalized by setting the units of distance, time and mass, so that λr = 4, B⊥ = u⊥ = 1, and µ0¯ρ = 1 where the ¯ρ is the unperturbed density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The initial magnetic and velocity fields are By = −uy = cos(2πr/λr) and Bz = Springer Nature 2021 LATEX template Formation of Magnetic Switchbacks 7 −uz = − sin(2πr/λr), which correspond to the Alfv´en wave propagating in the +R direction relative to the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' There are only four free dimensionless parameters: the relative strength of the unperturbed guide field ¯Br/B⊥ (which also determines ¯ur = −¯vA = − ¯Br to make the wave standing), the plasma beta ¯β = p/¯pB that defines the pressure p, and the two parameters, v1/u⊥ and λy/λr, for the wave velocity perturbation v1 sin(2πy/λy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' We can perturb either ur, Br or ρ to change the wave speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The size of the domain in the r direction is λr, while in the y direction a multiple of λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The simulations are performed with the BATS-R-US code [12, 13] on a fine grid (cell size ∆r = ∆y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='04 = λr/100) with a fifth order accurate scheme [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The left panels of Figure 2 show the solution for the long wavelength case, with the perturbation applied to Br, in the part of the domain where the shear is near maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The result is a distorted wave, similar to the analytic description, with large switchbacks (left bottom panel) that look remarkably similar to the observations in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The right panels show the solution for a case when the wave length of the perturbation λy is comparable to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The solution shows complex structures that do not resemble a circularly polarized wave, still the Alfv´enic relations, −By ≈ uy, −Bz ≈ uz and −Br ≈ ur − v1 sin(2πy/λy), hold (subtracting the initial perturbation from ur removes the background variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' In this case the y = 0 cuts show more complicated switchback structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Finally, we show that the mechanism also works in the radially expand- ing solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' We use physical units for easier comparison with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The 2D computational domain is a spherical wedge extending from r = 25 Rs to 40 Rs and the azimuthal angle goes from −5◦ to 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The 2D computa- tional grid consists of 4, 000 × 1, 600 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The boundaries are periodic in the azimuthal direction and outflow condition is applied at r = 40 Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The circu- larly polarized Alfv´en waves enter at r = 25 Rs with amplitude B⊥ = 80 nT and wavelength λr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='1 Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The number density, the radial velocity and the temperature are 800 cm−3, 300 km/s, and 350,000 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The radial field is Br = −120 + 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='8 sin(2πy/λy) nT and λy = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='18 Rs, which is half of the width of the domain at the inflow boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Figure 4 shows the solution at t = 10 hours, which is enough for the solar wind to propagate from 25 Rs to 40 Rs with 300 km/s speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The figures shows that switchbacks develop with their characteristic asymmetric shapes and the Alfv´enic relationship between magnetic and velocity fields are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' This simulation was set up to demonstrate the formation of switchbacks in an idealized solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The real solar wind is 3-dimensional with a spectrum of Alfv´en waves that become turbulent due to the spherical expansion [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' According to previous theoretical and numerical studies [7, 8] the turbulence will preserve the spherically polarized Alfv´en waves and further enhance their amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' This paper focused on explaining the puzzling observations by PSP, but the interaction of wave velocity shear with circularly polarized Alfv´en waves can play an important role in the physics of the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The interaction Springer Nature 2021 LATEX template 8 Formation of Magnetic Switchbacks Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' 4 Formation and propagation of spherically polarized Alfv´en waves in the spherically expanding solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The top panel shows the three components of the magnetic field and its magnitude in part of the 2D computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The circularly polarized Alfv´en waves enter through the left boundary at R = 25 Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The incoming radial field is perturbed along the Y direction, which causes a shear in the Alfv´en wave speed and the development of spherical polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Switchbacks with Br > 0 form at r > 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='5 Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The total magnetic field (top right) is relatively smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The black curve indicates a possible PSP trajectory at r ≈ 29 Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The bottom left panel shows the magnetic field and velocity components as well as the magnetic pressure pB, the density, the thermal pressure p, and the total pressure p + pB along the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' All quantities are comparable to PSP observations during the first encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The gradients of the total pressure are small, but not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Density variations are also substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The bottom right panel compares the magnetic (solid lines) and velocity (dotted lines) perturbations around a switchback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' The magnetic field components are converted to Alfv´en velocity components: VA = B/√ρµ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' For the radial components the background variation is removed with a smoothing over 100 grid cells (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='28 Rs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' All components satisfy the Alfv´enic relationship to a high accuracy similar to PSP observations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' can create mode conversion from Alfv´en turbulence to compressive turbulence heating and accelerating the solar wind [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' T´oth and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' van der Holst are supported by NSF grant PHY-2027555 and NASA grant 80NSSC22K0892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' PSP data was obtained through NASA CDAWeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Simulations were performed on the Pleiades supercomputer at NASA Ames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' BATSRUS is open source at BR[nT] BT [nT] BN [nT] B [nT] 200 2 100 100 0 150 50 50 1 [Rs] -50 0 Jo 0E 100 Y -10 -1 -50 -50 50 -150 -2 -106 -106 252627 28 30 25 26 2728 29 ¥30 25 26 2728 32930 25 26 2728 29 30 R [Rs] R [Rs] R [Rs] R [Rs] BR [nT] UR [km/s] dVAR & dUr [km/s] 100 E 340 80E 20 320 300 60 280 40 20 BT [nT] UT [km/s] 60 E -20 日 40 20 0 20 -20 VA,r & Ur [km/s] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='40 -40 100 E BN [nT] UN [km/s] 60 50 E 40 20 0 -20 -40 -50 PB [nPa] Density [cm3] -100 E 1000 VA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='n & Un [km/s] 800 100 F 600 400 50F p+PB [nPa] p [nPa] 50 -100E -2 -1 1 2 -2 -1 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='2 0 Y [Rs] Y [Rs] Y [Rs]Springer Nature 2021 LATEX template Formation of Magnetic Switchbacks 9 http://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='com/MSTEM-QUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' We thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' Tamas Gombosi at the University of Michigan for excellent comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=' References [1] Kasper, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content=', 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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} +page_content='3847/2041-8213/ac913d' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfsAEn/content/2301.02572v1.pdf'} diff --git a/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf b/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b30e40faaa1e4bc2806b620c8275d6b563986087 --- /dev/null +++ b/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8736cc487a0edfdec699db8fdcded6d27ae14ee44022dd7bfe101920be335ecb +size 471405 diff --git a/uNAzT4oBgHgl3EQfPvtT/content/tmp_files/2301.01188v1.pdf.txt b/uNAzT4oBgHgl3EQfPvtT/content/tmp_files/2301.01188v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..34372164c25b9ccda7f3aff0235b05c6f536d214 --- /dev/null +++ b/uNAzT4oBgHgl3EQfPvtT/content/tmp_files/2301.01188v1.pdf.txt @@ -0,0 +1,17330 @@ +DeepR Programming +Marek Gagolewski +v0.1.12 (draft) + +Dr habil. Marek Gagolewski +Deakin University, Australia +Systems Research Institute, Polish Academy of Sciences +Warsaw University of Technology, Poland +https://www.gagolewski.com +Copyright (C) 2022–2023 by Marek Gagolewski. Some rights reserved. +This open-access textbook is an independent, non-profit project. It is licensed under +the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International +License (CC BY-NC-ND 4.0). Please spread the word about it. +This project received no funding, administrative, technical, or editorial support from +Deakin University, Warsaw University of Technology, Polish Academy of Sciences, or +any other source. +Product and company names mentioned herein may be the trademarks of their +respective owners. Rather than use a trademark symbol with every occurrence of +a trademarked name, the names are used in an editorial fashion to the benefit of the +trademark owner, with no intention of infringement of the trademark. +Weird is the world we live in, but the following had to be written. +Everyefforthasbeenmadeinthepreparationofthisbooktoensuretheaccuracyofthe +information presented. However, the information contained in this book is provided +withoutwarranty,eitherexpressorimplied.Theauthorwillofcoursenotbeheldliable +for any damages caused or alleged to be caused directly or indirectly by this book. +Anyway, any bug reports/corrections/feature requests are welcome. To make this text- +book even better, please file them at https://github.com/gagolews/deepr. +Typeset with XeLATEX. Please be understanding: it was an algorithmic process, hence +the results are ∈ [good enough, perfect). +Homepage: https://deepr.gagolewski.com/ +Datasets: https://github.com/gagolews/teaching-data +Release: v0.1.12 (draft) (2022-12-29T10:59:45+1100) +ISBN: 978-0-6455719-2-9 (reserved) (vX.Y.Z; 2023; Melbourne: Marek Gagolewski) + +Contents +Preface +xi +0.1 +To R, or not to R +. . . . . . . . . . . . . . . . . . . . . . . . . . +xi +0.2 +R as a Language and an Environment . . . . . . . . . . . . . . . . +xi +0.3 +Aims, Scope, and Design Philosophy +. . . . . . . . . . . . . . . . +xii +0.4 +￿ Classification of R Data Types and Book Structure +. . . . . . . . +xiv +0.5 +About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . +xvi +0.6 +Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . +xvi +I +Deep +1 +1 +Introduction +3 +1.1 +Hello, World! . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +1.2 +Setting up the Development Environment +. . . . . . . . . . . . . +4 +1.2.1 +Installing R . . . . . . . . . . . . . . . . . . . . . . . . . +4 +1.2.2 +Interactive Mode +. . . . . . . . . . . . . . . . . . . . . . +4 +1.2.3 +Batch Mode: Working with R Scripts (**) +. . . . . . . . . . +5 +1.2.4 +Weaving: Automatic Report Generation (**) . . . . . . . . . +5 +1.2.5 +Semi-Interactive Modes (Jupyter Notebooks, Sending Code to +an Associated R Console, etc.) . . . . . . . . . . . . . . . . +6 +1.3 +Atomic Vectors at a Glance +. . . . . . . . . . . . . . . . . . . . . +8 +1.4 +Getting Help +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +1.5 +Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +2 +Numeric Vectors +13 +2.1 +Creating Numeric Vectors +. . . . . . . . . . . . . . . . . . . . . +13 +2.1.1 +Numeric Constants . . . . . . . . . . . . . . . . . . . . . +13 +2.1.2 +Concatenating Vectors with c . . . . . . . . . . . . . . . . +14 +2.1.3 +Repeating Entries with rep +. . . . . . . . . . . . . . . . . +14 +2.1.4 +Generating Arithmetic Progressions with seq and `:` . . . . +16 +2.1.5 +Generating Pseudorandom Numbers . . . . . . . . . . . . +17 +2.1.6 +Reading Data with scan . . . . . . . . . . . . . . . . . . . +19 +2.2 +Creating Named Objects +. . . . . . . . . . . . . . . . . . . . . . +21 +2.3 +Vectorised Mathematical Functions . . . . . . . . . . . . . . . . . +23 +2.3.1 +abs and sqrt . . . . . . . . . . . . . . . . . . . . . . . . . +23 +2.3.2 +Rounding . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +2.3.3 +Natural Exponential Function and Logarithm . . . . . . . . +24 +2.3.4 +Probability Distributions (*) . . . . . . . . . . . . . . . . . +26 + +IV +CONTENTS +2.3.5 +Special Functions (*) +. . . . . . . . . . . . . . . . . . . . +29 +2.4 +Arithmetic Operations +. . . . . . . . . . . . . . . . . . . . . . . +30 +2.4.1 +Vectorised Arithmetic Operators +. . . . . . . . . . . . . . +30 +2.4.2 +Recycling Rule +. . . . . . . . . . . . . . . . . . . . . . . +31 +2.4.3 +Operator Precedence +. . . . . . . . . . . . . . . . . . . . +32 +2.4.4 +Accumulating . . . . . . . . . . . . . . . . . . . . . . . . +33 +2.4.5 +Aggregating . . . . . . . . . . . . . . . . . . . . . . . . . +35 +2.5 +Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +37 +3 +Logical Vectors +39 +3.1 +Creating Logical Vectors +. . . . . . . . . . . . . . . . . . . . . . +39 +3.2 +Comparing Elements . . . . . . . . . . . . . . . . . . . . . . . . +40 +3.2.1 +Vectorised Comparison Operators . . . . . . . . . . . . . . +40 +3.2.2 +Testing for NA, NaN, and Inf +. . . . . . . . . . . . . . . . . +40 +3.2.3 +Dealing with Floating Point Round-Off Errors (*) +. . . . . . +41 +3.3 +Logical Operations +. . . . . . . . . . . . . . . . . . . . . . . . . +44 +3.3.1 +Vectorised Logical Operators +. . . . . . . . . . . . . . . . +44 +3.3.2 +Operator Precedence Revisited +. . . . . . . . . . . . . . . +45 +3.3.3 +Dealing with Missingness . . . . . . . . . . . . . . . . . . +45 +3.3.4 +Aggregating with all, any, and sum +. . . . . . . . . . . . . +46 +3.3.5 +Simplifying Predicates +. . . . . . . . . . . . . . . . . . . +47 +3.4 +Choosing Elements with ifelse . . . . . . . . . . . . . . . . . . . +48 +3.5 +Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +50 +4 +Lists and Attributes +53 +4.1 +Type Hierarchy and Conversion . . . . . . . . . . . . . . . . . . . +53 +4.1.1 +Explicit Type Casting . . . . . . . . . . . . . . . . . . . . +54 +4.1.2 +Implicit Conversion (Coercion) +. . . . . . . . . . . . . . . +54 +4.2 +Lists +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +56 +4.2.1 +Creating Lists . . . . . . . . . . . . . . . . . . . . . . . . +56 +4.2.2 +Coercing to and from Lists +. . . . . . . . . . . . . . . . . +58 +4.3 +NULL +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +59 +4.4 +Object Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . +59 +4.4.1 +Developing Perceptual Indifference to Most Attributes . . . . +60 +4.4.2 +But There Are Some Use Cases . . . . . . . . . . . . . . . . +61 +4.4.3 +Special Attributes . . . . . . . . . . . . . . . . . . . . . . +62 +4.4.4 +Labelling Vector Elements with the names Attribute +. . . . . +63 +4.4.5 +Altering and Removing Attributes . . . . . . . . . . . . . . +66 +4.5 +Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +67 +5 +Vector Indexing +69 +5.1 +head and tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . +69 +5.2 +Subsetting of and Extracting from Vectors +. . . . . . . . . . . . . +70 +5.2.1 +Nonnegative Indexes +. . . . . . . . . . . . . . . . . . . . +70 +5.2.2 +Negative Indexes . . . . . . . . . . . . . . . . . . . . . . +72 +5.2.3 +Logical Indexer . . . . . . . . . . . . . . . . . . . . . . . +73 +5.2.4 +Character Indexer . . . . . . . . . . . . . . . . . . . . . . +74 + +CONTENTS +V +5.3 +Replacing Elements . . . . . . . . . . . . . . . . . . . . . . . . . +76 +5.3.1 +Modifying Atomic Vectors . . . . . . . . . . . . . . . . . . +76 +5.3.2 +Modifying Lists . . . . . . . . . . . . . . . . . . . . . . . +77 +5.3.3 +Inserting New Elements . . . . . . . . . . . . . . . . . . . +79 +5.4 +Functions Related to Indexing +. . . . . . . . . . . . . . . . . . . +80 +5.4.1 +Matching of Elements in Another Vector . . . . . . . . . . . +80 +5.4.2 +Assigning Numbers into Intervals . . . . . . . . . . . . . . +81 +5.4.3 +Splitting Vectors into Subgroups +. . . . . . . . . . . . . . +81 +5.4.4 +Ordering Elements . . . . . . . . . . . . . . . . . . . . . +84 +5.4.5 +Identifying Duplicates +. . . . . . . . . . . . . . . . . . . +87 +5.4.6 +Counting Index Occurrences +. . . . . . . . . . . . . . . . +87 +5.5 +Preserving and Losing Attributes . . . . . . . . . . . . . . . . . . +88 +5.5.1 +c +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +88 +5.5.2 +as.something . . . . . . . . . . . . . . . . . . . . . . . . +89 +5.5.3 +Subsetting +. . . . . . . . . . . . . . . . . . . . . . . . . +89 +5.5.4 +Vectorised Functions . . . . . . . . . . . . . . . . . . . . +89 +5.6 +Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +90 +6 +Character Vectors +95 +6.1 +Creating Character Vectors . . . . . . . . . . . . . . . . . . . . . +95 +6.1.1 +Inputting Individual Strings +. . . . . . . . . . . . . . . . +95 +6.1.2 +Many Strings, One Object . . . . . . . . . . . . . . . . . . +98 +6.1.3 +Concatenating Character Vectors . . . . . . . . . . . . . . +98 +6.1.4 +Formatting Objects . . . . . . . . . . . . . . . . . . . . . +99 +6.1.5 +Reading Text Data from Files +. . . . . . . . . . . . . . . . +99 +6.2 +Pattern Searching +. . . . . . . . . . . . . . . . . . . . . . . . . +100 +6.2.1 +Comparing Whole Strings . . . . . . . . . . . . . . . . . . +100 +6.2.2 +Partial Matching +. . . . . . . . . . . . . . . . . . . . . . +100 +6.2.3 +Matching Anywhere Within a String . . . . . . . . . . . . . +101 +6.2.4 +Using Regular Expressions (*) . . . . . . . . . . . . . . . . +102 +6.2.5 +Locating Pattern Occurrences . . . . . . . . . . . . . . . . +102 +6.2.6 +Replacing Pattern Occurrences +. . . . . . . . . . . . . . . +105 +6.2.7 +Splitting Strings into Tokens +. . . . . . . . . . . . . . . . +106 +6.3 +Other String Operations +. . . . . . . . . . . . . . . . . . . . . . +106 +6.3.1 +Extracting Substrings . . . . . . . . . . . . . . . . . . . . +106 +6.3.2 +Translating Characters +. . . . . . . . . . . . . . . . . . . +107 +6.3.3 +Ordering Strings +. . . . . . . . . . . . . . . . . . . . . . +108 +6.4 +Other Atomic Vector Types (*) . . . . . . . . . . . . . . . . . . . . +108 +6.4.1 +Integer Vectors (*) . . . . . . . . . . . . . . . . . . . . . . +109 +6.4.2 +Raw Vectors (*) . . . . . . . . . . . . . . . . . . . . . . . +110 +6.4.3 +Complex Vectors (*) . . . . . . . . . . . . . . . . . . . . . +110 +6.5 +Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +110 +7 +Functions +113 +7.1 +Creating and Invoking Functions . . . . . . . . . . . . . . . . . . +115 +7.1.1 +Anonymous Functions +. . . . . . . . . . . . . . . . . . . +115 +7.1.2 +Named Functions . . . . . . . . . . . . . . . . . . . . . . +115 + +VI +CONTENTS +7.1.3 +Passing Arguments To Functions +. . . . . . . . . . . . . . +116 +7.1.4 +Grouping Expressions with Curly Braces, `{` +. . . . . . . . +117 +7.2 +Functional Programming . . . . . . . . . . . . . . . . . . . . . . +120 +7.2.1 +Functions are Objects . . . . . . . . . . . . . . . . . . . . +120 +7.2.2 +Calling on Precomputed Arguments with do.call . . . . . . +122 +7.2.3 +Common Higher-Order Functions +. . . . . . . . . . . . . +122 +7.2.4 +Vectorising Functions with Map +. . . . . . . . . . . . . . . +123 +7.3 +Accessing Third-Party Functions +. . . . . . . . . . . . . . . . . . +126 +7.3.1 +Using R Packages . . . . . . . . . . . . . . . . . . . . . . +126 +Default Packages . . . . . . . . . . . . . . . . . . . . . . +128 +Source vs Binary Packages (*) . . . . . . . . . . . . . . . . +128 +7.3.2 +Managing Dependencies (*) . . . . . . . . . . . . . . . . . +129 +7.3.3 +Calling External Programs +. . . . . . . . . . . . . . . . . +130 +7.3.4 +A Note on Interfacing C, C++, Python, Java, etc. (*) +. . . . . +131 +7.4 +Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +132 +8 +Flow of Execution +137 +8.1 +Conditional Evaluation . . . . . . . . . . . . . . . . . . . . . . . +137 +8.1.1 +Return Value +. . . . . . . . . . . . . . . . . . . . . . . . +138 +8.1.2 +Nested ifs +. . . . . . . . . . . . . . . . . . . . . . . . . +139 +8.1.3 +Condition: Either True of False +. . . . . . . . . . . . . . . +140 +8.1.4 +Short-Circuit Evaluation +. . . . . . . . . . . . . . . . . . +141 +8.2 +Exception Handling +. . . . . . . . . . . . . . . . . . . . . . . . +142 +8.3 +Repeated Evaluation +. . . . . . . . . . . . . . . . . . . . . . . . +144 +8.3.1 +while . . . . . . . . . . . . . . . . . . . . . . . . . . . . +144 +8.3.2 +for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +145 +8.3.3 +break and next +. . . . . . . . . . . . . . . . . . . . . . . +147 +8.3.4 +return +. . . . . . . . . . . . . . . . . . . . . . . . . . . +149 +8.3.5 +A Note on Time and Space Complexity of Algorithms (*) . . . +149 +8.4 +Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +152 +II +Deeper +155 +9 +Designing Functions +157 +9.1 +Principles of Sustainable Design +. . . . . . . . . . . . . . . . . . +157 +9.1.1 +To Write or to Abstain . . . . . . . . . . . . . . . . . . . . +157 +9.1.2 +To Pamper or to Challenge +. . . . . . . . . . . . . . . . . +158 +9.1.3 +To Build or to Reuse . . . . . . . . . . . . . . . . . . . . . +159 +9.2 +Managing Data Flow +. . . . . . . . . . . . . . . . . . . . . . . . +160 +9.2.1 +Checking Input Data Integrity and Argument Handling . . . +160 +9.2.2 +Putting Outputs into Context . . . . . . . . . . . . . . . . +164 +9.3 +Organising and Maintaining Functions . . . . . . . . . . . . . . . +167 +9.3.1 +Function Libraries +. . . . . . . . . . . . . . . . . . . . . +167 +9.3.2 +Writing R Packages . . . . . . . . . . . . . . . . . . . . . +167 +9.3.3 +Documenting R Packages . . . . . . . . . . . . . . . . . . +168 +9.3.4 +Assuring Quality Code +. . . . . . . . . . . . . . . . . . . +169 +Managing Changes and Working Collaboratively +. . . . . . +169 + +CONTENTS +VII +Test-driven Development and Continuous Integration . . . . +170 +Debugging . . . . . . . . . . . . . . . . . . . . . . . . . +170 +Profiling +. . . . . . . . . . . . . . . . . . . . . . . . . . +171 +9.4 +Special Functions: Syntactic Sugar +. . . . . . . . . . . . . . . . . +171 +9.4.1 +A Note on Backticks . . . . . . . . . . . . . . . . . . . . . +171 +9.4.2 +Curly Braces, `{` +. . . . . . . . . . . . . . . . . . . . . . +172 +9.4.3 +`if` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +172 +9.4.4 +Operators are Functions Too +. . . . . . . . . . . . . . . . +173 +Calling Built-in Operators as Functions . . . . . . . . . . . +173 +Creating Own Binary Operators . . . . . . . . . . . . . . . +174 +9.4.5 +Replacement Functions . . . . . . . . . . . . . . . . . . . +174 +Creating Own Replacement Functions . . . . . . . . . . . . +174 +Substituting Parts of Vectors +. . . . . . . . . . . . . . . . +175 +Replacing Attributes +. . . . . . . . . . . . . . . . . . . . +176 +Compositions of Replacement Functions +. . . . . . . . . . +177 +9.5 +Arguments and Local Variables +. . . . . . . . . . . . . . . . . . . +180 +9.5.1 +Pass by “Value” +. . . . . . . . . . . . . . . . . . . . . . . +180 +9.5.2 +Variable Scope +. . . . . . . . . . . . . . . . . . . . . . . +180 +9.5.3 +Closures (*) . . . . . . . . . . . . . . . . . . . . . . . . . +181 +9.5.4 +Default Arguments . . . . . . . . . . . . . . . . . . . . . +182 +9.5.5 +Lazy Evaluation . . . . . . . . . . . . . . . . . . . . . . . +183 +9.5.6 +Ellipsis, `...` . . . . . . . . . . . . . . . . . . . . . . . . +183 +9.5.7 +Metaprogramming (*) . . . . . . . . . . . . . . . . . . . . +185 +9.6 +Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +187 +10 S3 Classes +189 +10.1 Object Type vs Class +. . . . . . . . . . . . . . . . . . . . . . . . +190 +10.2 Generics and Method Dispatching +. . . . . . . . . . . . . . . . . +193 +10.2.1 +Generics, Default, and Custom Methods +. . . . . . . . . . +193 +10.2.2 Creating Own Generics . . . . . . . . . . . . . . . . . . . +195 +10.2.3 +Built-in Generics . . . . . . . . . . . . . . . . . . . . . . +197 +10.2.4 Dispatching Only on One Argument and Calling S3 Methods +Directly . . . . . . . . . . . . . . . . . . . . . . . . . . . +199 +10.2.5 +Multi-class-ness +. . . . . . . . . . . . . . . . . . . . . . +202 +10.2.6 Operator Overloading . . . . . . . . . . . . . . . . . . . . +203 +10.3 Common Built-in S3 Classes +. . . . . . . . . . . . . . . . . . . . +206 +10.3.1 +Date, Time, etc. . . . . . . . . . . . . . . . . . . . . . . . +206 +10.3.2 +Formulae (*) +. . . . . . . . . . . . . . . . . . . . . . . . +208 +10.3.3 +Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . +209 +10.3.4 +Ordered Factors . . . . . . . . . . . . . . . . . . . . . . . +212 +10.4 Argument Checking Revisited +. . . . . . . . . . . . . . . . . . . +213 +10.5 (Over)using the Forward-pipe Operator, `|>` (*) . . . . . . . . . . . +215 +10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +217 +11 Matrices and Other Arrays +219 +11.1 +Creating Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . +219 +11.1.1 +matrix and array +. . . . . . . . . . . . . . . . . . . . . . +219 + +VIII +CONTENTS +11.1.2 +Promoting and Stacking Vectors +. . . . . . . . . . . . . . +221 +11.1.3 +Simplifying Lists +. . . . . . . . . . . . . . . . . . . . . . +222 +11.1.4 +Beyond Numeric Arrays . . . . . . . . . . . . . . . . . . . +224 +11.1.5 +Internal Representation . . . . . . . . . . . . . . . . . . . +225 +11.2 +Array Indexing +. . . . . . . . . . . . . . . . . . . . . . . . . . . +228 +11.2.1 +Arrays Are Built upon Basic Vectors . . . . . . . . . . . . . +228 +11.2.2 +Selecting Individual Elements . . . . . . . . . . . . . . . . +228 +11.2.3 +Selecting Rows and Columns +. . . . . . . . . . . . . . . . +229 +11.2.4 +Dropping Dimensions +. . . . . . . . . . . . . . . . . . . +229 +11.2.5 +Selecting Submatrices . . . . . . . . . . . . . . . . . . . . +230 +11.2.6 +Selecting Elements Based on Logical Vectors +. . . . . . . . +231 +11.2.7 +Selecting Based on Two-Column Numeric Matrices . . . . . +232 +11.2.8 +Higher-Dimensional Arrays . . . . . . . . . . . . . . . . . +233 +11.2.9 +Replacing Elements . . . . . . . . . . . . . . . . . . . . . +234 +11.3 +Common Operations . . . . . . . . . . . . . . . . . . . . . . . . +234 +11.3.1 +Matrix Transpose . . . . . . . . . . . . . . . . . . . . . . +234 +11.3.2 +Vectorised Mathematical Functions . . . . . . . . . . . . . +235 +11.3.3 +Aggregating Rows and Columns . . . . . . . . . . . . . . . +235 +11.3.4 +Binary Operators . . . . . . . . . . . . . . . . . . . . . . +236 +11.4 +Numerical Matrix Algebra (*) . . . . . . . . . . . . . . . . . . . . +239 +11.4.1 +Matrix Multiplication . . . . . . . . . . . . . . . . . . . . +239 +11.4.2 +Solving Systems of Linear Equations +. . . . . . . . . . . . +241 +11.4.3 +Norms and Metrics . . . . . . . . . . . . . . . . . . . . . +241 +11.4.4 +Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . +242 +11.4.5 +QR Decomposition . . . . . . . . . . . . . . . . . . . . . +244 +11.4.6 +SVD Decomposition +. . . . . . . . . . . . . . . . . . . . +245 +11.5 +S4 Classes (*) . . . . . . . . . . . . . . . . . . . . . . . . . . . . +246 +11.5.1 +Defining S4 Classes . . . . . . . . . . . . . . . . . . . . . +247 +11.5.2 +Accessing Slots . . . . . . . . . . . . . . . . . . . . . . . +248 +11.5.3 +Defining Methods . . . . . . . . . . . . . . . . . . . . . . +249 +11.5.4 +Defining Constructors +. . . . . . . . . . . . . . . . . . . +250 +11.5.5 +Inheritance . . . . . . . . . . . . . . . . . . . . . . . . . +251 +11.5.6 +A Note on the Matrix Package . . . . . . . . . . . . . . . . +252 +11.6 +Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +253 +12 Data Frames +257 +12.1 +Creating Data Frames +. . . . . . . . . . . . . . . . . . . . . . . +258 +12.1.1 +data.frame and as.data.frame . . . . . . . . . . . . . . . . +258 +12.1.2 +cbind.data.frame and rbind.data.frame . . . . . . . . . . . +261 +12.1.3 +Reading Data Frames . . . . . . . . . . . . . . . . . . . . +264 +12.1.4 +Interfacing Relational Databases and Querying with SQL (*) . +265 +12.1.5 +Strings as Factors? +. . . . . . . . . . . . . . . . . . . . . +266 +12.1.6 +Internal Representation . . . . . . . . . . . . . . . . . . . +268 +12.2 Data Frame Subsetting . . . . . . . . . . . . . . . . . . . . . . . +270 +12.2.1 +Data Frames are Lists . . . . . . . . . . . . . . . . . . . . +270 +12.2.2 +Data Frames are Matrix-like . . . . . . . . . . . . . . . . . +273 +12.3 Common Operations . . . . . . . . . . . . . . . . . . . . . . . . +276 + +CONTENTS +IX +12.3.1 +Ordering Rows +. . . . . . . . . . . . . . . . . . . . . . . +276 +12.3.2 +Handling Duplicated Rows +. . . . . . . . . . . . . . . . . +279 +12.3.3 +Joining (Merging) Data Frames +. . . . . . . . . . . . . . . +279 +12.3.4 +Aggregating and Transforming Columns +. . . . . . . . . . +280 +12.3.5 +Handling Missing Values +. . . . . . . . . . . . . . . . . . +282 +12.3.6 +Reshaping Data Frames . . . . . . . . . . . . . . . . . . . +282 +12.3.7 +Aggregating Data in Groups . . . . . . . . . . . . . . . . . +285 +12.3.8 +Transforming Data in Groups . . . . . . . . . . . . . . . . +293 +12.3.9 +Metaprogramming-Based Techniques (*) . . . . . . . . . . +296 +12.3.10 A Note on the dplyr (tidyverse) and data.table Packages (*) . +299 +12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +300 +13 ￿ Graphics +307 +13.1 +￿ Placeholders for Plots Referred to Elsewhere +. . . . . . . . . . . +307 +III +Deepest +309 +14 ￿ Interfacing Compiled Code +311 +14.1 +￿ R/C API +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +311 +14.2 ￿ External Pointers . . . . . . . . . . . . . . . . . . . . . . . . . +311 +14.3 ￿ RCpp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +311 +15 ￿ Expressions +313 +15.1 +￿ The Dollar Operator, `$` +. . . . . . . . . . . . . . . . . . . . . +313 +15.2 ￿ Formulae, `~` . . . . . . . . . . . . . . . . . . . . . . . . . . . +313 +16 ￿ Environments +315 +16.1 +￿ Classification of R Data Types (Revisited) . . . . . . . . . . . . . +315 +16.2 ￿ Copying on Demand +. . . . . . . . . . . . . . . . . . . . . . . +315 +16.3 ￿ A Note on Reference Classes (*) . . . . . . . . . . . . . . . . . . +315 +17 ￿ Evaluating Expressions +317 +18 ￿ Evaluating Functions +319 +18.1 +￿ Evaluation of Default Arguments . . . . . . . . . . . . . . . . . +319 +18.2 ￿ Ellipsis Revisited . . . . . . . . . . . . . . . . . . . . . . . . . +319 +18.3 ￿ S3 Method Lookup by UseMethod +. . . . . . . . . . . . . . . . . +319 +18.4 ￿ Overloading S3 Group Generics . . . . . . . . . . . . . . . . . . +319 +18.5 ￿ Package Namespaces . . . . . . . . . . . . . . . . . . . . . . . +319 +IV +Appendix +321 +A +Changelog +323 +References +325 + +X +CONTENTS +DeepRProgramming is a comprehensive course on one of the most popular languages +in data science (statistical computing, graphics, machine learning, data wrangling +and analytics). It introduces the base language in-depth and is aimed at ambitious +students,practitioners,and researcherswho wouldliketobecome independentusers +of this powerful environment. +This early draft is distributed in the hope that it will be useful. +For many students around the world, educational resources are hardly affordable. +Therefore, I have decided that this book should remain an independent, non-profit, +open-access project (available both in PDF1 and HTML2 forms). Whilst, for some +people, the presence of a “designer tag” from a major publisher might still be a proxy +forquality,itismyhopethatthispublicationwillproveusefultothosewhoseekknow- +ledge for knowledge’s sake. +Please spread the news about it by sharing the above URLs with your mates, peers, or +students. Thank you. +Also, check out my other book, Minimalist Data Wrangling with Python3 [20]. +Anybug/typosreports/fixes4 areappreciated.Althoughavailableonline,thisisawhole +course, and should be read from the beginning to the end. Please refer to the Preface +for general introductory remarks and design philosophy. +Consider citing this book as: Gagolewski M. (2023), DeepRProgramming, Zenodo, Mel- +bourne, ISBN: 978-0-6455719-2-9, URL: https://deepr.gagolewski.com/. +1 https://deepr.gagolewski.com/deepr.pdf +2 https://deepr.gagolewski.com/ +3 https://datawranglingpy.gagolewski.com/ +4 https://github.com/gagolews/deepr/issues + +0 +Preface +0.1 +To R, or not to R +R [50] has been named the eleventh most dreaded programming language in the 2022 +StackOverflow Developer Survey5. +Also, it is a free app, so there must be something wrong with it, right? +But whatever, R is deprecated anyway; the “modern” way is to use tidyverse. Or we +should all just switch to Python6. +Well, not really7. +0.2 +R as a Language and an Environment +Let us get one thing straight: R is not just a statistical package. It is a general-purpose, +high-level programming language, that just happens to be very powerful for any kind +of numerical, data-intense computing. It offers extensive support for statistical, ma- +chine learning, data analysis, data wrangling, and data visualisation applications, but +there is a lot more. +Initially, R was written “for statisticians by statisticians”. Therefore, it may be thought +ofasafreeyetmorecapablealternativetoStata,SAS,SPSS,Statistica,Minitab,Weka, +etc. Unlike some of them, however, a spreadsheet-like GUI is not the main gateway for +performing computations on data. In R, a user must write code to get things actually +done. Despite the learning curve’s being a little steeper for non-programmers, in the +long run, it empowers their users because they are not limited only to the most com- +mon scenarios. If some functionality is missing or does not suit their needs, they can +easily implement everything themselves. +Itisthusveryconvenientforrapidprototyping.Ithelpsturnourideasintooperational +code that can be tested, extended, polished, run in production, and otherwise enjoyed +overall. As an interpreted language, it can be run not only in an interactive read-eval- +5 https://survey.stackoverflow.co/2022/ +6 https://datawranglingpy.gagolewski.com/ +7 Or, as Aussies would say, yeah, nah. + +XII +PREFACE +print loop (command–result, question–answer, …), but also in batch mode (running +whole, standalone scripts). +Thus, we would rather position R amongst such tools/languages for numerical or sci- +entific computing as Python with the NumPy ecosystem, Julia, GNU Octave, Scilab, +MATLAB, etc. However, it is more specialised in data science applications than all of +them. Hence, it provides a much smoother experience. This is why, over the years, R +has become the de facto standard in statistics and many other related fields. +Important R is a whole ecosystem (environment). It not only consists of the R lan- +guage interpreter, but also features advanced: +• graphical capabilities (see Chapter 13), +• a help system (Section 1.4), +• ways for convenient interfacing with compiled code (Chapter 14), +• apackagesystemandcentralisedpackagerepositories(suchasCRANandBiocon- +ductor; Section 7.3.1), +• a lively community of users and developers – curious and passionate people, just +like you and me. +Note R’s predecessor is the popular S system designed in the 1980s by John M. Cham- +bersandhiscolleaguesatBellLabsS:[3,4,8,40].RiscalledGNUS,afree,open-source +version of its commercial counterpart developed in the mid-1990s8by Robert Gentle- +man and Ross Ihaka of the Statistics Department, University of Auckland, and a large +number of contributors; see [7, 31] for some historical notes. +R has a C language-like syntax that involves the use of {curly braces}. Still, in principle, +it is a beautiful, functional programming language: its design was heavily inspired by +Scheme (see [1] and Chapter 17 for more details). It is also somewhat object-oriented +(Chapter 10). +0.3 +Aims, Scope, and Design Philosophy +Many users have been introduced to R by means of some very advanced operations +involving data frames, formulas, and functions that rely on nonstandard evaluation +(metaprogramming), like: +8 R version 0.49 released in April 1997 (the first for which source code? is available on CRAN), was already +quitefeature-rich(e.g.,implementedS3methods,formulae,anddataframesintroducedinthe1991version +of S [8]). +8 https://cloud.r-project.org/src/base/R-0/ + +PREFACE +XIII +lm( +Ozone~Solar.R+Temp, +data=subset(airquality, Temp>60, select=-(Month:Day)) +) |> summary() +This is horrible. +Another group has been isolated from the base R through a thick layer of third-party +packagesthatfeatureanoverwhelmingnumberoffunctions(everyoperation,regard- +less of its complexity, has a different name), often duplicating the core functionality, +and sometimes being quite incompatible with our traditional system. +Both families should be fine — as long as they limit themselves to solving only the +simplest and most common data processing problems. +But we yearn for more. We do not want hundreds of prefabricated recipes for popular +dishes that we can mindlessly apply without much understanding. +Our aim is to learn baseR, which is supposedtobe the common language (lingua franca) +to all R users. We want to be able to write code that everybody should be able to under- +stand, and which will be likely to work without modifications ten years from now (no +slang!). +We want to be able to tackle any data-intense problem. Furthermore, we want to de- +velop skills that are transferable, so that learning new tools such as Julia or Python with +NumPy and Pandas will be much easier later (because R is not the only notable envir- +onment out there). +Anyway, enough preaching. This graduate9-level textbook is for independent readers +who do not mind a slightly steeper learning curve at the beginning, but who are able +to appreciate a more cohesively and comprehensively10 organised material. +Some will benefit from it as a first introduction to R (but without all the pampering). +For others11, this will be a good course from intermediate to advanced (do not skip the +first chapters, though). +Either way, do not forget to solve all the prescribed exercises. +Good luck. +9 The author taught similar courses for his wonderfully ambitious undergraduate data/computer sci- +ence and maths students at Warsaw University of Technology, where such an approach has proven not dif- +ficult at all. It requires a more independent, curious, and motivated mindset, though. And that’s the way +to go, in the long run. +10 Yours truly is neither a historian, a stenographer, nor a grammarian. We allow ourselves to make a few +noninvasive idealisations for didactic purposes. Languages evolve over time, R now is different than it used +to be, and we can shape it (slowly, because we value its stable API) to become something even better in the +future. +11 It might also happen that for some, this will not be a good course at all, either at this stage of their +career (come back later) or in general (no dramas). This is a non-profit, open-access project, but it does not +mean it is ideal for everyone – in such a case, give other sources a try, e.g., [5, 10, 35, 41, 42, 43, 49], etc. Some +of them are also freely available. + +XIV +PREFACE +0.4 +￿ Classification of R Data Types and Book Structure +RDataTypes +Basic +Atomic +NULL +logical +numeric +character +list +function +... +Compound +factor +matrix +array +data.frame +formula +Date +kmeans +... +Figure 1: An overview of the most prevalent R data types (see Figure 16.1 for a more +comprehensive list) +The most commonly used R data types can be classified as follows; see also Figure 1. +1. Basic types – which we discuss in the first part of this book – internal or built-in +types, upon which more complex ones are hinged: +• atomic vectors – represent whole sequences of values, where every element is +of the same type: +– logical (Chapter 3) – includes items that are TRUE (“yes”, “present”), +FALSE (“no”, “absent”), or NA (“not available”, “missing”); +– numeric(Chapter2)–featuresrealnumbers,suchas1,3.14,-0.0000001, +etc.; +– character (Chapter 6) – contains strings of characters, e.g., "groß", +"123", or “Добрий день”; +• function (Chapter 7) – used to group a series of expressions (code lines) so +that they can be applied on different kinds of input data to generate the +(hopefully) desired outcomes, for instance, cat, print, plot, sample, and sum; +• list (Chapter 4) a.k.a. a generic vector – can store elements of mixed types; +The above will be complemented with a discussion on vector indexing (Chapter 5) +and ways to control the program flow (Chapter 8). + +PREFACE +XV +2. Compound types – discussed in the second part – wrappers around objects of basic +types that might behave differently from the underlying primitives thanks to the +dedicated operations overloaded for them. They are +• factor (Section 10.3.3) – a vector-like object that represents qualitative data +(on a nominal or an ordered scale); +• matrix (Chapter 11) – stores tabular data, i.e., arranged into rows and +columns, where each cell is usually of the same type; +• data.frame (Chapter 12) – also used for depositing tabular data, but this time +such that each column can be of different type; +• and many more, which we or third-parties can define arbitrarily using, +amongst others, the principles of S3-style object orientated-programming +(Chapter 10). +In this part of the book, we also discuss the principles of sustainable coding +(Chapter 9) as well as introduce the basic ways to prepare publication-quality +graphics (Chapter 13). +3. ￿ Some more advanced material that, in most cases, we can easily do without, but +which is still essential to gain a full understanding of and control over the environ- +ment, is discussed in the first part. This includes, amongst others, the following +data types: +• externalptr (sec:to-do); +• environment (sec:to-do); +• symbol (name), call, expression (sec:to-do); +• formula (Section 15.2) – used by some functions to specify supervised learn- +ing models or define operations to be performed within data subgroups, +amongst others; +￿ Also, we will discuss other things, but this is an early draft of this book, so right +now, we only provide a placeholder therefor (sec:to-do). Please come back later. +Note The above classification is just a first approximation to the complete type clas- +sification that we will discuss in Section 16.1; see also Figure 16.1. +Also, we should not be surprised that above we do not see any of the data types defined +by a few very popular12 third-party packages. We will later see that we can most often +do without them. If that is not the case, we will become skilled enough to learn them +easily ourselves. +12 Which does not automatically mean good. For instance, sugar, salt, and some drugs are very popular, +but it does not make them healthy. + +XVI +PREFACE +0.5 +About the Author +I, Marek Gagolewski13 (pronounced like Ma’rek Gong-olive-ski), am currently a Senior +Lecturer in Applied AI at Deakin University in Melbourne, VIC, Australia and an +Associate Professor in Data Science at the Systems Research Institute of the Polish +Academy of Sciences. +My research interests are related to data science, in particular: modelling complex +phenomena, developing usable, general-purpose algorithms, studying their analyt- +ical properties, and finding out how people use, misuse, understand, and misunder- +stand methods of data analysis in research, commercial, and decision-making set- +tings. I’m an author of 90+ publications, including journal papers in outlets such as +Proceedings of the National Academy of Sciences (PNAS), Information Fusion, International +Journal of Forecasting, Statistical Modelling, Journal of Statistical Software, Information Sci- +ences, Knowledge-Based Systems, IEEE Transactions on Fuzzy Systems, and Journal of Infor- +metrics. +In my “spare” time, I write books for my students (also check out my Minimalist Data +Wrangling with Python14 [20]) and develop open-source (libre) data analysis software, +such as stringi15 (one of the most often downloaded R packages), genieclust16 (a fast +and robust clustering algorithm in both Python and R), and many others17. +0.6 +Acknowledgements +DeepRProgramming is based on my experience as an author of a quite successful Polish +textbook ProgramowaniewjęzykuR (see [19]) which was published by PWN (1st ed. 2014, +2nd ed. 2016). The current book is an entirely different work. However, its predecessor +served as an excellent testbed for many ideas conveyed here. +Theteachingstyleexercisedinthisbookhasprovensuccessfulinmanysimilarcourses +that yours truly has been responsible for, including at Warsaw University of Techno- +logy, Data Science Retreat (Berlin), and Deakin University (Melbourne). I thank all my +students and colleagues for the feedback given over the last 10+ years. +We describe R version 4.2.2 Patched (2022-11-10 r83330). However, we expect 99.9% of +material covered here to be valid in future releases (consider filing a bug report if you +discover that this is not the case). +13 https://www.gagolewski.com +14 https://datawranglingpy.gagolewski.com/ +15 https://stringi.gagolewski.com +16 https://genieclust.gagolewski.com +17 https://github.com/gagolews + +PREFACE +XVII +This book was prepared in a Markdown superset called MyST18, Sphinx19, and TeX +(XeLaTeX). Code chunks were processed with the R package knitr [44]. All fig- +ures were plotted with the low-level graphics package using the author’s own style +template. A little help from Makefiles, custom shell scripts, and Sphinx plugins +(sphinxcontrib-bibtex20, sphinxcontrib-proof21) dotted the j’s and crossed the f ’s. +The Ubuntu Mono22 font is used for the display of code. Typesetting of the main text +relies upon the Alegreya23 and Lato24 typefaces. +This book received no funding, administrative, technical, or editorial support from +Deakin University, Warsaw University of Technology, Polish Academy of Sciences, or +any other source. +18 https://myst-parser.readthedocs.io/en/latest/index.html +19 https://www.sphinx-doc.org/ +20 https://pypi.org/project/sphinxcontrib-bibtex/ +21 https://pypi.org/project/sphinxcontrib-proof/ +22 https://design.ubuntu.com/font/ +23 https://www.huertatipografica.com/en +24 https://www.latofonts.com/ + + +Part I +Deep + + +1 +Introduction +1.1 +Hello, World! +Traditionally, every programming journey starts with the printing of a “Hello, World”- +like greeting. Let’s then get it over with asap: +cat("My hovercraft is full of eels.") +## My hovercraft is full of eels. +By calling the cat function, we printed out a given character string that we enclosed +in double quote characters. +Documenting code is a good development practice. It is thus worth knowing that any +text followed by a hash sign (that is not part of a string) is a comment, ignored by the +interpreter. +# This is a comment. +# This is another comment. +cat("I cannot wait", "till lunchtime.") +# two arguments (another comment) +## I cannot wait till lunchtime. +cat("# I will not buy this record.\n# It is scratched.") +# `\n` == newline +## # I will not buy this record. +## # It is scratched. +By convention, in this book, the textual outputs generated by R itself are always pre- +ceded by two hashes. This makes copy-pasting all code chunks easier in the case where +the kind reader would like to experiment with them by themself (which is always +highly encouraged). +Whenever a call to some function is to be made, the round brackets are oblig- +atory. All objects within the parentheses (they are separated by commas) con- +stitute the input data to be consumed by the operation. Thus, the syntax is: +some_function_to_be_called(argument1, argument2, etc.). + +4 +I DEEP +1.2 +Setting up the Development Environment +1.2.1 +Installing R +Itisquitenaturaltopinefortheabilitytoexecutetheabovecodeourselves–wecannot +learn programming without getting our hands dirty. +The official precompiled binary distributions of R can be downloaded from https:// +cran.r-project.org/. +For serious programming work1, we recommend, sooner rather than later, switching +to2 one of the Unix-like operating systems. This includes the free, open-source (== +good) variants of GNU/Linux, amongst others, or the proprietary (== very far from +good) m**OS. The users thereof might employ their favourite package manager (e.g., +apt, dnf, pacman, or Homebrew) to install R. +Users of other operating systems (such as Wi***ws) might consider installing +Anaconda or Miniconda if they require some level of interoperability with the Py- +thon environment, e.g., they would like to work with the Jupyter environment (Sec- +tion 1.2.5). +Below we review several ways in which we can write and execute R code. It is up to +the benign reader to research, setup, and learn the development environment that +suits their needs. As usual in real life, there is no single universal approach that always +works best in all the scenarios. +1.2.2 +Interactive Mode +R’s read-eval-print loop (REPL) can give us instant gratification whenever we would like +to compute something quickly, e.g., determine basic aggregates of a few numbers +entered by hand or evaluate a mathematical expression like “2+2”. +How to start the R console varies from system to system, e.g., users of Unix-like boxes +can simply execute R from the terminal (shell). Wi***ws folks can fire up the RGui from +the Start menu. +Important When working interactively, the default3 command prompt, “>”, means: +I am awaiting an order. Moreover, “+” denotes: Please continue. In such a case, we should +either complete the unfinished expression, or cancel the operation by pressing ESC or +CTRL+C (depends on the operating system). +> cat("And now +(continues on next page) +1 For instance, when an easy interoperability with other programming languages/environments is re- +quired or when we think about scheduling jobs on Linux-based computing/container clusters. +2 Or at least trying out – by installing a copy of GNU/Linux on a virtual machine (VM). +3 It can be changed; see help("options"). + +1 INTRODUCTION +5 +(continued from previous page) ++ for something ++ completely different ++ ++ ++ it is an unfinished expression... ++ awaiting another double quote character and then the closing bracket... ++ ++ press ESC or CTRL+C to abort input +> +For readability, we never print out the command prompt characters in this book. +1.2.3 +Batch Mode: Working with R Scripts (**) +The interactive mode of operation is unsuitable for more complicated tasks, though. +The users of Unix-like operating systems will be interested in another extreme, which +involves writing standalone R scripts that can be executed one by line, without any +user intervention. +To do so, in the terminal (command line, shell), we can invoke: +Rscript file.R +where file.R is the path to some source file. +Exercise 1.1 (**) In your favourite text editor (e.g., Notepad++, Kate, vi, Emacs, RStudio, or +VSCodium), create a file named test.R. Write a few calls to the cat function. Then, execute this +script from the terminal by invoking the Rscript program. +1.2.4 +Weaving: Automatic Report Generation (**) +Reproducibledataanalysis4 requiresustokeeptheresults(text,tables,plots,auxiliary +files) synchronised with their generating code and data. +utils::Sweave (the Sweave function from the utils package) and knitr [44] are two +example template processors that evaluate R code chunks within documents written +in LaTeX, HTML, or other markup languages. The chunks are replaced by the outputs +they yield. +This book is a showcase of such an approach – all the results, including Figure 2.3 and +the above “Hello, World”, were generated programmatically. Thanks to its being writ- +teninthehighlyuniversalMarkdown5 language,itcouldbeeasilyconvertedtoasingle +4 The idea dates back to Knuth’s literate programming concept; see [32]. +5 https://daringfireball.net/projects/markdown/ + +6 +I DEEP +PDF document6 as well as the whole website7. Tools like pandoc and docutils facilitate +such operations. +Exercise 1.2 (**) Install the knitr package by calling install.packages("knitr") from +within an R session. Then, create a text file named test.Rmd with the following content: +# Hello, Markdown! +This is my first automatically generated report, +where I print stuff. +```{r} +print("G'day!") +print(2+2) +``` +Thank you for your attention. +Assuming that the file is located in the current working directory (compare Section 7.3.3), call +knitr::knit("test.Rmd") from within the R console or run the following in the terminal: +Rscript -e 'knitr::knit("test.Rmd")' +Then, inspect the generated Markdown file, test.md. +Furthermore, if you have the pandoc tool installed, to generate a standalone HTML file, execute +in the terminal: +pandoc test.md --standalone -o test.html +Alternatively, for ways to call external programs from R, see Section 7.3.3. +1.2.5 +Semi-Interactive Modes (Jupyter Notebooks, Sending Code to an As- +sociated R Console, etc.) +The nature of the most frequent use cases of R encourages a semi-interactive work- +flow, where we progress with prototyping fast by trial-and-error. +In this mode, we write a series of short code fragments inside a standalone R script. +Each fragment implements a simple, well-defined task, such as the loading of data +files, data cleansing, feature visualisation, computations of some information ag- +gregates, etc. +Importantly, any code chunk can be sent to the associated R console and executed +therein. This way, we can inspect the results it generates at any time. If we are not +happy with the outcome, we can apply any corrections that are necessary. +6 https://deepr.gagolewski.com/deepr.pdf +7 https://deepr.gagolewski.com + +1 INTRODUCTION +7 +There are quite a few integrated development environments (IDEs; sometimes re- +quiring additional plugins) that enable such a workflow, including JupyterLab, Emacs, +RStudio, and VSCodium. +Executing an individual code line or a whole text selection is usually done by pressing +a (configurable) keyboard shortcut such as Ctrl+Enter or Shift+Enter. +Exercise 1.3 (*) JupyterLab8 isadevelopmentenvironmentthatrunsinawebbrowser.Itwas +programmed in Python, but supports many programming languages. Thanks to IRkernel9, we +can use it with R. +1. Install JupyterLab and IRkernel (for instance, if you use Anaconda, run conda install +-c r r-essentials). +2. From the File menu, select Create a new R source file and save it as, e.g., test.R. +3. From the File menu, select Create a new console for editor running the R kernel. +4. Type some print “Hello, World”-like calls. +5. Press Shift+Enter (whilst working in the editor) to send different code fragments onto the +console and execute them. Inspect the results. +See Figure 1.1 for an illustration. +Figure 1.1: JupyterLab: A source file editor and the associated R console, where we can +run arbitrary code fragments +Example 1.4 (*) The Jupyter project, whose JupyterLab is part of, also supports the handling +of dedicated Notebooks. There, editable and executable code chunks and results they generate can +8 https://jupyterlab.readthedocs.io/en/stable/ +9 https://irkernel.github.io/ + +File +Edit +View +Kernel +Settings +Help +Run +Tabs +C +三 test.R +×+ +OPEN TABS +Close All +test.R +1 +# +# +2 +test.R +3 +#Press shift+Enter to executecurrent lineor selection +4 +# in the associated console below +KERNELS +Shut Down All +5 +test.R +plot(rnorm(1000), rnorm(1000), main="G'day!") +5 +> +TERMINALS +Shut Down All + test.R +X +[1]: plot(rnorm(1000)， rnorm(1000)， main="G'day!") +G'day! +3 +O +C +2 +08 +000 +1000) +O +norm(1 +O +098 +88 +:[]8 +I DEEP +be kept together in a single .ipynb (JSON) file; see Figure 1.2 for an illustration and Chapter 1 of +[20] for a quick introduction (from the Python language kernel perspective). +This environment is quite convenient for live coding (e.g., for teachers) or performing explorat- +ory data analyses. However, for more serious programming work, the code can get quite messy +(luckily, there is always an option to export a notebook to an executable, plain text R script). +Figure 1.2: An example Jupyter Notebook, where we can keep the code and the results +together +1.3 +Atomic Vectors at a Glance +After the printing of the “Hello, World” message, a typical programming course would +normally proceed with the discussion on basic data types for storing individual nu- +meric or logical values. Next, we would be introduced to arithmetic and comparison +operations on such scalars, followed by the definition of whole arrays or other collec- +tions of such values, complemented by the methods to iterate over them, one element +after another. +In R, no separate types representing individual values have been defined. Instead, +what seems to be a single datum, is already a vector (sequence, array) of length 1. +2.71828 +# input a number (here: the same as print(2.71828)) +## [1] 2.7183 +length(2.71828) +# it is a vector featuring one element +## [1] 1 + +Jupyter +Welcome (unsaved changes) +R +Logout +File +Edit +View +Insert +Cell +Kernel +Widgets +Help +Trusted +RO +Example Jupyter Notebook +In [1l:plot(rnorm(1000), rnorm(1000), main="G'day!") +G'day! +00 +8 +8 +00 +2 +8 +80 +00 +morm(1000) +O +8 +00 +QQ +Q +2 +0% +8 +8 +O +?- +4 +4 +-2 +0 +2 +morm(1000)1 INTRODUCTION +9 +To create a vector of any length, we can call the c function, which combines given ar- +guments into a single sequence: +c(1, 2, 3) +# three vectors of length 1 +-> +one vector of length 3 +## [1] 1 2 3 +length(c(1, 2, 3)) +## [1] 3 +In Chapter 2, Chapter 3, and Chapter 6, we will discuss the most prevalent types of +atomic vectors: numeric, logical, and character ones, respectively. +c(0, 1, -3.14159, 12345.6) +# four numbers +## [1] +0.0000 +1.0000 +-3.1416 12345.6000 +c(TRUE, FALSE) +# two logical values +## [1] +TRUE FALSE +c("spam", "spam", "bacon and spam") +# three character strings +## [1] "spam" +"spam" +"bacon and spam" +We call them atomic, because they can only group together values of the same type. +Lists, which we will discuss in Chapter 4, are, on the other hand, referred to as generic +vectors – they can be used for storing items of mixed types – other lists as well. +Note Not having separate scalar types greatly simplifies the programming of numer- +ical computing tasks. Vectors are prevalent in our main areas of interest – statistics, +simulations, data science, machine learning, and all other data-oriented computing. +For example, columns and rows in tables (values of different features describing cli- +ents,ratingsofitemsgivenbyusers)ortimeseries(stockmarketprices,readingsfrom +temperature sensors) are all best represented by means of such sequences. +Moreover, the fact that vectors are the core part of the R language makes their use +very natural – as opposed to the languages that require special add-ons for vector +processing, e.g., numpy for Python [29]. By learning different ways to process them as +a whole, instead of one element at a time, we will assure that our ideas can quickly be +turnedintoworkingcode(rapidprototyping).Forinstance,computingsummarystat- +isticssuchas,say,themeanabsolutedeviationofsomesequence x,willbeaseffortless +as writing mean(abs(x-mean(x))). Such a code is not only easy to read and maintain, +but it is also fast to run. +1.4 +Getting Help +Our aim is to become independent, advanced R programmers. +Independent, however, does not mean omniscient. The R help system is the authorit- + +10 +I DEEP +ative source of knowledge about specific functions or more general topics. To open a +help page, we call: +help("topic") +# equivalently: ?"topic" +Exercise 1.5 Sight (without going into detail) the manual on the length function by calling +help("length"). Note that most help pages are structured as follows: +1. Header: “package:base” means that the function is a base one (see Section 7.3.1 for more +details on the R package system); +2. Title; +3. Description: a short description of what the function does; +4. Usage: the list of formal arguments (parameters) to the function; +5. Arguments: the meaning of each formal argument explained; +6. Details: technical information; +7. Value: return value explained; +8. References: further reading; +9. See Also: links to other help pages; +10. Examples: R code that is worth to run and study by yourself. +We can also search within all the installed help pages by calling: +help.search("vague topic") +# equivalently: ??"vague topic" +Oftentimes, this way we will be able to find answers to our questions more reliably +than when asking DuckDuckGo or G**gle (which commonly feature many low qual- +ity/irrelevant/distracting results that can make us lose the sacred code writer’s flow). +Important All code chunks, including code comments and textual outputs, form an +integral part of this book’s text. They should not be skipped by the reader. On the con- +trary, they should become objects of our intense reflection and thorough investiga- +tion. +For instance, whenever we introduce a few function, it may be a good idea to look it +up in the help system. Moreover, playing with the presented code (running, modify- +ing, experimenting, etc.) is also very beneficial. We should develop the habit of asking +ourselves questions like “what would happen if…”, and then finding the answers on +our own. +Wearenowreadytodiscussthemostimportantoperationsonnumericvectors,which +constitute the main theme of the next chapter. See you there. + +1 INTRODUCTION +11 +1.5 +Exercises +Exercise 1.6 What are the three most important types of atomic vectors? +Exercise 1.7 According to the classification of the R data types we introduced in the previous +chapter, are atomic vectors basic or compound types? + + +2 +Numeric Vectors +In this chapter, we discuss the uttermost common operations on numeric vectors. +They are so fundamental that we will also find them in other scientific computing en- +vironments, including Python with NumPy or TensorFlow, Julia, MATLAB, GNU Octave, +or Scilab. +At first blush, the number of functions we are going to explore may seem quite large. +Still, the reader is kindly asked to place some trust (a rare thing these days) in yours +truly. It is because our selection is comprised only of the most representative and edu- +cational amongst the plethora of possible choices. More complex building blocks can +either be reduced to a creative combination of the former or be easily found – should +the need arise – in a number additional packages or libraries (e.g., the GNU GSL [23]). +A solid understanding of base R programming is necessary for the effective dealing +with the popular packages (such as data.table, dplyr, or caret). Most importantly, +base R’s API is stable, hence the code we write today will most likely work the same way +in 10 years. This is often not the case when we rely on third-party add-ons. +In the sequel, we will be advocating a minimalistic, keep-it-simple approach to the art +of programming of data processing pipelines, one that is a good balance between “do- +ing it all by oneself”, “minimising the information overload”, “being lazy”, and “stand- +ing on the shoulders of giants”. +Note +The exercises that we suggest below are all self-contained, unless explicitly +stated otherwise. The use of language constructs that are yet to be formally intro- +duced (in particular, if, for, and while which we will explain in Chapter 8) is not only +unnecessary, but discouraged. Moreover, we recommend against taking shortcuts by +looking up partial solutions on the internet. Rather, to get the most out of this course, +the reader should be seeking relevant information within the current and preceding +chapters as well as the R help system. +2.1 +Creating Numeric Vectors +2.1.1 +Numeric Constants +The simplest numeric vectors are those of length one: + +14 +I DEEP +-3.14 +## [1] -3.14 +1.23e-4 +## [1] 0.000123 +The latter is in what we call the scientificnotation which is convenient means of entering +numbers of very large or small order of magnitude. Here, “e” stands for “… times 10 to +the power of…”. Therefore, 1.23e-4 is equal to 1.23×10−4 = 0.000123. In other words, +given 1.23, we move the decimal separator by 4 digits towards the left. +In real life, some information items may be inherently or temporarily missing, un- +known, or Not Available. R is data processing-oriented, hence it is equipped with a +special indicator: +NA_real_ +# numeric NA (missing value) +## [1] NA +This is similar to the Null marker in database query languages such as SQL. Note that +NA_real_ is displayed simply as “NA”, chiefly for readability. +Moreover, Inf denotes the infinity (∞; a value that is larger than the largest represent- +abledoubleprecision–64bit–floatingpointnumber)and NaNstandsfornot-a-number +(it is returned as the result of some illegal operations, e.g., 0/0 or ∞ − ∞). +2.1.2 +Concatenating Vectors with c +Letusprovidesomewaystocreatenumericvectorswithpossiblymorethan1element. +First, the c function we introduced in the previous chapter, can be used to combine +(concatenate) many numeric vectors, each of any length, so as to form a single object: +c(1, 2, 3) +# 3 vectors of length 1 +-> +1 vector of length 3 +## [1] 1 2 3 +c(1, c(2, NA_real_, 4), 5, c(6, c(7, Inf))) +## [1] +1 +2 +NA +4 +5 +6 +7 Inf +Note +Running help("c"), we will see that its usage is like “c(...)”. In the current +context, this means that the c function takes an arbitrary number of arguments. In +Section 9.5.6 we will study the dot-dot-dot (ellipsis) parameter in more detail. +2.1.3 +Repeating Entries with rep +Second, rep replicates the elements in a given vector a given number of times. + +2 NUMERIC VECTORS +15 +rep(1, 5) +## [1] 1 1 1 1 1 +rep(c(1, 2, 3), 4) +## +[1] 1 2 3 1 2 3 1 2 3 1 2 3 +In the second case, the whole vector (1, 2, 3) has been recycled (tiled) four times. Inter- +estingly, if the second argument was a vector of the same length as the first one, the +behaviour would be quite different: +rep(c(1, 2, 3), c(2, 1, 4)) +## [1] 1 1 2 3 3 3 3 +rep(c(1, 2, 3), c(4, 4, 4)) +## +[1] 1 1 1 1 2 2 2 2 3 3 3 3 +Here, each element is repeated the corresponding number of times. +If we call help("rep"), we will come across the notion like “rep(x, ...)” in the Usage +section. Unfortunately, it is rather peculiar, but reading further we discover the dot- +dot-dot stands for one of the following further parameters (see the Arguments section): +• times, +• length.out, +• each. +So far, we have been playing with times, which is listed second in the parameter list +(after x – the vector whose elements are to be repeated). +Important It turns out that the following function calls are all equivalent: +rep(c(1, 2, 3), 4) +# positional matching of arguments: `x`, then `times` +rep(c(1, 2, 3), times=4) +# `times` is the second argument +rep(x=c(1, 2, 3), times=4) +# keyword arguments of the form name=value +rep(times=4, x=c(1, 2, 3)) +# keyword arguments can be given in any order +rep(times=4, c(1, 2, 3)) +# mixed positional and keyword arguments +Wecanalsopasseachorlength.out(adothasnospecialmeaninginR;seeSection2.2), +but their names should be mentioned explicitly: +rep(c(1, 2, 3), length.out=7) +## [1] 1 2 3 1 2 3 1 +rep(c(1, 2, 3), each=3) +## [1] 1 1 1 2 2 2 3 3 3 +rep(c(1, 2, 3), length.out=7, each=3) +## [1] 1 1 1 2 2 2 3 + +16 +I DEEP +Note Whether it was a good programming practice to actually implement a range of +variedbehavioursinsideasinglefunctionisamatteroftaste.Ontheonehand,inallof +the examples above, we do repeat the input elements somehow, so remembering just +one function name is really convenient. Nevertheless, a drastic change in the repeti- +tion pattern depending, e.g., on the length of the times argument can be bug-prone. +Anyway, we have been warned1. +Zero-length vectors are also possible: +rep(c(1, 2, 3), 0) +## numeric(0) +Even though their handling might be a little tricky (compare Chapter 9), we will see +later that they are useful in contexts like “create an empty data frame with a specific +column structure”. +2.1.4 +Generating Arithmetic Progressions with seq and `:` +Third, we can call the seq function to create a sequence of equally-spaced numbers (on +a linear scale, i.e., an arithmetic progression). +seq(1, 15, 2) +## [1] +1 +3 +5 +7 +9 11 13 15 +Reading the function’s help page, we note that it has the following parameters: from, +to, by, length.out, amongst others. +Thus, the above call is equivalent to: +seq(from=1, to=15, by=2) +## [1] +1 +3 +5 +7 +9 11 13 15 +Note that to actually means “up to”: +seq(from=1, to=16, by=2) +## [1] +1 +3 +5 +7 +9 11 13 15 +We can also pass length.out instead of by. In such a case, the increments or decre- +ments will be computed via the formula ((to - from)/(length.out - 1)); this default +value is reported in the Usage section in help("seq"). +1 Some“caring”Rusersmightbetemptedtointroducetwonewfunctionsnow,oneforgenerating(1,2,3, +1, 2, 3, …) only and the other outputting patterns like (1, 1, 1, 2, 2, 2, …). They would most likely wrap them in a +new package and announce that on Twitter. But this is nothing else than a multiplication of entities without +actual necessity; we would end up with three functions: the original one, rep, which everyone should know +anyway because it is so basic and has been and will be used everywhere by almost everybody so far, and +the two redundant ones, whose user-friendliness is only illusory. See also Chapter 9 for discussion on the +design of functions. + +2 NUMERIC VECTORS +17 +seq(1, 0, length.out=5) +## [1] 1.00 0.75 0.50 0.25 0.00 +Also, this: +seq(length.out=5) +# default `from` is 1 +## [1] 1 2 3 4 5 +Arithmetic progressions with step equal to 1 or -1 can also be generated via the `:` +operator. +1:10 +# seq(1, 10) or seq(1, 10, 1) +## +[1] +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +-1:10 +# seq(-1, 10) or seq(-1, 10, 1) +## +[1] -1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +-1:-10 +# seq(-1, -10) or seq(-1, -10, -1) +## +[1] +-1 +-2 +-3 +-4 +-5 +-6 +-7 +-8 +-9 -10 +Note the order of precedence of this operator: “-1:10” means “(-1):10” and not +“-(1:10)”; compare Section 2.4.3. +Exercise 2.1 Takealookatthemanualpageofseq_alongandseq_lenanddeterminewhether +they can easily be done without, having seq2 at hand. +2.1.5 +Generating Pseudorandom Numbers +Wecanalsogeneratesequencesdrawnindependentlyfromarangeofunivariateprob- +ability distributions. +runif(7) +# uniform U(0, 1) +## [1] 0.287578 0.788305 0.408977 0.883017 0.940467 0.045556 0.528105 +rnorm(7) +# normal N(0, 1) +## [1] +1.23950 -0.10897 -0.11724 +0.18308 +1.28055 -1.72727 +1.69018 +These correspond to seven pseudorandom deviates following the uniform distribu- +tion on the unit interval (i.e., (0, 1)) and the standard normal distribution (i.e., with +expectation 0 and standard deviation 1), respectively; compare Figure 2.3. +For more named distribution classes (frequently occurring in various real-world stat- +istical modelling exercises), see Section 2.3.4. +Another useful function samples a number of values from a given vector, either with +or without replacement: +2 Also note that certain configurations of seq and its variants might return vectors of type integer in- +stead of double, some of them in a compact (ALTREP) form; see Section 6.4.1. + +18 +I DEEP +sample(1:10, 20, replace=TRUE) +# 20 with replacement (allow repetitions) +## +[1] +3 +3 10 +2 +6 +5 +4 +6 +9 10 +5 +3 +9 +9 +9 +3 +8 10 +7 10 +sample(1:10, 5, replace=FALSE) +# 5 without replacement (do not repeat) +## [1] 9 3 4 6 1 +Thedistributionofthesampledvaluesdoesnotneedtobeuniform;the probargument +may be fed with a vector of the corresponding probabilities. For example, here are 20 +independent realisations of the random variable 𝑋 such that Pr(𝑋 = 0) = 0.9 (the +probability that we obtain 0 is equal to 90%) and Pr(𝑋 = 1) = 0.1: +sample(0:1, 20, replace=TRUE, prob=c(0.9, 0.1)) +## +[1] 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 +Note If n is a single number (a numeric vector of length 1), then sample(n, ...) is +equivalent to sample(1:n, ...). Similarly, seq(n) is a synonym for seq(1, n) or seq(1, +length(n)), depending on the length of n. This is a dangerous behaviour which can +occasionally backfire and lead to bugs (check what happens when n is, e.g., 0). Non- +etheless, we have been warned and from now on are going to be extra careful (but are +we really?). Read more at help("sample") and help("seq"). +Let us stress that the numbers we obtain are merely pseudorandom, because they are +generated algorithmically. R uses the Mersenne-Twister MT19937 method [36] by de- +fault; see help("RNG") and [16, 24, 33]. By setting the seed of the random number gener- +ator, i.e., re-setting its state to a given one, we can obtain results that are reproducible. +set.seed(12345) +# seeds are specified with integers +sample(1:10, 5, replace=TRUE) # a,b,c,d,e +## [1] +3 10 +8 10 +8 +sample(1:10, 5, replace=TRUE) # f,g,h,i,j +## [1] +2 +6 +6 +7 10 +Setting the seed to the one used previously gives: +set.seed(12345) +sample(1:10, 5, replace=TRUE) # a,b,c,d,e +## [1] +3 10 +8 10 +8 +We did not(?) expect that! And now for something completely different: +set.seed(12345) +sample(1:10, 10, replace=TRUE) # a,b,c,d,e,f,g,h,i,j +## +[1] +3 10 +8 10 +8 +2 +6 +6 +7 10 +Reproducibilityisacrucialfeatureofeachtrulyscientificexperiment.Thesameinitial +condition (here: the same seed), leads to exactly the same outcomes. + +2 NUMERIC VECTORS +19 +Note Some claim that the only unsuspicious seed is 42, but each programmer can +have their own picks. Yours truly, for example, uses 123, 1234, and 12345 as well. When +performing many runs of Monte Carlo experiments, it may be a good idea to call set. +seed(i) in the i-th iteration of a simulation we are trying to program. +Anyhow, we should make sure that our seed settings are applied consistently across all +ourscripts.Otherwise,wemightbeaccusedoftamperingwithevidence.Forinstance, +here is the ultimate proof that we are very lucky today: +set.seed(1679619) +# totally unsuspicious, right? +sample(0:1, 20, replace=TRUE) +# so random +## +[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 +This is exactly why reproducible scripts and auxiliary data should be published along- +side all research reports or papers. Only open, transparent science can be fully trust- +worthy. +If set.seed is not called explicitly, and therandomstate is not restoredfromthe previ- +ously saved R session (see Chapter 16), then the random generator is initialised based +on the current wall time and the identifier of the running R instance (PID). This may +give the impression that the numbers we generate are surprising. +In order to understand the “pseudo” part of the said randomness better, in Section 8.3, +we will build a very simple random generator ourselves. +2.1.6 +Reading Data with scan +The example text file named euraud-20200101-20200630.csv3 gives the EUR to AUD +exchange rates (how many Australian Dollars can one buy for 1 Euro) from 1 January +to 30 June 2020 (remember COVID-19?). Let us preview the first couple of lines: +# EUR/AUD Exchange Rates +# Source: Statistical Data Warehouse of the European Central Bank System +# https://www.ecb.europa.eu/stats/policy_and_exchange_rates/ +# (provided free of charge) +NA +1.6006 +1.6031 +NA +The four first lines that begin with “#” merely serve as comments for us, humans; they +should be ignored by the interpreter. The first “real” value, NA corresponds to 1 January +(Wednesday; New Years Day; Forex markets were closed, hence a missing observa- +tion). +3 https://github.com/gagolews/teaching-data/raw/master/marek/euraud-20200101-20200630.csv + +20 +I DEEP +The scan function can be used to read all the inputs and convert them to a single nu- +meric vector: +scan(paste0("https://github.com/gagolews/teaching-data/raw/", +"master/marek/euraud-20200101-20200630.csv"), comment.char="#") +## +[1] +NA 1.6006 1.6031 +NA +NA 1.6119 1.6251 1.6195 1.6193 1.6132 +## [11] +NA +NA 1.6117 1.6110 1.6188 1.6115 1.6122 +NA +NA 1.6154 +## [21] 1.6177 1.6184 1.6149 1.6127 +NA +NA 1.6291 1.6290 1.6299 1.6412 +## [31] 1.6494 +NA +NA 1.6521 1.6439 1.6299 1.6282 1.6417 +NA +NA +## [41] 1.6373 1.6260 1.6175 1.6138 1.6151 +NA +NA 1.6129 1.6195 1.6142 +## [51] 1.6294 1.6363 +NA +NA 1.6384 1.6442 1.6565 1.6672 1.6875 +NA +## [61] +NA 1.6998 1.6911 1.6794 1.6917 1.7103 +NA +NA 1.7330 1.7377 +## [71] 1.7389 1.7674 1.7684 +NA +NA 1.8198 1.8287 1.8568 1.8635 1.8226 +## [81] +NA +NA 1.8586 1.8315 1.7993 1.8162 1.8209 +NA +NA 1.8021 +## [91] 1.7967 1.8053 1.7970 1.8004 +NA +NA 1.7790 1.7578 1.7596 +## +[ reached getOption("max.print") -- omitted 83 entries ] +We used the paste0 function to concatenate two long (too long to fit a single line of +code) strings to form a single URL; see Section 6.1.3. +We can also read the files located on our computer, for example: +scan("~/teaching-data/marek/euraud-20200101-20200630.csv", +comment.char="#") +uses an absolute file path that starts at the user’s home directory, denoted “~”: yours +truly’s case is /home/gagolews/. +Note For portability reasons, we should use slashes, “/”, as path separators (but see +help("file.path") and help(".Platform")). These are not only recognised by all Unix- +like boxes but also other popular operating systems. Note that URLs (such as https: +//www.r-project.org/) feature slashes too. +Paths can also be relative to the current working directory, denoted “.”. It can be read +viaacallto getwd.Usually,itisthedirectoryfromwheretheRsessionhasbeenstarted. +For instance, if the current working directory was /home/gagolews/teaching-data/ +marek,wecouldhavewrittenthefilepathequivalentlyas./euraud-20200101-20200630. +csv or even euraud-20200101-20200630.csv. +On as side note, ../ would denote the parent directory of the current work- +ing directory. For instance, ../r/iris.csv would be equivalent to /home/gagolews/ +teaching-data/r/iris.csv. +Exercise 2.2 Read the help page about scan. Take note of the following formal arguments and +their meaning: dec, sep, what, comment.char, and na.strings. +Later we will discuss the read.table and read.csv, which are wrappers around scan + +2 NUMERIC VECTORS +21 +that can be used to read tabular data. Note that write can be used to export an atomic +vector’s contents to a text file. +Example 2.3 Figure2.1showsthegraphoftheaforementionedexchangerates,whichwasgen- +erated by calling: +plot(scan(paste0("https://github.com/gagolews/teaching-data/raw/", +"master/marek/euraud-20200101-20200630.csv"), comment.char="#"), +xlab="Day", ylab="EUR/AUD") +0 +50 +100 +150 +1.60 +1.65 +1.70 +1.75 +1.80 +1.85 +Day +EUR/AUD +Figure 2.1: EUR/AUD exchange rates from 2020-01-01 (day 1) to 2020-06-30 (day 182) +Somewhat misleadingly (and for the reasons that will become apparent later), the document- +ation of plot can be accessed by calling help("plot.default"). Read about, and experiment +with,differentvaluesofthemain,xlab,ylab,type,col,pch,cex,lty,andlwdarguments.More +plotting routines will be discussed in Chapter 13. +2.2 +Creating Named Objects +Often,theobjectswebringforthwillneedtobememorisedsothattheycanbereferred +to in further computations. The assignment operator, `<-`, can be used for this very +purpose: +x <- 1:3 +# creates a numeric vector and binds the name `x` to it +The now-named object can be recalled and dealt with as we please: + +22 +I DEEP +print(x) +# or just `x` in the R console +## [1] 1 2 3 +sum(x) +# example operation: compute the sum of all elements in `x` +## [1] 6 +Important In R, all names are case-sensitive. Hence, x and X can coexist peacefully: +when set, they refer to two different objects. Also, if we tried to call Print(x) above, we +would get an error. +Typically, we will be using what we refer to as syntactic names (see Section 9.4.1 +for an exception though). In the R help system (see help("make.names") and also +help("Quotes")), we read: A syntactically valid name consists of letters, numbers and the +dot or underline characters and starts with a letter or the dot not followed by a number. Names +such as .2way are not valid, and neither are the reserved words. For the list of the latter, see +help("Reserved"). +A good name is self-explanatory and thus reader-friendly: patients, mean, and aver- +age_scoresarewaybetter(iftheyreallyarewhattheyclaimtheyare)than xyz123, crap, +or spam. Also, it might not be such a bad idea to get used to denoting: +• vectors with x, y, z, +• matrices (and matrix-like objects) with A, B, …, X, Y, Z, +• integer indexes with letters i, j, k, l, +• object sizes with n, m, d, p or nx, ny, etc., +especially when they are only of temporary nature (for storing some auxiliary results, +iterating over collections of objects, etc.). +There are numerous naming conventions that we can adopt, but most often they are +a matter of taste; snake_case, lowerCamelCase, UpperCamelCase, flatcase, or dot.case +are equally good as long as they are used coherently (for instance, some use snake_case +for vectors and UpperCamelCase for functions). It may even be the case that we have +little choice but to adhere to the naming conventions agreed upon in the project we +are about to contribute to. +Note Let us stress that a dot, “.”, has no special meaning (however, see Chapter 10 +and Chapter 16 for some asterisks); na.omit is as good a name as na_omit, naOmit, NA- +OMIT, naomit, and NaOmit. Users coming from some other (C, C++, Java, Python, etc.) +programming languages will need to habituate themselves to this convention. +R, as a dynamic language, allows for introducing new variables at any time. Moreover, +existing names can be re-bound to new values. For instance: + +2 NUMERIC VECTORS +23 +(y <- c(1, 10, 100)) +# bracketed expression - printing not suppressed +## [1] +1 +10 100 +x <- y +print(x) +## [1] +1 +10 100 +Now x refers to a verbatim copy of y. +Note Objects are automatically destroyed when there are no more names bound with +them. In particular, by now the garbage collector should have got rid of the 1:3 vector +begotten above (to which the name x was bound previously). See sec:to-do for more +details on memory management. +2.3 +Vectorised Mathematical Functions +Mathematically, we will be denoting a given vector 𝒙 of length n as (𝑥1, 𝑥2, … , 𝑥𝑛). In +other words, its i-th element is equal to 𝑥𝑖. +Let us review some ubiquitous operations in numerical computing. +2.3.1 +abs and sqrt +R implements vectorised versions of the most popular mathematical functions, e.g., +abs (absolute value, |𝑥|) and sqrt (square root, √𝑥). +abs(c(2, -1, 0, -3, NA_real_)) +## [1] +2 +1 +0 +3 NA +Here, vectorised means that instead of being defined to act on a single numeric value, +the function of interest is applied on each element in a vector. The i-th resulting item +is a transformed version of the i-th input. If an input is a missing value, the corres- +ponding output will be marked as “don’t know” as well. +Another example: +x <- c(4, 2, -1) +(y <- sqrt(x)) +## Warning in sqrt(x): NaNs produced +## [1] 2.0000 1.4142 +NaN +To attract our attention to the fact that computing the square root of a negative value +yields a not-a-number, R generated an informative warning. A warning is not an error +though: the result is being reckoned as usual. + +24 +I DEEP +Also the fact that the irrational √2 is displayed as 1.4142 does not mean that it is such a +crudeapproximationto1.41421356237309504880168872420969807856967187537694 …; +it is only rounded when printing, for aesthetic reasons. In fact, in Section 3.2.3 we will +point out that the computer’s floating-point arithmetic allows for roughly 16 decimal +digits precision (but we shall see that the devil is in the detail). +print(y, digits=16) +# display more significant figures +## [1] 2.000000000000000 1.414213562373095 +NaN +2.3.2 +Rounding +The following functions get rid of all or portions of fractional parts of numbers: +• floor(x) (rounds down to the nearest integer, denoted ⌊𝑥⌋), +• ceiling(x) (rounds up, denoted ⌈𝑥⌉), +• trunc(x) (rounds towards zero), and +• round(x, digits=0) (rounds to the nearest number with digits decimal digits). +For instance: +x <- c(7.0001, 6.9999, -4.3149, -5.19999, 123.4567, -765.4321, 0.5, 1.5, 2.5) +floor(x) +## [1] +7 +6 +-5 +-6 +123 -766 +0 +1 +2 +ceiling(x) +## [1] +8 +7 +-4 +-5 +124 -765 +1 +2 +3 +trunc(x) +## [1] +7 +6 +-4 +-5 +123 -765 +0 +1 +2 +Note If we call help("round"), we will read that its usage is like round(x, digits=0), +which means that the digits parameter is equipped with the defaultvalue of 0. In other +words, if rounding to 0 decimal digits is what we need, the second argument can be +omitted. +round(x) +# the same as round(x, 0) +## [1] +7 +7 +-4 +-5 +123 -765 +0 +2 +2 +round(x, 1) +## [1] +7.0 +7.0 +-4.3 +-5.2 +123.5 -765.4 +0.5 +1.5 +2.5 +round(x, -2) +## [1] +0 +0 +0 +0 +100 -800 +0 +0 +0 +2.3.3 +Natural Exponential Function and Logarithm +Moreover: + +2 NUMERIC VECTORS +25 +• exp(x)outputsthenaturalexponentialfunction,𝑒𝑥,wheretheEuler’snumber𝑒 ≃ +2.718, +• log(x, +base=exp(1)) computes, by default, the natural logarithm of 𝑥, log𝑒 𝑥 +(which is most often denoted simply as log 𝑥). +Recall that if 𝑥 = 𝑒𝑦, then log𝑒 𝑥 = 𝑦, i.e., one is the inverse of the other. +log(c(0, 1, 2.7183, 7.3891, 20.0855)) +# grows slowly +## [1] -Inf +0 +1 +2 +3 +exp(c(0, 1, 2, 3)) +# grows fast +## [1] +1.0000 +2.7183 +7.3891 20.0855 +Note These functions enjoy a number of very useful identities and inequalities, in- +cluding: +• log(𝑥 ⋅ 𝑦) = log 𝑥 + log 𝑦, +• log(𝑥𝑦) = 𝑦 log 𝑥, +• 𝑒𝑥+𝑦 = 𝑒𝑥 ⋅ 𝑒𝑦. +For more properties like these, take a glance at Chapter 4 of the freely available hand- +book [38]. +For the logarithm to a different base, say log10 𝑥, we can call: +log(c(0, 1, 10, 100, 1000, 1e10), 10) +# or log(..., base=10) +## [1] -Inf +0 +1 +2 +3 +10 +Note that if log𝑏 𝑥 = 𝑦, then 𝑥 = 𝑏𝑦, for any 1 ≠ 𝑏 > 0. +Note +Commonly, a logarithmic scale is used for variables that grow rapidly when +expressed as functions of each other; see Figure 2.2. +x <- seq(0, 10, length.out=1001) +par(mfrow=c(1, 2)) +# two plots in one figure (1 row, 2 columns) +plot(x, exp(x), type="l") +plot(x, exp(x), type="l", log="y") +# log-scale on the y-axis + +26 +I DEEP +0 +2 +4 +6 +8 +10 +0 +5000 +10000 +15000 +20000 +x +exp(x) +0 +2 +4 +6 +8 +10 +1 +10 +100 +1000 +10000 +x +exp(x) +Figure 2.2: Linear- vs log-scale on the y-axis +Note that 𝑒𝑥 on the log-scale is just a straight line. Also, keep in mind that such a trans- +formation of the axes can only be applied in the case of values strictly greater than 0. +2.3.4 +Probability Distributions (*) +It should come as no surprise that R offers an extensive support for many univariate +probability distribution families, including: +• continuousdistributions,whichtakevaluesbeingarbitraryrealnumbers(overthe +whole possible range or in some interval): +– *unif (uniform), +– *norm (normal), +– *exp (exponential), +– *gamma (gamma, Γ), +– *beta (beta, B), +– *lnorm (log-normal), +– *t (Student), +– *cauchy (Cauchy–Lorentz), +– *chisq (chi-squared, 𝜒2), +– *f (Snedecor–Fisher), + +2 NUMERIC VECTORS +27 +– *weibull (Weibull); +with the prefix “*” being one of: +– “d” (probability density function, PDF), +– “p” (cumulative distribution function, CDF; or survival function, SF), +– “q” (quantile function, being the inverse of the CDF), +– “r” (generation of random deviates; already mentioned); +• discrete distributions, i.e., those whose possible outcomes can be easily enumer- +ated (e.g., some integers). +– *binom (binomial), +– *geom (geometric), +– *pois (Poisson), +– *hyper (hypergeometric), +– *nbinom (negative binomial); +here, prefixes “p” and “r” have the same meaning as above, however: +– “d” now gives the probability mass function (PMF), +– “q”yieldsthequantilefunction,butonethatisdefinedasageneralisedinverse +of the CDF. +Each distribution is characterised by a set of underlying parameters. For instance, a +normal distribution N(𝜇, 𝜎) can be pinpointed by setting its expected value 𝜇 ∈ ℝ +and standard deviation 𝜎 > 0. In R, these two have been named mean and sd, respect- +ively; see help("dnorm"). +Note The parametrisations assumed in R can be subtly different from what we know +from statistical textbooks or probability courses. For example, the normal distribu- +tion can be parameterised based on either standard deviation or variance, and the ex- +ponential distribution can be defined via its expected value or the reciprocal thereof. +We thus advise the reader to study carefully the documentation of help("dnorm"), +help("dunif"), help("dexp"), help("dbinom"), and the like. +It is also worth to know the typical use cases of each of the distribution listed, e.g., +a Poisson distribution can describe the probability of observing the number of in- +dependent events in a fixed time interval (e.g., the number of users downloading a +copy of R from CRAN per hour), and an exponential distribution can model the time +between such events; compare [17]. +Exercise 2.4 Acalltohist(x)drawsahistogram,whichcanserveasanestimatoroftheunder- +lyingcontinuousprobabilitydensityfunctionofagivensample;seeFigure2.3foranillustration. + +28 +I DEEP +par(mfrow=c(1, 2)) +# 2 plots in 1 figure +# Uniform U(0, 1) +hist(runif(10000, 0, 1), col="white", probability=TRUE, main="") +x <- seq(0, 1, length.out=101) +lines(x, dunif(x, 0, 1), lwd=2) +# draw the true density function (PDF) +# Normal N(0, 1) +hist(rnorm(10000, 0, 1), col="white", probability=TRUE, main="") +x <- seq(-4, 4, length.out=101) +lines(x, dnorm(x, 0, 1), lwd=2) +# draw the PDF +runif(10000, 0, 1) +Density +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +rnorm(10000, 0, 1) +Density +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +Figure 2.3: Example histograms of some pseudorandom samples and the true under- +lying probability density functions: the uniform distribution on the unit interval (left) +and the standard normal distribution (right) +Draw a histogram of some random samples of different sizes n from the following distributions: +• rnorm(n, µ, σ) — normal N(𝜇, 𝜎) with expected values 𝜇 ∈ {−1, 0, 5} (i.e., 𝜇 being +equal to either −1, 0, or 5; read “∈” as “belongs to the given set” or “in”) and standard devi- +ations 𝜎 ∈ {0.5, 1, 5}; +• runif(n, a, b) — uniform U(𝑎, 𝑏) on the interval (𝑎, 𝑏) with 𝑎 = 0 and 𝑏 = 1 as well +as 𝑎 = −1 and 𝑏 = 1; +• rbeta(n, α, β) — beta B(𝛼, 𝛽) with 𝛼, 𝛽 ∈ {0.5, 1, 2}; +• rexp(n, λ) — exponential E(𝜆) with rates 𝜆 ∈ {0.5, 1, 10}; +Moreover,readaboutandplaywiththe breaks, main, xlab, ylab, xlim, ylim,and colparamet- +ers; see help("hist"). +Example 2.5 Werollasix-sideddice12times.Let𝐶bearandomvariabledenotingthenumber + +2 NUMERIC VECTORS +29 +of cases wherethe “1” face is thrown. 𝐶 follows a binomial distribution Bin(𝑛, 𝑝) with paramet- +ers 𝑛 = 12 (the number of Bernoulli trials) and 𝑝 = 1/6 (the probability of success in a single +roll). +Theprobabilitiesthatthenumberof“1”srolledwillbeequalto0,1,…,4,i.e.,𝑃(𝐶 = 0),𝑃(𝐶 = +1),…,𝑃(𝐶 = 4),respectively,canbecomputedbasedontheprobabilitymassfunction(dbinom): +dbinom(0:4, 12, 1/6) +# probability mass function at 5 different points +## [1] 0.112157 0.269176 0.296094 0.197396 0.088828 +On the other hand, the probability that we throw more than three “1”s, 𝑃(𝐶 > 3) = 1 − +𝑃(𝐶 ≤ 3), can be determined by means of the cumulative distribution function (pbinom) or +survival function (pbinom(..., lower.tail=FALSE)): +1-pbinom(3, 12, 1/6) +# pbinom(3, 12, 1/6, lower.tail=FALSE) +## [1] 0.12518 +The smallest 𝑐 such that 𝑃(𝐶 ≤ 𝑐) ≥ 0.95 can be computed based on the quantile function: +qbinom(0.5, 12, 1/6) +## [1] 2 +pbinom(3:4, 12, 1/6) +# for comparison - 0.95 is in-between +## [1] 0.87482 0.96365 +In other words, at least 95% of the time we will be observing no more than 4 successes. +Also here are some pseudorandom realisations of 𝐶 – the number of “1”s in 30 simulations of 12 +independent dice rolls each: +rbinom(30, 12, 1/6) +## +[1] 1 3 2 4 4 0 2 4 2 2 4 2 3 2 0 4 1 0 1 4 4 3 2 6 2 3 2 2 1 1 +2.3.5 +Special Functions (*) +Within mathematical formulae and across assorted application areas, certain func- +tions appear more frequently than others. Hence, for the sake of notational brevity +and computational precision, many of them have been assigned special names. For +instance, the following may be mentioned in the definitions related to some of the +probability distributions listed above: +• gamma(x) for 𝑥 > 0 computes Γ(𝑥) = ∫ +∞ +0 𝑡𝑥−1𝑒−𝑡 𝑑𝑡, +• beta(a, b) for 𝑎, 𝑏 > 0 yields 𝐵(𝑎, 𝑏) = Γ(𝑎)Γ(𝑏) +Γ(𝑎+𝑏) = ∫ +1 +0 𝑡𝑎−1(1 − 𝑡)𝑏−1 𝑑𝑡. +Whydo wehave beta if itis merely amixof gammas?Aspecific,tailoredfunctionshould +be faster and more precise than its DIY version; its underlying implementation does +not have to involve any calls to gamma at all. + +30 +I DEEP +beta(0.25, 250) +# okay +## [1] 0.91213 +gamma(0.25)*gamma(250)/gamma(250.25) +# not okay +## [1] NaN +The Γ function grows so rapidly that already gamma(172) yields Inf. It is due to the fact +that a computer’s arithmetic is not infinitely precise; compare Section 3.2.3. +Special functions are plentiful; see the open-access [38] for one of the most definitive +references (and also [2] for its predecessor). R package gsl [28] provides a vectorised +interface to the famous GNU GSL [23] library, which implements many of them. +Exercise 2.6 The Pochhammer symbol, (𝑎)𝑥 = Γ(𝑎 + 𝑥)/Γ(𝑎), can be computed via a call to +gsl::poch(a, x) (i.e., the poch function from the gsl package; see Section 7.3.1): +# call install.packages("gsl") first +library("gsl") +# load the package +poch(10, 3:6) +# calls gsl_sf_poch() from GNU GSL +## [1] +1320 +17160 +240240 3603600 +Read the documentation of the corresponding gsl_sf_poch function in the GNU GSL manual +available here4. +And since you are there, do not hesitate to go through the list of all the other functions, including +those related to statistics, permutations, combinations, and so forth. +Manyfunctionsalsohavetheirlogarithm-ofversions;see,e.g., lgammaand lbeta.Also, +for instance, dnorm and dbeta has the log parameter. Its classical use case is the (nu- +merical) maximum likelihood estimation, which involves the sums of the logarithms +of densities. +2.4 +Arithmetic Operations +2.4.1 +Vectorised Arithmetic Operators +R features the following arithmetic operators: +• `+` (addition) and `-` (subtraction), +• `*` (multiplication) and `/` (division), +• `%/%` (integer division) and `%%` (modulo, division remainder), +• `^` (exponentiation; synonym: `**`). +They are all vectorised: they take two vectors on input and yield another vector in result. +4 https://www.gnu.org/software/gsl/doc/html/ + +2 NUMERIC VECTORS +31 +c(1, 2, 3) * c(10, 100, 1000) +## [1] +10 +200 3000 +We note that the multiplication was performed in an elementwise fashion: the 1st ele- +ment in the left vector was multiplied by the corresponding element in the right vector +and the result has been stored in the 1st element of the output, then the 2nd element +in the left… all right, we get the point. +Other operators are vectorised in the same manner: +0:10 + seq(0, 1, 0.1) +## +[1] +0.0 +1.1 +2.2 +3.3 +4.4 +5.5 +6.6 +7.7 +8.8 +9.9 11.0 +0:7 / rep(3, length.out=8) +# division by 3 +## [1] 0.00000 0.33333 0.66667 1.00000 1.33333 1.66667 2.00000 2.33333 +0:7 %/% rep(3, length.out=8) # integer division +## [1] 0 0 0 1 1 1 2 2 +0:7 %% rep(3, length.out=8) +# division remainder +## [1] 0 1 2 0 1 2 0 1 +Note that operations involving missing values also yield NAs: +c(1, NA_real_, 3, NA_real_) + c(NA_real_, 2, 3, NA_real_) +## [1] NA NA +6 NA +2.4.2 +Recycling Rule +Some of the above statements can be written more concisely. When the operands are +of different lengths, the shorter one is recycled (think: rep(y, length.out=length(x))) +as many times as necessary. +0:7 / 3 +## [1] 0.00000 0.33333 0.66667 1.00000 1.33333 1.66667 2.00000 2.33333 +1:10 * c(-1, 1) +## +[1] -1 +2 -3 +4 -5 +6 -7 +8 -9 10 +2 ^ (0:10) +## +[1] +1 +2 +4 +8 +16 +32 +64 +128 +256 +512 1024 +We call this the recycling rule. +Ifanoperandcannotberecycledinitsentirety,awarning5 isgenerated,buttheoutput +is still available. +5 A few built-in functions do not warn at all when incomplete recycling is performed (e.g., paste) or can +even give an error (e.g., as.data.frame.list). Consider this inconsistency an annoying bug and hope it will +be fixed in the next decade or so. + +32 +I DEEP +c(1, 10, 100) * 1:8 +## Warning in c(1, 10, 100) * 1:8: longer object length is not a multiple of +## shorter object length +## [1] +1 +20 300 +4 +50 600 +7 +80 +Note Some functions are also deeply vectorised, i.e., with respect to multiple argu- +ments. For example, +runif(3, c(10, 20, 30), c(11, 22, 33)) +## [1] 10.288 21.577 31.227 +generates three random numbers uniformly distributed over the intervals (10, 11), +(20, 22), and (30, 33), respectively. +Also, pmin and pmax return the parallel minimum and maximum of the corresponding +elements of the input vectors: +pmin(c(1, 2, 3, 4), c(4, 2, 3, 1)) +## [1] 1 2 3 1 +pmin(3, 1:5) +## [1] 1 2 3 3 3 +pmax(0, pmin(1, c(0.25, -2, 5, -0.5, 0, 1.3, 0.99))) +# clipping to [0, 1] +## [1] 0.25 0.00 1.00 0.00 0.00 1.00 0.99 +Note Vectorisationand the recyclingrule areperhapsmost useful whenapplying bin- +aryoperatorsonsequencesofidenticallengthsorwhenperformingvector-scalar(i.e., +a sequence vs a single value) operations. However, there is much more: schemes like +“every k-th element” appear in Taylor series expansions (multiply by c(-1, 1)), k-fold +cross validation, etc.; see also Section 11.3.4 for use cases in matrix/tensor processing. +2.4.3 +Operator Precedence +Apart from the seven binary arithmetic operators, other noteworthy, already men- +tioned ones include: the unary `-` (change of sign), `:` (sequence generation), and +`<-` (assignment). +Expressions involving multiple operations need a set of rules governing the order +of computations (unless we enforce it using round brackets). We have said that +“-1:10” means “(-1):10” rather than “-(1:10)”. But what about, say, “1+1+1+1+1*0” or +“3*2^0:5+10”? +Let us list the aforementioned operators in their order of precedence, from the least +to the most binding (see also help("Syntax")): + +2 NUMERIC VECTORS +33 +1. `<-` (right-to-left), +2. `+` and `-`, +3. `*` and `/`, +4. `%%` and `%/%`, +5. `:`, +6. `+` and `-` (unary), +7. `^` (right-to-left). +Hence, “-2^2/3+3*4” means “((-(2^2))/3)+(3*4)” and not, for example, -((2^(2/ +(3+3)))*4). +Note that `+` and `-`, `*` and `/`, as well as `%%` and `%/%` have the same priority. +Expressions involving a series of operations in the same group, are evaluated left-to- +right, with the exception of `^` and `<-`, which are performed from right to left. +Therefore: +• “2*3/4*5” is equivalent to “((2*3)/4)*5”, +• “2^3^4” is the same as “2^(3^4)” (which, mathematically, we would write as 234 = +281), +• “x <- y <- 4*3%%8/2” binds both y and x with 6 and not x with the previous value +of y. +And let us remember: when in doubt, we can always bracket a subexpression to make +sure it is executed in the intended order (which can also increase readability of the +code). +2.4.4 +Accumulating +The `+` and `*` operators as well as the pmin and pmax functions implement element- +wise operations that are applied on the corresponding elements taken from two given +vectors: +⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜ +⎝ +𝑥1 +𝑥2 +𝑥3 +⋮ +𝑥𝑛 +⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟ +⎠ ++ +⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜ +⎝ +𝑦1 +𝑦2 +𝑦3 +⋮ +𝑦𝑛 +⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟ +⎠ += +⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜ +⎝ +𝑥1 + 𝑦1 +𝑥2 + 𝑦2 +𝑥3 + 𝑦3 +⋮ +𝑥𝑛 + 𝑦𝑛 +⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟ +⎠ +. +However, we can also scan through all the values in a single vector and combine the +successive elements that we inspect using the corresponding operation: +• cumsum(x) gives the cumulative sum of the elements in a vector, +• cumprod(x) computes the cumulative product, +• cummin(x) yields the cumulative minimum, + +34 +I DEEP +• cummax(x) generates the cumulative maximum. +The i-th element in the output vector will consist of the sum/product/min/max of the +first i inputs: +cumsum +⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜ +⎝ +𝑥1 +𝑥2 +𝑥3 +⋮ +𝑥𝑛 +⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟ +⎠ += +⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜ +⎝ +𝑥1 +𝑥1 + 𝑥2 +𝑥1 + 𝑥2 + 𝑥3 +⋮ +⋱ +𝑥1 + 𝑥2 + 𝑥3 + ⋯ + 𝑥𝑛 +⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟ +⎠ +. +For example: +cumsum(1:8) +## [1] +1 +3 +6 10 15 21 28 36 +cumprod(1:8) +## [1] +1 +2 +6 +24 +120 +720 +5040 40320 +cummin(c(3, 2, 4, 5, 1, 6, 0)) +## [1] 3 2 2 2 1 1 0 +cummax(c(3, 2, 4, 5, 1, 6, 0)) +## [1] 3 3 4 5 5 6 6 +If we are interested only in the last cumulant, summarising all the inputs, we have the +following functions at our disposal: +• sum(x) computes the sum of elements in a vector, ∑𝑛 +𝑖=1 𝑥𝑖 = 𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛, +• prod(x) outputs the product of all elements, ∏𝑛 +𝑖=1 𝑥𝑖 = 𝑥1𝑥2 ⋯ 𝑥𝑛, +• min(x) computes the minimum, +• max(x) reckons the greatest value. +For example: +sum(1:8) +## [1] 36 +prod(1:8) +## [1] 40320 +min(c(3, 2, 4, 5, 1, 6, 0)) +## [1] 0 +max(c(3, 2, 4, 5, 1, 6, 0)) +## [1] 6 +Note In Chapter 7, we will discuss the Reduce function, which generalises the above +by allowing any binary operation to be propagated over a given vector. +Example 2.7 diff can be considered an inverse to cumsum: it computes the iterative difference. + +2 NUMERIC VECTORS +35 +Namely, it subtracts the first two elements, then the 2nd from the 3rd one, the 3rd from the 4th, +and so on. In other words, diff(x) gives 𝒚 such that 𝑦𝑖 = 𝑥𝑖+1 − 𝑥𝑖. +x <- c(-2, 3, 6, 2, 15) +diff(x) +## [1] +5 +3 -4 13 +cumsum(diff(x)) +## [1] +5 +8 +4 17 +cumsum(c(-2, diff(x))) +# recreates x +## [1] -2 +3 +6 +2 15 +Thanks to diff, we can compute the daily changes to the EUR/AUD forex rates; see Figure 2.4. +aud <- scan(paste0("https://github.com/gagolews/teaching-data/raw/", +"master/marek/euraud-20200101-20200630.csv"), comment.char="#") +aud_all <- na.omit(aud) +# remove all missing values +plot(diff(aud_all), type="s", ylab="Daily change [EUR/AUD]") +abline(h=0, lty="dotted") +# draw a horizontal line at y=0 +0 +20 +40 +60 +80 +100 +120 +-0.04 +-0.02 +0.00 +0.02 +0.04 +Index +Daily change [EUR/AUD] +Figure 2.4: Iterative differences of the exchange rates (non-missing values only) +2.4.5 +Aggregating +The above functions form the basis for some popular summary statistics6 (sample ag- +gregates), such as: +• mean(x) gives the arithmetic mean, sum(x)/length(x), +6 Actually, var and median, amongst others, are defined by the stats package, but this one is automatic- +ally loaded by default, so let us not make a fuss about it now. + +36 +I DEEP +• var(x) yields the (unbiased) sample variance, sum((x-mean(x))^2)/(length(x)-1), +• sd(x) is the standard deviation, sqrt(var(x)), +• median(x) computes the sample median, i.e., the middle value in the sorted ver- +sion of x. +For instance7: +x <- runif(1000) +c(min(x), mean(x), median(x), max(x), sd(x)) +## [1] 0.00046535 0.49727780 0.48995025 0.99940453 0.28748391 +Exercise 2.8 Let 𝒙 be any vector of length 𝑛 with positive elements. Compute its geometric and +harmonic mean, which are given by, respectively, +𝑛 +√√√ +⎷ +𝑛 +∏ +𝑖=1 +𝑥𝑖 = 𝑒 +1 +𝑛 ∑𝑛 +𝑖=1 log 𝑥𝑖 +and +𝑛 +∑𝑛 +𝑖=1 +1 +𝑥𝑖 +. +Whensolvingexerciseslikethisone,itdoesnotreallymatterwhatdatayouapplythesefunctions +on (see, however, Section 9.3.4 for discussion). We are being abstract in the sense that the 𝒙 vec- +tor can be anything: from the one that features very accurate financial predictions that will help +minimiseinequityandmakethisworldlessmiserable,throughthedatayouhavebeencollecting +for the last the years in relation to your definitely-super-important PhD research, whatever your +company asked you to crunch today, to something related to your hobby project that you enjoy +doing after hours. Therefore, just test the above on something like “x <- runif(10)”, and move +on. +All the aforementioned functions return a missing value if any of the input elements +is unavailable. Luckily, they are equipped with the na.rm parameter on behalf of which +we can request the removal of NAs. +aud <- scan(paste0("https://github.com/gagolews/teaching-data/raw/", +"master/marek/euraud-20200101-20200630.csv"), comment.char="#") +c(min(aud), mean(aud), max(aud)) +## [1] NA NA NA +c(min(aud, na.rm=TRUE), mean(aud, na.rm=TRUE), max(aud, na.rm=TRUE)) +## [1] 1.6006 1.6775 1.8635 +Note In the documentation, we read that the usage of some of the aforementioned +functions is like sum(..., na.rm=FALSE). prod, min, and max are defined similarly. They +acceptanynumberofinputvectors,eachofthemcanbeofarbitrarylength.Therefore, +min(1, 2, 3), min(c(1,2,3)) as well as min(c(1,2),3) all return the same result. +However, we can also read that we have mean(x, trim=0, na.rm=FALSE, ...). This +7 Notethat min, median,and maxisaspecialcaseof quantile,whichwewilldiscussmuchfurther,namely, +in Section 4.4.3. This is because it returns a named vector. + +2 NUMERIC VECTORS +37 +time, only one vector can be aggregated and any further arguments (except trim and +na.rm) are ignored. +The extra flexibility (which we do not have to rely upon, ever) of the former group is +due their being associative operations: it holds, e.g., (2+3)+4 = 2+(3+4). Hence, +the operations can be performed in any order, in any groups. +Also note that they are more primitive operations: it is mean that is based on sum, not +vice versa. +2.5 +Exercises +Exercise 2.9 Answer the following questions: +• What is the meaning of the dot-dot-dot parameter in the definition of the c function? +• We say that the round function is vectorised: what does that mean? +• What do we mean by saying that multiplication operates element-by-element? +• How does the recycling rule work when applying `+`? +• How to (and why) set the seed of the pseudorandom number generator? +• What is the difference between NA_real_ and NaN? +• How are default arguments specified in the manual of, e.g., the round function? +• Is a call to rep(times=4, x=1:5)” equivalent to rep(4, 1:5)? +• List a few ways to generate a sequence like (-1, -0.75, -0.5, …, 0.75, 1). +• Is“-3:5”thesameas "-(3:5)"?Whatabouttheprecedenceofoperatorsinexpressionssuch +as “2^3/4*5^6”, “5*6+4/17%%8”, and “1+-2^3:4”? +• If x is a numeric vector of length 𝑛 (for some 𝑛 ≥ 0), how many values will sample(x) +output? +• Does scan support reading directly from compressed archives, e.g., .csv.gz files? +When in doubt, refer back to the material discussed in this chapter and/or the R manual. +Exercise 2.10 The following code generates an example graph of arcsine and arccosine, whose +preparation – thanks to vectorisation – is quite straightforward. +x <- seq(-1, 1, length.out=11) +# increase length.out for a smoother curve +plot(x, asin(x), +# asin() computed for 11 points +type="l", +# lines +ylim=c(-pi/2, pi), +# y axis limits like c(y_min, y_max) +ylab="asin(x), acos(x)") +# y axis label +(continues on next page) + +38 +I DEEP +(continued from previous page) +lines(x, acos(x), col="red", lty="dashed") +# adds to the current plot +legend("topright", c("asin(x)", "acos(x)"), +lty=c("solid", "dashed"), col=c("black", "red"), bg="white") +Inspired by the above, plot the following functions: | sin 𝑥2|, |sin |𝑥||, √⌊𝑥⌋, and 1/(1 + 𝑒−𝑥). +Recall that the documentation of plot can be viewed by calling help("plot.default"). +Exercise 2.11 It can be shown that: +4 +𝑛 +∑ +𝑖=1 +(−1)𝑖+1 +2𝑖 − 1 += 4 (1 +1 − 1 +3 + 1 +5 − 1 +7 + ⋯) +slowly converges to 𝜋 as 𝑛 approaches ∞. Compute the above for 𝑛 = 1,000,000 and 𝑛 = +1,000,000,000 using the vectorised functions and operators discussed in this chapter, making +use of the recycling rule as much as possible. +Exercise 2.12 Let x and y be two vectors of identical lengths 𝑛, say: +x <- rnorm(100) +y <- 2*x+10+rnorm(100, 0, 0.5) +Compute the Pearson linear correlation coefficient given by: +𝑟 = +∑𝑛 +𝑖=1 (𝑥𝑖 − 1 +𝑛 ∑𝑛 +𝑗=1 𝑥𝑗) (𝑦𝑖 − 1 +𝑛 ∑𝑛 +𝑗=1 𝑦𝑗) +√∑𝑛 +𝑖=1 (𝑥𝑖 − 1 +𝑛 ∑𝑛 +𝑗=1 𝑥𝑗) +2 √∑𝑛 +𝑖=1 (𝑦𝑖 − 1 +𝑛 ∑𝑛 +𝑗=1 𝑦𝑗) +2 . +To make sure you have come up with a correct implementation, compare your result to a call to +the built-in cor(x, y). +Exercise 2.13 (*) Look up on the internet an R package that features functions to compute the +5-day moving (rolling) average and median of a given vector. Apply them on the EUR/AUD cur- +rency exchange data and plot thus obtained smoothened versions of the time series. +Exercise 2.14 (**)Computethe𝑘-movingaverageusingacalltoconvolve(..., type="filter"). +In the next chapter we will study operations that involve logical values. + +3 +Logical Vectors +There are three logical constants in R. Wait… how many? +3.1 +Creating Logical Vectors +R defines three logical constants: TRUE, FALSE, and NA – meant to represent “yes”, “no”, +and “???”, respectively. Each of them, when instantiated, is an atomic vector of length +one. +Some of the functions we introduced in the previous chapter can be used to generate +logical vectors as well: +c(TRUE, FALSE, FALSE, NA, TRUE, FALSE) +## [1] +TRUE FALSE FALSE +NA +TRUE FALSE +rep(c(TRUE, FALSE, NA), each=2) +## [1] +TRUE +TRUE FALSE FALSE +NA +NA +sample(c(TRUE, FALSE), 10, replace=TRUE, prob=c(0.8, 0.2)) +## +[1] +TRUE +TRUE +TRUE FALSE FALSE +TRUE +TRUE FALSE +TRUE +TRUE +Note “T” is a synonym for TRUE and “F” stands for FALSE. However, these are not re- +servedkeywordsandcanbere-assignedanyothervalues.Therefore,weadviseagainst +relying on them and hence we will never use them throughout the course of this +course. +Also note that the logical missing value is spelled simply as “NA” and not “NA_logical_”. +The fact that both the logical “NA” and the numeric "NA_real_" are, for the sake of +our mental well-being, both printed as "NA" on the R console, does not mean they are +identical; see Section 4.1 for discussion. + +40 +I DEEP +3.2 +Comparing Elements +3.2.1 +Vectorised Comparison Operators +Logical vectors frequently come into being as results of various testing activities. +In particular, the binary operators: +• `<` (less than), +• `<=` (less than or equal), +• `>` (greater than), +• `>=` (greater than or equal) +• `==` (equal), +• `!=` (not equal), +comparethecorrespondingelementsoftwonumericvectorsandoutputalogicalvector. +1 < 3 +## [1] TRUE +c(1, 2, 3, 4) == c(2, 2, 3, 8) +## [1] FALSE +TRUE +TRUE FALSE +1:10 <= 10:1 +## +[1] +TRUE +TRUE +TRUE +TRUE +TRUE FALSE FALSE FALSE FALSE FALSE +Thus, they operate in an elementwise manner. Moreover, the recycling rule is applied +if necessary: +3 < 1:5 +# c(3, 3, 3, 3, 3) < c(1, 2, 3, 4, 5) +## [1] FALSE FALSE FALSE +TRUE +TRUE +c(1, 4) == 1:4 +# c(1, 4, 1, 4) == c(1, 2, 3, 4) +## [1] +TRUE FALSE FALSE +TRUE +Therefore, we can say that they are vectorised in the same manner as the arithmetic +operators `+`, `*`, etc.; compare Section 2.4.1. +3.2.2 +Testing for NA, NaN, and Inf +Comparisons against missing values and not-numbers yield NAs. Therefore, instead +of the incorrect x == NA_reals_ or x == NaN, testing for missingness should rather be +performed via a call to the vectorised is.na function. +is.na(c(NA_real_, Inf, -Inf, NaN, -1, 0, 1)) +## [1] +TRUE FALSE FALSE +TRUE FALSE FALSE FALSE +is.nan(c(NA_real_, Inf, -Inf, NaN, -1, 0, 1)) +(continues on next page) + +3 LOGICAL VECTORS +41 +(continued from previous page) +## [1] FALSE FALSE FALSE +TRUE FALSE FALSE FALSE +is.na(c(TRUE, FALSE, NA, TRUE)) +# works for logical vectors too +## [1] FALSE FALSE +TRUE FALSE +Moreover, is.finite is noteworthy, because it returns FALSE on Infs, NA_real_s and +NaNs. +is.finite(c(NA_real_, Inf, -Inf, NaN, -1, 0, 1)) +## [1] FALSE FALSE FALSE FALSE +TRUE +TRUE +TRUE +See also the more specific is.nan and is.infinite. +3.2.3 +Dealing with Floating Point Round-Off Errors (*) +In mathematics, real numbers are merely an idealisation. In practice, however, it is +impossibletostorethemwithinfiniteprecision(think𝜋 = 3.1415926535897932384626433...): +computer memory is limited and our time is precious. +Therefore, a widely agreed upon consensus had to be reached. In R, we rely on the so- +called double-precision floating point format. Floating point means that the numbers can +be both small (close to zero) and large: ±2.23 × 10−308 and ±1.79 × 10308 are both +acceptable. +Note +2.23e-308 == 0.00000000000000000000000000000000000000000000000000 +00000000000000000000000000000000000000000000000000 +00000000000000000000000000000000000000000000000000 +00000000000000000000000000000000000000000000000000 +00000000000000000000000000000000000000000000000000 +000000000000000000000000000000000000000000000000000000000223 +1.79e308 == 17900000000000000000000000000000000000000000000000000000000 +00000000000000000000000000000000000000000000000000 +00000000000000000000000000000000000000000000000000 +00000000000000000000000000000000000000000000000000 +00000000000000000000000000000000000000000000000000 +00000000000000000000000000000000000000000000000000 +These two are quite distant from each other. +Everynumericvaluetakes8bytes(orequivalently64bits)ofmemory.Weare,however, +able to store only about 15-17 decimal digits: + +42 +I DEEP +print(0.12345678901234567890123456789012345678901234, digits=22) +# 22 is max +## [1] 0.1234567890123456773699 +whichlimitstheprecisionofourcomputations.Theabout partis–unfortunately–due +to the numbers’ being written in the computer-friendly binary, not human-aligned +decimal, base. This can lead to some unexpected outcomes. +In particular: +• 0.1 cannot be represented exactly, because it cannot be written as a finite series of +reciprocals of powers of 2 (it holds 0.1 = 2−4 + 2−5 + 2−8 + 2−9 + …). This leads +to surprising results such as: +0.1 + 0.1 + 0.1 == 0.3 +## [1] FALSE +Despite the fact that what follows does not show anything suspicious: +c(0.1, 0.1 + 0.1 + 0.1, 0.3) +## [1] 0.1 0.3 0.3 +Printing involves rounding, hence, in the above context, is misleading. Above, we +have something more like: +print(c(0.1, 0.1 + 0.1 + 0.1, 0.3), digits=22) +## [1] 0.1000000000000000055511 0.3000000000000000444089 +## [3] 0.2999999999999999888978 +• All integers between −253 and 253 all stored exactly – this is good news. However, +the next integer is beyond the representable range: +2^53 + 1 == 2^53 +## [1] TRUE +• The above suggests that, more generally, the order of operations may matter, in +particular, the associativity property may be violated when dealing with numbers +of different orders of magnitude: +2^53 + 2^-53 - 2^53 - 2^-53 +# should be == 0.0 +## [1] -1.1102e-16 +• Some numbers may just be just too large, too small, or too close to zero to be rep- +resented exactly: +c(sum(2^((1023-52):1023)), sum(2^((1023-53):1023))) +## [1] 1.7977e+308 +Inf +c(2^(-1022-52), 2^(-1022-53)) +## [1] 4.9407e-324 +0.0000e+00 + +3 LOGICAL VECTORS +43 +Important The double-precision floating point format (IEEE 754) is not specific to R: +it is used by most other computing environments, including Python and C++. +For discussion, see [27, 30, 33] ([26] can be of particular interest to the general statist- +ical/data analysis audience). +Can we do anything about these issues? +First, when dealing with integers of reasonable order of magnitude (a frequent case +wherewearedealingvariousresourceorcaseIDsinourdatasets),restassuredthatwe +aresafe:theircomparison,addition,subtraction,andmultiplicationisalwaysprecise. +In all other cases (including applying other operations on integers, e.g., division or +sqrt), we need to be very careful with comparisons, especially involving testing for +equality, `==`. +The sole fact that sin 𝜋 = 0, mathematically speaking, does not mean that we should +expect that: +sin(pi) == 0 +## [1] FALSE +Instead, they are so close to each other that we can treat the difference between them as +negligible. Thus, in practice, instead of testing if 𝑥 = 𝑦, we will be considering: +• |𝑥 − 𝑦| (absolute error) or +• +|𝑥−𝑦| +|𝑦| +(relative error; which takes the order of magnitude of the numbers into ac- +count but obviously cannot be applied if 𝑦 is very close of 0), +and determining if these are less than some assumed error margin, 𝜀 > 0, say, 10−8 +or 2−26. +For example: +abs(sin(pi) - 0) < 2^-26 +## [1] TRUE +Note Note that rounding can sometimes have a similar effect as testing for almost- +equality in terms of the absolute error. +round(sin(pi), 8) == 0 +## [1] TRUE +Important Our recommendations are valid for the most popular applications of R, + +44 +I DEEP +i.e., statistical and, more generally, scientific computing1. The datasets we handle on +a daily basis do not represent accurate measurements themselves, bah, the World it- +self is far from ideal, therefore we do not have to lose sleep over our not being able to +precisely pinpoint the exact solution. +3.3 +Logical Operations +3.3.1 +Vectorised Logical Operators +The comparison operators such as `==` and `>` accept only two arguments. Their +chaining is forbidden; a test which we would mathematically write as 0 ≤ 𝑥 ≤ 1 +(or 𝑥 ∈ [0, 1]) cannot be expressed as “0<=x<=1” in R. +Therefore, we need a way to combine two logical conditions so as to be able to state +that “𝑥 ≥ 0 and, at the same time, 𝑥 ≤ 1”. +In such situations, the following logical operators and functions come in handy: +• `!` (not, negation; unary), +• `&` (and, conjunction; are both predicates true?), +• `|` (or, alternation; is at least one true?), +• xor (exclusive-or, exclusive disjunction, either-or; is one and only one of the pre- +dicates true?). +They again act elementwisely and implement the recycling rule if necessary (and ap- +plicable). +x <- c(-10, -1, -0.25, 0, 0.5, 1, 5, 100) +(x >= 0) & (x <= 1) +## [1] FALSE FALSE FALSE +TRUE +TRUE +TRUE FALSE FALSE +(x < 0) | (x > 1) +## [1] +TRUE +TRUE +TRUE FALSE FALSE FALSE +TRUE +TRUE +!((x < 0) | (x > 1)) +## [1] FALSE FALSE FALSE +TRUE +TRUE +TRUE FALSE FALSE +xor(x >= -1, x <= 1) +## [1] +TRUE FALSE FALSE FALSE FALSE FALSE +TRUE +TRUE +1 However,infinancialapplications,weshouldratherrelyonbase-10numbers(comparethe0.1problem +above). Also note that there exist some libraries implementing higher precision floating-point numbers or +even interval arithmetic that keeps track of error propagation operation chains. + +3 LOGICAL VECTORS +45 +Important The vectorised `&` and `|` operators should not be confused with their +scalar, short-circuit counterparts, `&&` and `||`, which we discuss in Section 8.1.4. +3.3.2 +Operator Precedence Revisited +The operators introduced in this chapter have lower precedence than the arithmetic +ones. In particular, the binary `+` and `-`. Calling help("Syntax") reveals that we can +extend our listing from Section 2.4.3 as follows: +1. `<-` (right-to-left; least binding), +2. `|`, +3. `&`, +4. `!` (unary), +5. `<`, `>`, `<=`, `>=`, `==`, and `!=`, +6. `+` and `-`, +7. `*` and `/`, +8. … +3.3.3 +Dealing with Missingness +Operations involving missing values follow the principles of the Łukasiewicz’s three- +valued logic, which is based on common sense. For instance, “NA | TRUE” is TRUE, be- +cause or needs at least one argument to be TRUE to generate such a result. On the other +hand, “NA | FALSE” is NA, because the result would be different depending on what we +substituted NA for. +Let us take a moment to contemplate the operations’ truth tables for all the possible +combinations of inputs: +u <- c(TRUE, FALSE, NA, +TRUE, +FALSE, NA, +TRUE, FALSE, NA) +v <- c(TRUE, TRUE, TRUE, FALSE, FALSE, FALSE, NA, +NA, +NA) +!u +## [1] FALSE +TRUE +NA FALSE +TRUE +NA FALSE +TRUE +NA +u & v +## [1] +TRUE FALSE +NA FALSE FALSE FALSE +NA FALSE +NA +u | v +## [1] +TRUE +TRUE +TRUE +TRUE FALSE +NA +TRUE +NA +NA +xor(u, v) +## [1] FALSE +TRUE +NA +TRUE FALSE +NA +NA +NA +NA + +46 +I DEEP +3.3.4 +Aggregating with all, any, and sum +Just like in the case of numeric vectors, we can summarise the contents of logical se- +quences. +all tests whether every element in a logical vector is equal to TRUE and any determines +if there exists an element that is TRUE. +x <- runif(10000) +all(x <= 0.2) +# are all values in x <= 0.2? +## [1] FALSE +any(x <= 0.2) +# is there at least one element in x that is <= 0.2? +## [1] TRUE +Note The all function will frequently be used in conjunction with “==”. This is because +the latter, as we have said above, is itself vectorised: it does not test whether a vector +as a whole is equal to another one. +z <- c(1, 2, 3) +z == 1:3 +# elementwise equal +## [1] TRUE TRUE TRUE +all(z == 1:3) +# elementwise equal summarised +## [1] TRUE +However, let us keep in mind the warning about the testing for exact equality of +floating-point numbers stated in Section 3.2.3. Sometimes, considering absolute or +relative errors might be more appropriate. +z <- sin((0:10)*pi) +# sin(0), sin(pi), sin(2*pi), ..., sin(10*pi) +all(z == 0.0) +# danger zone! please don't... +## [1] FALSE +all(abs(z - 0.0) < 1e-9) +# are the absolute errors negligible? +## [1] TRUE +We can also call sum on a logical vector. Taken into account that it interprets TRUE as +numeric 1 and FALSE as 0 (more on this in Section 4.1), it will give us the number of +elements equal to TRUE. +sum(x <= 0.2) +# how many elements in x are <= 0.2? +## [1] 1998 +Also, by computing sum(x)/length(x), we can obtain the proportion (fraction) of val- +ues equal to TRUE in x. Equivalently: +mean(x <= 0.2) +# proportion of elements <= 0.2 +## [1] 0.1998 + +3 LOGICAL VECTORS +47 +Naturally, we expect mean(runif(n) <= 0.2)” to be equal to 0.2 (20%), but with ran- +domness we can never be sure. +3.3.5 +Simplifying Predicates +Eachaspiringprogrammerneeds to becomefluentwiththerulesgoverningthetrans- +formations of logical conditions, for example, that the negation of “(x >= 0) & (x < +1)” is equivalent to “(x < 0) | (x >= 1)”. +Each such rule is called a tautology. Here are some of them: +• !(!p) is equivalent to p (double negation), +• !(p & q) holds if and only if !p | !q (De Morgan’s law), +• !(p | q) is !p & !q (another De Morgan’s law), +• all(p) is equivalent to !any(!p). +Various combinations thereof are of course possible. Some further simplifications are +enabled by other properties of the binary operations: +• commutativity (symmetry), e.g., 𝑎 + 𝑏 = 𝑏 + 𝑎, 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎, +• associativity, e.g., (𝑎 + 𝑏) + 𝑐 += +𝑎 + (𝑏 + 𝑐), max(max(𝑎, 𝑏), 𝑐) += +max(𝑎, max(𝑏, 𝑐)), +• distributivity, e.g., 𝑎 ∗ 𝑏 + 𝑎 ∗ 𝑐 = 𝑎 ∗ (𝑏 + 𝑐), min(max(𝑎, 𝑏), max(𝑎, 𝑐)) = +max(𝑎, min(𝑏, 𝑐)), +and relations, including: +• transitivity, e.g., if 𝑎 ≤ 𝑏 and 𝑏 ≤ 𝑐 then surely 𝑎 ≤ 𝑐. +Exercise 3.1 Assuming that a, b, and c are numeric vectors, simplify the following expressions: +• !(b>a & b=b & b>=c & a>=c), +• a>b & ad, +• a>b | a<=b, +• a<=b & a>c | a>b & a<=c, +• a<=b & (a>c | a>b) & a<=c, +• !all(a > b & b < c). + +48 +I DEEP +3.4 +Choosing Elements with ifelse +The ifelsefunctionisavectorisedversionofthescalar if…elseconditionalstatement +which we will do without for as long as until Chapter 8. +It allows us to select an element from either one or another vector based on some lo- +gical condition. +A call to ifelse(l, t, f), where l is a logical vector, returns a vector y such that: +𝑦𝑖 = { 𝑡𝑖 +if 𝑙𝑖 is TRUE , +𝑓𝑖 +if 𝑙𝑖 is FALSE . +In other words, the 𝑖-th element of the result vector is equal to 𝑡𝑖 if 𝑙𝑖 is TRUE and to 𝑓𝑖 +otherwise. +For example: +(z <- rnorm(6)) +# example vector +## [1] -0.560476 -0.230177 +1.558708 +0.070508 +0.129288 +1.715065 +ifelse(z >= 0, z, -z) +# like abs(z) +## [1] 0.560476 0.230177 1.558708 0.070508 0.129288 1.715065 +or: +(x <- rnorm(6)) +# example vector +## [1] +0.46092 -1.26506 -0.68685 -0.44566 +1.22408 +0.35981 +(y <- rnorm(6)) +# example vector +## [1] +0.40077 +0.11068 -0.55584 +1.78691 +0.49785 -1.96662 +ifelse(x >= y, x, y) +# like pmax(x, y) +## [1] +0.46092 +0.11068 -0.55584 +1.78691 +1.22408 +0.35981 +By now, we should not be surprised that the recycling rule is fired up if necessary: +ifelse(x > 0, x^2, 0) +# squares of positive xs and 0 otherwise +## [1] 0.21244 0.00000 0.00000 0.00000 1.49838 0.12947 +Note Keep in mind that all arguments are evaluated in their entirety before decid- +ing on which element should be selected. Therefore, the following call will generate a +warning: +ifelse(z >= 0, log(z), NA_real_) +## Warning in log(z): NaNs produced +## [1] +NA +NA +0.44386 -2.65202 -2.04571 +0.53945 +This is because with log(z), we are computing the logarithms of negative values any- +way. To fix this, we can write: + +3 LOGICAL VECTORS +49 +log(ifelse(z >= 0, z, NA_real_)) +## [1] +NA +NA +0.44386 -2.65202 -2.04571 +0.53945 +The calls to ifelse can naturally be nested in the case where we yearn for an if…else +if…else-type expression. +Example 3.2 A version of pmax(pmax(x, y), z) can be written as: +ifelse(x >= y, +ifelse(z >= x, z, x), +ifelse(z >= y, z, y) +) +## [1] 0.46092 0.11068 1.55871 1.78691 1.22408 1.71506 +However,determiningthe threeintermediatelogicalvectorsis notnecessary;wecansaveone call +to `>=` by introducing an auxiliary variable: +xy <- ifelse(x >= y, x, y) +ifelse(z >= xy, z, xy) +## [1] 0.46092 0.11068 1.55871 1.78691 1.22408 1.71506 +Exercise 3.3 Figure 3.1 depicts a realisation of the mixture 𝑍 = 0.2𝑋 + 0.8𝑌 of two normal +distributions 𝑋 ∼ N(−2, 0.5) and 𝑌 ∼ N(3, 1). +n <- 100000 +z <- ifelse(runif(n) <= 0.2, rnorm(n, -2, 0.5), rnorm(n, 3, 1)) +hist(z, breaks=101, probability=TRUE, main="", col="white") +In other words, we generated a variate from the normal distribution that has expected value of -2 +with probability 20% and from the one with expectation of 3 otherwise. +Inspired by the above, generate the following Gaussian mixtures: +• +2 +3𝑋 + 1 +3𝑌, where 𝑋 ∼ N(100, 16) and 𝑌 ∼ N(116, 8), +• 0.3𝑋 + 0.4𝑌 + 0.3𝑍, where 𝑋 ∼ N(−10, 2), 𝑌 ∼ N(0, 2), and 𝑍 ∼ N(10, 2). +(*) On a side note, knowing that if 𝑋 follows N(0, 1), then the scaled-shifted 𝜎𝑋 + 𝜇 is distrib- +uted N(𝜇, 𝜎), the above can be equivalently written as: +w <- (runif(n) <= 0.2) +z <- rnorm(n, 0, 1)*ifelse(w, 0.5, 1) + ifelse(w, -2, 3) + +50 +I DEEP +z +Density +-4 +-2 +0 +2 +4 +6 +8 +0.00 +0.10 +0.20 +0.30 +Figure 3.1: A mixture of two Gaussians generated with ifelse +3.5 +Exercises +Exercise 3.4 Answer the following questions: +• Whythestatement“Earthisflatorthesmallpoxvaccineisproveneffective”isobviouslytrue? +• What is the difference between NA and NA_real_? +• Why is “FALSE & NA” equal to FALSE, but “TRUE & NA” is NA? +• Whyhas “ifelse(x>=0, sqrt(x), NA_real_)”atendencytogeneratewarningsand how +to rewrite it so as to prevent that from happening? +• What is the interpretation of “mean(x >= 0 & x <= 1)”? +• For some integer 𝑥 and 𝑦, how to verify whether 0 < 𝑥 < 100, 0 < 𝑦 < 100, and 𝑥 < 𝑦, +all at the same time? +• Mathematically, for all real 𝑥, 𝑦 > 0, it holds log 𝑥𝑦 = log 𝑥 + log 𝑦. Why then +“all(log(x*y) == log(x)+log(y))” can sometimes return FALSE? How to fix this? +• Is “x/y/z” always equal to “x/(y/z)”? How to fix this? +• What is the purpose of very specific functions such as log1p and expm1 (see their help page) +and many other ones listed in, e.g., the GNU GSL library [23]? Is our referring to them a +violation of the beloved “let us be minimalistic” approach? +• If we know that 𝑥 may be subject to error, how to test whether 𝑥 > 0 in a robust manner? +• Is “y<-5” the same as “y <- 5” or rather “y < -5”? + +3 LOGICAL VECTORS +51 +Exercise 3.5 Compute the cross-entropy loss between a numeric vector 𝒑 with values in the in- +terval (0, 1) and a logical vector 𝒚, both of length 𝑛 (you can generate them randomly or manu- +ally, it does not matter, it is just an exercise): +ℒ(𝒑, 𝒚) = 1 +𝑛 +𝑛 +∑ +𝑖=1 +ℓ𝑖, +where +ℓ𝑖 = { − log 𝑝𝑖 +if 𝑦𝑖 is TRUE , +− log(1 − 𝑝𝑖) +if 𝑦𝑖 is FALSE . +Interpretation: in classification problems, 𝑦𝑖 ∈ {FALSE, TRUE} denotes the true class of the +𝑖-th object (say, whether the 𝑖-th hospital patient is symptomatic) and 𝑝𝑖 ∈ (0, 1) a machine +learningalgorithm’sconfidencethat𝑖belongstoclassTRUE(e.g.,howsureadecisiontreemodel +is that the corresponding person is unwell). Ideally, if 𝑦𝑖 is TRUE, 𝑝𝑖 should be close to 1 and to 0 +otherwise. The cross-entropy loss quantifies by how much a classifier differs from the omniscient +one. The use of the logarithm penalises strong beliefs in the wrong answer. +By the way! If you have solved any of the exercises encountered so far by referring to if +statements, for loops, vector indexing like x[...], or any external R package, please +go back and re-write your code. Let us keep it simple (effective, readable) by using the +base R’s vectorised operations that we have introduced. + + +4 +Lists and Attributes +After two brain-teasing chapters, it is time to cool it down a little. In this more tech- +nical part, we will introduce lists, which serve as universal containers for R objects +of any size and type. Moreover, we will also show that each R object can be equipped +with a number of optional attributes, thanks to which we will not only be able to label +elements in any vector, but also – later – introduce new complex data types such as +matrices and data frames. +4.1 +Type Hierarchy and Conversion +So far, we were dealing with three types of atomic vectors: +1. logical (Chapter 3), +2. numeric (Chapter 2), +3. character (which we have barely touched upon yet, but rest assured that they will +be covered in detail very soon; see Chapter 6). +To determine the type of an object programmatically, we can call the typeof function. +typeof(c(1, 2, 3)) +## [1] "double" +typeof(c(TRUE, FALSE, TRUE, NA)) +## [1] "logical" +typeof(c("spam", "spam", "bacon", "gluten-free spam")) +## [1] "character" +It turns out that we can easily convert between these types, either on our explicit de- +mand (typecasting), or on-the-fly (coercion, when we perform an operation that expects +something different from the kind of input it was fed with). +Note (*)Numericvectorsarereportedasbeingeitheroftype double(double-precision +floating-point numbers) or integer (32-bit; it is a subset of double); see Section 6.4.1. +In most practical cases, this is a technical detail which we can safely ignore; compare +also the mode function. + +54 +I DEEP +4.1.1 +Explicit Type Casting +We can use functions such as as.logical, as.numeric, and as.character to coerce (con- +vert) given objects to the corresponding types. +as.numeric(c(TRUE, FALSE, NA, TRUE, NA, FALSE)) +## [1] +1 +0 NA +1 NA +0 +as.logical(c(-2, -1, 0, 1, 2, 3, NA_real_, -Inf, NaN)) +## [1] +TRUE +TRUE FALSE +TRUE +TRUE +TRUE +NA +TRUE +NA +Important It is easily seen that the rules are: +• TRUE → 1, +• FALSE → 0, +• NA → NA_real_, +and: +• 0 → FALSE, +• NA_real_ and NaN → NA, +• anything else → TRUE. +The distinction between zero and non-zero is commonly applied in other program- +ming languages as well. +Moreover, in the case of the conversion involving character strings, we have: +as.character(c(TRUE, FALSE, NA, TRUE, NA, FALSE)) +## [1] "TRUE" +"FALSE" NA +"TRUE" +NA +"FALSE" +as.character(c(-2, -1, 0, 1, 2, 3, NA_real_, -Inf, NaN)) +## [1] "-2" +"-1" +"0" +"1" +"2" +"3" +NA +"-Inf" "NaN" +as.logical(c("TRUE", "True", "true", "T", +"FALSE", "False", "false", "F", +"anything other than these", NA_character_)) +## +[1] +TRUE +TRUE +TRUE +TRUE FALSE FALSE FALSE FALSE +NA +NA +as.numeric(c("0", "-1.23e4", "pi", "2+2", "NaN", "-Inf", NA_character_)) +## Warning: NAs introduced by coercion +## [1] +0 -12300 +NA +NA +NaN +-Inf +NA +4.1.2 +Implicit Conversion (Coercion) +Recall that we referred to the three vector types as atomic ones: they can only be used +to store elements of the same type. +If we make an attempt at composing an object of mixed types with c, the commontype + +4 LISTS AND ATTRIBUTES +55 +will be determined in such a way that storing the data is done without information +loss: +c(-1, FALSE, TRUE, 2, "three", NA) +## [1] "-1" +"FALSE" "TRUE" +"2" +"three" NA +c("zero", TRUE, NA) +## [1] "zero" "TRUE" NA +c(-1, FALSE, TRUE, 2, NA) +## [1] -1 +0 +1 +2 NA +Hence, we see that logical is the least, whereas character – the most general of the +three. +Note The logical NA is converted to NA_real_ and NA_character_ in the above examples. +Ruserstendtorelyonimplicittypeconversionwhentheywrite c(1, 2, NA, 4)instead +of the more explicit c(1, 2, NA_real_, 4). In most cases, this is fine. +However, occasionally, it will be wiser to be more unequivocal. For instance, +rep(NA_real_, 1e9) pre-allocates a long numeric vector, instead of a logical one. +Some functions that expect vectors of specific types can apply coercion by themselves +(or act as if they do so): +c(NA, FALSE, TRUE) + 10 # implicit conversion logical -> numeric +## [1] NA 10 11 +c(-1, 0, 1) & TRUE +# implicit conversion numeric -> logical +## [1] +TRUE FALSE +TRUE +sum(c(TRUE, TRUE, FALSE, TRUE, FALSE)) +# same as sum(as.numeric(...)) +## [1] 3 +cumsum(c(TRUE, TRUE, FALSE, TRUE, FALSE)) +## [1] 1 2 2 3 3 +cummin(c(TRUE, TRUE, FALSE, TRUE, FALSE)) +## [1] 1 1 0 0 0 +Exercise 4.1 Inoneofthepreviousexercises,wehavecomputedthecross-entropylossbetweena +logical vector 𝒚 ∈ {0, 1}𝑛 and a numeric vector 𝒑 ∈ (0, 1)𝑛. This measure can be equivalently +defined as: +ℒ(𝒑, 𝒚) = −1 +𝑛 +⎛⎜ +⎝ +𝑛 +∑ +𝑖=1 +𝑦𝑖 log(𝑝𝑖) + (1 − 𝑦𝑖) log(1 − 𝑝𝑖)⎞⎟ +⎠ +. +Implement the above formula (using vectorised operations, but not relying on ifelse this time) +and compute the cross-entropy loss between, say, “y <- sample(c(FALSE, TRUE), n)” and “p +<- runif(n)”forsomen.NotehowseamlesslywearetranslatingbetweenFALSE/TRUEsand0/1s +in the above equation (in particular, where we let 1 − 𝑦𝑖 mean the logical negation of 𝑦𝑖). + +56 +I DEEP +4.2 +Lists +Lists are generalised vectors. They can be comprised of R objects of any kind, also other +lists. This is why we classify them as recursive (and not atomic) objects. They are espe- +cially useful wherever there is a need to handle some multitude as a single entity. +4.2.1 +Creating Lists +The most straightforward way to create a list is by means of the list function: +list(1, 2, 3) +## [[1]] +## [1] 1 +## +## [[2]] +## [1] 2 +## +## [[3]] +## [1] 3 +Notice that the above is not the same as “c(1, 2, 3)”. We got a sequence that wraps +threenumericvectors,eachoflengthone.Also,howoverlytalkativeRiswhenprinting +out lists! +list(c(1, 2, 3), 4, c(TRUE, FALSE, FALSE, NA, TRUE), "and so forth") +## [[1]] +## [1] 1 2 3 +## +## [[2]] +## [1] 4 +## +## [[3]] +## [1] +TRUE FALSE FALSE +NA +TRUE +## +## [[4]] +## [1] "and so forth" +list(list(c(TRUE, FALSE, NA, TRUE), letters), runif(5)) +# a list of lists +## [[1]] +## [[1]][[1]] +## [1] +TRUE FALSE +NA +TRUE +## +## [[1]][[2]] +## +[1] "a" "b" "c" "d" "e" "f" "g" "h" "i" "j" "k" "l" "m" "n" "o" "p" "q" +## [18] "r" "s" "t" "u" "v" "w" "x" "y" "z" +(continues on next page) + +4 LISTS AND ATTRIBUTES +57 +(continued from previous page) +## +## +## [[2]] +## [1] 0.28758 0.78831 0.40898 0.88302 0.94047 +However, the str function can be used to print R objects in a more concise fashion: +str(list(list(c(TRUE, FALSE, NA, TRUE), letters), runif(5))) +## List of 2 +## +$ :List of 2 +## +..$ : logi [1:4] TRUE FALSE NA TRUE +## +..$ : chr [1:26] "a" "b" "c" "d" ... +## +$ : num [1:5] 0.288 0.788 0.409 0.883 0.94 +Note In Section 4.1, we said that the c function, when fed with arguments of mixed +types, tries to determine the common type that retains the sense of data. If a coercion +to an atomic vector is not possible, the result will be a list. +c(1, "two", sd) +# `sd` is an object of type `function` +## [[1]] +## [1] 1 +## +## [[2]] +## [1] "two" +## +## [[3]] +## function (x, na.rm = FALSE) +## sqrt(var(if (is.vector(x) || is.factor(x)) x else as.double(x), +## +na.rm = na.rm)) +## +## +Thus, the c function can also be used to concatenate lists: +c(list(1), list(2), list(3)) +# 3 lists -> 1 list +## [[1]] +## [1] 1 +## +## [[2]] +## [1] 2 +## +## [[3]] +## [1] 3 + +58 +I DEEP +Lists can be repeated using rep: +rep(list(1:11, LETTERS), 2) +## [[1]] +## +[1] +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 +## +## [[2]] +## +[1] "A" "B" "C" "D" "E" "F" "G" "H" "I" "J" "K" "L" "M" "N" "O" "P" "Q" +## [18] "R" "S" "T" "U" "V" "W" "X" "Y" "Z" +## +## [[3]] +## +[1] +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 +## +## [[4]] +## +[1] "A" "B" "C" "D" "E" "F" "G" "H" "I" "J" "K" "L" "M" "N" "O" "P" "Q" +## [18] "R" "S" "T" "U" "V" "W" "X" "Y" "Z" +4.2.2 +Coercing to and from Lists +The conversion of an atomic vector to a list of length-1 vectors can be done via a call to +as.list: +as.list(c(1, 2, 3)) +# vector of length 3 -> list of 3 length-1 vectors +## [[1]] +## [1] 1 +## +## [[2]] +## [1] 2 +## +## [[3]] +## [1] 3 +Unfortunately, calling, say as.numeric on a list (even if it a list comprised of numeric +vectors only) will result in an error. However, we can try to flatten a list to an atomic +vector, provided that it is possible, by calling unlist. +unlist(list(list(1, 2), list(3, list(4:8)), 9)) +## [1] 1 2 3 4 5 6 7 8 9 +unlist(list(list(1, 2), list(3, list(4:8)), "spam")) +## [1] "1" +"2" +"3" +"4" +"5" +"6" +"7" +"8" +"spam" +Note (*)InChapter11,wewillmentionthe simplify2arrayfunctionwhichgeneralises +unlist in a way that can sometimes result in a matrix. + +4 LISTS AND ATTRIBUTES +59 +4.3 +NULL +The NULL object (the one and only object of type “NULL”) can be used as a placeholder for +any other R object or designate the absence of such. +list(NULL, NULL, month.name) +## [[1]] +## NULL +## +## [[2]] +## NULL +## +## [[3]] +## +[1] "January" +"February" +"March" +"April" +"May" +## +[6] "June" +"July" +"August" +"September" "October" +## [11] "November" +"December" +NULL is different from a vector of length zero, because the latter has a type. +However, NULL sometimes behaves as a 0-length vector. In particular, length(NULL) re- +turns 0. Also, c called with no arguments returns NULL. +Testing for NULL-ness can be done with a call to is.null. +Important NULLisnotalike NA(oritisother-typedvariants);thelattercanbeemplaced +in an atomic vector. +c(1, NA, 3, NULL, 5) +# NULL behaves as a 0-length vector here +## [1] +1 NA +3 +5 +They both have very distinct semantics (no value vs a missing value). +Later we will see that some functions return NULL, invisibly, because they actually have +nothing interesting to yield. This is the case of print or plot, which are called because +of their side effects (printing and plotting). +Also, in some contexts, replacing content with NULL (e.g., when subsetting a list) will +actually result in its removal. +4.4 +Object Attributes +Lists can be used to wrap many objects and form a single, ordered collection thereof. + +60 +I DEEP +Attributes, on the other hand, give means to inject some extra data into an object of +any type (except NULL). +Attributes are (unordered) key=value pairs, where key in an arbitrary single charac- +ter string and value is any R object except NULL. They can be introduced by calling, +amongst others1, the structure function: +x_simple <- 1:10 +x <- structure( +x_simple, +# the object to be equipped with attributes +attribute1="value1", +attribute2=c(6, 100, 324) +) +4.4.1 +Developing Perceptual Indifference to Most Attributes +Let us see how the above x is reported on the console: +print(x) +## +[1] +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +## attr(,"attribute1") +## [1] "value1" +## attr(,"attribute2") +## [1] +6 100 324 +Note that the object of concern, “1:10”, was displayed first. We need to get used to +that; most of the time, we should treat the “attr…” parts of the display as if they were +printed in tiny font. +Equipping an object with attributes does not change its very nature (see, however +Chapter 10 for some exceptions). For example, the above x, despite featuring some ex- +tra data (metadata), is still treated as an ordinary sequence of numbers by most func- +tions: +sum(x) +# the same as sum(1:10), sum() does not care about any attributes +## [1] 55 +typeof(x) +# just a numeric vector, but with some perks +## [1] "integer" +Important Attributes are generally ignored by most functions unless they have spe- +cifically been programmed to pay attention to them. +1 Other ways include the replacement versions of the attr and attributes functions; see Section 9.4.5. + +4 LISTS AND ATTRIBUTES +61 +4.4.2 +But There Are Some Use Cases +Some R functions add attributes to the return value to sneak extra information that +might be useful, just in case. +For instance, na.omit, whose main aim is to remove missing values from an atomic +vector, yields: +y <- c(10, 20, NA, 40, 50, NA, 70) +(y_na_free <- na.omit(y)) +## [1] 10 20 40 50 70 +## attr(,"na.action") +## [1] 3 6 +## attr(,"class") +## [1] "omit" +We can enjoy the NA-free version of y in any further computations: +mean(y_na_free) +## [1] 38 +However, the na.action attribute (we ignore the class part until Chapter 10) tells us +where the missing observations were: +attr(y_na_free, "na.action") +# read the attribute value +## [1] 3 6 +## attr(,"class") +## [1] "omit" +As another example, gregexpr can be used to search for a given pattern in a character +vector (for more details, see Chapter 6): +needle <- "spam|gluten" +# pattern to search for: spam OR gluten +haystack <- c("spam, spam, bacon, and gluten-free spam", "spammer") +# text +(pos <- gregexpr(needle, haystack)) +## [[1]] +## [1] +1 +7 24 36 +## attr(,"match.length") +## [1] 4 4 6 4 +## attr(,"index.type") +## [1] "chars" +## attr(,"useBytes") +## [1] TRUE +## +## [[2]] +## [1] 1 +## attr(,"match.length") +(continues on next page) + +62 +I DEEP +(continued from previous page) +## [1] 4 +## attr(,"index.type") +## [1] "chars" +## attr(,"useBytes") +## [1] TRUE +Wesoughtalloccurrencesofthepatternwithintwocharacterstrings.Astheirnumber +may vary from string to string, to wrap the results in a list was a good design choice. +Each list element gives the starting positions where matches can be found (there are +four and one match(es), respectively). +Eachvectorofpositionsalsofeaturesitsown match.lengthattribute(amongstothers), +in case we need it. +Exercise 4.2 Create a list with EUR/AUD, EUR/GBP, and EUR/USD exchange rates read +from the euraud-*.csv, eurgbp-*.csv, and eurusd-*.csv files in our data repository2. Each +of its three elements should be a numeric vector storing the currency exchange rates. Further- +more, equip them with currency_from, currency_to, date_from, and date_to attributes, for +example: +## +[1] +NA 1.6006 1.6031 +NA +NA 1.6119 1.6251 1.6195 1.6193 1.6132 +## [11] +NA +NA 1.6117 1.6110 1.6188 1.6115 1.6122 +NA +## attr(,"currency_from") +## [1] "EUR" +## attr(,"currency_to") +## [1] "AUD" +## attr(,"date_from") +## [1] "2020-01-01" +## attr(,"date_to") +## [1] "2020-06-30" +Note that such additional information could of course be stored in a few separate variables (other +vectors), but then it would not be as convenient to use as the above representation. +4.4.3 +Special Attributes +Attributes have a great potential which is somewhat wasted by R users due to their +rarely knowing: +• that attributes exist (pessimistic scenario) or +• how to handle them (realistic scenario). +But we now know. +What is more, some attributes have been predestined to play a fundamental role in R. +Namely, the most prevalent amongst the special attributes are: +2 https://github.com/gagolews/teaching-data/tree/master/marek + +4 LISTS AND ATTRIBUTES +63 +• names, row.names, and dimnames are used to label the elements of atomic and gen- +eric vectors (see below), and also rows and columns in matrices (Chapter 11) and +data frames (Chapter 12), +• dim allows for turning flat vectors into matrices and other tensors (Chapter 11), +• levels labels the underlying integer codes in factor objects (Section 10.3.3), +• class can be used to bring forth new complex data structures based on basic types +(Chapter 10). +We call them special, because: +• they cannot be assigned arbitrary values; for instance, we will soon see that names +canonlybemappedtoacharactervectorofthelengthequaltothatofthesequence +it is labelling, +• they can be accessed via designated functions, e.g., names, class, dim, dimnames, +levels, etc., +• they are widely recognised by many base and third-party R functions. +However, in spite of the above, special attributes can still be managed as any other +(ordinary) ones. +Exercise 4.3 comment is perhaps the most rarely used special attribute. Create an object +(whatever) equipped with the comment attribute. Verify that assigning to it anything other than +a character vector leads to an error. Read its value by calling the comment function. Display the +objectequippedwithcomment.Notethattheprintfunctionignoresitsexistencewhatsoever:this +is how special it is. +Important (*) The accessor functions such as names or class might return meaningful +values event if the corresponding attribute is not set explicitly; see, e.g., Section 11.1.5 +for an example. +4.4.4 +Labelling Vector Elements with the names Attribute +A special attribute called names can be used to label the elements of atomic vectors and +lists. +(x <- structure(c(13, 2, 6), names=c("spam", "sausage", "celery"))) +## +spam sausage +celery +## +13 +2 +6 +The labels may improve the expressivity and readability of our code and data. +Exercise 4.4 Verify that the above x is still an ordinary numeric vector by calling typeof and +sum on it. +Note that we can ignore the names attribute whatsoever. If we apply any operation dis- + +64 +I DEEP +cussedinChapter2,wewillstillgarnerthesameresultnomatterifsuchextrainform- +ation is present or not. +It is just the print function that changed its behaviour slightly (it is a special attribute +after all). Instead of reporting: +## [1] 13 +2 +6 +## attr(,"names ") +## [1] "spam" +"sausage" "celery" +we got a nicely formatted table-like display. Non-special attributes are still printed in +a standard way. +## +spam sausage +celery +## +13 +2 +6 +## attr(,"additional_attribute") +## +[1] +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +Note In Chapter 5, we will also see that some operations (such as indexing) can gain +extra features in the presence of the names attribute. +This attribute can be read by calling: +attr(x, "names") +# just like any other attribute +## [1] "spam" +"sausage" "celery" +names(x) +# because it is so special +## [1] "spam" +"sausage" "celery" +Named vectors can be easily created with the c and list functions as well: +c(a=1, b=2) +## a b +## 1 2 +list(a=1, b=2) +## $a +## [1] 1 +## +## $b +## [1] 2 +c(a=c(x=1, y=2), b=3, c=c(z=4)) +# this is smart +## a.x a.y +b c.z +## +1 +2 +3 +4 +Letuscontemplateforawhilehowanamedlistlookslikewhenprintedontheconsole. +Again, it is still a list, but with some extras. + +4 LISTS AND ATTRIBUTES +65 +Exercise 4.5 A whole lot of functions return named vectors. Evaluate the following expressions +and read the corresponding pages in the documentation: +• quantile(runif(100)) (note that it generalises min, median, and max), +• hist(runif(100), plot=FALSE), +• options (on a side note, take note of the digits, scipen, max.print, and width options), +• capabilities. +Note (*) Most of the time, lists are used merely as containers for other R objects. This +is a boring yet essential role. However, let us just mention here that each data frame is +in fact a generic vector (see Chapter 12): each column thereof corresponds to a named +list element: +(df <- head(iris)) +# some data frame +## +Sepal.Length Sepal.Width Petal.Length Petal.Width Species +## 1 +5.1 +3.5 +1.4 +0.2 +setosa +## 2 +4.9 +3.0 +1.4 +0.2 +setosa +## 3 +4.7 +3.2 +1.3 +0.2 +setosa +## 4 +4.6 +3.1 +1.5 +0.2 +setosa +## 5 +5.0 +3.6 +1.4 +0.2 +setosa +## 6 +5.4 +3.9 +1.7 +0.4 +setosa +typeof(df) +# it is just a list (with extras that'll be discussed later) +## [1] "list" +unclass(df) +# how it is represented exactly (without the extras) +## $Sepal.Length +## [1] 5.1 4.9 4.7 4.6 5.0 5.4 +## +## $Sepal.Width +## [1] 3.5 3.0 3.2 3.1 3.6 3.9 +## +## $Petal.Length +## [1] 1.4 1.4 1.3 1.5 1.4 1.7 +## +## $Petal.Width +## [1] 0.2 0.2 0.2 0.2 0.2 0.4 +## +## $Species +## [1] setosa setosa setosa setosa setosa setosa +## Levels: setosa versicolor virginica +## +## attr(,"row.names") +## [1] 1 2 3 4 5 6 + +66 +I DEEP +Therefore, the functions we discuss in this chapter are of use in the processing of such +structured data as well. +4.4.5 +Altering and Removing Attributes +Weknowthatasingleattributecanbereadbycallingattr.Theirwholelistisgenerated +with a call to attributes. +(x <- structure(c("some", "object"), names=c("X", "Y"), +attribute1="value1", attribute2="value2", attribute3="value3")) +## +X +Y +## +"some" "object" +## attr(,"attribute1") +## [1] "value1" +## attr(,"attribute2") +## [1] "value2" +## attr(,"attribute3") +## [1] "value3" +attr(x, "attribute1") +# reads a single attribute, returns NULL if unset +## [1] "value1" +attributes(x) +# returns a named list with all attributes of an object +## $names +## [1] "X" "Y" +## +## $attribute1 +## [1] "value1" +## +## $attribute2 +## [1] "value2" +## +## $attribute3 +## [1] "value3" +We can alter an attribute’s value or add further attributes, by referring to the struc- +ture function once again. Moreover setting an attribute’s value to NULL gets rid of it +completely. +structure(x, attribute1=NULL, attribute4="added", attribute3="modified") +## +X +Y +## +"some" "object" +## attr(,"attribute2") +## [1] "value2" +## attr(,"attribute3") +## [1] "modified" +(continues on next page) + +4 LISTS AND ATTRIBUTES +67 +(continued from previous page) +## attr(,"attribute4") +## [1] "added" +As far as the names attribute is concerned, we may generated an un-named copy of an +object by calling: +unname(x) +## [1] "some" +"object" +## attr(,"attribute1") +## [1] "value1" +## attr(,"attribute2") +## [1] "value2" +## attr(,"attribute3") +## [1] "value3" +In Section 9.4.5, we will discuss the so-called replacement functions which will also +enable us to modify or remove an object’s attribute in-place, by calling “attr(x, +"some_attribute") <- new_value”. +Moreover, in Section 5.5 we note that certain operations (such as vector indexing, ele- +mentwise arithmetic operations, coercion) might not preserve all attributes of the ob- +jects that were given as their inputs. +4.5 +Exercises +Exercise 4.6 Answer the following. +• That is the meaning of “c(TRUE, FALSE) * 1:10”? +• What does “sum(as.logical(x))” compute when x is a numeric vector? +• We said that atomic vectors of type character are the most general ones. Therefore, is “as. +numeric(as.character(x))” the same as “as.numeric(x)”, regardless of the type of x? +• What is the meaning of “as.logical(x+y)” if x and y are logical vectors? What about “as. +logical(x*y)”, “as.logical(1-x)”, and “as.logical(x!=y)”? +• Let x be a named numeric vector, e.g., “x <- quantile(runif(100))”. What is the result +of “2*x”, “mean(x)”, and round(x, 2)? +• Give two ways to create a named character vector. +• Givetwoways(discussedabove;therearemore)toremovethenamesattributefromanobject. +Exercise 4.7 There are a few peculiarities when joining or coercing lists. Compare the results +generated by the following pairs of expressions: + +68 +I DEEP +# 1) +as.character(list(list(1, 2), list(3, list(4)), 5)) +as.character(unlist(list(list(1, 2), list(3, list(4)), 5))) +# 2) +as.numeric(list(list(1, 2), list(3, list(4)), 5)) +as.numeric(unlist(list(list(1, 2), list(3, list(4)), 5))) +# 3) +unlist(list(list(1, 2), sd)) +list(1, 2, sd) +# 4) +c(list(c(1, 2), 3), 4, 5) +c(list(c(1, 2), 3), c(4, 5)) +Exercise 4.8 Given numeric vectors x, y, z, and w, how to combine x, y, and list(z, w) so as +to obtain list(x, y, z, w)? More generally, given a set of atomic vectors and lists of atomic +vectors, how to combine them to get a single list that features all atomic vectors as its elements +(not a list of atomic vectors and lists, not atomic vectors unwound, etc.)? +Exercise 4.9 What is the meaning of the following when x is a logical vector? +• cummin(x) and cummin(!x), +• cummax(x) and cummax(!x), +• cumsum(x) and cumsum(!x), +• cumprod(x) and cumprod(!x). +Exercise 4.10 readRDS allows for serialising R objects and writing their snapshots to disk, so +that they can be later restored very quickly via a call to saveRDS. Verify whether this function +preserves object attributes. +Exercise 4.11 (*) Use jsonlite::fromJSON to read some JSON file in the form of a named list. +In the extremely unlikely event of us finding the current chapter boring, let us rejoice: +some of the exercises and remarks that we will encounter in the next part – devoted +to vector indexing – will definitely be deliciously stimulating! + +5 +Vector Indexing +We now know plenty of ways to process vectors in their entirety, but how to extract and +replace specific parts thereof? We will be referring to such activities collectively as in- +dexing, because they are often performed through the index operator, `[`. +5.1 +head and tail +Let us begin with something more lightweight, though. The head function can be used +to fetch a few elements from the beginning of a vector. +x <- 1:10 +head(x) +# head(x, 6) +## [1] 1 2 3 4 5 6 +head(x, 3) +# get first 3 +## [1] 1 2 3 +head(x, -3) +# skip last 3 +## [1] 1 2 3 4 5 6 7 +Similarly, tail extracts a few elements from the end of a sequence. +tail(x) +# tail(x, 6) +## [1] +5 +6 +7 +8 +9 10 +tail(x, 3) +# get last 3 +## [1] +8 +9 10 +tail(x, -3) +# skip first 3 +## [1] +4 +5 +6 +7 +8 +9 10 +Both functions work on lists too1. They are useful, e.g., when we wish to preview the +contents of a big object. +1 head and tail are actually S3 generics defined in the utils package. We will be able to call them on +matrices and data frames as well; see Chapter 10. + +70 +I DEEP +5.2 +Subsetting of and Extracting from Vectors +Given a vector x, “x[i]” returns its subset comprised of elements indicated by the in- +dexer i, which can be a single vector of: +• nonnegative integers (gives the positions of elements to retrieve), +• negative integers (gives the positions to omit), +• logical values (states whether the corresponding element should be fetched or +skipped), +• character strings (locates the elements with specific names). +5.2.1 +Nonnegative Indexes +Consider the following example vectors: +(x <- seq(10, 100, 10)) +## +[1] +10 +20 +30 +40 +50 +60 +70 +80 +90 100 +(y <- list(1, 11:12, 21:23)) +## [[1]] +## [1] 1 +## +## [[2]] +## [1] 11 12 +## +## [[3]] +## [1] 21 22 23 +The first element in a vector is at index 1. Hence: +x[1] +# the first element +## [1] 10 +x[length(x)] +# the last element +## [1] 100 +Important We might have wondered why “[1]” is being displayed each time we print +out an atomic vector on the console: +print((1:51)*10) +## +[1] +10 +20 +30 +40 +50 +60 +70 +80 +90 100 110 120 130 140 150 160 170 +## [18] 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 +## [35] 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 + +5 VECTOR INDEXING +71 +It is merely a visual hint indicating which vector element we output first in each line. +Vectorisation is a universal feature of R. Hence, it comes as no surprise that the in- +dexer can be also of length greater than one. +x[c(1, length(x))] +# the first and the last +## [1] +10 100 +x[1:3] +# the first three +## [1] 10 20 30 +Take note of some boundary cases: +x[c(1, 2, 1, 0, 3, NA_real_, 1, 11)] +# repeated, 0, missing, out of bound +## [1] 10 20 10 30 NA 10 NA +x[c()] +# indexing by an empty vector +## numeric(0) +Important Subsetting with `[` yields an object of the same type. +When applied on lists, the index operator always returns a list as well, even if we ask +for a single element: +y[2] +# a list that includes the 2nd element +## [[1]] +## [1] 11 12 +y[c(1, 3)] +# note that this is not the same as x[1, 3] (a different story) +## [[1]] +## [1] 1 +## +## [[2]] +## [1] 21 22 23 +If we wish to extract a component, i.e., to dig into what is inside a list at a specific +location, we can refer to `[[`: +y[[2]] +# extract the 2nd element +## [1] 11 12 +This is exactly why R displays “[[1]]”, “[[2]]”, etc. when printing out lists on the con- +sole. +Note +Calling “x[[i]]” on an atomic vector, where i is a single value has almost2 +2 See also Section 5.5 for the discussion on the preservation of object attributes. + +72 +I DEEP +the same effect as “x[i]”. However, `[[` generates an error if the subscript is out of +bounds. +Note (*) `[[` allows an indexer comprised of more than one number. +y[[c(1, 3)]] +## Error in y[[c(1, 3)]]: subscript out of bounds +Its meaning is different from y[c(1, 3)], though; we are about to extract a single +value,remember?Here,indexingisappliedrecursively.Namely,theaboveisequivalent +to y[[1]][[3]] – we got an error because y[[1]] is of length smaller than three. +More examples: +y[[c(3, 1)]] +# y[[3]][[1]] +## [1] 21 +list(list(7))[[c(1, 1)]] +# 7, not list(7) +## [1] 7 +5.2.2 +Negative Indexes +The indexer can also be a vector of negative integers. This way, we can exclude the ele- +ments at given positions: +y[-1] +# all but the first +## [[1]] +## [1] 11 12 +## +## [[2]] +## [1] 21 22 23 +x[-(1:3)] +## [1] +40 +50 +60 +70 +80 +90 100 +x[-c(1, 0, 2, 1, 1, 8:100)] +# 0, repeated, out of bound indexes +## [1] 30 40 50 60 70 +Note Negative and positive indexes cannot be mixed. +x[-1:3] +# recall that -1:3 == (-1):3 +## Error in x[-1:3]: only 0's may be mixed with negative subscripts +Also, NA indexes are not allowed amongst negative ones. + +5 VECTOR INDEXING +73 +5.2.3 +Logical Indexer +A vector can also be subsetted by means of a logical vector. If they both are of identical +lengths, the consecutive elements in the latter indicate whether the corresponding +elements of the indexed vector are supposed to be selected (TRUE) or omitted (FALSE). +# +1*** +2 +3 +4 +5*** +6*** +7 +8*** +9? +10*** +x[c(TRUE, FALSE, FALSE, FALSE, TRUE, TRUE, FALSE, TRUE, NA, +TRUE)] +## [1] +10 +50 +60 +80 +NA 100 +In other words, x[l], where l is a logical vector, returns all x[i] with i such that l[i] +is TRUE. Above, we extracted the elements at indexes 1, 5, 6, 8, and 10. +Recall that in Chapter 3, we have discussed ample vectorised operations that gener- +ate logical vectors. Anything that yields a logical vector of the same length as x can be +passed as an indexer. +x > 60 +# yes, it is a perfect indexer candidate +## +[1] FALSE FALSE FALSE FALSE FALSE FALSE +TRUE +TRUE +TRUE +TRUE +x[x > 60] +# select elements in x that are greater than 60 +## [1] +70 +80 +90 100 +x[x < 30 | 70 < x] +# elements not between 30 and 70 +## [1] +10 +20 +80 +90 100 +x[x < mean(x)] +# elements smaller than the mean +## [1] 10 20 30 40 50 +x[x^2 > 7777 | log10(x) <= 1.6] +# indexing based on a transformed version of x +## [1] +10 +20 +30 +90 100 +(z <- round(runif(length(x)), 2)) +# ten pseudorandom numbers +## +[1] 0.29 0.79 0.41 0.88 0.94 0.05 0.53 0.89 0.55 0.46 +x[z <= 0.5] +# indexing based on z, not x — not a problem +## [1] +10 +30 +60 100 +Theindexerisalwaysevaluatedfirstandthenpassedtothesubsettingoperation–this +operation does not care how such a logical vector was generated. +Furthermore, recycling rule is of course applied when necessary: +x[c(FALSE, TRUE)] +# every second element +## [1] +20 +40 +60 +80 100 +y[c(TRUE, FALSE)] +# interestingly, there is no warning here +## [[1]] +## [1] 1 +## +## [[2]] +## [1] 21 22 23 +Exercise 5.1 Considerasimpledatabaseaboutsixpeople,theirmostfavouritedishes,andbirth +years. + +74 +I DEEP +name <- c("Graham", "John", "Terry", "Eric", +"Michael", "Terry") +food <- c("bacon", +"spam", "spam", +"eggs", +"spam", +"beans") +year <- c( 1941, +1939, +1942, +1943, +1943, +1940 +) +The consecutive elements in different vectors correspond to each other, e.g., Graham was born in +1941 and his favourite food was bacon. +• List the names of people born in 1941 or 1942. +• List the names of those who like spam. +• List the names of those who like spam and were born after 1940. +• Compute the average birth year of the lovers of spam. +• Give the average age, in 1969, of those who didn’t find spam utmostly delicious. +The answers to the above must be provided programmatically, i.e., we do not just write "Eric" +and "Graham". The codemustbe genericenough sothat itworksin the caseofanyotherdatabase +of this kind, no matter its size. +Exercise 5.2 Removemissingvaluesfromagivenvectorwithoutreferringtothe na.omitfunc- +tion. +5.2.4 +Character Indexer +If a vector is equipped with the names attribute, such as this one: +x <- structure(x, names=letters[1:10]) +# add names +print(x) +## +a +b +c +d +e +f +g +h +i +j +## +10 +20 +30 +40 +50 +60 +70 +80 +90 100 +These labels can be referred to for the purpose of extracting the elements. To do this, +we use an indexer which is a character vector: +x[c("a", "f", "a", "g", "z")] +## +a +f +a +g +## +10 +60 +10 +70 +NA +Important We have said that special object attributes add extra functionality on top +of the existing ones. Therefore, indexing by means of positive, negative, and logical +vectors is of course still available: +x[1:3] +## +a +b +c +## 10 20 30 +x[-(1:5)] +(continues on next page) + +5 VECTOR INDEXING +75 +(continued from previous page) +## +f +g +h +i +j +## +60 +70 +80 +90 100 +x[x > 70] +## +h +i +j +## +80 +90 100 +Lists can also be subsetted this way. +(y <- structure(y, names=c("first", "second", "third"))) +## $first +## [1] 1 +## +## $second +## [1] 11 12 +## +## $third +## [1] 21 22 23 +y[c("first", "second")] +## $first +## [1] 1 +## +## $second +## [1] 11 12 +y["third"] +# result is a list +## $third +## [1] 21 22 23 +y[["third"]] +# result is the specific element unwrapped +## [1] 21 22 23 +Important Labels do not have to be unique. When we have repeated names, the first +matching element is extracted: +structure(1:3, names=c("a", "b", "a"))["a"] +## a +## 1 +There is no direct way to select all but given names, just like with negative integer in- +dexers. For a workaround, see Section 5.4.1. +Note (*) The dollar operator, `$`, can also be used to extract a single element from a +named list in some contexts; see Section 15.1 for discussion. If label is a syntactically +valid name, then x$label does the same job as x[["label"]]. We are minimalistic-by- + +76 +I DEEP +design here, hence we will tend to avoid this operator, as it does not really increase the +expressive power of our function repertoire. Also, it does not work on atomic vectors +nor on matrices. +Exercise 5.3 Rewrite the solution to the above spam-lovers exercise assuming that we have the +three features wrapped inside a list now: +(people <- list( +Name=c("Graham", "John", "Terry", "Eric", +"Michael", "Terry", "Steve"), +Food=c("bacon", +"spam", "spam", +"eggs", +"spam", +"beans", "spam"), +Year=c( 1941, +1939, +1942, +1943, +1943, +1940, +NA_real_) +)) +## $Name +## [1] "Graham" +"John" +"Terry" +"Eric" +"Michael" "Terry" +"Steve" +## +## $Food +## [1] "bacon" "spam" +"spam" +"eggs" +"spam" +"beans" "spam" +## +## $Year +## [1] 1941 1939 1942 1943 1943 1940 +NA +Do not refer to name, food, and year directly. Instead, use the full people[["Name"]] etc. ac- +cessors. There is no need to pout, it is just tiny bit of extra work. Also note that we now have Steve +amongst us. +5.3 +Replacing Elements +5.3.1 +Modifying Atomic Vectors +There are also replacement versions of the above indexing schemes. They allow us to +substitute some new content for the old one. +(x <- 1:12) +## +[1] +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +x[length(x)] <- 42 +# modify the last element +print(x) +## +[1] +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 42 +The principles of vectorisation, recycling rule, and implicit coercion are all in place: +x[c(TRUE, FALSE)] <- c("a", "b", "c") +print(x) +## +[1] "a" +"2" +"b" +"4" +"c" +"6" +"a" +"8" +"b" +"10" "c" +"42" + +5 VECTOR INDEXING +77 +Long story long: first, to make sure that the new content can be poured into old wine- +skin, R needed to convert the numeric vector to a character one; compare Section 4.1. +Then, every second element therein, a total of six items, was replaced by a recycled +version of the replacement sequence of length 3. Finally, the name “x” was re-bound +to such a brought-forth object and the previous one became forgotten. +Note For more details on replacement functions in general, see Section 9.4.5. Such +operations alter the state of the object they are called on (quite a rare behaviour in +functional languages). +Exercise 5.4 Replacemissingvaluesinagivennumericvectorwiththearithmeticmeanofwell- +defined observations therein. +5.3.2 +Modifying Lists +List contents can be altered as well. For modifying individual elements, the safest op- +tion is to use the replacement version of the `[[` operator: +y <- list(a=1, b=1:2, c=1:3) +y[[1]] <- 100:110 +y[["c"]] <- -y[["c"]] +print(y) +## $a +## +[1] 100 101 102 103 104 105 106 107 108 109 110 +## +## $b +## [1] 1 2 +## +## $c +## [1] -1 -2 -3 +The replacement version of `[` modifies a whole sub-list: +y[1:3] <- list(1, c("a", "b", "c"), c(TRUE, FALSE)) +print(y) +## $a +## [1] 1 +## +## $b +## [1] "a" "b" "c" +## +## $c +## [1] +TRUE FALSE +Moreover: + +78 +I DEEP +y[1] <- list(1:10) +# replace 1 element with 1 object +y[-1] <- 10:11 +# replace 2 elements with 2 vectors of length 1 +print(y) +## $a +## +[1] +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +## +## $b +## [1] 10 +## +## $c +## [1] 11 +Note Let idxbeavectorofpositiveindexesofelementstobemodified.Overall,calling +“y[idx] <- z” behaves as if we wrote: +1. y[[idx[1]]] <- z[[1]], +2. y[[idx[2]]] <- z[[2]], +3. y[[idx[3]]] <- z[[3]], +and so forth. +Furthermore, z (but not idx) will be recycled if necessary, i.e., we take z[[j +%% +length(z)]] for consecutive js from 1 to the length of idx. +Exercise 5.5 Reflect upon the results of the following expressions: +• y[1] <- c("a", "b", "c"), +• y[[1]] <- c("a", "b", "c"), +• y[[1]] <- list(c("a", "b", "c")), +• y[1:3] <- c("a", "b", "c"), +• y[1:3] <- list(c("a", "b", "c")), +• y[1:3] <- "a", +• y[1:3] <- list("a"), +• y[c(1, 2, 1)] <- c("a", "b", "c"), +Important Setting a list item to NULL removes it from the list completely. +y <- list(1, 2, 3, 4) +y[1] <- NULL +# removes the 1st element (i.e., 1) +y[[1]] <- NULL +# removes the 1st element (i.e., now 2) +y[1] <- list(NULL) # sets the 1st element (i.e., now 3) to NULL +(continues on next page) + +5 VECTOR INDEXING +79 +(continued from previous page) +print(y) +## [[1]] +## NULL +## +## [[2]] +## [1] 4 +The same notation conventionis used fordroppingobject attributes; see Section 9.4.5. +5.3.3 +Inserting New Elements +New elements can be pushed at the end of the vector quite easily3. +(x <- 1:5) +## [1] 1 2 3 4 5 +x[length(x)+1] <- 6 +# insert at the end +print(x) +## [1] 1 2 3 4 5 6 +x[10] <- 10 +# insert at the end but add more items +print(x) +## +[1] +1 +2 +3 +4 +5 +6 NA NA NA 10 +The elements to be inserted can be named as well: +x["a"] <- 11 +# still inserts at the end +x["z"] <- 12 +x["c"] <- 13 +x["z"] <- 14 +# z is already there; replace +print(x) +## +a +z +c +## +1 +2 +3 +4 +5 +6 NA NA NA 10 11 14 13 +Notethat xwasnotequippedwiththe namesattributebefore–theunlabelledelements +were assigned blank labels (empty strings). +Note It is not possible to insert new elements at the beginning or in the middle of a +sequence, at least not with the index operator. By writing “x[3:4] <- 1:5” we do not +replace two elements in the middle by five other ones. However, we can always use the +c function to slice parts of the vector and intertwine them with some new content: +3 And often cheaply; see Section 8.3.5 for some performance notes. Still, a warning can be generated on +each size extension if the "check.bounds" flag is set; see help("options"). + +80 +I DEEP +x <- seq(10, 100, 10) +x <- c(x[1:2], 1:5, x[5:7]) +print(x) +## +[1] 10 20 +1 +2 +3 +4 +5 50 60 70 +5.4 +Functions Related to Indexing +Let us review some operations which pinpoint interesting elements in a vector (or +functions based on these). +5.4.1 +Matching of Elements in Another Vector +We know that the `==` operator acts in an elementwise manner. It compares each ele- +ment in a vector on the lefthand side to the corresponding element in a vector on the +right side. Thus, missing values and the recycling rule aside, if z <- (x == y), then +z[i] is TRUE if and only if x[i] == y[i]. +The `%in%` operator4 is vectorised differently: it checks whether each element on the +lefthand side matches one of the elements on the right. Given z <- (x %in% y), z[i] +is TRUE whenever x[i] == y[j] for some j. +c("spam", "bacon", "spam", "eggs", "spam") %in% c("eggs", "spam", "ham") +## [1] +TRUE FALSE +TRUE +TRUE +TRUE +Example 5.6 Here is how we can remove the elements of a vector that have been assigned spe- +cified labels. +(x <- structure(1:12, names=month.abb)) +# example vector +## Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec +## +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +x[!(names(x) %in% c("Jan", "May", "Sep", "Oct"))] +# get rid of some elements +## Feb Mar Apr Jun Jul Aug Nov Dec +## +2 +3 +4 +6 +7 +8 +11 +12 +More generally, match(x, y) gives us the index of the element in y that matches each +x[i]. +match(c("spam", "bacon", "spam", "eggs", "spam"), c("eggs", "spam", "ham")) +## [1] +2 NA +2 +1 +2 +match(month.abb, c("Jan", "May", "Sep", "Oct")) +# is the month on the list? +(continues on next page) +4 A fantastic name; see Section 9.4.4. + +5 VECTOR INDEXING +81 +(continued from previous page) +## +[1] +1 NA NA NA +2 NA NA NA +3 +4 NA NA +match(c("Jan", "May", "Sep", "Oct"), month.abb) +# which month is it? +## [1] +1 +5 +9 10 +NA_real_ denotes (by default) a no-match. +Exercise 5.7 Checkoutthedocumentationof`%in%`toseehowthisoperatorisreducedtoacall +to match. Also, verify that it treats missing values as well-defined ones. +If the elements in y are not unique, the smallest index j such that x[i] == y[j] is +returned. Therefore, for example, match(TRUE, l) can be used to fetch the index of the +first occurrence of a positive value in a logical vector l. +(x <- round(runif(10), 2)) +# example vector +## +[1] 0.29 0.79 0.41 0.88 0.94 0.05 0.53 0.89 0.55 0.46 +match(TRUE, x>0.8) +# index of the first value > 0.8 (from the left) +## [1] 4 +5.4.2 +Assigning Numbers into Intervals +findInterval can come in handy where the assigning of numeric values into real in- +tervals is needed. Namely, z <- findInterval(x, y) for increasing y gives z[i] being +theindex jsuchthat x[i]isbetween y[j](bydefault,inclusive)and y[j+1](bydefault, +exclusive). +For example, a sequence of five knots 𝒚 = (−∞, 0.25, 0.5, 0.75, ∞) yields a division of +the real line to the following four intervals: +[−∞, 0.25) +[0.25, 0.5) +[0.5, 0.75) +[0.75, ∞) +(1) +(2) +(3) +(4) +Hence, for instance: +findInterval(c(0, 0.2, 0.25, 0.4, 0.66, 1), c(-Inf, 0.25, 0.5, 0.75, Inf)) +## [1] 1 1 2 2 3 4 +Exercise 5.8 Refer to the manual of findInterval to verify the function’s behaviour when we +do not include ±∞ as end points and how to make ∞ classified as a member of the 4th interval. +Exercise 5.9 Using a call to findInterval, write a statement that generates a logical vector +whose i-th element indicateswhether x[i] is in the interval [0.25, 0.5]. Was this easier towrite +than an expression involving `<=` and `>=`? +5.4.3 +Splitting Vectors into Subgroups +split(x, z) can take the output of match or findInterval (and many other operations) +and divide the elements in a vector x into subgroups corresponding to identical zs. + +82 +I DEEP +For instance, we can assign people into groups determined by their favourite dish: +name <- c("Graham", "John", "Terry", "Eric", +"Michael", "Terry") +food <- c("bacon", +"spam", "spam", +"eggs", +"spam", +"beans") +split(name, food) +# group names with respect to food +## $bacon +## [1] "Graham" +## +## $beans +## [1] "Terry" +## +## $eggs +## [1] "Eric" +## +## $spam +## [1] "John" +"Terry" +"Michael" +The result is a named list with labels determined by the unique elements in the 2nd +vector. +Another example: here are some numbers pigeonholed into the four previously men- +tioned intervals: +x <- c(0, 0.2, 0.25, 0.4, 0.66, 1) +split(x, findInterval(x, c(-Inf, 0.25, 0.5, 0.75, Inf))) +## $`1` +## [1] 0.0 0.2 +## +## $`2` +## [1] 0.25 0.40 +## +## $`3` +## [1] 0.66 +## +## $`4` +## [1] 1 +Missing values in the second argument will result in the corresponding values in the +firstargumentignored.Also,unsurprisingly,recyclingruleisappliedwhennecessary. +We can also split x into groups defined by a combination of levels of two or more vari- +ables z1, z2, etc., by calling split(x, list(z1, z2, ...)). +Example 5.10 Thebuilt-in ToothGrowthisanamedlist(withsomeextraattributesthatmakes +us rather call it a data frame; see Chapter 12) represents the results of an experimental study in- +volving 60 guinea pigs. The experiment’s aim was to measure the effect of different vitamin C +supplement types and doses on the growth of the rodents’ teeth lengths: + +5 VECTOR INDEXING +83 +ToothGrowth <- as.list(ToothGrowth) +# it is a list, but with extra attribs +ToothGrowth[["supp"]] <- as.character(ToothGrowth[["supp"]]) +# was: factor +print(ToothGrowth) +## $len +## +[1] +4.2 11.5 +7.3 +5.8 +6.4 10.0 11.2 11.2 +5.2 +7.0 16.5 16.5 15.2 17.3 +## [15] 22.5 17.3 13.6 14.5 18.8 15.5 23.6 18.5 33.9 25.5 26.4 32.5 26.7 21.5 +## [29] 23.3 29.5 15.2 21.5 17.6 +9.7 14.5 10.0 +8.2 +9.4 16.5 +9.7 19.7 23.3 +## [43] 23.6 26.4 20.0 25.2 25.8 21.2 14.5 27.3 25.5 26.4 22.4 24.5 24.8 30.9 +## [57] 26.4 27.3 29.4 23.0 +## +## $supp +## +[1] "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" +## [15] "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" "VC" +## [29] "VC" "VC" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" +## [43] "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" "OJ" +## [57] "OJ" "OJ" "OJ" "OJ" +## +## $dose +## +[1] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 +## [18] 1.0 1.0 1.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 0.5 0.5 0.5 0.5 +## [35] 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 +## [52] 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 +Wecansplit lenwithrespecttothecombinationsof suppand dose(alsocalledinteractions)by +calling: +split(ToothGrowth[["len"]], ToothGrowth[c("supp", "dose")], sep="_") +## $OJ_0.5 +## +[1] 15.2 21.5 17.6 +9.7 14.5 10.0 +8.2 +9.4 16.5 +9.7 +## +## $VC_0.5 +## +[1] +4.2 11.5 +7.3 +5.8 +6.4 10.0 11.2 11.2 +5.2 +7.0 +## +## $OJ_1 +## +[1] 19.7 23.3 23.6 26.4 20.0 25.2 25.8 21.2 14.5 27.3 +## +## $VC_1 +## +[1] 16.5 16.5 15.2 17.3 22.5 17.3 13.6 14.5 18.8 15.5 +## +## $OJ_2 +## +[1] 25.5 26.4 22.4 24.5 24.8 30.9 26.4 27.3 29.4 23.0 +## +## $VC_2 +## +[1] 23.6 18.5 33.9 25.5 26.4 32.5 26.7 21.5 23.3 29.5 +Other synonyms are of course possible, e.g., split(ToothGrowth[[1]], ToothGrowth[-1]), + +84 +I DEEP +split(ToothGrowth[[1]], list(ToothGrowth[[2]], ToothGrowth[[3]])), etc. However, +we should meditate upon our conscious use of double vs single square brackets here. +Functions such as Map described in Section 7.2 will enable us to compute any summary statistics +withingroups(e.g.,thewithin-groupaverageslikewith“SELECT AVG(len) FROM ToothGrowth +GROUP BY supp, dose” in SQL). We are in no hurry. However, as an appetiser, let us feed the +boxplot function with a list of vectors; see Figure 5.1. +boxplot(split(ToothGrowth[["len"]], ToothGrowth[c("supp", "dose")], sep="_")) +OJ_0.5 +VC_0.5 +OJ_1 +VC_1 +OJ_2 +VC_2 +5 +10 +15 +20 +25 +30 +35 +Figure 5.1: Box-and-whisker plots of len split by supp and dose (the ToothGrowth data- +set) +Note unsplit can be used to revoke the effects of split. In particular, later we will get +used to calling unsplit(Map(some_transformation, split(x, z)), z) to modify the +values in x independently in each group defined by z (e.g., standardise the variables +within each class separately). +5.4.4 +Ordering Elements +The order function finds the ordering permutation of a given vector, i.e., a sequence +of indexes which leads to a sorted version thereof. +x <- c(1024, 7, 42, 666, 0, 32787) +(o <- order(x)) +# the ordering permutation of x +## [1] 5 2 3 4 1 6 +(continues on next page) + +5 VECTOR INDEXING +85 +(continued from previous page) +x[o] +# ordered version of x +## [1] +0 +7 +42 +666 +1024 32787 +Note that o[1] is the index of the smallest element in x, o[2] is the position of the 2nd +smallest, …, and o[length(o)] is the index of the greatest value. Hence, e.g., x[o[1]] +is equivalent to min(x). +Another example: +x <- c("b", "a", "abs", "bass", "aaargh", "aargh", "aaaargh") +(o <- order(x)) +## [1] 2 7 5 6 3 1 4 +x[o] +## [1] "a" +"aaaargh" "aaargh" +"aargh" +"abs" +"b" +"bass" +Here, as x is a character vector, the ordering is lexicographical (like in a dictionary), +because this is exactly how `<=` on strings works. +Note The ordering permutation that order returns is unique (that is why we call it the +permutation) even for inputs containing duplicated elements. Owing to the use of a +stable sorting algorithm, ties (repeated elements) will be listed in the order of occur- +rence. +order(c(10, 20, 40, 10, 10, 30, 20, 10, 10)) +## [1] 1 4 5 8 9 2 7 6 3 +Above we have, e.g., five 10s at positions 1, 4, 5, 6, 9. These five indexes are guaranteed +to be listed in this very order. +Ordering can also be performed in a nonincreasing manner: +x[order(x, decreasing=TRUE)] +## [1] "bass" +"b" +"abs" +"aargh" +"aaargh" +"aaaargh" "a" +Note A call to sort(x) is equivalent to x[order(x)], but the former function can be +faster in some scenarios. For instance, one of its arguments can induce a partially sor- +ted vector which can be useful if we only seek a few order statistics (e.g., the seven +smallest values). Speed is rarely a bottleneck in the case of sorting (when it is, we have +a problem!), this is why we will not bother ourselves with such topics until the last part +of this pleasant book. Currently, we aim at expanding our repertoire of skills and abil- +ities, so that we can implement anything we can think of (rapid prototyping with the +least footprint). +Exercise 5.11 is.unsorted(x) can be used to determine if the elements in a given vector are… + +86 +I DEEP +notsortedwithrespectto`<=`.WriteanRexpressionthatgeneratesthesameresultbyreferring +to the order function. Also, assuming that x is numeric, do the same by means of a call to diff. +Note Looking at help("order"), we see that it also accepts one or more arguments via +the dot-dot-dot parameter, “...”. This way, we can sort a vector with respect to many +criteria. If there are ties (equal observations) in the first variable, they will be resolved +by the order of elements in the second variable. This is most useful for rearranging the +rows of a data frame, which we will exercise in Chapter 12. +x +<- c( 10, +20, +30, +40, +50, +60) +y1 <- c("a", "b", "a", "a", "b", "b") +y2 <- c("w", "w", "v", "u", "u", "v") +x[order(y1)] +## [1] 10 30 40 20 50 60 +x[order(y2)] +## [1] 40 50 30 60 10 20 +x[order(y1, y2)] +## [1] 40 30 10 50 60 20 +x[order(y2, y1)] +## [1] 40 50 30 60 10 20 +Note (*) Calling order on a permutation (a vector that is an arbitrary arrangement of +n consecutive natural numbers) determines its inverse. +x <- c(10, 30, 40, 20, 10, 10, 50, 30) +order(x) +## [1] 1 5 6 4 2 8 3 7 +order(order(x)) +# inverse of the above permutation +## [1] 1 5 7 4 2 3 8 6 +(x[order(x)])[order(order(x))] +# we get x again +## [1] 10 30 40 20 10 10 50 30 +Note that order(order(x)) can be considered as a way to rank all the elements in x. For +instance, the 3rd value in x, 40, is assigned rank 7: it is the 7th smallest value in this +vector. Note that this breaks the ties at a first-come-first-served basis. But we can also +write: +order(order(x, runif(length(x)))) +# ranks with ties broken at random +## [1] 2 5 7 4 3 1 8 6 +For different variations of these, see the rank function. +Exercise 5.12 Recall that sample(x) returns a pseudorandom permutation of elements of a + +5 VECTOR INDEXING +87 +given vector unless x is a single positive number. Write an expression that always yields a proper +rearrangement, regardless of the size of x. +5.4.5 +Identifying Duplicates +Whether any element in a vector was already listed in the sequence, can be verified by +calling: +x <- c(10, 20, 30, 20, 40, 50, 50, 50, 20, 20, 60) +duplicated(x) +## +[1] FALSE FALSE FALSE +TRUE FALSE FALSE +TRUE +TRUE +TRUE +TRUE FALSE +This can be used to remove repeated observations; see also unique. Note that the value +that this function returns is not guaranteed to be sorted (unlike in some other lan- +guages/libraries). +Exercise 5.13 What can be the use case of a call to match(x, unique(x))? +Exercise 5.14 Given two named lists x and y which we treat as key-value pairs, determine their +set-theoretic union (with respect to the keys), for example: +x <- list(a=1, b=2) +y <- list(c=3, a=4) +z <- ...to.do... +# combine x and y +str(z) +## List of 3 +## +$ a: num 4 +## +$ b: num 2 +## +$ c: num 3 +5.4.6 +Counting Index Occurrences +tabulate takes a vector of values from a set of small positive integers (e.g., indexes) +and determines their number of occurrences: +x <- c(2, 4, 6, 2, 2, 2, 3, 6, 6, 3) +tabulate(x) +## [1] 0 4 2 1 0 3 +In other words, there are 0 ones, 4 twos, …, and 3 sixes. +Exercise 5.15 Usinga callto tabulate (amongstothers),returnanamedvectorwiththenum- +ber of occurrences of each unique element in a character vector. For example: +y <- c("a", "b", "a", "c", "a", "d", "e", "e", "g", "g", "c", "c", "g") +result <- ...to.do... +print(result) +(continues on next page) + +88 +I DEEP +(continued from previous page) +## a b c d e g +## 3 1 3 1 2 3 +5.5 +Preserving and Losing Attributes +As attributes are conceived as extra data, it is up to a function’s authors what they will +decide to do with them. Generally, it is safe to assume that much thought has been put +into the design of base R functions. Oftentimes, they behave quite reasonably. This is +why we are going to spend some time now exploring their approaches to the handling +of attributes. +Namely, for functions and operators that aim at transforming vectors passed as their +inputs, the assumed strategy may be to: +• ignore the input attributes completely, +• equip the output object with the same set of attributes, or +• take care only of some special attributes such as names, if that makes sense. +Below we explore some common patterns; see also Section 1.3 in [48]. +5.5.1 +c +First, c drops5 all attributes except names: +(x <- structure(1:4, names=c("a", "b", "c", "d"), attrib1="<3")) +## a b c d +## 1 2 3 4 +## attr(,"attrib1") +## [1] "<3" +c(x) +# only `names` are preserved +## a b c d +## 1 2 3 4 +Wecanthereforeendupcallingthisfunctionchieflyforthisnicesideeffect.Alsorecall +that unname drops the labels. +unname(x) +## [1] 1 2 3 4 +## attr(,"attrib1") +## [1] "<3" +5 To be precise, we mean the default S3 method of c here; compare Section 10.2.4. + +5 VECTOR INDEXING +89 +5.5.2 +as.something +as.vector, as.numeric, and similar drop all attributes in the case where the output is +an atomic vector, but it might not necessarily do so in other cases (because they are S3 +generics; see Chapter 10). +as.vector(x) +# drops all attributes if x is atomic +## [1] 1 2 3 4 +5.5.3 +Subsetting +Subsetting with `[` (except where the indexer is not given) drops all attributes but +names (as well as dim and dimnames; see Chapter 11), which is adjusted accordingly: +x[1] +# subset of labels +## a +## 1 +x[[1]] +# this always drops the labels +## [1] 1 +The replacement version of the index operator can be used to modify the values in an +existing vector whilst preserving all the attributes. In particular, skipping the indexer +will allow us to replace all the elements: +y <- x +y[] <- c("u", "v") +# note that c("u", "v") has no attributes at all +print(y) +## +a +b +c +d +## "u" "v" "u" "v" +## attr(,"attrib1") +## [1] "<3" +5.5.4 +Vectorised Functions +Vectorised unary functions tend to copy all the attributes. +round(x) +## a b c d +## 1 2 3 4 +## attr(,"attrib1") +## [1] "<3" +Binary operations should get the attributes from the longer input or both (with the +first argument preferred to the second) if they are of equal sizes. +y <- structure(c(1, 10), names=c("f", "g"), attrib1=":|", attrib2=":O") +(continues on next page) + +90 +I DEEP +(continued from previous page) +y * x +# x is longer +## +a +b +c +d +## +1 20 +3 40 +## attr(,"attrib1") +## [1] "<3" +y[c("h", "i")] <- c(100, 1000) +# add two new elements at the end +y * x +## +f +g +h +i +## +1 +20 +300 4000 +## attr(,"attrib1") +## [1] ":|" +## attr(,"attrib2") +## [1] ":O" +x * y +## +a +b +c +d +## +1 +20 +300 4000 +## attr(,"attrib1") +## [1] "<3" +## attr(,"attrib2") +## [1] ":O" +Also, refer to Section 9.4.5 for a way to copy all the attributes from one object to an- +other. +Important Even in base R the above rules are not enforced strictly. We consider them +bugs that should be, for the time being, treated as features (with which we need to +learn to live as they have not been fixed for years). But there is still hope. +As far as third-party extension packages are concerned, suffice it to say that a lot of +R programmers do not know what attributes are at all! It is always best to refer to the +documentation,performsomeexperiments,and/ormanuallyassurethepreservation +of the data we care about. +5.6 +Exercises +Exercise 5.16 Answerthefollowingquestions(contemplatefirst,thenuseRtofindtheanswer): +• What is the result of “x[c()]?” Is it the same as “x[]”? +• Is “x[c(1, 1, 1)]” equivalent to “x[1]”? +• Is “x[1]” equivalent to “x["1"]”? + +5 VECTOR INDEXING +91 +• Is “x[c(-1, -1, -1)]” equivalent to “x[-1]”? +• What does “x[c(0, 1, 2, NA)]” do? +• What does “x[0]” return? +• What does “x[1, 2, 3]” do? +• What about “x[c(0, -1, -2)]” and “x[c(-1, -2, NA)]”? +• Why “x[NA]” is so significantly different from “x[c(1, NA)]”? +• What is “x[c(FALSE, TRUE, 2)]”? +• What will we obtain by calling “x[xy] and x[which(x>y)]? +What about which.min(x) vs which(x == min(x))? +Exercise 5.25 Given two equal-length vectors x and y, fetch the value from the former that cor- +responds to the smallest value in the latter. Write three versions of such an expression, each deal- +ing with potential ties in y differently, for example: +x <- c("a", "b", "c", "d", "e", "f") +y <- c( +3, +1, +2, +1, +1, +4) +should choose either the first ("b"), last ("e"), or random ("b", "d", "e" with equal probability) +element from x fulfilling the above property. Make sure your code works for x being of type char- +acter or numeric as well as an empty vector. +Exercise 5.26 Implementanexpressionthatyieldsthesameresultas duplicated(x)foranu- +meric vector x, but using diff and order. +Exercise 5.27 Based on match and unique, implement your own versions of union(x, y), in- +tersect(x, y), setdiff(x, y), is.element(x, y), and setequal(x, y) for x and y being +non-empty numeric vectors. + + +6 +Character Vectors +Text is a universal, portable, economic, and efficient means of interacting between +humans and computers as well as exchanging data between programs or APIs. This +book is 99% made of text. And, wow, how much useful knowledge is in it, innit? +6.1 +Creating Character Vectors +6.1.1 +Inputting Individual Strings +Specific character strings are delimited either by a pair of double quotes or a pair of +single quotes (apostrophes). +"a string" +## [1] "a string" +'another string' +# and of course neither 'like this" nor "like this' +## [1] "another string" +The only difference between these two lies in the fact that we cannot directly include, +e.g., an apostrophe in a single quote-delimited string. On the other hand, "'tis good +ol' spam" and 'I "love" bacon' are both okay. +However, we may always use escape sequences to embrace characters whose inclusion +might otherwise be difficult or impossible. +R uses the backslash, “\”, as the escape character, in particular: +• \" inputs the double quote character, +• \' – single quote, +• \\ – backslash, +• \n – new line. +(x <- "I \"love\" bacon\n\\\"/") +## [1] "I \"love\" bacon\n\\\"/" +The print function (which was implicitly called to display the above object) does not +reveal the special meaning of the escape sequences. Rather, print outputs strings in + +96 +I DEEP +the very way which we ourselves would follow when inputting them. The number of +characters in x is 18, and not 23: +nchar(x) +## [1] 18 +To display the string as-it-really-is, we call: +cat(x) +## I "love" bacon +## \"/ +Raw character constants, where the backslash character’s special meaning is dis- +abled, can be entered using the notation like r"(...)", r"{...}", r"[...]", r"----(.. +.)----", etc.; see help("Quotes"). These can be useful when inputting regular expres- +sions (see below). +x <- r"(spam\n\\\"maps)" +print(x) +## [1] "spam\\n\\\\\\\"maps" +cat(x) +## spam\n\\\"maps +… and of course the string version of the missing value marker is “NA_character_”. +Note (*) Some output devices may support the following codes that control the posi- +tion of the caret (text cursor): +• \b – backspace (move cursor one column to the left), +• \t – tab (advance to the next tab stop, e.g., a multiply of 8), +• \r – carriage return (move to the beginning of the current line). +cat("abc\bd\tef\rg\nhij") +## gbd +ef +## hij +These can be used on unbuffered outputs (e.g., stderr; see Section 8.3.5) to display the +status of the current operation (a simple “animated” progress bar, the print-out of the +ETA, or the % completed). +Further, certain terminals can also understand the ECMA-48/ANSI-X3.64 escape se- +quences1 of the form “\u001b[...” to further control the cursor’s position, text colour, +and even style. For example, “\u001b[1;31m” outputs red bold text and “\u001b[0m” re- +1 https://en.wikipedia.org/wiki/ANSI_escape_code + +6 CHARACTER VECTORS +97 +sets the settings to default. Give, e.g., “cat("\u001b[1;31mspam\u001b[0m")” or “cat("\ +u001b[5;36m\u001b[Abacon\u001b[Espam\u001b[0m")” a try. +Note +(*) The Unicode standard 15.0 (version dated September 2022) defines over +149,186 characters, i.a., letters from different scripts, mathematical symbols, and +emojis. Each of them is assigned a unique numeric identifier; see the Unicode Char- +acter Code Charts2. For example, the invertedexclamationmark (see the Latin-1 Supple- +ment section therein) has been mapped to hexadecimal code 0xA1 (or 161 decimally). +Knowing this magic number allows us to specify a Unicode code point using one of +the following escape sequences: +• \uxxxx – codes using four hexadecimal digits, +• \Uxxxxxxxx – codes using eight hexadecimal digits. +For instance: +cat("!\u00a1!\U000000a1!") +## !¡!¡! +All R installations allow for working with Unicode strings (more precisely, UTF-8) +– a super-encoding which is native to most Unix-like boxes (including GNU/Linux +and m**OS). Other operating systems may use some 8-bit encoding as the system +one (e.g., latin1 or cp1252), but they can be mixed with Unicode seamlessly. See +help("Encoding"), help("iconv"), and [21] for discussion. +Nevertheless, certain output devices (web browsers, LaTeX renderers, text terminals) +might be unable to display each and every Unicode character, e.g., due to some fonts +missing. As far as the processing of character data is concerned, though, this does not +matter: R does it with its eyes closed. +For example, in the PDF version3 of this joyful book, none of the following Unicode +glyphs are displayed properly, because yours cordially did not care about installing +appropriate fonts in his XeLaTeX distribution. However, its HTML variant4, which is +generated from exactly the same source files as the former, will likely be rendered by +the kind reader’s web browser as intended. +cat("\U0001f642\u2665\u0bb8\U0001f923\U0001f60d\u2307") +## ￿￿￿￿￿￿ +2 https://www.unicode.org/charts/ +3 https://deepr.gagolewski.com/deepr.pdf +4 https://deepr.gagolewski.com + +98 +I DEEP +6.1.2 +Many Strings, One Object +Less trivial character vectors (meaning, of length greater than one) can be constructed +by means of, e.g., c or rep5. +(x <- c(rep("spam", 3), "bacon", NA_character_, "spam")) +## [1] "spam" +"spam" +"spam" +"bacon" NA +"spam" +Thus,acharactervectorisinfactasequenceofsequencesofcharacters.Thetotalnum- +ber of strings can be fetched, as usual, with the length function. However, the length +of each individual string may be read via the vectorised nchar. +length(x) +# how many strings? +## [1] 6 +nchar(x) +# the number of code points in each string +## [1] +4 +4 +4 +5 NA +4 +6.1.3 +Concatenating Character Vectors +paste can be used to concatenate (join) the corresponding elements of two or more +character vectors: +paste(c("a", "b", "c"), c("1", "2", "3")) +# sep=" " by default +## [1] "a 1" "b 2" "c 3" +paste(c("a", "b", "c"), c("1", "2", "3"), sep="") +# see also paste0 +## [1] "a1" "b2" "c3" +The function is deeply vectorised: +paste(c("a", "b", "c"), 1:6, c("!", "?")) +# implicit coercion of numbers +## [1] "a 1 !" "b 2 ?" "c 3 !" "a 4 ?" "b 5 !" "c 6 ?" +We can also collapse (flatten, aggregate) a sequence of strings into a single string: +paste(c("a", "b", "c", "d"), collapse=",") +## [1] "a,b,c,d" +paste(c("a", "b", "c", "d"), 1:2, sep="", collapse="") +## [1] "a1b2c1d2" +Unfortunately (perhaps for the so-called convenience), paste does not treat missing +values just like most other vectorised elementwise functions: +paste(c("A", NA_character_, "B"), "!", sep="") +## [1] "A!" +"NA!" "B!" +5 Internally, thereis a string cache (a hash table), so that multiple clones of the same string do not occupy +more RAM than it is necessary. + +6 CHARACTER VECTORS +99 +6.1.4 +Formatting Objects +Strings can also come into being by turning other R objects into text. For example, +the quite customisable (see Chapter 10) format can be used for pretty-printing data in +dynamically generated reports. +x <- c(123456.789, -pi, NaN) +format(x) +## [1] "123456.7890" " +-3.1416" " +NaN" +cat(format(x, digits=8, scientific=FALSE, drop0trailing=TRUE), sep="\n") +## 123456.789 +## +-3.1415927 +## +NaN +Moreover, sprintf is a workhorse for turning possibly many atomic vectors to strings. +The numbers’ precision, strings’ widths and justification, etc., can be fully controlled. +Its first argument is a format string; special escape sequences starting with percent +sign, “%”, serve as placeholders for the actual values. For instance, “%s” is meant to be +replaced with a corresponding string and “%f” with a floating point value. Additional +options are available, e.g., “%10.2f” is a number that, when converted to text, will oc- +cupy ten text columns6, with two decimal digits of precision. Also, e.g., “%1$s”, “%2$s”, +… will insert the 1st, 2nd, … argument as text. +sprintf("%.5f", pi) +## [1] "3.14159" +sprintf("%s%s", "a", c("X", "Y", "Z")) +# like paste(...) +## [1] "aX" "aY" "aZ" +sprintf("key=%s, value=%.1f", c("spam", "eggs"), c(100000, 0)) +## [1] "key=spam, value=100000.0" "key=eggs, value=0.0" +sprintf("%.*f", 1:5, pi) +# variable precision +## [1] "3.1" +"3.14" +"3.142" +"3.1416" +"3.14159" +sprintf("%1$s, %2$s, %1$s, and %1$s", "spam", "bacon") +# numbered argument +## [1] "spam, bacon, spam, and spam" +See help("sprintf") for more details. I recommend. Marek Gagolewski. +6.1.5 +Reading Text Data from Files +Given a raw text file, readLines can load it into memory so that it is represented as a +character vector, with each line stored in a separate string. +f <- readLines( +"https://github.com/gagolews/teaching-data/raw/master/README.md" +) +(continues on next page) +6 Actually, this is only true for 8-bit native encodings. See also sprintf from the stringx package which +takes the text width, and not the number of bytes, into account. + +100 +I DEEP +(continued from previous page) +print(head(f)) +## [1] "# [Marek](https://www.gagolewski.com)'s Teaching and Training Data" +## [2] "" +## [3] "> *See the comment lines within the files themselves for" +## [4] "> a detailed description of each dataset.*" +## [5] "" +## [6] "*Good* datasets are actually hard to find!" +writeLines is its counterpart. There is also an option to read or write parts of files at a +time, which me mention in Section 8.3.5. Also, cat(..., append=TRUE) can be used to +create a text file incrementally. +6.2 +Pattern Searching +6.2.1 +Comparing Whole Strings +Wehavealready revieweda coupleof ways tocomparestringsasa whole.Forinstance, +the `==` operator implements elementwise testing: +c("spam", "spam", "bacon", "eggs") == c("spam", "eggs") +# recycling rule +## [1] +TRUE FALSE FALSE +TRUE +Moreover, in Section 5.4.1, we have introduced the match function and its derivative, +the `%in%` operator, which are vectorised in a different way: +match(c("spam", "spam", "bacon", "eggs"), c("spam", "eggs")) +## [1] +1 +1 NA +2 +c("spam", "spam", "bacon", "eggs") %in% c("spam", "eggs") +## [1] +TRUE +TRUE FALSE +TRUE +Note +We should stress that these are simple, bytewise comparisons of the cor- +responding code points and as such they might not be valid in, for example, nat- +ural language processing activities; compare [13]. In particular, German word groß is +not deemed equal to gross, although we expect that should be the case, at least in a +German language setting. Moreover, in the rare situations where we read Unicode- +unnormalised data (say, not in the NFC form; see [12]), canonically equivalent strings +may be considered as different. +6.2.2 +Partial Matching +When only a consideration of the initial part of each string is required, we can call: + +6 CHARACTER VECTORS +101 +startsWith(c("s", "spam", "spamtastic", "spontaneous", "spoon"), "spam") +## [1] FALSE +TRUE +TRUE FALSE FALSE +Both the above and endsWith are applied elementwisely in case of many search pre- +fixes/suffixes, just like in `==`. +Partial matching of strings can be performed with charmatch. This is a each-vs-all ver- +sion of startsWith: +charmatch(c("s", "sp", "spam", "spams", "eggs", "bacon"), c("spam", "eggs")) +## [1] +1 +1 +1 NA +2 NA +charmatch(c("s", "sp", "spam", "spoo", "spoof"), c("spam", "spoon")) +## [1] +0 +0 +1 +2 NA +Note that 0 designates that there was an ambiguity in the matching of a string to a +given table. +Note (*) In Chapter 18, we discuss the more-advanced match.arg which is (unfortu- +nately)frequentlycalledfromwithinotherRfunctions,andinChapter15,wemention +the (discouraged) partial matching of argument names in function calls. +6.2.3 +Matching Anywhere Within a String +Fixedpatternscanbealsosearchedforanywherewithin characterstringsusing grepl: +x <- c("spam", "y spammite spam", "yummy SPAM", "sram") +grepl("spam", x, fixed=TRUE) +# fixed patterns, as opposed to regexes below +## [1] +TRUE +TRUE FALSE FALSE +Important Note that the order of arguments is like grepl(needle, haystack), not +the other way around. Also, this function is not vectorised with respect to the first +argument. +Exercise 6.1 Determine how can a call to grep(y, +x, +value=FALSE) and grep(y, +x, +value=TRUE) be implemented based on grepl and other operations that we are already famil- +iar with. +Note As a curiosity, agrepl performs approximate matching based on Levenshtein’s +edit distance. This can account for a small number of “typos”. +agrepl("spam", x) +## [1] +TRUE +TRUE FALSE +TRUE +(continues on next page) + +102 +I DEEP +(continued from previous page) +agrepl("ham", x, ignore.case=TRUE) +## [1] TRUE TRUE TRUE TRUE +6.2.4 +Using Regular Expressions (*) +Setting perl=TRUE allows for identifying occurrences of patterns specified by the +PCRE2 regular expressions (regexes). +grepl("^spam", x, perl=TRUE) +# strings that begin with `spam` +## [1] +TRUE FALSE FALSE FALSE +grepl("(?i)^spam|spam$", x, perl=TRUE) +# begin or end; case ignored +## [1] +TRUE +TRUE +TRUE FALSE +Note For more details on regular expressions in general, see, e.g., [18]. The ultimate +reference for PCRE2 pattern syntax is the man7 page pcre2pattern(3). R also gives ac- +cess to ERE-like TRE library (see help("regex")), which is the default one. However, +we discourage its use, because it is feature-poorer. +Exercise 6.2 The list.files function generates the list of file names in a given directory that +matchagivenregularexpression.Forinstance,thefollowinggivesallCSVfilesinsomedirectory. +list.files("../../Projects/teaching-data/r/", r"(\.csv$)") +# or "\\.csv$" +## [1] "air_quality_1973.csv" "anscombe.csv" +"iris.csv" +## [4] "titanic.csv" +"tooth_growth.csv" +"trees.csv" +## [7] "world_phones.csv" +Writeasingleregularexpressionthatmatchesfilenamesendingwith“.csv”or“.csv.gz”.Also, +write a regex that matches CSV files whose names do not begin with “eurusd”. +6.2.5 +Locating Pattern Occurrences +regexpr finds the first occurrence of a pattern in each string: +regexpr("spam", x, fixed=TRUE) +## [1] +1 +3 -1 -1 +## attr(,"match.length") +## [1] +4 +4 -1 -1 +## attr(,"index.type") +## [1] "chars" +(continues on next page) +7 http://www.pcre.org/current/doc/html/pcre2pattern.html + +6 CHARACTER VECTORS +103 +(continued from previous page) +## attr(,"useBytes") +## [1] TRUE +In particular, there is a pattern occurrence starting at the 3th code point of the 2nd +string in x. Moreover, there is no pattern match in the last string (denoted with -1). +The match.length attribute is generally more worthwhile when searching with regular +expressions. +To locate all the matches, i.e., globally, we use gregexpr: +# `spam` followed by 0 or more letters, case insensitively +gregexpr("(?i)spam\\p{L}*", x, perl=TRUE) +## [[1]] +## [1] 1 +## attr(,"match.length") +## [1] 4 +## attr(,"index.type") +## [1] "chars" +## attr(,"useBytes") +## [1] TRUE +## +## [[2]] +## [1] +3 12 +## attr(,"match.length") +## [1] 8 4 +## attr(,"index.type") +## [1] "chars" +## attr(,"useBytes") +## [1] TRUE +## +## [[3]] +## [1] 7 +## attr(,"match.length") +## [1] 4 +## attr(,"index.type") +## [1] "chars" +## attr(,"useBytes") +## [1] TRUE +## +## [[4]] +## [1] -1 +## attr(,"match.length") +## [1] -1 +## attr(,"index.type") +(continues on next page) + +104 +I DEEP +(continued from previous page) +## [1] "chars" +## attr(,"useBytes") +## [1] TRUE +As we have noted in Section 4.4.2, wrapping the results in a list was a clever choice as +the number of matches can obviously vary between strings. +In Section 7.2, we will take a look at the Map function, which, along with substring +introduced below, can aid in getting the most out of such data. Meanwhile, let us just +mention that regmatches extracts the matching substrings: +regmatches(x, gregexpr("(?i)spam\\p{L}*", x, perl=TRUE)) +## [[1]] +## [1] "spam" +## +## [[2]] +## [1] "spammite" "spam" +## +## [[3]] +## [1] "SPAM" +## +## [[4]] +## character(0) +Note (*) Let us consider what happens when a regular expression contains parenthes- +ised subexpressions (capture groups). +r <- "(?[^. ]+)\\.(?[^ ]*)" +The above regex consists of two such parts. The first one is labelled “basename” and is +comprised of a number of arbitrary characters except for the space and the dot. The +second group, named “extension” is a substring of anything but the space. These two +are separated by a dot. +Such a pattern can be used for unpacking space-delimited lists of file names. +z <- "dataset.csv.gz something_else.txt spam" +regexpr(r, z, perl=TRUE) +## [1] 1 +## attr(,"match.length") +## [1] 14 +## attr(,"index.type") +## [1] "chars" +## attr(,"useBytes") +## [1] TRUE +(continues on next page) + +6 CHARACTER VECTORS +105 +(continued from previous page) +## attr(,"capture.start") +## +basename extension +## [1,] +1 +9 +## attr(,"capture.length") +## +basename extension +## [1,] +7 +6 +## attr(,"capture.names") +## [1] "basename" +"extension" +gregexpr(r, z, perl=TRUE) +## [[1]] +## [1] +1 16 +## attr(,"match.length") +## [1] 14 18 +## attr(,"index.type") +## [1] "chars" +## attr(,"useBytes") +## [1] TRUE +## attr(,"capture.start") +## +basename extension +## [1,] +1 +9 +## [2,] +16 +31 +## attr(,"capture.length") +## +basename extension +## [1,] +7 +6 +## [2,] +14 +3 +## attr(,"capture.names") +## [1] "basename" +"extension" +The capture.* attributes give us access to the matches to the individual capture +groups, i.e., the basename and the extension. +Exercise 6.3 (*) Check out the difference between the results generated by regexec and reg- +expr as well as gregexec and gregexpr. +6.2.6 +Replacing Pattern Occurrences +sub and gsub can replace first and all, respectively, matches to a pattern: +x <- c("spam", "y spammite spam", "yummy SPAM", "sram") +sub("spam", "ham", x, fixed=TRUE) +## [1] "ham" +"y hammite spam" "yummy SPAM" +"sram" +gsub("spam", "ham", x, fixed=TRUE) +## [1] "ham" +"y hammite ham" "yummy SPAM" +"sram" + +106 +I DEEP +Note (*) If a regex features some capture groups, matches thereto can be mentioned +not only in the pattern itself, but also in the replacement string: +gsub("(\\p{L})\\p{L}\\1", "\\1", "aha egg gag NaN spam", perl=TRUE) +## [1] "a egg g N spam" +The above matches a letter (it is a capture group), another letter, and the former letter +again. Each such palindrome of length 3 is replaced with just the repeated letter. +Exercise 6.4 (*)Displaythesourcecodeof glob2rxbycalling print(glob2rx)andstudyhow +this function converts wildcards such as file???.* or *.csv to regular expressions that can be +passed to, e.g., list.files. +6.2.7 +Splitting Strings into Tokens +strsplit divides each string in a character vector into chunks. This time, though, the +search pattern, specifying the token delimiter, is given as the second argument: +strsplit(c("spam;spam;eggs;;bacon", "spam"), ";", fixed=TRUE) +## [[1]] +## [1] "spam" +"spam" +"eggs" +"" +"bacon" +## +## [[2]] +## [1] "spam" +6.3 +Other String Operations +6.3.1 +Extracting Substrings +substring extracts parts of strings between given character position ranges. +substring("spammity spam", 1, 4) +# from 1st to 4th character +## [1] "spam" +substring("spammity spam", 10) +# from 10th to end +## [1] "spam" +substring("spammity spam", c(1, 10), c(4, 14)) +# vectorisation +## [1] "spam" "spam" +substring(c("spammity spam", "bacon and eggs"), 1, c(4, 5)) +## [1] "spam" +"bacon" +Note There is also a replacement (compare Section 9.4.5) version of the above: + +6 CHARACTER VECTORS +107 +x <- "spam, spam, bacon, and spam" +substring(x, 7, 11) <- "eggs" +print(x) +## [1] "spam, eggs, bacon, and spam" +Unfortunately, the number of characters in the replacement string should not exceed +the length of the part being substituted (try “chickpeas” instead of “eggs”). However, +substring replacement can be written as a composition of substring extraction and +concatenation: +paste(substring(x, 1, 6), "chickpeas", substring(x, 11), sep="") +## [1] "spam, chickpeas, bacon, and spam" +Exercise 6.5 Taketheoutputgeneratedbyregexprandapplysubstringtoextractthepattern +occurrences. If there is no match in some string, the corresponding output should be NA. +6.3.2 +Translating Characters +tolower and toupper can be used to convert between lower and upper case: +toupper("spam") +## [1] "SPAM" +Note Just like many other string operations in base R, these functions perform very +simple character substitutions and they might not be valid in natural language pro- +cessing tasks. For instance, groß is not converted to GROSS, which is the correct case +folding in German. +Moreover, chartr translates individual characters: +chartr("\\", "/", "c:\\windows\\system\\cmd.exe") +# chartr(old, new, x) +## [1] "c:/windows/system/cmd.exe" +chartr("([S", ")]*", ":( :S :[") +## [1] ":) :* :]" +In the first line, we replace each backslash with slash. The second example replaces “(”, +“[”, and “S” with “)”, “]”, and “*”, respectively. + +108 +I DEEP +6.3.3 +Ordering Strings +We have previously mentioned that operators such as `<` and `>=` as well as func- +tions like sort, order, rank, but also xtfrm8 are based on the lexicographic ordering of +strings. +sort(c("chłodny", "hardy", "chladný", "hladný")) +## [1] "chladný" "chłodny" "hardy" +"hladný" +It is worth noting that the ordering is dependent on the currently selected locale, see +Sys.getlocale("LC_COLLATE"). For instance, in the Slovak language setting, we would +obtain "hardy" < "hladný" < "chladný" < "chłodny". +Note +Many “structured” data items can be displayed or transmitted as human- +readablestrings.Inparticular,weknowthat as.numericcanbeusedtoconvertastring +to a number. Moreover, in Section 10.3.1 we will discuss date-time objects such as +"1970-01-01 00:00:00 GMT". We will be processing them with specialised functions +such as strptime and strftime. +Important (*) As we have mentioned, many string operations in base R are not neces- +sarily portable. The stringx package [22] defines drop-in, “fixed” replacements there- +for.TheyarebasedontheInternationalComponentsforUnicode(ICU9)library,which +is a de facto standard for the processing of Unicode text, and the R package stringi; +see [21]. +# call install.packages("stringx") first +suppressPackageStartupMessages(library("stringx")) +# load the package +sort(c("chłodny", "hardy", "chladný", "hladný"), locale="sk_SK") +## [1] "hardy" +"hladný" +"chladný" "chłodny" +toupper("gro\u00DF") +# compare base::toupper("gro\u00DF") +## [1] "GROSS" +detach("package:stringx") +# unload the package +6.4 +Other Atomic Vector Types (*) +We have discussed four vector types: logical, double, character, and list (the lat- +ter being a generic-recursive vector). To get the complete picture of the sequence-like +8 See Section 12.3.1 for a use case. +9 http://site.icu-project.org/ + +6 CHARACTER VECTORS +109 +types in R, let us briefly mention integer, complex, and raw atomic types, so that we +are not surprised when we encounter them. +6.4.1 +Integer Vectors (*) +Integer scalars can be input manually by using the L suffix: +(x <- c(1L, 2L, -1L, NA_integer_)) +# looks like numeric +## [1] +1 +2 -1 NA +typeof(x) +# but is integer +## [1] "integer" +Some functions return them in certain contexts10: +typeof(1:10) +# seq(1, 10) as well, but not seq(1, 10, 1) +## [1] "integer" +as.integer(c(-1.1, 0, 1.9, 2.1)) +# truncate/round towards 0 +## [1] -1 +0 +1 +2 +In the vast majority of expressions, integer vectors behave like numeric ones, and are +silently coerced to double if need be, so there is no real practical reason to distinguish +between them (they are of internal interest, e.g., when writing C/C++ extensions; see +Chapter 14). For example: +1L/2L +# like 1/2 == 1.0/2.0 +## [1] 0.5 +Note (*) R integers are 32-bit signed types. The double type can store more integers +than them (with the maximal contiguously representable integer being 253 vs 231 − 1 +in the former case; see Section 3.2.3): +as.integer(2^31-1) + 1L +# 32-bit integer overflow +## Warning in as.integer(2^31 - 1) + 1L: NAs produced by integer overflow +## [1] NA +as.integer(2^31-1) + 1 == 2^31 # integer+double == double – OK +## [1] TRUE +(2^53 - 1) + 1 == 2^53 +# OK +## [1] TRUE +(2^53 + 1) - 1 == 2^53 +# lost due to FP rounding, left result is 2^53 - 1 +## [1] FALSE +10 Actually, 1:10returnsanintegervectorinacompact(ALTREP,see[39])form;comparetheresultsofthe +call to “.Internal(inspect(1:10))” and “.Internal(inspect(seq(1, 10, 1)))”. This way, the whole vector +does not have to be allocated which saves memory and time. At the R level, though, it behaves as any other +integer (numeric) sequence. + +110 +I DEEP +Note Since R 3.0, there is support for vectors longer than 231 − 1 elements. As there +areno64-bitintegersinR,theseareindexedbydoublesanyway(aswehavebeendoing +all this time). Interestingly, x[1.9] is the same as x[1] and x[-1.9] means x[-1] (trun- +cation of the fractional part). This is why the notation like x[length(x)*0.2] works re- +gardless of whether the length of x is a multiple of 5 or not, which is neat. +6.4.2 +Raw Vectors (*) +Vectors of type raw can store bytes, i.e., unsigned 8-bit integers, whose range is 0-255 +(there are no raw NAs). For example: +as.raw(c(-1, 0, 1, 2, 0xc0, 254, 255, 256, NA)) +## Warning: out-of-range values treated as 0 in coercion to raw +## [1] 00 00 01 02 c0 fe ff 00 00 +They are displayed as two-digit hexadecimal (base-16) numbers. Also note that we may +enter such numbers using the “0x” prefix. +There are only few functions that deal with such vectors: e.g., readBin, charToRaw, and +rawToChar. +6.4.3 +Complex Vectors (*) +We can also play with vectors of type complex, with “1i” representing the imaginary +unit, √−1. Complex numbers appear in quite a few engineering or scientific applic- +ations, e.g., in physics, electronics, or signal processing and are (at least: should be) +part of many university-level subjects or textbooks in mathematics11. +c(0, 1i, pi+pi*1i, NA_complex_) +## [1] 0.0000+0.0000i 0.0000+1.0000i 3.1416+3.1416i +NA +Apart from the basic operators, mathematical and aggregation functions, procedures +like fft, solve, qr, or svd can be fed with or produce such data. For more details, see +help("complex") and some matrix examples in Chapter 11. +6.5 +Exercises +Exercises marked with (*) might require tinkering with regular expressions or third- +party R packages. +11 Even the statistics/machine learning oriented ones, because of their heavy use of numerical comput- +ing, e.g., [14, 25]. + +6 CHARACTER VECTORS +111 +Exercise 6.6 Answer the following questions: +• How many characters are there in the string "ab\n\\\t\\\\\""? What about "-{ab\n\\\ +t\\\\\"-)}-"? +• What is the result of calling “paste(NA, 1:5, collapse="")”? +• Whatisthemeaningofthefollowing sprintfformatstrings: "%s", "%20s", "%-20s", "%f", +"%g", "%e", "%5f", "%5.2f%%", "%.2f", "%0+5f", and "[%+-5.2f]"? +• What is the difference between regexpr and gregexpr? What does “g” in the latter name +stand for? +• What is the result of a call to “grepl(c("spam", "spammity spam", "aubergines"), +"spam")”? +• Is it always the case that “"Aaron" < "Zorro"”? +• If x is a character vector, is “x == x” always equal to TRUE? +• If x and y are character vectors of lengths n and m, respectively, what is the length of the +output of “match(x, y)”? +• If x is a named vector, why there is a difference between “x[NA]” and “x[NA_character_]”? +• What is the difference between “x == y” and “x %in% y”? +Exercise 6.7 Let x, y, and z be atomic vectors and a and b be single strings. Generate the same +results as “pastena(x, collapse=b)”, “pastena(x, y, sep=a)”, “pastena(x, y, sep=a, +collapse=b)”, “pastena(x, y, z, sep=a)”, “pastena(x, y, z, sep=a, collapse=b)”, +assuming that pastena is a version of paste (which we do not have) that handles missing data +in a way consistent with most other functions. +Exercise 6.8 Based on list.files and glob2rx, generate the list of all PDFs on your com- +puter. Then, using file.size filter out the files smaller than 10 MiB. +Exercise 6.9 Read a text file that stores a long paragraph of some banal prose. Concatenate +all the lines to form a single, long string. Using strwrap and cat, output the paragraph on the +console, nicely formatted to fit an aesthetic width, say, 60 text columns. +Exercise 6.10 (*) Implement your own, simplified version of basename and dirname. +Exercise 6.11 (*) Implement an operation similar to trimws using the functions introduced in +this chapter. +Exercise 6.12 (*) Write a regex that extracts all words from each string in a given character +vector. +Exercise 6.13 (*) Write a regex that extracts, from each string in a character vector, all: +• integers numbers (signed or unsigned), +• floating-point numbers, +• numbers of any kind (including those in scientific notation), +• #hashtags, + +112 +I DEEP +• email@addresses, +• hyperlinks of the form http://… and https://…. +Exercise 6.14 (*) What does 42i, 42L, and 0x42 stand for? +Exercise 6.15 (*) Check out stri_sort in the stringi package (or sort.character in +stringx) for a way to obtain an ordering like "a1" < "a2" < "a10" < "a11" < "a100". +Exercise 6.16 (*) In sprintf, the formatter "%20s" means that if a string is less than 20 bytes +long,theremainingbyteswillbereplacedwithspaces.OnlyforASCIIcharacters(Englishletters, +digits, some punctuation marks, etc.) it is true that one character is represented by 1 byte. Other +Unicode code points can take up between 2 and 4 bytes. +cat(sprintf("..%6s..", c("abc", "1!<", "aßc", "ąß©")), sep="\n") +# aligned? +## .. +abc.. +## .. +1!<.. +## .. +aßc.. +## ..ąß©.. +Use the stri_pad function from the stringi package to align the strings aesthetically. Altern- +atively, check out sprintf from stringx. +Exercise 6.17 (*) Implement an operation similar to stri_pad from stringi using the func- +tions introduced in this chapter. + +7 +Functions +R is a functional language, where functions play first fiddle. Each action we perform +reduces itself to a call to some function, or a combination thereof. +So far we have been tinkering with dozens of available functions which arepart of base +R, with only few exceptions. They constitute the essential vocabulary that everyone +must be able to speak fluently. +Any operation, be it sum, sqrt, or paste, when fed with a number of arguments, gen- +erates some (hopefully useful) return value. +sum(1:10) +# invoking `sum` on a specific argument +## [1] 55 +From a user’s perspective, each function is merely a tool. To achieve a goal at hand, we +do not really have to care about what is going on under its hood, i.e., how the inputs +are actually being transformed so that, after a couple of nanoseconds or hours, we +can enjoy what has been yielded. This is very convenient: all we need to know is the +function’s specification which can be stated, for example, informally, in plain Polish +or Malay, in its help page. +In this chapter, we will learn how to write our own functions. The use of this skill is a +good development practice when we expect that some operations are to be executed +many times but perhaps on different data. +Also, some R functions are meant to invoke other functions, for instance on every ele- +ment in a list or every section of a data frame grouped by a qualitative variable, so it +is good to learn know how we can specify a custom operation to be propagated there- +over. +Example 7.1 Given some objects (whatever): +x1 <- runif(16) +x2 <- runif(32) +x3 <- runif(64) +when we want to apply the same action on different data, say, compute the root mean square, +instead of re-typing almost identical expressions (or a bunch of them) over and over again: +sqrt(mean(x1^2)) +## [1] 0.6545 +(continues on next page) + +114 +I DEEP +(continued from previous page) +sqrt(mean(x2^2)) +# the same second time - borderline okay +## [1] 0.56203 +sqrt(mean(x3^2)) +# tedious, barbarous, and error-prone +## [1] 0.57206 +we can generalise the operation to any object like x: +rms <- +# bound what follows to name `rms` +function(x) +# a function that takes one parameter, `x` +sqrt(mean(x^2)) +# expression to transform the input to yield output +and then re-use it on different concrete data instances: +rms(x1) +## [1] 0.6545 +rms(x2) +## [1] 0.56203 +rms(x3) +## [1] 0.57206 +or even combine it with other function calls: +rms(sqrt(c(x1, x2, x3)))^2 +## [1] 0.50824 +Important +Does writing your own functions equal reinventing the wheel? Can +everything be found on the internet these days (including on Stack Overflow, GitHub, +or CRAN)? +Luckily, this is not the case. Otherwise, data analysts’, researchers’, and developers’ +lives could be considered monotonous, dreary, and uninspiring. Plus, sometimes it is +much quicker to write a function from scratch than to get through the whole garbage +dump from where, only occasionally, we can dig out some pearls. Not to mention the +self-educative side: we become better programmers by crunching those exercises. We +are advocating for minimalism here, remember? +This and many more other important issues in function design will be reflected upon +in Chapter 9. + +7 FUNCTIONS +115 +7.1 +Creating and Invoking Functions +7.1.1 +Anonymous Functions +Functions are usually created by means of the following notation: +function(args) body +First, args is a (possibly empty) list of comma-separated parameter names which are +supposed to act as input variables. +Second, body is a single R expression which will be evaluated when the function is +called. The value that this expression yields will constitute the function’s output. +For example, here is a definition of a function which takes no inputs and generates a +constant output: +function() 1 +## function() 1 +Wethuscreatedafunctionobject.However,ithasdisappearedimmediatelythereafter, +as we have not used it at all. +Any function, say, f can be invoked, i.e., evaluated on concrete data, by using the nota- +tion f(arg1, ..., argn), where “arg1, ..., argn” are the arguments to be passed to +f. +(function() 1)() +# invoking f like f(); here, no arguments are expected +## [1] 1 +Only now we have obtained a return value. +Note (*) Calling typeof on a function object will report "closure" (for user-defined +functions), "builtin", or "primitive" (for some built-in, base ones), for reasons that +we explain in Section 9.5.3. +typeof(function() 1) +## [1] "closure" +7.1.2 +Named Functions +Function objects can be bound with names so that they can be referred to multiple +times: + +116 +I DEEP +one <- function() 1 +# one <- (function() 1) +We created an object named one (we use bold font to indicate that it is of type function, +because functions are so important in R). We are very familiar with such a notation, +as not since yesterday we are used to writing “x <- 1” etc. +Invoking one, which can be done by writing one(), will yield a return value: +one() +# (function() 1)() +## [1] 1 +This output can be used in further computations, for instance: +0:2 - one() +# 0:2 - (function() 1)(), i.e., 0:2 - 1 +## [1] -1 +0 +1 +7.1.3 +Passing Arguments To Functions +Functions with no arguments are kind of boring, thus let us distil a more serious op- +eration: +concat <- function(x, y) paste(x, y, sep="") +Here we have created a mapping whose aim is to concatenate two objects by means of +a specialised call to paste. Yours faithfully pleads guilty to multiplying entities need- +lessly, because it should not be a problem for anyone to write paste(x, y, sep="") each +time. Yet, ‘tis merely an illustration. +The concat function has two parameters, “x” and “y”. Hence, calling it will require the +provision of two arguments, which we put within round brackets and separate from +each other by commas. +u <- 1:5 +concat("spam", u) +# i.e., concat(x="spam", y=1:5) +## [1] "spam1" "spam2" "spam3" "spam4" "spam5" +Important Notice the distinction: parameters (also called formal arguments) are ab- +stract, general, or symbolic; “something, anything that will be put in place of x when +the function is invoked”. By contrast, arguments (a.k.a. actual parameters) are con- +crete, specific, and real. +During the above call, x in the function’s body is precisely "spam", and nothing else. +Also, the u object from the caller’s environment is seen under the name y there. Most +of the time (however, see Chapter 18), it is best to think of the function as being fed not +with u per se, but the value that u is bound to, i.e., “1:5”. + +7 FUNCTIONS +117 +Also: +x <- 1:5 +y <- "spam" +concat(y, x) +# concat(x="spam", y=1:5) +## [1] "spam1" "spam2" "spam3" "spam4" "spam5" +This is still a call to equivalent to concat(x=y, y=x). The argument x is being assigned +with the value of y from the calling environment, "spam". Yes, one x is not the same +as the other x, and which is which is unambiguously defined by the context. Under- +standing and being able to manipulate such abstractions is basic logic and common +sense that everyone should master. +Exercise 7.2 Write a function called standardise that takes a numeric vector x as argument +and returns its standardised version, i.e., from each element in x subtract the sample arithmetic +mean and then divide it by the standard deviation. +Note Recall from Section 2.1.3 that, syntactically speaking, the following are perfectly +valid alternatives to the positionally-matched call concat("spam", u). +concat(x="spam", y=u) +concat(y=u, x="spam") +concat("spam", y=u) +concat(u, x="spam") +concat(x="spam", u) +concat(y=u, "spam") +However,thelasttwoshouldparticularlybeavoided,forthesakeofthereaders’sanity. +It is best to provide positionally-matched arguments before the keyword-based ones. +Also, in Section 10.5, we introduce the (overused) forward-pipe operator, `|>`, which +enables the above to be written as “"spam" |> concat(u)”. +7.1.4 +Grouping Expressions with Curly Braces, `{` +We have been informed that a function’s body is a single R expression whose evalu- +ated value is passed to the user as its output. This may sound restrictive and contrast +with what we have experienced so far. Rarely are we faced with such simple comput- +ing tasks and we have already seen R functions performing quite sophisticated oper- +ations. +It turns out that, grammatically, a single R expression can be arbitrarily complex +(Chapter 15); we can use curly braces to group many calls that are to be evaluated one +after another. +For instance: + +118 +I DEEP +{ +cat("first expression\n") +cat("second expression\n") +# ... +cat("last expression\n") +} +## first expression +## second expression +## last expression +Notethatweusedfourspacestovisuallyindenttheconstituentsforgreaterreadability +(somedevelopersprefertabsoverspaces,othersfindtwoorthreespacesmoreurbane, +but we do not). This single (compound) expression can now play a role of a function’s +body. +Important The last expression evaluated in a curly-braces delimited block will be con- +sidered its the output value. +x <- { +1 +2 +3 +# <--- last expression: will be taken as the output value +} +print(x) +## [1] 3 +Note (*)Theabovecodeblockcanalsobewrittenmoreconciselybyreplacingnewlines +with semicolons, although with perhaps some loss in readability: +{1; 2; 3} +## [1] 3 +In Section 9.4, we will give a few more details about `{`. +Example 7.3 Here is a version of the above concat function which takes care of a more Chapter +2-style missing values’ propagation: +concat <- function(a, b) +{ +z <- paste(a, b, sep="") +z[is.na(a) | is.na(b)] <- NA_character_ +z +# last expression in the block – return value +} + +7 FUNCTIONS +119 +Example calls: +concat("a", 1:3) +## [1] "a1" "a2" "a3" +concat(NA_character_, 1:3) +## [1] NA NA NA +concat(1:6, c("a", NA_character_, "c")) +## [1] "1a" NA +"3c" "4a" NA +"6c" +Letusappreciatethefactthatwecouldkeepthecodebriefthanksto pasteand`|`implementing +the recycling rule. +Exercise 7.4 Write a function called normalise that takes a numeric vector x and returns its +version shifted and scaled to the [0, 1] interval. To do so, from each element subtract the sample +minimumandthendivideitbytherange,i.e.,thedifferencebetweenthemaximumandthemin- +imum. Avoid computing min(x) twice. +Exercise 7.5 Write a function that applies the robust standardisation of a numeric vector: sub- +tract the median and divide it by the median absolute deviation, 1.4826 times the median of the +absolute differences between the values and their median. +Note +R is an open-source (free, libre) project – users are not only encouraged to +run the software for whatever the purpose, but also study and modify its source +code without any restrictions. This applies both to functions that we have authored +ourselves: +print(concat) +## function(a, b) +## { +## +z <- paste(a, b, sep="") +## +z[is.na(a) | is.na(b)] <- NA_character_ +## +z +# last expression in the block – return value +## } +## +and to the routines that are part of base R or any other extension packages: +print(union) +## function (x, y) +## { +## +u <- as.vector(x) +## +v <- as.vector(y) +## +unique(c(u, v)) +## } +## +## +Nevertheless, some functionality might be implemented in a compiled programming + +120 +I DEEP +language such as C, C++, or Fortran; notice a call to .Internal in the source code of +paste, .Primitive in list, or .Call in runif. Therefore, we will sometimes have to dig +a little bit deeper to access the underlying source code; see Chapter 14 for more details. +7.2 +Functional Programming +R is a functional programming language. As such, it shares a number of common fea- +tures with other languages that emphasise on the role of function manipulation in +software development (e.g., Common Lisp, Scheme, OCaml, Haskell, Clojure, F#). Let +us explore them now. +7.2.1 +Functions are Objects +R functions were given the right to a fair go; they are what we refer to as first-class cit- +izens. In other words, our interaction with them is not limited to their invocation; we +treat them as any other language objects. Namely, they can be: +• stored inside list objects: +list(identity, nrow, sum) +# a list with three elements of type function +## [[1]] +## function (x) +## x +## +## +## +## [[2]] +## function (x) +## dim(x)[1L] +## +## +## +## [[3]] +## function (..., na.rm = FALSE) +.Primitive("sum") +This is possible owing to the fact that lists, as we recall, can embrace R objects of +any kind. +• created and then called inside another function’s body: +euclidean_distance <- function(x, y) +{ +square <- function(z) z^2 +# auxiliary/internal/helper function +(continues on next page) + +7 FUNCTIONS +121 +(continued from previous page) +sqrt(sum(square(x-y))) +# square root of the sum of squares +} +euclidean_distance(c(0, 1), c(1, 0)) +# example call +## [1] 1.4142 +This is why we tend to classify functions as representatives of recursive types (com- +pare is.recursive). +• passed as arguments to other operations: +# Replaces missing values with a given aggregate +# of all non-missing elements: +fill_na <- function(x, filler_fun) +{ +missing_ones <- is.na(x) +# otherwise, we'd call is.na twice +replacement_value <- filler_fun(x[!missing_ones]) +x[missing_ones] <- replacement_value +x +} +fill_na(c(0, NA_real_, NA_real_, 2, 3, 7, NA_real_), mean) +## [1] 0 3 3 2 3 7 3 +fill_na(c(0, NA_real_, NA_real_, 2, 3, 7, NA_real_), median) +## [1] 0.0 2.5 2.5 2.0 3.0 7.0 2.5 +We call these higher-order functions. +Note More advanced techniques, which we will discuss later (i.e., closures, lazy eval- +uation, metaprogramming, etc.), will let the functions be: +• returned as other function’s outputs (sec:to-do), +• equipped auxiliary data (sec:to-do), +• generated programmatically on the fly (sec:to-do), and +• modified at runtime (sec:to-do). +Below we review some noteworthy higher-order functions, in particular: do.call and +Map. Many other ones will be introduced in due course or are left as an educative exer- +cise. + +122 +I DEEP +7.2.2 +Calling on Precomputed Arguments with do.call +The notation like f(arg1, ..., argn) has no monopoly over how we are supposed to +call a function on a specific sequence of comma-delimited arguments: the latter do +not have to be hardcoded. +Here is an alternative. We can first prepare a number of objects to be passed as f’s +inputs, wrap them in a list l, and then invoke do.call(f, l) to get the same result. +words <- list( +c("spam", +"bacon", +"eggs"), +c("buckwheat", "quinoa", "barley"), +c("ham", +"spam", +"spam") +) +do.call(paste, words) +# paste(words[[1]], words[[2]], words[[3]]) +## [1] "spam buckwheat ham" "bacon quinoa spam" +"eggs barley spam" +do.call(cbind, words) +# column-bind; returns a matrix (explained later) +## +[,1] +[,2] +[,3] +## [1,] "spam" +"buckwheat" "ham" +## [2,] "bacon" "quinoa" +"spam" +## [3,] "eggs" +"barley" +"spam" +do.call(rbind, words) +# row-bind (explained later) +## +[,1] +[,2] +[,3] +## [1,] "spam" +"bacon" +"eggs" +## [2,] "buckwheat" "quinoa" "barley" +## [3,] "ham" +"spam" +"spam" +Note that the length and content of the list passed as the 2nd argument of do.call can +be arbitrary (possibly unknown at the time of writing the code). See Section 12.1.2 for +more use cases, e.g., ways to concatenate a list of data frames (perhaps produced by +some complex chain of commands) into a single data frame. +If elements of the list are named, they will be matched to the corresponding keyword +arguments. +x <- 2^(seq(-2, 2, length.out=101)) +plot_opts <- list(col="red", lty="dashed", type="l") +do.call(plot, c(list(x, log2(x), xlab="x", ylab="log2(x)"), plot_opts)) +## (the displaying of the plot has been suppressed) +Note that, e.g., plot_opts can now be reused in further calls to graphical functions. +This is very convenient as it avoids repetitions. +7.2.3 +Common Higher-Order Functions +There is an important class of higher-order functions that allow us to apply custom +operations on consecutive elements of sequences without relying on loop-like state- +ments, at least explicitly. They can be found in all functional programming languages + +7 FUNCTIONS +123 +(e.g., Lisp, Haskell, Scala) and have been ported to various add-on libraries (functools +in Python, more recent versions of the C++ Standard Library, etc.) or frameworks +(Apache Spark and the like). Their presence reflects the obvious truth that some kinds +of operations occur more frequently than other ones. +In particular: +• Map calls a function on each element of a sequence in order to transform: +– their individual components (just like sqrt, round, or the unary `!` operator +in R), or +– the corresponding elements of many sequences so as to vectorise a given op- +eration elementwisely (compare the binary `+` or paste), +• Reduce (also called accumulate) applies a binary operation to combine consecutive +elementsinasequence,e.g.,togeneratetheaggregates,like,totally(compare sum, +prod, all, max) or cumulatively (compare cumsum, cummmin), +• Filter creates a subset of a sequence that is comprised of elements that enjoy a +given property (which we typically achieve in R by means of the `[` operator), +• Find locates the first element that fulfils some logical condition (compare which), +and so forth. +Below we will only focus on the Map function. The inspection of the remaining ones is +left as an exercise. This is because, oftentimes, we can be better-off with their more +R-ish versions (e.g., using the subsetting operator, `[`). +7.2.4 +Vectorising Functions with Map +In data-centric computing, we are frequently faced with tasks that involve processing +eachandeveryelementinasequenceindependently,oneafteranother.Suchusecases +can benefit from vectorised operations like those discussed in Chapter 2, Chapter 3, +and Chapter 6. +Most of the functions that we have introduced in the preceding parts, unfortunately, +cannot be applied on lists. For instance, if we try calling sqrt on a list, we will get an +error, even if it is a list of numeric vectors only. One way to compute the square root of +all elements would be to invoke sqrt(unlist(...)). It is a go-to approach if we wish +to treat all the list’s elements as one sequence. But this comes at a price of losing the +list’s structure. +Wehavealsodiscussedsomeoperationsthatarenotvectorisedwithrespecttoalltheir +arguments, even though they could have been designed this way, e.g., grepl. +The Map function1 applies an operation on each element in a vector or the correspond- +ing elements in a number of vectors. In many situations, it may be used as a more +elegant alternative to for loops that we will introduce in the next chapter. +1 Yes, the author is aware that Map was implemented using the slightly more primitive mapply, but we are +not fond of the latter’s having the SIMPLIFY argument set to TRUE by default. + +124 +I DEEP +First2, a call to Map(f, x) yields a list whose i-th element is equal to f(x[[i]]) (recall +that `[[` works on atomic vectors too). +For example: +x <- list( +# an example named list +x1=1:3, +x2=seq(0, 1, by=0.25), +x3=c(1, 0, NA_real_, 0, 0, 1, NA_real_) +) +Map(sqrt, x) +# x is named, hence the result will be named too +## $x1 +## [1] 1.0000 1.4142 1.7321 +## +## $x2 +## [1] 0.00000 0.50000 0.70711 0.86603 1.00000 +## +## $x3 +## [1] +1 +0 NA +0 +0 +1 NA +Map(length, x) +## $x1 +## [1] 3 +## +## $x2 +## [1] 5 +## +## $x3 +## [1] 7 +unlist(Map(mean, x)) +# compute three aggregates, convert to an atomic vector +## +x1 +x2 +x3 +## 2.0 0.5 +NA +Map(function(n) round(runif(n, -1, 1), 1), c(2, 4, 6)) +# x is atomic now +## [[1]] +## [1] 0.4 0.8 +## +## [[2]] +## [1] +0.5 +0.8 -0.1 -0.7 +## +## [[3]] +## [1] -0.3 +0.0 +0.5 +1.0 -0.9 -0.7 +Next, we can vectorise a given function over a number of parameters. A call to, e.g., +Map(f, x, y, z) results in a list whose i-th element is equal to f(x[[i]], y[[i]], +z[[i]]). Just like in case of, e.g., paste, recycling rule will be applied if necessary. +2 This use case scenario can also be programmed using lapply; lapply(x, f, ...) is equivalent to Map(f, +x, MoreArgs=list(...)). + +7 FUNCTIONS +125 +For example, the following generates list(seq(1, 6), seq(11, 13), seq(21, 29)): +Map(seq, c(1, 11, 21), c(6, 13, 29)) +## [[1]] +## [1] 1 2 3 4 5 6 +## +## [[2]] +## [1] 11 12 13 +## +## [[3]] +## [1] 21 22 23 24 25 26 27 28 29 +Moreover, we can get list(seq(1, 40, length.out=10), seq(11, 40, length.out=5), +seq(21, 40, length.out=10), seq(31, 40, length.out=5)) by calling: +Map(seq, c(1, 11, 21, 31), 40, length.out=c(10, 5)) +## [[1]] +## +[1] +1.0000 +5.3333 +9.6667 14.0000 18.3333 22.6667 27.0000 31.3333 +## +[9] 35.6667 40.0000 +## +## [[2]] +## [1] 11.00 18.25 25.50 32.75 40.00 +## +## [[3]] +## +[1] 21.000 23.111 25.222 27.333 29.444 31.556 33.667 35.778 37.889 40.000 +## +## [[4]] +## [1] 31.00 33.25 35.50 37.75 40.00 +Note +If we have some additional arguments to be passed to the function applied +(which the function does not have to be vectorised over), we can wrap them inside +a separate list and toss it via the MoreArgs argument (à la do.call). +unlist(Map(mean, x, MoreArgs=list(na.rm=TRUE))) +# mean(..., na.rm=TRUE) +## +x1 +x2 +x3 +## 2.0 0.5 0.4 +Alternatively, we can always construct a custom anonymous function: +unlist(Map(function(xi) mean(xi, na.rm=TRUE), x)) +## +x1 +x2 +x3 +## 2.0 0.5 0.4 + +126 +I DEEP +Exercise 7.6 Hereisanexamplelistoffiles(seeourteachingdatarepository3)withdailyForex +rates: +file_names <- c( +"euraud-20200101-20200630.csv", +"eurgbp-20200101-20200630.csv", +"eurusd-20200101-20200630.csv" +) +Call Map to read each dataset with scan and determine the minimal, mean, and maximal value +in each series. +Exercise 7.7 Implement your own version of the Filter function based on a call to Map. +7.3 +Accessing Third-Party Functions +When we indulge in the writing of a software piece, a few questions naturally arise. Is +the problem we are facing fairly complex? Has it already been successfully addressed +in its entirety? If not, can it, or its parts, be split into manageable chunks? Can it be +constructed based on some readily available nontrivial components? +A smart developer is independent, but knows when to stand on the shoulders to cry +on. Let us explore some ways in which we can reuse the existing function libraries. +7.3.1 +Using R Packages +Most contributed R extensions come in the form of the so-called add-on packages, +which can include: +• reusable code (e.g., new functions), +• data (which we can exercise on), +• documentation (manuals, vignettes, etc.); +see Section 9.3.2 for some more and [45] for all the details. +Most packages are published in the moderated repository that is part of the Compre- +hensive R Archive Network (CRAN4). However, there are also other popular sources such +as Bioconductor5 which specialises in bioinformatics. +To fetch a package pkg from a repository (CRAN by default; see, however, the repos +argument), we call install.packages("pkg"). +3 https://github.com/gagolews/teaching-data/tree/master/marek +4 https://cloud.r-project.org/ +5 https://bioconductor.org/ + +7 FUNCTIONS +127 +A call to library("pkg") loads an indicated package and makes its exported objects +available to the user (i.e., attaches it on the search list; see sec:to-do). +For instance, in one of the previous chapters, we have mentioned the gsl package: +# call install.packages("gsl") first +library("gsl") +# load the package +poch(10, 3:6) +# calls gsl_sf_poch() from GNU GSL +## [1] +1320 +17160 +240240 3603600 +Here, poch is an object exported by package gsl. If we did not call library("gsl"), +trying to access the former would result in an error. +We could also have accessed the above function without attaching it onto the object +search list by using the pkg::object syntax, i.e., gsl::poch. +Exercise 7.8 Use the find function to determine which packages define the following objects: +mean, var, find, and Map. Recall from Section 1.4 where such information can be found in these +objects’ manual pages. +Note For more information about any R extension, call help(package="pkg"). Also, +it is a good idea to visit the package’s CRAN entry at an address like https://CRAN. +R-project.org/package=pkg to access some additional information (e.g., vignettes; +see also vignette(package="pkg")). Why waste our time and energy by querying a web +search engine that will lead us to some (usually low-quality) middleman when you can +acquire authoritative knowledge directly from the source? +Moreover, it is worth exploring various CRAN Task Views6 that group the packages +into topics such as Genetics, Graphics, and Optimisation. These are edited by experts in +their relevant fields. +Important Frequently, R packages are written in their respective authors’ free time, +many of whom are volunteers/public servants/enthusiasts who are neither paid for +doing this nor it is part of the so-called their job. You can show appreciation for their +generosity by, e.g., spreading the word about their software by citing them in public- +ations (see citation(package="pkg")), talking about them during lunch time, or men- +tioning them in (un)social media. You can also help them improve the existing code +base by reporting bugs, polishing documentation, proposing new features, or clean- +ing up the redundant fragments of their APIs. Some readers will become one of them +someday (when they will come up with something useful for our community). +6 https://cloud.r-project.org/web/views/ + +128 +I DEEP +Default Packages +Note that the always-on package base is a must-have that provides us with the most +crucial functions (vector addition, c, Map, library). Certain other packages are also +loaded by default: +getOption("defaultPackages") +## [1] "datasets" +"utils" +"grDevices" "graphics" +"stats" +## [6] "methods" +Although this list can – technically speaking – be changed, in this book we assume +that the above are always attached, because it is reasonable to do so. This is why in +Section 2.4.5, there was no need to call, for example, library("stats") before refer- +ring to the var and sd functions. +On a side note, grDevices and graphics will be discussed in sec:to-do and methods +will be mentioned in sec:to-do. datasets brings a few example R objects that we can +exercise our skills on. Functions from utils, graphics, and stats, on the other hand, +already appeared here and there. +Source vs Binary Packages (*) +R is a free and open project, therefore its packages are published primarily in the +source form – so that anyone can study how they work and improve them or reuse +parts thereof in different projects. +If we call install.packages("path", repos=NULL, type="source"), we should be able +toinstallapackagefromsources: pathcaneitherbepinpointingadirectoryorasource +tarball (see help("untar"), most often as a compressed pkg_version.tar.gz file). +Notethattype="source"isthedefaultunlessoneisonW****wsorsomem**OSboxes; +see getOption("pkgType"). This is because these two might require additional build +tools to be present in the system, especially if a package features C, C++, or Fortran +code; see Chapter 14 and Section C.3 of [47]: +• Rtools7 on W****ws, +• Xcode Command Line Tools8 on m**OS. +Because of these systems’ being less developer-oriented, as a courtesy to their users, +CRAN also distributes the platform-specific binary versions of the packages (.zip or +.tgz files). install.packages will try to fetch them by default. +Example 7.9 It is very easy to fetch a package’s source directly from GitLab or GitHub, which +arequitepopularhostingplatformsthesedays.Atthetimeofwritingthis,therelevantlinkswere, +respectively: +• https://gitlab.com/user/repo/-/archive/branch/repo-branch.zip +• https://github.com/user/repo/archive/branch.zip +7 https://cran.r-project.org/bin/windows/Rtools/ +8 https://developer.apple.com/xcode/resources/ + +7 FUNCTIONS +129 +For example, to download the contents of the master branch in the repository rpackagedemo +owned by gagolews, we can call: +f <- tempfile() +# temporary file name - download destination +download.file("https://github.com/gagolews/rpackagedemo/archive/master.zip", +destfile=f) +Next, the contents can be extracted with unzip: +t <- tempdir() +# temporary directory to extract the files to +(d <- unzip(f, exdir=t)) +# returns extracted file paths +The path where the files were extracted can be passed to install.packages: +install.packages(dirname(d)[1], repos=NULL, type="source") +file.remove(c(f, d)) +# clean up +Exercise 7.10 Use the git2r package to clone the git repository located at https://github.com/ +gagolews/rpackagedemo.git and install the package published therein from within the current +R session. +7.3.2 +Managing Dependencies (*) +The currently-installed add-on packages may be upgraded to their most recent ver- +sions available on CRAN (or other indicated repository) by calling update.packages. +As a general rule, the more experienced developers we become, the less excited we get +aboutthenew.Sure,bugfixesandsomewell-thoughtofadditionalfeaturesareusually +welcome, but just we wait until an updated package API for the n-th time, 𝑛 ≥ 2, +breaks our program that used to work flawlessly for so long. +Hence, when designing software projects (see Chapter 9 for more details), it is essen- +tial that we ask ourselves the ultimate question: do we really need to import that pack- +age with lots of dependencies from which we will just use only about 3–5 functions? +Wouldn’t it be better to write our own version of some functionality (and learn some- +thing new, exercise our brain, etc.) or call a mature terminal-based tool? +Otherwise, as all the historical versions of all the packages are archived on CRAN9, +some software dependency management can easily be conducted by storing differ- +ent version of packages in different directories (only one version of a package can be +loaded at a time though). This way, wecan createsome sort of an isolated environment +for the add-ons. +To fetch the locations where packages are sought (in this very order), call: +.libPaths() +## [1] "/home/gagolews/R/x86_64-pc-linux-gnu-library/4.2" +(continues on next page) +9 https://cran.r-project.org/src/contrib/Archive/ + +130 +I DEEP +(continued from previous page) +## [2] "/usr/local/lib/R/site-library" +## [3] "/usr/lib/R/site-library" +## [4] "/usr/lib/R/library" +Thesamefunctioncanbeusedtoaddnewfolderstothesearchpath;seealsotheenvir- +onment variable R_LIBS_USER (e.g., help("Sys.setenv")). The install.packages func- +tion will honour them as target directories, see its lib parameter for more details. +Moreover,thepackagesmaydepositsomeauxiliarydataontheuser’smachine.There- +fore, it might be a good idea to set the following directories (via the corresponding +environment variables) as relative to the current project: +tools::R_user_dir("pkg", "data") +# R_USER_DATA_DIR +## [1] "/home/gagolews/.local/share/R/pkg" +tools::R_user_dir("pkg", "config") # R_USER_CONFIG_DIR +## [1] "/home/gagolews/.config/R/pkg" +tools::R_user_dir("pkg", "cache") +# R_USER_CACHE_DIR +## [1] "/home/gagolews/.cache/R/pkg" +7.3.3 +Calling External Programs +Many tasks can naturally be accomplished by calling external programs. Such an ap- +proachisparticularlynaturalonUnix-likesystems,whichclassicallyfollowamodular, +minimalistic design patterns: there are many tools at a developer’s hand and each tool +is specialised at solving a single, well-defined problem. +Apart from the many standard Unix commands10, we can consider, for example: +• pandoc11 converts documents between markup formats, e.g., Markdown, reStruc- +turedText, LaTeX, LibreOffice Writer, EPUB; +• pdflatex, xelatex, and lualatex compile LaTeX documents to PDF; +• convert (from ImageMagick12) applies various operations on bitmap graphics +(scaling, cropping, conversion between formats); +• graphviz13 and PlantUML14 can be used to create various graphs and diagrams; +• jupyter-nbconvert converts Jupyter15 notebooks (see Section 1.2.5) to other +formats such as LaTeX, HTML, Markdown, etc.; +• python,{command}perl,…canbecalledtoperformtasksthatcanbeexpressedmore +easily in languages other than R; +10 https://en.wikipedia.org/wiki/List_of_Unix_commands +11 https://pandoc.org/ +12 https://imagemagick.org/ +13 https://graphviz.org/ +14 https://plantuml.com/ +15 https://jupyter.org/ + +7 FUNCTIONS +131 +and so forth. +Good news is that R not only can be called from the shell (in an interactive or batch +mode; see Section 1.2), but also it can serve well as a glue language itself. +The system2 function can be used to invoke any system command. Communication +between such programs can be done by means of, e.g., intermediate text, JSON, CSV, +XML, or any other files. The stdin, stdout, and stderr arguments can be used to con- +trol the redirection of the standard I/O streams. +system2("pandoc", "-s input.md -o output.html") +system2("bash", "-c 'for i in `seq 1 2 10`; do echo $i; done'", stdout=TRUE) +## [1] "1" "3" "5" "7" "9" +system2("python3", "-", stdout=TRUE, +input=c( +"import numpy as np", +"print(repr(np.arange(5)))" +)) +## [1] "array([0, 1, 2, 3, 4])" +Note that the current working directory can be read and changed by means of a call to +getwd and setwd, respectively. It is the directory from where the current R session was +started. +Important Relying on system2 assumes that the commands referred to are available +onthetargetplatform.Hence,itmightnotbeportable,unlessadditionalassumptions +are made (e.g., that a user runs some Unix system, that certain libraries are installed +therein). We strongly recommend GNU/Linux or FreeBSD for both software devel- +opment and production use, as they are free, open, developer-friendly, user-loving, +reliable, ethical, and sustainable. +7.3.4 +A Note on Interfacing C, C++, Python, Java, etc. (*) +Most stand-alone data processing algorithms are implemented in compiled, slightly +lower-level programming languages. This usually makes them faster and more re- +usable in other environments. For instance, it is often the case that an industry- +standard library is written in very portable C, C++, or Fortran and has some bindings +availableforeasieraccess fromwithin R,Python, Julia,etc.This isthe casewith FFTW, +LIBSVM, mlpack, OpenBLAS, ICU, and GNU GSL, amongst many others. +For basic ways to interact with such compiled code, see Chapter 14. +Also, the rJava package can be used to dynamically create JVM objects and access their +fields and methods. Similarly, reticulate can be used to access Python objects, in- +cluding numpyarraysand pandasdataframes(butseealsothe rpy2packageforPython). +Important +We should not feel obliged to use R in all the parts of a data pro- + +132 +I DEEP +cessing pipeline. Some activities can be expressed more naturally in other lan- +guages/environments (e.g., parse raw data and create an SQL database in Python, but +visualise it in R). We can use other tools as the glue language (including R, Python, or +Bash) that will steer the data flow in the right direction. +7.4 +Exercises +Exercise 7.11 Answer the following questions: +• What is the result of “x <- 2; x <- function(x) x^2; x(x)”? +• How to write a function that returns two objects? +• What is a higher-order function? +• What are the use cases of do.call? +• Why a call to Map is not necessary in the expression “Map(paste, x, y, z)”? +• What is the difference between Map(mean, x, na.rm=TRUE) and Map(mean, x, More- +Args=list(na.rm=TRUE))? +• What do we mean when we write stringx::sprintf? +• How to get access to the vignettes (tutorials, FAQs, etc.) of the data.table and dplyr pack- +ages? Why perhaps 95% of R users would just googleit and what is sub-optimal about this +strategy? +• What is the difference between a source and a binary package? +• How to update the base package? +• How to assure that we will always run an R session with only specific versions of a set of +packages? +Exercise 7.12 Write a function that computes the Gini index of a vector of positive integers x, +which, assuming 𝑥1 ≤ 𝑥2 ≤ … ≤ 𝑥𝑛, is equal to: +𝐺(𝑥1, … , 𝑥𝑛) = ∑𝑛 +𝑖=1(𝑛 − 2𝑖 + 1)𝑥𝑖 +(𝑛 − 1) ∑𝑛 +𝑖=1 𝑥𝑖 +. +Exercise 7.13 Implement a function between(x, a, b) that verifies whether each element +in x is in the [a, b] interval or not. Return a logical vector of the same length as x. Make sure +the function is correctly vectorised with respect to all the arguments and handles missing data +correctly. +Exercise 7.14 Write your own version of the strrep function called dup. + +7 FUNCTIONS +133 +dup <- ...to.do... +dup(c("a", "b", "c"), c(1, 3, 5)) +## [1] "a" +"bbb" +"ccccc" +dup("a", 1:3) +## [1] "a" +"aa" +"aaa" +dup(c("a", "b", "c"), 4) +## [1] "aaaa" "bbbb" "cccc" +Exercise 7.15 Given a list x, generate its sublist with all the elements equal to NULL removed. +Exercise 7.16 Implement your own version of the built-in sequence function. +Exercise 7.17 Using Map, how can we generate window indexes like: +## [[1]] +## [1] 1 2 3 +## +## [[2]] +## [1] 2 3 4 +## +## [[3]] +## [1] 3 4 5 +## +## [[4]] +## [1] 4 5 6 +Writeafunction windows(k, n)thatyieldskindexwindowswithelementsbetween1andn(the +above example is for k=3 and k=6). +Exercise 7.18 Implement a function movstat(f, x, k) that computes, using Map, a given ag- +gregate f of each k consecutive elements in x. For instance: +movstat <- ...to.do... +x <- c(1, 3, 5, 10, 25, -25) +# example data +movstat(mean, x, 3) +# 3-moving mean +## [1] +3.0000 +6.0000 13.3333 +3.3333 +movstat(median, x, 3) +# 3-moving median +## [1] +3.0000 +6.0000 13.3333 +3.3333 +Exercise 7.19 Write a function to extract all q-grams, q ≥ 1, from a given character vector. +Return a list of character vectors. For examples, 2-grams (bigrams) in "abcd" are: "ab", "bc", +“cd”`. +Exercise 7.20 Recodeacharactervectorwithasmallnumberofdistinctvaluestoavectorwhere +each unique code is assigned a positive integer from 1 to k. Example calls and the corresponding +expected results: + +134 +I DEEP +recode <- ...to.do... +recode(c("a", "a", "a", "b", "b")) +## [1] 1 1 1 2 2 +recode(c("x", "z", "y", "x", "y", "x")) +## [1] 1 3 2 1 2 1 +Exercise 7.21 Implement a function that returns the number of occurrences of each unique ele- +mentinagivenatomicvector.Thereturnvalueshouldbeanumericvectorequippedwitha names +attribute. +count <- ...to.do... +count(c(5, 5, 5, 5, 42, 42, 954)) +## +5 +42 954 +## +4 +2 +1 +count(c("x", "z", "y", "x", "y", "x", "w", "x", "x", "y", NA_character_)) +## +w +x +y +z +## +1 +5 +3 +1 +1 +Hint: use match and tabulate. +Exercise 7.22 Implement a function that extends upon the built-in duplicated, indicating +which occurrence (starting from the beginning of the vector) of a repeated value a given value +constitutes. +duplicatedn <- ...to.do... +duplicatedn(c("a", "a", "a", "b", "b")) +## [1] 1 2 3 1 2 +duplicatedn(c("x", "z", "y", "x", "y", "x", "w", "x", "x", "y", "z")) +## +[1] 1 1 1 2 2 3 1 4 5 3 2 +Exercise 7.23 Based on a call to Map, implement a function my_split such that, given a vec- +tor x and an atomic vector y of the same length as x, my_split(x, y) yields the same result as +split(x, y). +Exercise 7.24 Extend my_split to handle the second argument being a list of the form +list(y1, y2, ...) that represents the product of many levels. If the ys are of different lengths, +apply the recycling rule. +Exercise 7.25 Implement my_unsplit being your own version of the built-in unsplit. Make +sure it holds my_unsplit(split(x, g), g) == x for x and g of the same lengths. +Exercise 7.26 Write a function that takes as arguments: (a) an integer n, (b) a numeric vector +x of length k and no duplicated elements, (c) a vector of probabilities p of length k; verify that +𝑝𝑖 ≥ 0 for all 𝑖 and ∑𝑘 +𝑖=1 𝑝𝑖 ≃ 1. Based on a random number generator from the uniform +distribution on the unit interval, generate n independent realisations of a random variable 𝑋 +such that Pr(𝑋 = 𝑥𝑖) = 𝑝𝑖 for 𝑖 = 1, … , 𝑘. Hint: to obtain a single value: +1. generate 𝑢 ∈ [0, 1], + +7 FUNCTIONS +135 +2. find 𝑚 ∈ {1, … , 𝑘} such that 𝑢 ∈ (∑𝑚−1 +𝑗=1 𝑝𝑗, ∑𝑚 +𝑗=1 𝑝𝑗], +3. the result is then 𝑥𝑚. +Exercise 7.27 Write a function that takes as arguments: (a) an increasingly sorted vector x of +length n, (b) any vector y of length n, (c) a vector z of length k and elements in [𝑥1, 𝑥𝑛). Let 𝑓 be +the piecewise linear spline that interpolates the points (𝑥1, 𝑦1), … , (𝑥𝑛, 𝑦𝑛). Return a vector w +of length k such that 𝑤𝑖 = 𝑓 (𝑧𝑖). +Exercise 7.28 (*) Write functions dpareto, ppareto, qpareto, and rpareto that implement +the basic functions related to the Pareto distribution; compare Section 2.3.4. + + +8 +Flow of Execution +The ifelse and Map functions are very powerful, but they allow us to process only the +consecutive elements in a vector. +Thus, let us (finally!) discuss different ways to alter a program’s control flow manually, +based on some criterion, and to evaluate the same expression a number of times, but +perhapsondifferentdata.Beforeproceedinganyfurther,letus,however,contemplate +on the fact that we have managed to do without them for such a long time – and the +data processing exercises we learnt to solve were far from trivial. +8.1 +Conditional Evaluation +Life is full of surprises, so we would be nice if we were able to adapt to whatever the +circumstances are going to be. +The following evaluates a given expression if and only if a logical condition is true. +if (condition) expression +When performing some other_expression is preferred rather than doing nothing in +the case of the condition’s being false, we can write: +if (condition) expression else other_expression +For instance: +(x <- runif(1)) +# to spice things up +## [1] 0.28758 +if (x > 0.5) cat("head") else cat("tail") +## tail +Many expressions can of course be grouped with curly braces, “{” +if (x > 0.5) { +cat("head") +x <- 1 +(continues on next page) + +138 +I DEEP +(continued from previous page) +} else { +cat("tail") +x <- 0 +} +## tail +print(x) +## [1] 0 +Important Atthetoplevel,weshouldnotputanewlinebefore else,otherwisewewill +get an error like Error: unexpected 'else' in "else". This is because the interpreter +enthusiastically executes the statements been read line by line as soon as it regards +them as stand-alone expressions. In this case, we first get an if without else, and +then, separately, a dangling else without the preceding if. +This does not happen when a conditional statement is part of an expression group, +because the latter is read in its entirety. +function (x) +{ +# opening bracket – start +if (x > 0.5) +cat("head") +else +# not dandling, because {...} is read as a whole +cat("tail") +} +# closing bracket – expression ends +As an exercise, try removing the curly braces and see what happens. +8.1.1 +Return Value +`if` is a function (compare Section 9.4), hence has a return value – the result of eval- +uating the conditional expression. +(x <- runif(1)) +## [1] 0.28758 +y <- if (x > 0.5) "head" +# no else +print(y) +## NULL +y <- if (x > 0.5) "head" else "tail" +print(y) +## [1] "tail" +This is particularly useful when a call to `if` is the last expression in the code block +constituting a function’s body. + +8 FLOW OF EXECUTION +139 +mint <- function(x) +{ +if (x > 0.5) +# the last expression (actually, the only one) +"head" +# this can be the return value... +else +"tail" +# or this one, depending on the condition +} +mint(x) +## [1] "tail" +unlist(Map(mint, runif(5))) +## [1] "tail" "head" "tail" "head" "head" +Example 8.1 Add-on packages can also be loaded using requireNamespace. Contrary to lib- +rary, the former does not fail when a package is not available. Also, it does not attach it on the +search list; see sec:to-do. +Instead, it returns a logical value indicating if the package is available for use. This can be use- +ful inside other functions where the availability of some additional features depends on the user +environment’s configuration: +process_data <- function(x) +{ +if (requireNamespace("some_extension_package", quietly=TRUE)) +some_extension_package::very_fast_method(x) +else +normal_method(x) +} +8.1.2 +Nested ifs +If more than two test cases are possible, i.e., when we need to go beyond either con- +dition or !condition, then we can use the following construction: +if (a) { +expression_a +} else if (b) { +expression_b +} else if (c) { +expression_c +} else { +expression_else +} +This evaluates all conditions a, b, … (in this order) until the first positive case is found, +and then executes the corresponding expression. + +140 +I DEEP +Note that the above is nothing else than a series of nested if statements: +if (a) { +expression_a +} else { +if (b) { +expression_b +} else { +if (c) { +expression_c +} else { +expression_else +} +} +} +but written in a less readable1 manner. +Exercise 8.2 Write a function named sign that determines if a given numeric value is "pos- +itive", "negative", or "zero". +8.1.3 +Condition: Either True of False +if expects a condition that is a single, well-defined logical value, either TRUE or FALSE. +Thence, problems may arise when this is not the case. +If the condition is of length not equal to one, we get an error: +if (c(TRUE, FALSE)) cat("spam") +## Error in if (c(TRUE, FALSE)) cat("spam"): the condition has length > 1 +if (logical(0)) cat("bacon") +## Error in if (logical(0)) cat("bacon"): argument is of length zero +We cannot pass a missing value either: +if (NA) cat("ham") +## Error in if (NA) cat("ham"): missing value where TRUE/FALSE needed +Important If we think that we are absolutely immune to the writing of code violating +the above constraints, just we wait until the condition becomes a function of data for +which there is no sanity-checking in place. +mint <- function(x) +if (x > 0.5) "H" else "T" +(continues on next page) +1 Somewhat related is the switch function which we study in sec:to-do. It relies on lazy evaluation of its +arguments. Still, it can always be replaced by a series of ifs. + +8 FLOW OF EXECUTION +141 +(continued from previous page) +mint(0.25) +## [1] "T" +mint(runif(5)) +## Error in if (x > 0.5) "H" else "T": the condition has length > 1 +mint(log(rnorm(1))) +# not obvious, only triggered sometimes +## Warning in log(rnorm(1)): NaNs produced +## Error in if (x > 0.5) "H" else "T": missing value where TRUE/FALSE needed +In Chapter 9, we will be particularly interested in ways to assure input data integrity, +so that situations such as above will either fail gracefully or succeed bombastically. +Here, we should probably make sure that x is a single finite numeric value. Alternat- +ively, we had rather test whether all(x > 0.5, na.rm=TRUE). +Interestingly, objects other that logical are accepted: they will be coerced if needed. +x <- 1:5 +if (length(x)) +# i.e., length(x) != 0, but way less readable +cat("length is not zero") +## length is not zero +Recall that coercion of numeric to logical yields FALSE if and only if the original value +is zero. +8.1.4 +Short-Circuit Evaluation +Specially for formulating logical conditions in if and while (see below), we have the +scalar `||` (alternative) and `&&` (conjunction) operators. +FALSE || TRUE +## [1] TRUE +NA || TRUE +## [1] TRUE +Contrary to their vectorised counterparts (`|` and `&`), the scalar operators are lazy +(Section 9.5.5) in the sense that they evaluate the first operand and then determine +if the computing of the second one is necessary (because, e.g., FALSE && whatever is +always FALSE anyway). +Therefore, +if (a && b) +expression +is equivalent to: + +142 +I DEEP +if (a) { +if (b) { +# compute b only if a is TRUE +expression +} +} +and: +if (a || b) +expression +corresponds to: +if (a) { +expression +} else if (b) { +# compute b only if a is FALSE +expression +} +For instance, “is.vector(x) && length(x) > 0 && x[[1]] > 0” is a safe test that +takes into account that “x[[1]]” has only the desired meaning for objects that are not +non-empty vectors. +Some other examples (recall that the expressions within the curly braces are evaluated +one after another and that the result is determined by the last value in the series): +{cat("spam"); FALSE} || {cat("ham"); TRUE} || {cat("cherries"); FALSE} +## spamham +## [1] TRUE +{cat("spam"); TRUE} && {cat("ham"); FALSE} && {cat("cherries"); TRUE} +## spamham +## [1] FALSE +Exercise 8.3 Study the source code of isTRUE and isFALSE and determine if these functions +could be useful in formulating the conditions within the if expressions. +8.2 +Exception Handling +Exceptionsareexceptional,buttheymayhappenandbreakthings.Forinstance,when +we try to download a file and the internet connection drops. Or an optimisation al- +gorithm fails to converge. Or we just have a bug in our code. Or: + +8 FLOW OF EXECUTION +143 +read.csv("/path/to/a/file/that/does/not/exist") +## Warning in file(file, "rt"): cannot open file '/path/to/a/file/that/does/ +## not/exist': No such file or directory +## Error in file(file, "rt"): cannot open the connection +Three types of conditions are frequently observed: +• errors – they stop the flow of execution, +• warnings – non critical, but can be turned into errors (see warn in option), +• messages – they transmit some diagnostic information. +These can be manually triggered by means of stop, warning, and message functions. +Errors (but warnings too) can be handled by means of the tryCatch function, amongst +others. +tryCatch({ +# block of expressions to execute, until an error occurs +cat("a\n") +stop("b") +# error – breaks the linear control flow +cat("c\n") +}, +error = function(e) { +# executed immediately upon an error +cat(sprintf("error: %s\n", e[["message"]])) +}, +finally = { +# always executed at the end, regardless of error occurrence +cat("finally!\n") +} +) +## a +## error: b +## finally! +The two other conditions can be ignored by calling suppressWarnings and suppress- +Messages. +log(-1) +## Warning in log(-1): NaNs produced +## [1] NaN +suppressWarnings(log(-1)) +# yeah, yeah, we know what we're doing +## [1] NaN +Exercise 8.4 Atthetimeofwritingofthisbook,the data.tablepackageemitsamessageupon +attachment. Call suppressMessages to silence it. Note that consecutive calls to library do not +reload an already loaded package, therefore the message will only be seen once per R session. +Related functions include stopifnot discussed in Section 9.2 and on.exit mentioned +in sec:to-do; see also Section 9.3.4 for some code debugging tips. + +144 +I DEEP +8.3 +Repeated Evaluation +And now for something completely different… time for the elephant in the room! +We have been able to do without loops so far (and will be quite all right in the second +part of the book too), because many data processing tasks can be written in terms of +vectorised operations such as `+`, sqrt, sum, `[`, Map, and Reduce. Oftentimes, com- +pared to their loop-based counterparts, they are not only much more readable but also +more efficient. We will explore this in the exercises below. +However, at times, using an explicit while or for loop might be the only natural way +of solving a problem, for instance, when processing chunks of data streams. Also, an +explicitly “looped” algorithm may occasionally have better2 time or memory complex- +ity. +8.3.1 +while +if considers a given logical condition and thus determines whether to execute a given +statement. On the other hand, +while (condition) +# single TRUE or FALSE, as in `if` +expression +evaluates a given expression as long as the logical condition is true. Therefore, it is ad- +visable to make the condition dependent upon some variable that can be modified by +the expression. +i <- 1 +while (i <= 3) { +cat(sprintf("%d, ", i)) +i <- i + 1 +} +## 1, 2, 3, +Nested loops are of course possible too: +i <- 1 +while (i <= 2) { +j <- 1 +while (j <= 3) { +cat(sprintf("%d %d, ", i, j)) +j <- j + 1 +} +(continues on next page) +2 But in such cases it will often benefit from a rewrite in C or C++; see Chapter 14. + +8 FLOW OF EXECUTION +145 +(continued from previous page) +cat("\n") +i <- i + 1 +} +## 1 1, 1 2, 1 3, +## 2 1, 2 2, 2 3, +Example 8.5 Implement a simple linear congruential pseudorandom number generator that, +given some seed 𝑋0 ∈ [0, 𝑚), outputs a sequence (𝑋1, 𝑋2, … ) defined by: +𝑋𝑖 = (𝑎𝑋𝑖−1 + 𝑐) +mod 𝑚, +with, e.g., 𝑎 = 75, 𝑐 = 74, and 𝑚 = 216 + 1 (here, mod is the division reminder, `%%`). Note +thatthisgeneratorhaspoorstatisticalpropertiesandshouldnotbeusedinpractice.Inparticular, +after some number of operations 𝑘, we will find a cycle such that 𝑋𝑘 = 𝑋1, 𝑋𝑘+1 = 𝑋2, …. +8.3.2 +for +The for-each loop: +for (name in vector) +expression +takes each element, from the beginning to the end, in a given vector, assigns it some +name, and evaluates the expression. +Example: +fridge <- c("spam", "spam", "bacon", "eggs") +for (food in fridge) +cat(sprintf("%s, ", food)) +## spam, spam, bacon, eggs, +One more: +for (i in 1:length(fridge)) +# better: seq_along(fridge), see below +cat(sprintf("%s, ", fridge[i])) +## spam, spam, bacon, eggs, +Just one more, promise: +for (i in 1:2) { +for (j in 1:3) +cat(sprintf("%d %d, ", i, j)) +cat("\n") +} +## 1 1, 1 2, 1 3, +## 2 1, 2 2, 2 3, + +146 +I DEEP +Note that the iterator still exists after the loop’s watch has ended: +print(i) +## [1] 2 +print(j) +## [1] 3 +Important Writing: +for (i in 1:length(x)) +print(x[i]) +is not necessarily safe, because if x is an empty vector, then: +x <- logical(0) +for (i in 1:length(x)) print(x[i]) +## [1] NA +## logical(0) +Recall from Chapter 5 that x[1] tries to access an out of bounds element here and x[0] +returns nothing. +Wegenerallysuggestreplacing1:length(x)withseq_along(x)orseq_len(length(x)). +wherever possible. +Note The model for loop above is roughly equivalent to: +name <- NULL +tmp_vector <- vector +tmp_iter <- 1 +while (tmp_iter <= length(tmp_vector)) { +name <- tmp_vector[[tmp_iter]] +expression +tmp_iter <- tmp_iter + 1 +} +Note that tmp_vector is determined before the loop itself. Hence, any changes to vec- +tor will not influence the execution flow. Also note that due to the use of `[[`, the loop +can be applied on lists as well. +Example 8.6 Let x be a list and f be a function. The following code generates the same result as +Map(f, x): +n <- length(x) +(continues on next page) + +8 FLOW OF EXECUTION +147 +(continued from previous page) +ret <- vector("list", n) +# a new list of length `n` +for (i in seq_len(n)) +ret[[i]] <- f(x[[i]]) +Example 8.7 Letxandybetwolistsandfbeafunction.HereisthemostbasicversionofMap(f, +x, y). Note that x and y might be of different lengths. +nx <- length(x) +ny <- length(y) +n <- max(nx, ny) +ret <- vector("list", n) +for (i in seq_len(n)) +ret[[i]] <- f(x[[((i-1)%%nx)+1]], y[[((i-1)%%ny)+1]]) +Feelfreetoupgradetheabovebyaddingawarninglikethe longer argument is not a multiple +of the length of the shorter one.Also,rewriteitwithouttheuseofthemodulooperators,`%%`. +8.3.3 +break and next +breakcanbeusedtoescapethecurrentloop. nextskipstheremainingexpressionsand +advances to the next iteration (to where the testing of the logical condition occurs). +Here is a rather random example: +x <- runif(1000) +s <- 0 +for (e in x) { +if (e > 0.1) +next +print(e) +if (e < 0.01) +break +s <- s + e +} +## [1] 0.045556 +## [1] 0.04206 +## [1] 0.024614 +## [1] 0.045831 +## [1] 0.094841 +## [1] 0.00062477 +print(s) +## [1] 0.2529 + +148 +I DEEP +Computes the sum of the elements in x that are less than or equal to 0.1 from the be- +ginning, stopping at the first element less than 0.01. +Note that we have used the frequently occurring design pattern: +for (e in x) { +if (condition) +next +many_statements... +} +which is equivalent to: +for (e in x) { +if (!condition) { +many_statements... +} +} +but avoids introducing a nested block of expressions. +Note (*) There is a third loop type, +repeat +expression +which is a shorthand for +while (TRUE) +expression +i.e., it is a possibly infinite loop. Such loops are useful when implementing situations +such as do-stuff-until-a-thing-happens, e.g., when we want to execute a command at +least once. +i <- 1 +repeat { +# while (TRUE) +# simulate dice casting until we throw "1" +if (runif(1) < 1/6) break +# not an infinite loop after all +i <- i+1 +} +print(i) +## [1] 6 +Exercise 8.8 What is wrong with the following code? + +8 FLOW OF EXECUTION +149 +j <- 1 +while (j <= 10) { +if (j %% 2 == 0) next +print(j) +j <- j + 1 +} +Exercise 8.9 What about this one? +j <- 1 +while (j <= 10); +j <- j + 1 +8.3.4 +return +return, when called from within a function, immediately yields a specified value and +goes back to the caller. +For example, here is a simple recursive function that flattens a given list: +my_unlist <- function(x) { +if (is.atomic(x)) +return(x) +# so if we are here, x is definitely not atomic +out <- NULL +for (e in x) +out <- c(out, my_unlist(e)) +out +# or return(out); it's the last expression anyway, so not necessary +} +my_unlist(list(list(list(1, 2), 3), list(4, list(5, list(6, 7:10))))) +## +[1] +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +Note that return is a function: the round brackets are obligatory, +8.3.5 +A Note on Time and Space Complexity of Algorithms (*) +Analysis of algorithms (e.g., [9, 34]), can give us a rough estimate of their run times or +memory consumption as a function of the input data size, especially for big data. +In scientific computing and data science, we most often deal with vectors (sequences) +or matrices/data frames (tabular data). Therefore, we might be interested in determ- +ining how many primitive operations need to be performed as a function of their length +n or the number of rows n and columns p, respectively. + +150 +I DEEP +The O (Big-Oh) notation, for instance, can express the upper bounds for time/resource +consumption in asymptotic cases. For instance, we say that the time complexity is +𝑂(𝑛2), if for large 𝑛, the number of operations to perform will be proportional to at +most the square of the vector size (more precisely, there exists 𝑚 and 𝐶 > 0 such that +for all 𝑛 > 𝑚, the number of operations is ≤ 𝐶𝑛2). +Therefore, if we have two algorithms that solve the same task, one that has 𝑂(𝑛2) time +complexity, and other of 𝑂(𝑛3), it is better to choose the former, because for large +problem sizes we expect it to be faster. +Moreover, whether time grows proportionally to log 𝑛, √𝑛, 𝑛, 𝑛 log 𝑛, 𝑛2, 𝑛3, or 2𝑛, +can be useful in predicting how big the data can be if we have a fixed deadline or not +too much space left on the disk. +Exercise 8.10 The hclust function determines a hierarchical clustering of a dataset. It is fed +withanobjectthatstoresthedistancebetweenallthepairsofinputpoints.Thereare𝑛(𝑛−1)/2 +(i.e., 𝑂(𝑛2)) unique point pairs for any given n. One numeric scalar (double type) takes 8 bytes +of storage. If you have 16 GB or RAM, what is the largest dataset that you can cluster on your +machine using this function? +Oftentimes, we can learn about the time or memory complexity of the functions we +use from their documentation; see, e.g., help("findInterval"). +Example 8.11 Acourseindatastructuresinalgorithms,whichthisoneisnot,willgiveusplenty +of opportunities to implement many algorithms ourselves. This way, we can gain a lot of insights +and intuitions. +For instance, this is a 𝑂(𝑛)-time algorithm: +for (i in seq_len(n)) +expression +and this is one runs in 𝑂(𝑛2)-time: +for (i in seq_len(n)) +for (j in seq_len(n)) +expression +as long as, of course, the expression is rather primitive (e.g., operations on scalar variables). +R is a very expressive language and hence quite complex and lengthy operations can look pretty +innocent (it is a glue language for rapid prototyping, after all). +For example: +for (i in seq_len(n)) +for (j in seq_len(n)) +z <- z + x[[i]] + y[[j]] +can be seen as 𝑂(𝑛3) if each element in the lists x and y as well as z itself are atomic vectors of +length n. + +8 FLOW OF EXECUTION +151 +Similarly, +Map(mean, x) +is 𝑂(𝑛2) if x is a list of n atomic vectors each of length n. +Note A quite common statistical scenario involves the generation of a data buffer of +a fixed size: +ret <- c() +for (i in n) +ret[[i]] <- generate_data(i) +# here: ret[[length(ret)+1]] <- ... +Thisnotation, however,involvesthe growingofthe retarrayineachiteration.Luckily, +sinceRversion3.4.0,eachsuchsizeextensionhasamortised𝑂(1)timeduetothefact +that some more memory is internally reserved for its prospective growth (see, e.g., +Chapter 17 of [9]). +However, it would still be better to pre-allocate the output vector and grant it the de- +sired, final size already upon creation. +Note that we can construct vectors of specific lengths and types in an efficient way +(more efficient than with rep) by calling: +numeric(3) +## [1] 0 0 0 +numeric(0) +## numeric(0) +logical(5) +## [1] FALSE FALSE FALSE FALSE FALSE +character(2) +## [1] "" "" +vector("numeric", 8) +## [1] 0 0 0 0 0 0 0 0 +vector("list", 2) +## [[1]] +## NULL +## +## [[2]] +## NULL +Note Not all data fit into memory, but it does not mean that we should start installing +Apache Hadoop and Spark immediately. Some datasets can be processed on a chunk- +by-chunk basis. + +152 +I DEEP +R enables data stream handling (some can be of infinite length) through file connec- +tions, for example: +f <- file(paste0("https://raw.githubusercontent.com/gagolews/teaching-data/", +"master/README.md"), open="r") +# a big file, the biggest file ever +i <- 0 +while (TRUE) { +few_lines <- readLines(f, n=4) +# read only four lines at a time +if (length(few_lines) == 0) break +i <- i + length(few_lines) +} +close(f) +print(i) +# the number of lines +## [1] 93 +Many functions support reading from/writing to already established connections of +different types, e.g., file, gzfile, textConnection, batch-by-batch. +A frequent scenario involves reading a very large CSV, JSON, or XML file only thou- +sands of lines/records at a time, parsing and cleansing them, and exporting to SQL +databases (which we will exercise in Chapter 12). +Also note that the always-open text connections stdout and stderr (for writing), and +stdin (for reading) are by default mapped to the “terminal/console” and “keyboard”, +respectively. Call scan, cat, and stop to interact with such sources/targets. +8.4 +Exercises +Note that, from now on, we should stay alert. Many, if not all, of the following tasks +can still be implemented without the explicit use of the R loops, but based only on the +operations covered in the previous chapters. If this is the case, try implementing both +the looped and loop-free version. Use microbenchmark::microbenchmark or proc.time +to compare the run-times3. +Exercise 8.12 Answer the following questions: +• Let x be a numeric vector. When does if(x > 0) ... yield a warning? When does it give an +error? How to prevent this? +• What is the dangling else? +• What happens if you put if as the last expression in a curly braces block within a function’s +body? +3 It might be the case that a for-based solution is faster (e.g., for larger objects) because of the use of a +more efficient algorithm. Such cases will especially benefit from a rewrite in C or C++ (Chapter 14). + +8 FLOW OF EXECUTION +153 +• Why do we say that `&&` and `||` are lazy? What are their use cases? +• What is the difference between `&&` and `&`? +• Can while always be replaced with for? What about the other way around? +Exercise 8.13 Verify which of the following can be safely used as logical conditions in if state- +ments. If that is not the case for all x, y, …, determine the additional conditions that should be +imposed in order to make them valid. +• x == 0, +• x[y] > 0, +• any(x>0), +• match(x, y), +• any(x %in% y). +Exercise 8.14 Whatcango wrongin thefollowingcodechunk,dependingonthetypeandform +of x? Consider as many scenarios as possible. +count <- 0 +for (i in 1:length(x)) +if (x[i] > 0) +count <- count + 1 +Exercise 8.15 Implement shift_left(x, n) and shift_right(x, n). The former function +getsridofthefirst nobservationsin xandaddsnmissingvaluesattheendoftheresultingvector, +e.g., shift_left(c(1, 2, 3, 4, 5), 2) is c(3, 4, 5, NA, NA). On the other hand, +shift_right(c(1, 2, 3, 4, 5), 2) is c(NA, NA, 1, 2, 3). +Exercise 8.16 Implement your own version of diff. +Exercise 8.17 Writea functionthat determinesthelongestincreasingtrendinagivennumeric +vector, i.e., the length of the longest subsequence of consecutive elements that are increasing. For +example, the input c(1, 2, 3, 2, 1, 2, 3, 4, 3) should yield 4. +Exercise 8.18 Implement the functions that round down and round up, to a number of decimal +digits, each element in a numeric vector. +This concludes the first part of this magnificent book. + + +Part II +Deeper + + +9 +Designing Functions +In Chapter 7, we learnt how to write our own functions. This skill is key to enforcing +the good development practice of avoiding the repetition of code: running the same +command sequence on different data. +This chapter is devoted to the designing of such reusable modules so that they are +easier to use, test, and maintain. We also provide some more technical details which +were not of the highest importance upon our first exposure to this topic, but which +our crucial to our better understanding of how R works. +9.1 +Principles of Sustainable Design +Good design is more art than science. As usual in real life, we will need to make many +compromises. This is because improving things with regard to one criterion some- +times makes them worse with respect to other aspects1 (also which we are not aware +of). Also, not everything that counts can and will be counted. Below are some obser- +vations, ideas, and food for thought. +9.1.1 +To Write or to Abstain +Functions that we write ourselves can oftentimes be considered merely creative com- +binations of the building blocks available in base R or a few high-quality add-on pack- +ages2. Some are simpler than others. Thus, there is a question if a new operation +should be introduced at all: whether we are faced with the case of multiplying entities +without necessity. +Ontheonehand,theDRY(don’trepeatyourself)principletellsusthatmostfrequently +used (say, at least 3 times) code chunks should be generalised in the form of a new +function. This is definitely a correct approach with regard to non-trivial operations. +On the other hand, not every generalisation is necessarily welcome. Let us say that we +are lazy and tired of writing g(f(x)) for the n-th time. Why don’t we therefore intro- +duce h defined as a combination of g and f? This might seem like a good idea, but let +1 Compare the notion of Pareto efficiency. +2 If some non-trivial operation is missing, we can always implement it at the C language level; see +Chapter 14. + +158 +II DEEPER +us not take it for granted: being tired might be an indication of our body and mind +needing a rest; being lazy can be a call for more self-discipline (not an overly popular +word these days, but still, an important trait). +Example 9.1 paste0 is a specialised version of paste, but having the sep argument hardcoded +to an empty string. +• Even if this might be the most often applied use case, is the introduction of a new function +justifiable? Is it so hard to write paste="" each time? +• Would changing paste’s default argument be better? That of course would harm backward +compatibility, but what strategies could we apply to make the transition as smooth as pos- +sible? +• Would it be better to introduce a new version of paste with sep defaulting to "", informing +the users that the old version is deprecated and to be removed in, say, two years? Or maybe +one year is better? Or five? +Example 9.2 In R 4.0, deparse1 has been introduced: it is merely a combination of deparse +(see below) and paste: +print(deparse1) +## function (expr, collapse = " ", width.cutoff = 500L, ...) +## paste(deparse(expr, width.cutoff, ...), collapse = collapse) +## +## +Letussaythiscovers90%ofusecases:wasintroducingitajustifiedideathen?Whatifthatnum- +ber was 99%? +Overall, more functions contribute to the information overload. We do not want our +users to be overwhelmed by too many choices. Luckily, nothing is cemented once and +for all. Had we done bad design choices resulting in our API’s being bloated, we can +always clean up those that no longer spark joy. +9.1.2 +To Pamper or to Challenge +Think about the kind of audience we would like to serve: is it our team only, students, +professionals, certain client groups, etc.? Do they have mathematical, programming, +engineering, or scientific background? Not everything that is appropriate for one co- +hort, will be valuable for another. And not everything that is good for some now, will +be beneficial for them in the long run. People (their skills, attitudes, etc.) change. +Example 9.3 Assumewearewritingafriendlyandinclusivepackagefornoviceswhowouldlike + +9 DESIGNING FUNCTIONS +159 +to grasp the basics of data analysis as quickly3 as possible. Without much effort, it would enable +them to solve 80–95% of the most common, easy problems. +Think of introducing the students to a function that returns five largest observations in a given +vector. Let us call it nlargest: so pleasant to use. It makes the students feel empowered quickly. +Still,whenfacedwiththeremaining5–20%tasks,theywillhavetolearnanother,moreadvanced, +generic,andpowerfultoolanyway(inourcase,thebaseRitself).Aretheydeterminedandskilled +enough to do that? Time will tell. The least we can do is to be explicit about it. +Recall that it took us some time to arrive at order and subsetting via `[`. Assuming that we read +this book from the beginning to the end and solve all the exercises, which we should, we are now +able to implement the said nlargest (and lots of other functions) ourselves, using a single line of +code. This will also pay off in many scenarios that we will be facing in the future, e.g., when we +consider matrices and data frames. +Yes, everyone will be reinventing their own nlargest this way. But this constitutes a great exer- +cise: by our being too nice, some might have lost an opportunity to learn a new, more universal +skill. +Although most of the users would really love to minimise the effort put into all their +activities, ultimately, they sometimes need to learn new things. Let us thus not be +afraid to teach them stuff. +Furthermore, we do not want to discourage experts (or experts to-be) by presenting +themwithoverlysimplifiedsolutionsthatkeeptheirhandstiedwhensomethingmore +ambitious needs to be done. +9.1.3 +To Build or to Reuse +In the short term, the failfast philosophy encourages us to build our applications using +prefabricatedcomponents.Thisisfantasticattheearlystageofitslifecycle.Ifwebuild +somethingreallysimpleorwhosepurposeismerelytoillustrateanidea,show-offhow +“awesome” we are, or to educate, let us be explicit about it so that other users do not +feel obliged to treat our product (exercise) seriously. +In the (not so likely, probabilistically speaking) event of its becoming successful, we +should start thinking about the project’s long-term stability and sustainability. After +all, relying on third-party functions, packages, or programs makes our software pro- +jects less… independent. This may be problematic, because: +• the dependencies might not be available on every platform or may behave differ- +ently across various system configurations, +3 We will leave the reflection upon whether this is at all feasible for another time. +Note that this strategy is employed by many companies (and drug dealers): make the introductory exper- +ience super-smooth and fun. At the same time, do not allow your users to become independent too easily. +Instead, make them rely on your product lines/proprietary solutions/payable services etc. +The free software movement with its do-it-yourself approach stresses on users’ becoming autonomous. +This does not contradict the user-friendliness (but that many open-source projects could benefit from be- +coming less exclusive is a different story, and this book tries to make a change in this area too). + +160 +II DEEPER +• they may be huge (and can depend on other external software too), +• their APIs may change which could result in our project’s not working anymore, +• their functionality can change which can lead to some unexpected behaviours. +Hence, it might be a good idea to rewrite some parts from scratch on our own. +Exercise 9.4 Identify some R packages on CRAN with many dependencies. Seewhat functions +do they import from other packages. How often it is just a few lines of code? +TheUnixphilosophyemphasises uponthebuildingandusingofminimalisticyetnon- +trivial, single-purpose, high quality pieces of software that can work as parts of larger, +custom pipelines. R serves as a glue language quite well. +In the long run, some of our software projects might converge to such a tool – it might +be a good idea to standardise our API (e.g., make it available from the command-line; +Section 1.2) so that the users of other languages can benefit from our work too. +Important If ourprojectis merelyamodified interface/front-endtoa largerprogram +developed by others, we should be humble about it and make sure it is not us who get +all the credit for other people’s work. +Also,weshouldstateveryclearlyhowcantheoriginaltoolsbeusedtoachievethesame +goals, e.g., when working from the command line. +9.2 +Managing Data Flow +A function, most of the time, can and should be treated as a black box: its callers do not +have to care what it hides inside. After all, they are supposed to use it: given some in- +puts,theyexpectawell-defined(read:explainedinverydetailinthefunction’smanual; +see Section 9.3.3) outputs. +9.2.1 +Checking Input Data Integrity and Argument Handling +A function takes R objects of any kind as arguments, but it does not mean that feeding +it with every- or any-thing is healthy for its guts. +When designing functions, it is best to handle the inputs in a manner similar to base +R’s behaviour. This will make our contributions easier to handle. +Unfortunately, base R functions frequently do not handle arguments of similar kind +100% consistently. Such variability might be due to many reasons and, in essence, is +not necessarily bad. Usually, there might be many different possible behaviours and +choosingoneoveranotherwillmakeafewusersunhappyanyway.Somechoicesmight +not be optimal, but they are for historical compatibility (e.g., with S). Of course, it + +9 DESIGNING FUNCTIONS +161 +might also happen (but the probability is low) that there is a bug or something is not +at all well designed. +Thisiswhyitisbetterto keepthevocabularyquiterestricted(andweadvocateforsuch +minimalism in this book): even if there are exceptions to the general rules, with fewer +functions, they are simply easier to remember. +Consider the following case study, illustrating that even the extremely simple scenario +where we deal with a single positive integer, is not necessarily straightforward. +Exercise 9.5 In mathematical notation, we usually denote the number of objects in a collection +with the famous “n”. +It is implicitly assumed that such n is a single natural number (although whether this includes +0 or not should be specified at some point). The functions runif, sample, seq, rep, strrep, and +class::knn take it arguments. However, nothing prevents their users from trying to challenge +them by passing: +• 2.5, -1, 0, 1-1e-16 (non-positive numbers, non-integers); +• NA_real_, Inf (not finite); +• 1:5 (not of length 1; after all, there are no scalars in R) +• numeric(0) (an empty vector); +• TRUE, NA, c(TRUE, FALSE, NA), "1", c("1", "2", "3") (non-numeric, but coercible to); +• list(1), list(1, 2, 3), list(1:3, 4) (non-atomic); +• "spam" (utter nonsense); +• as.matrix(1), factor(7), factor(c(3, 4, 2, 3)), etc. (compound types; see Chapter +10). +Read the aforementioned functions’ reference manuals and call them on different inputs, noting +how differently they handle such atypical arguments. +Sometimes we will rely on other functions to handle the data integrity checking for +us. +Example 9.6 Let us consider the following function that generates n pseudorandom numbers +from the unit interval rounded to d decimal digits. We strongly believe or hope (good faith and +high competence assumption) that its authors knew what they were doing when they wrote: +round_rand <- function(n, d) +{ +x <- runif(n) +# runif will check if `n` makes sense +round(x, d) +# round will determine the appropriateness of `d` +} +What constitutes correct n and d and how the function behaves when not provided with positive +integers is determined by the two underlying functions, runif and round: + +162 +II DEEPER +round_rand(4, 1) +# the expected use case +## [1] 0.3 0.8 0.4 0.9 +round_rand(4.8, 1.9) +# 4, 2 +## [1] 0.94 0.05 0.53 0.89 +round_rand(4, NA) +## [1] NA NA NA NA +round_rand(0, 1) +## numeric(0) +Ifwellthought-outandproperlydocumented,manysuchdesignchoicescanbedefen- +ded. Some programmers will opt for high uniformity/compatibility across numerous +tools, but there are cases where some exceptions/diversity do more good than harm. +Yet, we should keep in mind that the functions we write might be part of a more com- +plicated data flow pipeline, where some other function generates a value that we did +not expect (because of a bug therein or because we did not study its manual) and this +value is used as input to our function. In our case, this would correspond to the said n +or d being determined programmatically. +Example 9.7 Continuing the previous example, the following might be somewhat challenging +with regard to our being flexible and open minded: +round_rand(c(100, 42, 63, 30), 1) +# length(c(...)), 1) +## [1] 0.7 0.6 0.1 0.9 +round_rand("4", 1) +# as.numeric(...), 1 +## [1] 0.2 0.0 0.3 1.0 +Sure, it is quite convenient, but might lead to problems that are hard to diagnose. +Also note the not-really informative error messages in cases like: +round_rand(NA, 1) +## Error in runif(n): invalid arguments +round_rand(4, "1") +## Error in round(x, d): non-numeric argument to mathematical function +Hence, some defensive design mechanisms are not a bad idea, especially if they lead to +generating an informative error message. +Important stopifnot gives aconvenientmeanstoassert theenjoymentofourexpect- +ations with regard to a function’s arguments (or some intermediate values). A call to +stopifnot(cond1, cond2, ...) is more or less equivalent to: +if (!(is.logical(cond1) && !any(is.na(cond1)) && all(cond1))) +stop("`cond1` are not all TRUE") +if (!(is.logical(cond2) && !any(is.na(cond2)) && all(cond2))) +(continues on next page) + +9 DESIGNING FUNCTIONS +163 +(continued from previous page) +stop("`cond2` are not all TRUE") +... +Thus, if all the elements in the given logical vectors are TRUE, nothing happens and we +can safely move on. +Example 9.8 We can rewrite the above function as follows: +round_rand2 <- function(n, d) +{ +stopifnot( +is.numeric(n), length(n) == 1, +is.finite(n), n > 0, n == floor(n), +is.numeric(d), length(d) == 1, +is.finite(d), d > 0, d == floor(d) +) +x <- runif(n) +# runif will check if n makes sense +round(x, d) +# round will determine the appropriateness of d +} +round_rand2(5, 1) +## [1] 0.7 0.7 0.5 0.6 0.3 +round_rand2(5.4, 1) +## Error in round_rand2(5.4, 1): n == floor(n) is not TRUE +round_rand2(5, "1") +## Error in round_rand2(5, "1"): is.numeric(d) is not TRUE +Thisimplementsthestrictesttestfor“asinglepositiveinteger”possible.Inthecaseofanyviolation +of the underlying condition, we get a very informative error message. +Example 9.9 At other times, we might be interested in argument checking like: +if (!is.numeric(n)) +n <- as.numeric(n) +if (length(n) > 1) { +warning("only the first element will be used") +n <- n[1] +} +n <- floor(n) +stopifnot(is.finite(n), n > 0) +This way, "4" and c(4.9, 100) will all be accepted as 44. +We see that there is always a tension between being generous/flexible and pre- +4 Note that here we rely on S3 generics is.numeric and as.numeric; see Section 10.4. + +164 +II DEEPER +cise/restrictive. Also, for some functions, it will be better to behave differently than +the others, because of their particular use cases. Too much uniformity is as bad as +chaos. Overall, we should rely on common sense, but add some lightweight foolproof +mechanisms. +It is our duty to be explicit about all the assumptions we make or exceptions we allow +(by writing good documentation; see Section 9.3.3). +We will revisit this topic in Section 10.4. +Note Example exercises related to the improving of the consistency of base R’s hand- +ling of arguments in different domains include the vctrs and stringx packages5. Can +these contributions be justified? +Exercise 9.10 Reflect on how you would handle the following scenarios (and how base R and +other packages or languages you know deals with them): +• a vectorised mathematical function (empty vectors? non-numeric inputs? what if it is +equipped with the names attribute? what if it has other ones?); +• an aggregation function (what about missing values? empty vectors?); +• a function vectorised with regard to two arguments (elementwise vectorisation? recycling +rule? only scalar vs vector or vector vs vector of the same length allowed? what if one argu- +ment is a row vector and the other is a column vector); +• a function vectorised with regard to all arguments (really all? maybe some exceptions are +necessary?); +• afunctionvectorisedwithrespecttothefirstargumentbutnotthesecond(whysucharestric- +tion? when?). +Find a few functions that match each case. +9.2.2 +Putting Outputs into Context +The functions we write do not exist in a vacuum. We should put them into a much +wider context: how are they going to be used when combined with other tools? +As a general rule, our functions should generate outputs of predictable kind, so that +when we write and read the code chunks that utilise them, we can easily deduce what +is going to happen. +Example 9.11 Some base R functions do not adhere to this rule for the sake of (questionable) +users’ convenience. We will meet a few of them in Chapter 11 and Chapter 12. In particular, sap- +ply and the underlying simplify2array, can return a list, an atomic vector, or a matrix. +5 Yours truly is the author of the latter and thus is guilty of multiplying entities beyond necessity. + +9 DESIGNING FUNCTIONS +165 +simplify2array(list(1, 3:4)) +# list +## [[1]] +## [1] 1 +## +## [[2]] +## [1] 3 4 +simplify2array(list(1, 3)) +# vector +## [1] 1 3 +simplify2array(list(1:2, 3:4)) +# matrix +## +[,1] [,2] +## [1,] +1 +3 +## [2,] +2 +4 +Further, the index operator with drop=TRUE, which is the default, may output an atomic vector. +But it may as well yield a matrix or a data frame. +(A <- matrix(1:6, nrow=3)) +# an example matrix +## +[,1] [,2] +## [1,] +1 +4 +## [2,] +2 +5 +## [3,] +3 +6 +A[1, ] +# vector +## [1] 1 4 +A[1:2, ] +# matrix +## +[,1] [,2] +## [1,] +1 +4 +## [2,] +2 +5 +A[1, , drop=FALSE] +# matrix with 1 row +## +[,1] [,2] +## [1,] +1 +4 +We proclaim that the default functions’ behaviour should be to return the object of +the most generic kind possible (if there are other options), and then to either have a +further argument which must be explicitly set if we really wish to simplify the output, +or we should ask the user to call a simplifier explicitly. +In the latter case, the simplifier should probably fail issuing an error if it is unable +to neaten the object or at least apply some brute force solution (e.g., or “fill the gaps” +somehow itself, possibly with a warning). +Example 9.12 For instance: +as.numeric(A[1:2, ]) +# always returns a vector +## [1] 1 2 4 5 +stringi::stri_list2matrix(list(1, 3:4)) +# fills the gaps with NAs +## +[,1] [,2] +(continues on next page) + +166 +II DEEPER +(continued from previous page) +## [1,] "1" +"3" +## [2,] NA +"4" +Ideally, a function should perform one (and only one) well-defined task. If a function +tends to generate objects of different kinds, depending on the arguments provided, +maybe it is better to write two functions instead? +Exercise 9.13 Functionssuchas rep, seq,and sampledonotperformasingletask.Ordothey? +Note (*) In a purely functional programming language, we can assume the so-called +referential transparency: a call to a pure function can always safely be replaced with the +value it is supposed to generate. If this is true, then for the same set of argument val- +ues, the output is always the same. Furthermore, there are no side effects. In R, it is +not really the case: +• a call can introduce/modify/delete the variables in other environments (see +sec:to-do), e.g., the state of the random number generator, +• metaprogramming and lazy evaluation techniques lead to the functions’ being +free to interpret the argument forms (not only: values) freely (see sec:to-do), +• printing, plotting, file reading, database access have obvious consequences with +regard to the state of some external resources. +Important +Each function must return some value, but there are several instances +(e.g., plotting, printing), where this does not make sense. +Insuchacase,weshouldconsiderreturning invisible(NULL),a NULLwhosefirst print- +ing will be suppressed. +Compare the following: +(function() NULL)() +# anonymous function, called instantly +## NULL +(function() invisible(NULL))() +# printing suppressed +x <- (function() invisible(NULL))() +print(x) +# no longer invisible +## NULL +Take a look at the return value of the built-on cat. + +9 DESIGNING FUNCTIONS +167 +9.3 +Organising and Maintaining Functions +9.3.1 +Function Libraries +Definitions of frequently-used functions or datasets can be emplaced in separate +source files (.R extension) for further reference. +Such libraries can be executed by calling: +source("path_to_file.R") +Exercise 9.14 Create a source file (script) named mylib.R, where you define a function called +nlargest which returns a few largest elements in a given atomic vector. +From within another script, call source("mylib.R") (note that relative paths to refer to the cur- +rent working director; (compare Section 2.1.6) and then write a few lines of code where you test +nlargest on some example inputs. +9.3.2 +Writing R Packages +When a function library grows substantially, or when there is a need for equipping it +with the relevant manual pages6 (Section 9.3.3) or compiled code (Chapter 14), turning +it into an own R package (Section 7.3.1) might be a good idea (even if it is only for our +own or small team’s purpose). +A source package is merely a directory containing some special files and subdirectories: +• DESCRIPTION – a text file that gives the name of the package, its version, authors, +dependencies upon other packages, license, etc.; see Section 1.1.1 of [45]; +• NAMESPACE – a text file containing directives stating which objects are to be expor- +ted so that they are available to the package users, and which names are to be im- +ported from other packages; +• R – a directory with R scripts (.R files), which define, e.g., functions, example +datasets, etc.; +• man – a directory with R documentation files (.Rd), describing at least all the ex- +ported objects; see Section 9.3.3; +• src – optional; compiled code, see Chapter 14; +• tests – optional; tests to run on the package check, see Section 9.3.4; +see Section 1.5 of Writing R Extensions [45] for more details and other options: there is +no need for us to repeat the information from the official manual as everyone can read +it themself. +6 This should read: i.e., always. + +168 +II DEEPER +Important A source package can be built and installed from within an R session by +calling: +install.packages("pkg_directory", repos=NULL, type="source") +ThenitcanbeusedasanyotherRpackage(Section9.3.3).Inparticular,itcanbeloaded +and attached via a call to: +library("pkg") +This makes all the objects listed in its NAMESPACE file available to the user. +Exercise 9.15 Create your own package mypkg featuring some of the solutions to the exercises +you have solved whilst studying the material in the previous chapters. When in doubt, refer to +the official manual [45]. +Important Note that you do not have to publish your package on CRAN7. Many users +are tempted to submit whatever they have been tinkering around with for a while. +Have mercy on the busy CRAN maintainers and do not contribute to the information +overload, unless you have come up with something potentially useful for other R users +(make it less about you, and more about the community; thank you in advance). R +packages can always be hosted on and installed from, for instance, GitLab or GitHub. +Note (*) The building and installing of packages also be done from the command line: +R CMD build pkg_directory +R CMD INSTALL --build pkg_directory +Also, some users could potentially benefit from creating own Makefiles that help auto- +mate the processes of building, testing, checking, etc. +9.3.3 +Documenting R Packages +Documenting functions and commenting code thoroughly is very important, even if +we just write for ourselves. Most programmers sooner or later will note that they find +it hard to determine what a piece of code is doing after they took a break from it. In +some sense, we always write for external audience, which incudes our future self. +The help system is one of the stronger assets of the R environment. By far we should +have interacted with many man pages and got a good idea of what constitutes an in- +formative documentation piece. +7 And always consult the CRAN Repository Policy at https://cran.r-project.org/web/packages/policies. +html. + +9 DESIGNING FUNCTIONS +169 +From the technical side, R Documentation (.Rd) files should be emplaced in the man +subdirectory of a source package. All exported objects (e.g., functions) should be de- +scribed clearly. Additional topics can be covered too. +During the package install, the .Rd files are converted to various output formats, e.g., +HTML or plain text, and displayed upon a call to the well-known help function. +Documentation files use a LaTeX-like syntax, which looks quite obscure to an un- +trained eye. The relevant commands are explained in very detail in Section 2 of Writing +R Extensions [45]. +Note The process of writing .Rd files by hand might be quite tedious, especially keep- +ing track of the changes to the \usage and \arguments commands. Rarely do we re- +commend the use of third-party packages, because base R facilities are usually good +enough, but roxygen2 might be worth a try, because it really makes the developers’ +lives easier. Most importantly, it allows for documentation to be specified alongside +the functions’ definitions, which is much more natural. +Exercise 9.16 Add a few manual pages to your example R package. +9.3.4 +Assuring Quality Code +Below we mention some good development practices related to maintaining quality +code. This is an important topic, but writing about them is tedious to the same ex- +tent that reading about them is boring, because it is the less scientific part of software +engineering. After all, these are some heuristics that are learnt best by observing and +mimicking what the others are doing (and hence the exercises below will encourage +to do so). +Managing Changes and Working Collaboratively +It is a good idea to employ some source code version control system such as git to keep +track of the changes made to the software. +Note It is worth investing some time and effort to learn how to use git from the com- +mand line; see https://git-scm.com/doc. +There are a few hosting providers for git repositories, with GitLab and GitHub being +a popular choice amongst open-source software developers. +Notonlydotheysupportworkingcollaborativelyontheprojects,butalsoareequipped +with additional tools for reporting bugs, suggesting feature requests, etc. +Exercise 9.17 Find where the source code of some of your most favourite R packages or other +open-source projects are hosted. Explore the corresponding repositories, feature trackers, wi- +kis, discussion boards, etc. Note that each community is different and is governed by different +guidelines: after all, we are from all over the world. + +170 +II DEEPER +Test-driven Development and Continuous Integration +It is often hygienic to include some principles of test-driven development when writ- +ing own functions. +Exercise 9.18 Assume that, for some reasons, we were asked to write a function to compute the +root mean square (quadratic mean) of a given numeric vector. Before implementing the actual +routine, it is a good idea to reflect upon what we want to achieve, especially how we want our +function to behave in certain boundary cases. +stopifnot gives simple means to assure a given assertion is fulfilled. If that is the case, it will +move forward quietly. +Let us say we have come up with the following set of expectations: +stopifnot(all.equal(rms(1), 1)) +stopifnot(all.equal(rms(1:100), 58.16786054171151931769)) +stopifnot(all.equal(rms(rep(pi, 10)), pi)) +stopifnot(all.equal(rms(numeric(0)), 0)) +Write a function rms that fulfils the above assertions. +Exercise 9.19 Implement your own version of the sample function (assuming replace=TRUE), +using calls to runif. However, start by writing a few unit tests. +There are also a couple of R packages that support writing and executing of unit tests, +including testthat, tinytest (which is a lighter-weight version of the former), RUnit, +or realtest. However, in the most typical use cases, relying on stopifnot is powerful +enough. +Exercise 9.20 (*) Consult the Writing R Extensions manual [45] about where and how to +include unit tests in your example package. +Note +(*) R includes a built-in mechanism to check a couple of code quality areas: +running R CMD check pkg_directory from the command line (preferably using the +most recent version of R) can suggest a number of improvements. +Also, it is possible to use various continuous integration techniques that are automat- +ically triggered when pushing changes to our software repositories; see GitLab CI or +GitHub Actions. For instance, it is possible to run a package build, install, and check +process upon every git commit. Also, CRAN features some continuous integration +services, including checking the package on a range of different platforms. +Debugging +For all his life, the current author has been debugging all his programs mostly by +manually printing the state of suspected variables (printf and the like) in different +areas of the code. No shame in that. + +9 DESIGNING FUNCTIONS +171 +For an interactive debugger, see the browser function. Also, refer to Section 9 of [49] +for more details. +Some IDEs (e.g., RStudio) support this feature too; see their corresponding docu- +mentation. +Profiling +Typically,aprogramspendsrelativelylongtimeexecutingonlyasmallportionofcode. +The Rprof functioncanbe ahelpful tool toidentifywhich chunksmightneed arewrite, +for instance using a compiled language (Chapter 14). +Please remember, though, that not only implementations of algorithms that have +hight computational complexity can form a bottleneck, but also data input and out- +put (such as reading files from disk, printing messages, on the console, querying Web +APIs, etc.). +9.4 +Special Functions: Syntactic Sugar +Some functions, such as `*`, are somewhat special. They can be referred to using an +alternativesyntaxwhichforsomereasonmostofusacceptedasthedefaultone.Below +we will reveal, amongst others, that “5 * 9” reduces in fact to an ordinary function call: +`*`(5, 9) +# a call to `*` with 2 arguments, equivalent to 5 * 9 +## [1] 45 +9.4.1 +A Note on Backticks +In Section 2.2, we have mentioned that we can assign (as in `<-`) syntactically valid +names to our objects. Most identifiers comprised of letters, digits, dots, and under- +scores can be used directly in R code. +However, it is possible to name our objects whichever we like: non-syntactically valid +(nonstandard) names just need to be enclosed in backticks (back quotes, grave ac- +cents): +`42 a quite peculiar name :O lollolll` <- (-5):5 +mean(1/(1+exp(-`42 a quite peculiar name :O lollolll`))) +## [1] 0.5 +Of course, they are less convenient, but still: backticks lets us access them in any con- +text. For example: +x <- list(`1`="a", `2`="b") +# structure(list("a", "b"), names=c("1", "2")) +(continues on next page) + +172 +II DEEPER +(continued from previous page) +x$`1` +# x[["1"]] is still okay (and we prefer this syntax) +## [1] "a" +9.4.2 +Curly Braces, `{` +A block of statements grouped with curly braces, `{`, corresponds to a function call. +When we write: +{ +print(TRUE) +cat("two") +3 +} +## [1] TRUE +## two +## [1] 3 +The parser translates it to a call to: +`{`(print(TRUE), cat("two"), 3) +## [1] TRUE +## two +## [1] 3 +When the above is executed, every argument, one by one, is evaluated and then the +last value is returned in result of that call. +9.4.3 +`if` +if is a function too; as mentioned in Section 8.1, it returns the value corresponding to +the expression evaluated conditionally. Hence, we may write: +if (runif(1) < 0.5) "head" else "tail" +## [1] "head" +but also: +`if`(runif(1) < 0.5, "head", "tail") +## [1] "head" +Note A call like `if`(test, what_if_true, what_if_false) can only work properly +because of the lazy evaluation of function arguments; see Section 9.5.5. + +9 DESIGNING FUNCTIONS +173 +On a side note, while, for, repeat can also be called that way, but they return invis- +ible(NULL). +9.4.4 +Operators are Functions Too +Calling Built-in Operators as Functions +Every arithmetic, logical, and comparison operator is translated to a call to the cor- +responding function. For instance: +`<`(`+`(`*`(`-`(3), 4)), 5) +# 2+(-3)*4 < 5 +## [1] TRUE +Also, x[i] is equivalent to `[`(x, i) and x[[i]] maps to `[[`(x, i). +Knowing this will not only enable us to manipulate unevaluated R code (Chapter 15) +or access the corresponding manual pages (see, e.g., help("[")), but also write some +expressions in a more concise manner. For instance, +x <- list(1:5, 11:17, 21:23) +unlist(Map(`[`, x, 1)) +# 1 is a further argument passed to `[` +## [1] +1 11 21 +is equivalent to a call to Map(function(e) e[1], x). +Note Unsurprisingly, the assignment operator, `<-`, is a function too. It returns the +assigned value, invisibly. +Knowingthat`<-`bindsrighttoleft(compare help("Syntax")),thisiswhytheexpres- +sion “a <- b <- 1” results in both a and b being assigned 1: it is equivalent to “`<-`("a", +`<-`("b", 1))” and “`<-`("b", 1)” returns 1. +Owing to the pass-by-value semantics (Section 9.5.1) we can also expect that we will +alwaysbe(withtheexceptionofenvironments,Chapter16)assigningacopyofthevalue +on the righthand side. +x <- 1:6 +y <- x +# makes a copy (but delayed, on demand, for performance reasons) +y[c(TRUE, FALSE)] <- NA_real_ +# modify every 2nd element +print(y) +## [1] NA +2 NA +4 NA +6 +print(x) +# state of x has not changed — x and y are different objects +## [1] 1 2 3 4 5 6 +This is especially worth pointing out to Python (amongst others) programmers, where +the above assignment would mean that x and y both refer to the same (shared) object +in the computer’s memory. + +174 +II DEEPER +However, with no harm done to semantics, the actual copying of x is postponed until +absolutely necessary (Section 16.2). This is efficient both time- and memory-wise. +Creating Own Binary Operators +We can also introduce our own binary operators named like `%myopname%`: +`%:)%` <- function(e1, e2) (e1+e2)/2 +5 %:)% 1:10 +## +[1] 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 +Recall that `%%` and `%/%` are built-in operators denoting division remainder and in- +teger division. Rarely will we be defining our own operators, but when we encounter +a similar one next time, we will no longer be surprised. For instance, in Chapter 11 we +will learn about `%*%` which implements matrix multiplication. +Note In Chapter 10, we will note that most existing operators can be overloaded for +objects of different types. +9.4.5 +Replacement Functions +Functions generally do not change the state of their arguments. However, there is +some syntactic sugar that allows us to replace objects or parts thereof with new con- +tent. We call them replacement functions. +For instance, three of the following calls replace the input x with its modified version: +x <- 1:5 +# example input +x[3] <- 0 +# replace the 3rd element with 0 +length(x) <- 7 +# "replace" length +names(x) <- LETTERS[seq_along(x)] +# replace the names attribute +print(x) +# x is different than before +## +A +B +C +D +E +F +G +## +1 +2 +0 +4 +5 NA NA +Creating Own Replacement Functions +A replacement function is a mapping named like `name<-` with at least two paramet- +ers: +• x (the object to be modified), +• ... (possible further arguments), +• value (as the last parameter; the object on the righthand side of the `<-` operator). +Most often, we will be interacting with existing replacement functions, not creating + +9 DESIGNING FUNCTIONS +175 +our own ones. However, knowing how to do the latter is key to understanding this +language feature. +For example: +`add<-` <- function(x, where=TRUE, value) +{ +x[where] <- x[where] + value +x +# the modified object that will replace the original one +} +The above aims to add some value to a subset of the input vector x (by default, to each +element therein) and return its altered version that will replace the object it has been +called upon. +y <- 1:5 +# example vector +add(y) <- 10 +# calls `add<-`(y, value=10) +print(y) +# y has changed +## [1] 11 12 13 14 15 +add(y, 3) <- 1000 +# calls `add<-`(y, 3, value=1000) +print(y) +# y has changed again +## [1] +11 +12 1013 +14 +15 +We see that calling “add(y, w) <- v” works as if we have called “y <- `add<-`(y, w, +value=v)”. +Note (*) According to [49], a call “add(y, 3) <- 1000” is a syntactic sugar precisely for: +`*tmp*` <- y +# temporary substitution +y <- `add<-`(`*tmp*`, 3, value=1000) +rm("*tmp*") +# remove the named object from the current scope +This has at least twoimplications. First,in the unlikelyeventthat a variable`*tmp*` ex- +istedbeforethecalltothereplacementfunction,itwillbenomore,itwillceasetobe.It +will be an ex-variable. Second, the temporary substitution guarantees that y must ex- +ist before the call (a function’s body does not have to refer to all the arguments passed; +because of lazy evaluation, see Section 9.5.5, we could get away with it otherwise). +Substituting Parts of Vectors +The replacement versions of the subsetting operators are named as follows: +• `[<-` is used in substitutions like “x[i] <- value”, +• `[[<-` is called when we perform “x[[i]] <- value”, +• `$<-` is used whilst calling “x$i <- value”. +Here is a use case: + +176 +II DEEPER +x <- 1:5 +`[<-`(x, c(3, 5), NA_real_) +# returns a new object +## [1] +1 +2 NA +4 NA +print(x) +# does not change the original input +## [1] 1 2 3 4 5 +On a side note, `length<-` can be used to expand or shorten a given vector by calling +“length(x) <- new_length”, see also Section 5.3.3. +x <- 1:5 +x[7] <- 7 +length(x) <- 10 +print(x) +## +[1] +1 +2 +3 +4 +5 NA +7 NA NA NA +length(x) <- 3 +print(x) +## [1] 1 2 3 +Despite the fact that, semantically speaking, calling `[<-` results in the creation of a +new vector (a modified version of the original one), we may luckily expect some per- +formance optimisations happening behind our back (reference counting, modifica- +tion in-place; see sec:to-do). +Exercise 9.21 Write a function `extend<-` which pushes new elements at the end of a given +vector, modifying it in place. +`extend<-` <- function(x, value) ...to.do... +Example use: +x <- 1 +extend(x) <- 2 +# push 2 at the back +extend(x) <- 3:10 +# add 3, 4, ..., 10 +print(x) +## +[1] +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +Replacing Attributes +Many replacement functions deal with the re-setting of objects’ attributes (Sec- +tion 4.4). +In particular, for each special attribute, there is also its replacement version, e.g., +`names<-`, `class<-`, `dim<-`, `levels<-`, etc. +x <- 1:3 +names(x) <- c("a", "b", "c") +# change the `names` attribute +print(x) +# x has been altered +(continues on next page) + +9 DESIGNING FUNCTIONS +177 +(continued from previous page) +## a b c +## 1 2 3 +Individual (arbitrary, including non-special ones) attributes can be set using `attr<-` +and all of them can be established by means of a single call to `attributes<-`. +x <- "spam" +attributes(x) <- list(shape="oval", smell="meaty") +attributes(x) <- c(attributes(x), taste="umami") +attr(x, "colour") <- "rose" +print(x) +## [1] "spam" +## attr(,"shape") +## [1] "oval" +## attr(,"smell") +## [1] "meaty" +## attr(,"taste") +## [1] "umami" +## attr(,"colour") +## [1] "rose" +Also note that setting an attribute to NULL results, by convention, in its removal: +attr(x, "taste") <- NULL +# this is tasteless now +print(x) +## [1] "spam" +## attr(,"shape") +## [1] "oval" +## attr(,"smell") +## [1] "meaty" +## attr(,"colour") +## [1] "rose" +attributes(x) <- NULL +# remove all +print(x) +## [1] "spam" +Which can be useful in contexts such as: +x <- structure(c(a=1, b=2, c=3), some_attrib="value") +y <- `attributes<-`(x, NULL) +Here, x retains its attributes and y is a version of x with metadata removed. +Compositions of Replacement Functions +Updating only selected names like: + +178 +II DEEPER +x <- c(a=1, b=2, c=3) +names(x)[2] <- "spam" +print(x) +## +a spam +c +## +1 +2 +3 +is possible due to the fact that “names(x)[i] <- v” is equivalent to: +old_names <- names(x) +new_names <- `[<-`(old_names, i, value=v) +x <- `names<-`(x, value=new_names) +Important More generally, a composition of replacement calls “g(f(x, a), b) <- y” +yields a result equivalent to “x <- `f<-`(x, a, value=`g<-`(f(x, a), b, value=y))”. +Note that both f and `f<-` need to be defined, but having g is not necessary. +Exercise 9.22 (*) What is “h(g(f(x, a), b), c) <- y” equivalent to? +Exercise 9.23 Write a (actually very useful!) function `recode<-` which replaces specific ele- +ments in a character vector with some other ones, allowing the following interface: +`recode<-` <- function(x, value) ...to.do... +x <- c("spam", "bacon", "eggs", "spam", "eggs") +recode(x) <- c(eggs="best spam", bacon="yummy spam") +print(x) +## [1] "spam" +"yummy spam" "best spam" +"spam" +"best spam" +We see that the named character vector gives a few from="to" pairs, e.g., all eggs are to be re- +placed by best spam. +Now, determine which calls are equivalent to the following: +x <- c(a=1, b=2, c=3) +recode(names(x)) <- c(c="z", b="y") +# or equivalently = ... ? +print(x) +## a y z +## 1 2 3 +y <- list(c("spam", "bacon", "spam"), c("spam", "eggs", "cauliflower")) +recode(y[[2]]) <- c(cauliflower="broccoli") +# or = ... ? +print(y) +## [[1]] +## [1] "spam" +"bacon" "spam" +## +## [[2]] +## [1] "spam" +"eggs" +"broccoli" + +9 DESIGNING FUNCTIONS +179 +Exercise 9.24 (*) Consider the `recode<-` function from the previous exercise. +Hereisanexamplematrixwiththedimnamesattributewhosenamesattributeisset(moredetails +in Chapter 11): +(x <- Titanic["Crew", "Male", , ]) +## +Survived +## Age +No Yes +## +Child +0 +0 +## +Adult 670 192 +recode(names(dimnames(x))) <- c(Age="age", Survived="survived") +print(x) +## +survived +## age +No Yes +## +Child +0 +0 +## +Adult 670 192 +This changes the x object. For each of the following subtasks, write a single call which alters +names(dimnames(x)) without modifying x in-place but returning a recoded copy of: +• names(dimnames(x)), +• dimnames(x)), +• x. +Exercise 9.25 (*) Consider the `recode<-` function once again but now let an example object +be a data frame featuring a column of class factor: +x <- iris[c(1, 2, 51, 101), ] +recode(levels(x[["Species"]])) <- c( +setosa="SET", versicolor="VER", virginica="VIR" +) +print(x) +## +Sepal.Length Sepal.Width Petal.Length Petal.Width Species +## 1 +5.1 +3.5 +1.4 +0.2 +SET +## 2 +4.9 +3.0 +1.4 +0.2 +SET +## 51 +7.0 +3.2 +4.7 +1.4 +VER +## 101 +6.3 +3.3 +6.0 +2.5 +VIR +Without modifying x in-place, how to change levels(x[["Species"]]) and return an altered +copy of: +• levels(x[["Species"]]), +• x[["Species"]], +• x? + +180 +II DEEPER +9.5 +Arguments and Local Variables +9.5.1 +Pass by “Value” +As a general rule, functions cannot change the state of their arguments8. We can think +of them as being passed by value, i.e., as if their copy was made. +test_change <- function(y) +{ +y[1] <- 7 +y +} +x <- 1:5 +test_change(x) +## [1] 7 2 3 4 5 +print(x) +# same +## [1] 1 2 3 4 5 +If the above was not the case, the state of x would have been changed after the call. +9.5.2 +Variable Scope +Function arguments as well as any other variables we create inside a function’s body +are relative to each call to that function. +test_change <- function(x) +{ +x <- x+1 +z <- -x +z +} +x <- 1:5 +test_change(x*10) +## [1] -11 -21 -31 -41 -51 +print(x) +# x in the function's body was a different x +## [1] 1 2 3 4 5 +print(z) +# z was local +## Error in print(z): object 'z' not found +Both x and z are local variables and live only whilst our function is being executed. +8 With the exception of objects of type environment, which are passed by reference; see Chapter 16. Also, +the fact that we have access to unevaluated R expressions can cause further deviations to this rule (see be- +low). + +9 DESIGNING FUNCTIONS +181 +The former temporarily “overshadows”9 the object of the same name from the caller’s +context. +Important It is a good development practice to refrain from referring to objects not +created within the current function, especially to “global” variables. We can always +pass an object as an argument explicitly. +Note It is a function call as such, not curly braces per se that form a local scope. +Writing “x <- { y <- 1; y + 1 }”, y is not an auxiliary variable; it is an ordinary +named object created alongside x. +On the other hand, in “x <- (function() { z <- 1; z + 1 })()”, z will not be available +thereafter. +9.5.3 +Closures (*) +Most user-defined functions are in fact representatives of the so-called closures; see +Chapter 18 and [1]. They not only consist of an R expression to evaluate, but also can +carry some auxiliary data. +For instance, given two equal-length numeric vectors x and y, a call to approxfun(x, +y) returns a function that linearly interpolates between the consecutive points (𝑥1, 𝑦1), +(𝑥2, 𝑦2), and so forth, so that a corresponding 𝑦 can be determined for any 𝑥. +x <- seq(0, 1, length.out=11) +f1 <- approxfun(x, x^2) +f2 <- approxfun(x, x^3) +f1(0.75) +# check that it is quite close to the true 0.75^2 +## [1] 0.565 +f2(0.75) +# compare with 0.75^3 +## [1] 0.4275 +Inspecting, however, the source codes of the above functions: +print(f1) +## function (v) +## .approxfun(x, y, v, method, yleft, yright, f, na.rm) +## +## +print(f2) +(continues on next page) +9 In Chapter 18, we will discuss this topic in-depth; objects are bound to their names within environ- +ments. Moreover, R uses lexical (static) scoping, which is not necessarily intuitive, especially taking into +account that a function’s environment can always be changed. + +182 +II DEEPER +(continued from previous page) +## function (v) +## .approxfun(x, y, v, method, yleft, yright, f, na.rm) +## +## +we might wonder how can they produce different results — it is evident that they are +identical. It turns out, however, that they internally store some additional data that is +referred to upon their calls: +environment(f1)[["y"]] +## +[1] 0.00 0.01 0.04 0.09 0.16 0.25 0.36 0.49 0.64 0.81 1.00 +environment(f2)[["y"]] +## +[1] 0.000 0.001 0.008 0.027 0.064 0.125 0.216 0.343 0.512 0.729 1.000 +This and many more we will explore in great detail in the third part of this book. +9.5.4 +Default Arguments +We have already mentioned above that, when designing functions that perform com- +plex tasks, we will sometimes be faced with a design problem: how to find a sweet spot +betweenbeinggenerous/mindfulofthediverseneedsofourusersandmakingtheAPI +neither overwhelming nor oversimplistic. +We know that it is best if a function performs one, well-specified task, but also allows +its behaviour be tuned-up if one wishes to do so. This principle can be facilitated by +the use of default arguments. +For instance, log computes logarithms, by default the natural ones. +log(2.718) +# the same as log(2.78, base=exp(1)) — default base +## [1] 0.9999 +log(4, base=2) +# different base +## [1] 2 +Exercise 9.26 Study the documentation of the following functions and note the default values +that they define: round, hist, grep, and download.file. +We can easily define our own functions equipped with such recommended settings: +test_default <- function(x=1) x +test_default() +# use default +## [1] 1 +test_default(2) +# use something else +## [1] 2 +Most often, default arguments are just constants, e.g., 1. However, they can be any R + +9 DESIGNING FUNCTIONS +183 +expressions, also including a reference to other arguments passed to the same func- +tion; see more in Section 18.1. +Note that default arguments will most often appear and the end of the parameter list, +but see Section 9.4.5 (on replacement functions) for a well-justified exception. +9.5.5 +Lazy Evaluation +In some languages, function arguments are always evaluated prior to a call. In R, +though, they are only computed when actually needed. We call it lazy or delayed evalu- +ation. Recall that in Section 8.1.4 we introduced the short-circuit evaluation operators +`||` (or) and `&&` (and). They are able to do their job precisely thanks to this mechan- +ism. +Example 9.27 In the following example, we do not use the function’s argument at all: +lazy_test1 <- function(x) 1 +# x not used at all +lazy_test1({cat("and now for something completely different!"); 7}) +## [1] 1 +Otherwise, we would see a message being printed out on the console. +Example 9.28 Next, let us use x amidst other expressions in the body: +lazy_test2 <- function(x) +{ +cat("it's... ") +y <- x+x +# using x twice +cat(" a man with two noses") +y +} +lazy_test2({cat("and now for something completely different!"); 7}) +## it's... and now for something completely different! a man with two noses +## [1] 14 +Note that an argument is evaluated once and its value is stored for further reference. If that was +not the case, we would see two messages like and now.... +9.5.6 +Ellipsis, `...` +Let us start with an exercise. +Exercise 9.29 Note the presence of `...` in the parameter list of c, list, structure, cbind, +rbind, cat, Map (and the underlying mapply), lapply (a specialised version of Map), optimise, +optim, uniroot, integrate, outer, aggregate. What purpose does it serve, according to these +functions manual pages? + +184 +II DEEPER +We can create a variadic function by placing a dot-dot-dot (ellipsis; see help("dots")), +`...`, somewhere in its parameter list. The ellipsis serves as placeholder for all objects +passed to the function but not matched by any formal (named) parameters. +The easiest way to process arguments passed via `...` programmatically (see also Sec- +tion 18.2) is by redirecting them to list. +test_dots <- function(...) +list(...) +test_dots(1, a=2) +## [[1]] +## [1] 1 +## +## $a +## [1] 2 +Such a list can be processed just like… any other R list. What we can do with these +arguments is only limited by our creativity (in particular, recall from Section 7.2.2 the +very powerful do.call function). Still, there are two major use cases of the ellipsis10: +• create a new object by combining an arbitrary number of other objects: +c(1, 2, 3) +# 3 arguments +## [1] 1 2 3 +c(1:5, 6:7) +# 2 arguments +## [1] 1 2 3 4 5 6 7 +structure("spam") +# 0 additional arguments +## [1] "spam" +structure("spam", color="rose", taste="umami") +# 2 further arguments +## [1] "spam" +## attr(,"color") +## [1] "rose" +## attr(,"taste") +## [1] "umami" +cbind(1:2, 3:4) +## +[,1] [,2] +## [1,] +1 +3 +## [2,] +2 +4 +cbind(1:2, 3:4, 5:6, 7:8) +## +[,1] [,2] [,3] [,4] +## [1,] +1 +3 +5 +7 +## [2,] +2 +4 +6 +8 +sum(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 42) +## [1] 108 +10 Which is somewhat similar to Python’s *args and **kwargs in a function’s parameter list. + +9 DESIGNING FUNCTIONS +185 +• pass further arguments (as-is) to other methods : +lapply(list(c(1, NA, 3), 4:9), mean, na.rm=TRUE) +# mean(x, na.rm=TRUE) +## [[1]] +## [1] 2 +## +## [[2]] +## [1] 6.5 +integrate(dbeta, 0, 1, +shape1=2.5, shape2=0.5) +# dbeta(x, shape1=2.5, shape2=0.5) +## 1 with absolute error < 1.2e-05 +Example 9.30 The documentation of lapply (let us call help("lapply") now) states that this +function is defined as lapply(X, FUN, ...). Here, the ellipsis is a placeholder for a number of +optional arguments that can be passed to FUN. Hence, if we denote the i-th element of a vector X +by X[[i]], calling lapply(X, FUN, ...) will return a list whose i-th element will be equal to +FUN(X[[i]], ...). +Exercise 9.31 Usingasinglecallto lapply,generatealistwiththreenumericvectorsoflengths +3, 9, and 7, respectively, drawn from the uniform distribution on the unit interval. Then, upgrade +your code to get numbers sampled form the interval [−1, 1]. +9.5.7 +Metaprogramming (*) +Under the hood, lazy evaluation is a quite complicated mechanism that relies upon +the storing of unevaluated R expressions and special promises to instantiate them11. +It turns out that we have access to such expressions programmatically. In particular, +a call to the composition of deparse and substitute can convert them to a character +vector: +test_deparse_substitute <- function(x) +deparse(substitute(x)) +test_deparse_substitute(testing+1+2+3) +## [1] "testing + 1 + 2 + 3" +test_deparse_substitute(spam & spam^2 & bacon | grilled(spam)) +## [1] "spam & spam^2 & bacon | grilled(spam)" +Exercise 9.32 Check out the y-axis label generated by plot.default((1:100)^2). Inspect its +source code and note a call to the two aforementioned functions. +Similarly, call shapiro.test(log(rlnorm(100))) and take note of the data: field. +A function is free to do with such an expression whatever it likes. For instance, it can +manipulate it and evaluate it in a different context. Thanks to such a language feature, +11 Such an evaluation model has been heavily inspired by Scheme [31]. It will be explained in more detail +in sec:to-do. + +186 +II DEEPER +certain operations can be designed so that their users can express them much more +compactly. This is certainly (in theory) a very powerful tool but from practice we know +many instances where it has been over/misused and made the use of R confusing. +Example 9.33 (*) The built-in subset and transform use metaprogramming techniques to +specify basic data frame transformation techniques (see Section 12.3.9). For instance: +transform( +subset( +iris, +Sepal.Length>=7.7 & Sepal.Width >= 3.0, +select=c(Species, Sepal.Length:Sepal.Width) +), +Sepal.Length.mm=Sepal.Length/10 +) +## +Species Sepal.Length Sepal.Width Sepal.Length.mm +## 118 virginica +7.7 +3.8 +0.77 +## 132 virginica +7.9 +3.8 +0.79 +## 136 virginica +7.7 +3.0 +0.77 +Notethatnoneofthearguments–except iris–makessenseoutsideofthefunctioncallcontexts. +In particular, neither Sepal.Length nor Sepal.Width variables exist. +The two functions took the liberty to interpret the arguments passed as they felt like. They have +created their own virtual reality within our well-defined world. The reader must refer to their +documentation to discover the meaning of the special syntax used therein. +Note (*) Some functions have rather peculiar default arguments. For instance, in the +manualpageof prop.test,wereadthatthe alternativeparameterdefaultsto c("two. +sided", "less", "greater") but that "two.sided" is actually the default one. +If we call print(prop.test), we will find the code line responsible for this behaviour: +“alternative <- match.arg(alternative)”. Consider the following example: +test_match_arg <- function(x=c("a", "b", "c")) match.arg(x) +test_match_arg() +# missing argument — choose 1st +## [1] "a" +test_match_arg("c") +# one of the predefined options +## [1] "c" +test_match_arg("d") +# unexpected setting +## Error in match.arg(x): 'arg' should be one of "a", "b", "c" +In this setting, match.arg allows only an actual parameter amongst a given set of +choices, but selects the first option if the argument is missing. +Unfortunately,wehavetolearnthisbehaviourbyheart,becauseactuallylookingatthe +above source code gives us no clue about this being possible whatsoever. If such an ex- + +9 DESIGNING FUNCTIONS +187 +pression was normally evaluated, we would either be passing the default argument or +whatever the user passed as x (but then the function would not know about the range +of possible choices). A call to “match.arg(x, c("a", "b", "c"))” could guarantee the +desired functionality and would be much more readable. Instead, metaprogramming +techniques allowed match.arg to access the enclosing function’s default argument list +without our explicitly referring to them. +One may ask “why is it so” and the only sensible answer to this will be “because its +programmer decided it must be this way”. Let us contemplate this for a while. In +cases like this, we are dealing not with some base R language design choice that we +might like or dislike, but which we should normally just accept as an inherent feature. +Rather, we are struggling intellectually because of some R programmer’s (mis)use (in +good faith…) of R’s flexibility itself. They have introduced a slang/dialect on top of our +mother tongue, whose meaning is valid only within this function. Blame the middle- +man, not the environment, please. +We generally advocate for avoiding metaprogramming wherever possible (and will +elaborate on this later on, including formulas (`~`), built-in functions like subset or +transform, etc.). +9.6 +Exercises +Exercise 9.34 Answer the following questions: +• Will “stopifnot(1)” stop? What about “stopifnot(NA)”, “stopifnot(TRUE, FALSE)”, +and “stopifnot(c(TRUE, TRUE, NA))”? +• What does the `if` function return? +• Does `attributes<-`(x, NULL) modify x? +• When can we be interested in calling `[` and `[<-` as functions (and not as operators) dir- +ectly? +• How to define our own binary operator? Can it have some default arguments? +• What are the main use cases of `...`? +• What is wrong with transform, subset, and match.arg? +• When a call like “f(-1, do_something_that_takes_a_million_years())” does not ne- +cessarily have to be a bad idea? +Exercise 9.35 What is the return value of a call to “f(list(1, 2, 3))”? +f <- function(x) +{ +(continues on next page) + +188 +II DEEPER +(continued from previous page) +for (e in x) { +print(e) +} +} +Is it NULL, invisible(NULL), x[[length(x)]], or invisible(x[[length(x)]])? +Exercise 9.36 The split function also has its replacement version. Study its documentation to +learn how it works. +Exercise 9.37 A call to ls(envir=baseenv()) returns all objects defined in package base (see +Chapter 16). List the names corresponding to some replacement functions. +Important Apply the principle of test-driven development when solving the remain- +ing exercises (or those which you have skipped intentionally). +Exercise 9.38 Implement your own version of the Position and Find functions. Evaluation +should stop as soon as the first element fulfilling a given predicate has been found. +Exercise 9.39 Implement your own version of the Reduce function. +Exercise 9.40 Write a function slide(f, +x, +k, +...) which returns a list y of size +length(x)-k+1 such that y[[i]] = f(x[i:(i+k-1)], ...) +unlist(slide(sum, 1:5, 1)) +## [1] 1 2 3 4 5 +unlist(slide(sum, 1:5, 3)) +## [1] +6 +9 12 +unlist(slide(sum, 1:5, 5)) +## [1] 15 +Exercise 9.41 Using slide defined above, write another function that counts how many in- +creasing pairs of numbers are featured in a given numeric vector. For instance, in c(0,2,1,1, +0,1,6,0) there are three such pairs: (0,2), (0,1), (1,6). +Exercise 9.42 (*) Write your own version of tools::package_dependencies with re- +verse=TRUE based on information extracted by calling utils::available.packages. + +10 +S3 Classes +Let x be a randomly generated matrix with 1,000,000 rows and 1,000 columns, y be a +data frame with results from the latest survey indicating that things are not the way +most people (no matter the side of the many political spectra) think they are, and and +z be another matrix, this time with many zeroes. +Human brain is not capable of handling too much information which is too specific. +This is why we have a natural tendency to group different entities based on their sim- +ilarities so as to form some more abstract classes thereof. +Also, many of us are inherently lazy. Thus, oftentimes we will take shortcuts to min- +imise energy (at a price to be paid later). +Printing out a matrix, a data frame, and a time series are all still instances of the dis- +playing of things, although they surely differ in detail. Now that ad probably forgot- +ten which objects are hidden behind x, y, and z, being able to simply call “print(y)” +without having to recall that, yes, y is a data frame, might seem quite appealing. +This chapter introduces the so-called S3 classes [8], which provide a lightweight object +oriented programming (OOP) approach for automated dispatching of calls to generics +of the type “print(y)” to concrete methods such as “print.data.frame(y)”, based on the +class of the object they are invoked upon. +S3 classes in their essence are beautifully simple1. They are inspired2 by the well- +thought-through concepts present in other functional programming languages (such +as the Common Lisp Object System; see below). Ultimately, those generics and methods +are ordinary R functions (Chapter 7) and classes are merely additional object attributes +(Section 4.4). +Of course this does not mean that wrapping our heads around them will be effortless. +However, unlike other “class systems”3, S3 is ubiquitous in R programming: suffice it +1 However, some classes, even the built-in ones that we describe here, can be poorly designed (e.g, some +crucial methods might be missing, they can be not-well-interoperable with other classes, etc.). Do not +blame this messenger. Remember that the R environment is still very reliable. Also, there are cases where +changing the current behaviour in one place could lead to undesirable consequences elsewhere. +2 They were built on top of the ordinary (“old S”) R, hence have certain limitations what we discuss in the +sequel: classes cannot be formally defined (often we will use named lists for representing objects, and we +know we cannot be any more flexible than this), and the dispatching can only be based on the class of one +(usually the first, but, e.g., binary operators take both types into account) of the arguments. +3 Other class systems may give an impression that they are alien implants that were forcefully added to +our language to solve a specific, rather narrow class of problems; e.g., S4 (Section 11.5), Reference Classes +(Section 16.3), and other ones proposed by third-party packages + +190 +II DEEPER +to say that factors, matrices, and data frames discussed in the next chapters are quite +straightforward, S3-based extensions of the concepts we have introduced so far. +10.1 +Object Type vs Class +Recall that typeof (introduced in Section 4.1) returns the internal type of any R object. +Even though there are only few admissible cases thereof4, they open the world of end- +less possibilities5. +Thebasictypeswecoveredsofar(mostlyatomicandgenericvectors;compareFigure1) +provide a basis for more complex data structures. This is thanks to the fact that they +can be equipped with arbitrary attributes (Section 4.4). +typeof(NULL) +## [1] "NULL" +typeof(c(TRUE, FALSE, NA)) +## [1] "logical" +typeof(c(1, 2, 3, NA_real_)) +## [1] "double" +typeof(c("a", "b", NA_character_)) +## [1] "character" +typeof(list(list(1, 2, 3), LETTERS)) +## [1] "list" +typeof(function(x) x) +## [1] "closure" +The interesting fact is that most compound types, whose most prevalent instances are +constructed using the mechanisms discussed in this chapter6, only pretend they are +something different than what they actually are. They are often quite good at doing +their job, though, and hence might be useful. By knowing what is under their hood +we will demystify them and become able to manipulate their state outside of the pre- +scribed use cases. +Important Setting the class attribute might make some objects behave differently in +certain scenarios. +Example 10.1 Let us consider two identical objects equipped with different class attributes. +4 TheirlistishardcodedattheClanguagelevel;comparethelistof SEXPTYPEsin[48]andseealsoChapter +14. +5 In particular, later we mention externalptrs which are simply pointers to memory allocated on the +heap, so these might be any instances of C structs or C++ classes, etc. This makes R a very extensible lan- +guage. +6 But of course there is more; see the S4 and other systems discussed in Section 11.5. + +10 S3 CLASSES +191 +xt <- structure(123, class="POSIXct") +# POSIX calendar time +xd <- structure(123, class="Date") +Despite that both objects are being represented using numeric vectors: +c(typeof(xt), typeof(xd)) +## [1] "double" "double" +When printed, they are decoded quite differently: +print(xt) +## [1] "1970-01-01 10:02:03 AEST" +print(xd) +## [1] "1970-05-04" +In the former case, 123 is treated as the number of seconds since the so-called UNIX epoch, 1970- +01-01T00:00:00+0000. The latter is deciphered as the number of days since the said (quite +widely used in computer systems by the way) timestamp. +Wemayhencesuspect,andweareabsolutelyright,thatthereexistssomeunderlyingmechanism +that actually calls a version of print that is dependent on an object’s virtual class. +That this only depends on the class attribute, which might be set, unset, or reset quite freely, is +emphasised below: +attr(xt, "class") <- "Date" +# change class from POSIXct to Date +print(xt) +# same 123, but now interpreted as Date +## [1] "1970-05-04" +as.numeric(xt) +# drops all attributes +## [1] 123 +unclass(xd) +# drops the class attribute; `attr<-`(xd, "class", NULL) +## [1] 123 +We are having so much fun that one more illustration can only add to joy. +Example 10.2 Consider an example data frame: +x <- iris[1:3, 1:2] +# a subset of a built-in example data frame +print(x) +## +Sepal.Length Sepal.Width +## 1 +5.1 +3.5 +## 2 +4.9 +3.0 +## 3 +4.7 +3.2 +This is an object of the following class (an object whose class attribute is set to): +attr(x, "class") +## [1] "data.frame" + +192 +II DEEPER +Some may say, and they are absolutely right, that we have not covered data frames yet: this is +the topic of Chapter 12, which is still ahead of us. However, from the current perspective, we are +interestedinthefactthatanRdataframeismerelyalist(Chapter4)ofvectorsofthesamelengths +equipped with names and row.names attributes. +typeof(x) +## [1] "list" +attr(x, "class") <- NULL +# or x <- unclass(x) +print(x) +## $Sepal.Length +## [1] 5.1 4.9 4.7 +## +## $Sepal.Width +## [1] 3.5 3.0 3.2 +## +## attr(,"row.names") +## [1] 1 2 3 +Important Revealing how x is actually represented, enables us to process it (although +perhaps not in the most convenient or efficient manner) using the extensive skill set +that we have already7 developed by studying the material covered in the previous part +of our book (including solving all the exercises). This can be particularly useful, espe- +ciallybearinginmindthatsome(built-inorthird-party)datatypesarenotparticularly +well-designed. +Note again that attributes are simple additions to R objects. However, as we said in +Section 4.4.3, certain attributes are special, and class is one of them. +In particular, we can set class to be only a character vector (possibly of length greater +than one; see Section 10.2.5). +x <- 12345 +attr(x, "class") <- 1 +# character vectors only +## Error in attr(x, "class") <- 1: attempt to set invalid 'class' attribute +Furthermore, there exists the class function that can read the value of the class at- +tribute. Its replacement version is also available. +class(x) <- "Date" +# set; the same as attr(x, "class") <- "Date" +class(x) +# get; the same as attr(x, "class") +## [1] "Date" +Important +The class function always yields a value, even if the corresponding at- +7 For instance, consider once again the example from Section 5.4.3 that applies the split function on a +data frame reduced to a list. + +10 S3 CLASSES +193 +tribute is not set. We call it an implicit class. Compare between the following and the +outputs generated by typeof: +class(NULL) +# no `class` set, because NULL cannot have attributes at all +## [1] "NULL" +class(c(TRUE, FALSE, NA)) +# no attributes, so class is implicit (= typeof) +## [1] "logical" +class(c(1, 2, 3, NA_real_)) +# typeof yields "double" +## [1] "numeric" +class(c("a", "b", NA_character_)) +## [1] "character" +class(list(list(1, 2, 3), LETTERS)) +## [1] "list" +class(function(x) x) +# typeof yields "closure" +## [1] "function" +Also, in Chapter 11, we will note that any object equipped with the dim attribute also +has an implicit class: +(x <- as.matrix(c(1, 2, 3))) +## +[,1] +## [1,] +1 +## [2,] +2 +## [3,] +3 +attributes(x) +# `class` is not amongst the attributes +## $dim +## [1] 3 1 +class(x) +# implicit class +## [1] "matrix" "array" +typeof(x) +# it is still a numeric vector (under the hood) +## [1] "double" +10.2 +Generics and Method Dispatching +10.2.1 +Generics, Default, and Custom Methods +Let us inspect the source code of the print function: +print(print) +# sic! +## function (x, ...) +## UseMethod("print") +## +## + +194 +II DEEPER +Any function like the above8 we will call from now on an S3 (S version 3) generic. Its +only job is to invoke UseMethod("print")9. This dispatches the control flow to another +function, referred to as method, based on the class of the first argument. +Forexample,letusdefineanobjectofclass categorical(anamethatwehavejustcome +up with; we could have called it cat, CtGrCl, or SpanishInquisition as well), which will +be our own version of the famous built-in factor type that we discuss later. +x <- structure( +c(1, 3, 2, 1, 1, 1, 3), +levels=c("a", "b", "c"), +class="categorical" +) +We assume that such an object is a vector of small positive integers (codes) equipped +with the levels attribute being a character vector of length no less than the maximum +of the said integers. The first category will be used to decipher the meaning of code +“1”, for example. Hence, the above vector represents a sequence a, c, b, a, a, a, c. +We have not defined any special method for the printing of objects of class categor- +ical. Hence, when we call print, the default (fallback) method will be called: +print(x) +## [1] 1 3 2 1 1 1 3 +## attr(,"levels") +## [1] "a" "b" "c" +## attr(,"class") +## [1] "categorical" +This is the standard function for displaying numeric vectors that we are all well famil- +iar with. Its name is print.default, and we can always call it directly: +print.default(x) +# the default print method +## [1] 1 3 2 1 1 1 3 +## attr(,"levels") +## [1] "a" "b" "c" +## attr(,"class") +## [1] "categorical" +Wecan,however,introduceourownmethodforthecustomprintingofobjectsofclass +categorical, whose name must precisely be print.categorical: +print.categorical <- function(x, ...) +(continues on next page) +8 Note that some functions can have a version of UseMethod hidden at the C language level (internally); +see Section 10.2.3. +9 Which in this context is equivalent to UseMethod("print", x), with x being the first argument to the +function. + +10 S3 CLASSES +195 +(continued from previous page) +{ +x_character <- attr(x, "levels")[unclass(x)] +print(x_character) +# calls `print.default` +cat(sprintf("Categories: %s\n", +paste(attr(x, "levels"), collapse=", "))) +invisible(x) +# this is what all print methods do; see help("print") +} +Now, calling print automatically dispatches the control flow to the above method: +print(x) +## [1] "a" "c" "b" "a" "a" "a" "c" +## Categories: a, b, c +Of course, the default method can still be called; calling print.default(x) directly will +output the same result as before. +Note print.categorical has been equipped with the dot-dot-dot attribute, because +the generic print had one too; we should always ensure consistency ourselves10. +10.2.2 +Creating Own Generics +Introducing new S3 generics is as straightforward as defining a function that calls +UseMethod. +For instance, here is a dispatcher which allows for creating new objects of class cat- +egorical based on other objects: +as.categorical <- function(x, ...) +UseMethod("as.categorical") +We always need to define the default method: +as.categorical.default <- function(x, ...) +{ +x <- as.character(x) +xu <- unique(sort(x)) +# drops NAs +structure( +match(x, xu), +class="categorical", +levels=xu +) +} +10 In particular, the checking of S3 generic/method consistency is part of R package check. + +196 +II DEEPER +Testing: +as.categorical(c("a", "c", "a", "a", "d", "c")) +## [1] "a" "c" "a" "a" "d" "c" +## Categories: a, c, d +as.categorical(c(3, 6, 4, NA, 9, 9, 6, NA, 3)) +## [1] "3" "6" "4" NA +"9" "9" "6" NA +"3" +## Categories: 3, 4, 6, 9 +Note that print.categorical has been invoked twice here. The above is quite flexible +already, because it relies on the generic (Section 10.2.3) as.character, which handles +a wide variety of data types. Of course, it does not mean we cannot be more precise +about some particular ones. +Example 10.3 For instance, we might wantto forbidthe conversionfrom lists,becausethis does +not necessarily make sense: +as.categorical.list <- function(x, ...) +stop("conversion of lists to categorical is not supported") +The users can always be instructed in the method’s documentation that they are the ones re- +sponsible for an explicit conversion of list objects to something different prior to a call to as. +categorical. Whether this was a good design choice, time will tell. +Example 10.4 Note that the default method deals with logical vectors perfectly fine: +as.categorical(c(TRUE, FALSE, NA, NA, FALSE)) +# as.categorical.default +## [1] "TRUE" +"FALSE" NA +NA +"FALSE" +## Categories: FALSE, TRUE +However, we might still want to introduce a specialised version, because we know a slightly more +efficient algorithm (and we have nothing better to do) based on the fact that FALSE and TRUE con- +verted to numeric yield 0 and 1, respectively: +as.categorical.logical <- function(x, ...) +{ +x <- as.logical(x) +# or stopifnot(is.logical(x)) ? +structure( +x + 1, +# only 1, 2, and NAs will be generated +class="categorical", +levels=c("FALSE", "TRUE") +) +} +This yields the same result, but is a bit faster: +as.categorical(c(TRUE, FALSE, NA, NA, FALSE)) +# as.categorical.logical +(continues on next page) + +10 S3 CLASSES +197 +(continued from previous page) +## [1] "TRUE" +"FALSE" NA +NA +"FALSE" +## Categories: FALSE, TRUE +Note that we have performed some argument validation at the beginning, because a user is al- +ways able to call a method directly on an R object of any kind (which is a good thing!; see Sec- +tion 10.2.4). In other words, there is no guarantee that the argument x must be of type logical. +10.2.3 +Built-in Generics +Many functions and operators we have introduced so far are in fact S3 generics: print, +head, `[`, `+`, `<=`, as.character, as.list, round, log, sum, c, and na.omit, to name a +few. +Someofthemmightnotevencall UseMethodexplicitly;dispatchingcanbedoneintern- +ally, at the C language level11. Overall, the list of all S3 generics is somewhat difficult to +generate12, but at least the internal ones are enumerated in help("InternalMethods") +and help("groupGeneric"). +Example 10.5 Let us overload the as.character method. The default one does not make much +sense for the objects of our custom type: +as.character(x) +## [1] "1" "3" "2" "1" "1" "1" "3" +So: +as.character.categorical <- function(x, ...) +attr(x, "levels")[unclass(x)] +And now: +as.character(x) +## [1] "a" "c" "b" "a" "a" "a" "c" +Exercise 10.6 Overload the unique method for objects of class categorical. +Exercise 10.7 Overload the rep method for objects of class categorical. +Example 10.8 New types should be designed carefully. For instance, if we forget to consider +overloading the to-numeric converter, we might end up with some users being puzzled13 when +they see: +11 Which is quite unfortunate because it decreases transparency; we need to look this information up +somewhere in the documentation (instead of simply inspecting a function’s source code; see, e.g., cbind). +Also, it allows for some methods to be hardcoded at the C language level too and thus be unoverload- +able. Some of such design choices can somewhat be defended, though, as they increase execution speed +or memory consumption, but still: we are not fans thereof. +12 See also .knownS3Generics and .S3_methods_table which are related to the advanced topics we cover in +Section 18.3. +13 It is a different story if this is our conscious design choice and that this is the behaviour we really + +198 +II DEEPER +(x <- as.categorical(c(4, 9, 100, 9, 9, 100, 42, 666, 4))) +## [1] "4" +"9" +"100" "9" +"9" +"100" "42" +"666" "4" +## Categories: 100, 4, 42, 666, 9 +as.double(x) +# as.double.default(x) +## [1] 2 5 1 5 5 1 3 4 2 +Hence, we might want to introduce: +as.double.categorical <- function(x, ...) +{ +# as.double.default(as.character.categorical(x)) +as.double(as.character(x)) +} +Which now yields: +as.double(x) +# as.double.categorical(x) +## [1] +4 +9 100 +9 +9 100 +42 666 +4 +Note We can still use unclass to fetch the codes: +unclass(x) +## [1] 2 5 1 5 5 1 3 4 2 +## attr(,"levels") +## [1] "100" "4" +"42" +"666" "9" +This is because the above returns a class-free object, which is now guaranteed to be +handled by the default methods (print, subsetting, as.character, etc.). +Exercise 10.9 What would happen if we used as.numeric instead of unclass in print. +categorical and as.character.categorical? +Exercise 10.10 Update the above methods in such a way that we can also create named objects +of class categorical (i.e., equipped with the names attribute). +Exercise 10.11 Note that the levels of x are sorted lexicographically, not numerically. Introduce +a single method that would make the above code (when re-run without any alterations) generate +a more natural result. +want. If we document this thoroughly (see how help("factor") discusses the behaviour of a to-numeric +conversion), only a user’s ignorance will there be to blame when they still are confused about this behaviour. +Remember that we can never make an API totally foolproof and that there will always be someone who +will challenge/stress-test our ideas. Bad design is always bad, but being overprotective has its cons as well. +Choose your audience wisely. + +10 S3 CLASSES +199 +10.2.4 +DispatchingOnlyonOneArgumentandCallingS3MethodsDirectly +With S3, the dispatching is done based on the class of only one14 argument: by default, +the first one from the parameter list. +For example, the c function is a generic which dispatches on the class of the first argu- +ment. Let us overload it for categorical objects (or, more precisely, create a function +that will be dispatched to when the generic is called upon a series of objects such that +the first element is of the said class). +c.categorical <- function(...) +as.categorical( +unlist( +lapply(list(...), as.character) +) +) +It converts each argument to a character vector (relying on the generic as.character +to take care of the details) and makes use of the fact that unlist converts a list of such +atomic vectors to a single sequence of strings. +Calling c with the first argument being of class categorical dispatches to the above +method: +x <- c(9, 5, 7, 7, 2) +xc <- as.categorical(x) +c(xc, x) +# c.categorical +## +[1] "9" "5" "7" "7" "2" "9" "5" "7" "7" "2" +## Categories: 2, 5, 7, 9 +However, if the first argument is, say, unclassed, the default method will be consulted: +c(x, xc) +# default c +## +[1] 9 5 7 7 2 4 2 3 3 1 +This method ignores the class attribute and sees xc as-it-is, a barebone numeric vec- +tor: +`attributes<-`(xc, NULL) +# the underlying codes +## [1] 4 2 3 3 1 +This is not a bug! This is a well-documented (and now explained) behaviour. After all, +compound types (classed objects) are merely emulated through the basic ones. +14 This is R, so there are of course many exceptions to this rule which were made for the (debatable) sake +of the R users’ convenience. In particular, in Section 12.1.2 we mention that cbind and rbind will dispatch +to the data.frame method if at least one argument is a data frame (and other ones are unclassed). Also it +is worth noting that the S4 class system that we discuss in Section 11.5 allows for dispatching based on the +classes many arguments. + +200 +II DEEPER +Important In most cases, S3 methods can be called directly to get the desired out- +come: +c.categorical(x, xc) +# force a call to the specific method +## +[1] "9" "5" "7" "7" "2" "9" "5" "7" "7" "2" +## Categories: 2, 5, 7, 9 +Note We said “in most cases”, because some methods can be: +• hardcoded at the C language level (for instance, there is no c.default defined at +all15), +• hidden (defined in a package namespace but not exported); see Section 18.3, +• overloaded as a group; see Section 18.4. +Example 10.12 Just for fun, let us find a partition of the iris dataset into three clusters using +the k-means algorithm: +res <- kmeans(iris[-5], centers=3, nstart=10) +print(res) +## K-means clustering with 3 clusters of sizes 50, 62, 38 +## +## Cluster means: +## +Sepal.Length Sepal.Width Petal.Length Petal.Width +## 1 +5.0060 +3.4280 +1.4620 +0.2460 +## 2 +5.9016 +2.7484 +4.3935 +1.4339 +## 3 +6.8500 +3.0737 +5.7421 +2.0711 +## +## Clustering vector: +## +[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 +## [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 +## [71] 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 +## +[ reached getOption("max.print") -- omitted 51 entries ] +## +## Within cluster sum of squares by cluster: +## [1] 15.151 39.821 23.879 +## +(between_SS / total_SS = +88.4 %) +## +## Available components: +(continues on next page) +15 Also, dispatching can be done internally to internal methods: overloading `<.character` will have no +effect unless the base `<` is replaced with a custom one that makes an explicit call to UseMethod. Most often, +we can expect that the built-in types (e.g., atomic vectors), factors, data frames, and matrices and other +arrays might be treated specially. + +10 S3 CLASSES +201 +(continued from previous page) +## +## [1] "cluster" +"centers" +"totss" +"withinss" +## [5] "tot.withinss" "betweenss" +"size" +"iter" +## [9] "ifault" +The above is an object of class: +class(res) +## [1] "kmeans" +which in fact is a: +typeof(res) +## [1] "list" +The underlying list looks like: +unclass(res) +## $cluster +## +[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 +## [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 +## [71] 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 +## +[ reached getOption("max.print") -- omitted 51 entries ] +## +## $centers +## +Sepal.Length Sepal.Width Petal.Length Petal.Width +## 1 +5.0060 +3.4280 +1.4620 +0.2460 +## 2 +5.9016 +2.7484 +4.3935 +1.4339 +## 3 +6.8500 +3.0737 +5.7421 +2.0711 +## +## $totss +## [1] 681.37 +## +## $withinss +## [1] 15.151 39.821 23.879 +## +## $tot.withinss +## [1] 78.851 +## +## $betweenss +## [1] 602.52 +## +## $size +## [1] 50 62 38 +## +(continues on next page) + +202 +II DEEPER +(continued from previous page) +## $iter +## [1] 2 +## +## $ifault +## [1] 0 +and we already know that the above is displayed in a fancy way only because there is a print +method overloaded for objects of class kmeans. +But is there really? +print.kmeans +## Error in eval(expr, envir, enclos): object 'print.kmeans' not found +Even though the method is hidden in the stats package’s namespace, from Section 18.3 we +will learn that it can be accessed by calling getS3method("print", "kmeans") or referring to +stats:::print.kmeans (note the triple colon). +10.2.5 +Multi-class-ness +The class attribute can be instantiated as a character vector of any length. For ex- +ample: +(t1 <- Sys.time()) +## [1] "2022-12-27 20:49:37 AEDT" +(t2 <- strptime("2021-08-15T12:59:59+1000", "%Y-%m-%dT%H:%M:%S%z")) +## [1] "2021-08-15 12:59:59" +Let us inspect the two objects’ classes: +class(t1) +## [1] "POSIXct" "POSIXt" +class(t2) +## [1] "POSIXlt" "POSIXt" +When we discuss date-time classes in more detail later, we will take note that the +former is represented as a numeric vector, whilst the latter is a list. Hence, primarily, +these two should be seen as instances of two distinct types. However, both of them +have a lot in common, hence it was a wise design choice to also allow them to be seen +as the representatives of the same generic category of POSIX time objects. +Important +When calling a generic function16 f on an object x of classes17 class1, +16 The case of binary operators is handled differently; see Section 10.2.6. +17 UseMethod dispatches on the implicit class as determined by the class function (note that the class +attribute does not necessarily have to be set in order for class to return a sensible answer). + +10 S3 CLASSES +203 +class2, …, classK (in this order), UseMethod(f, x) dispatches to the method determ- +ined as follows: +1. if f.class1 is available18, call it; +2. otherwise, if f.class2 is available, call this one; +3. …; +4. otherwise, if f.classK is available, invoke it; +5. otherwise, refer to the fallback f.default. +Example 10.13 There is a method diff for objects of class POSIXt featuring a statement: +r <- if (inherits(x, "POSIXlt")) as.POSIXct(x) else x +This way, we can be handling both POSIXct and POSIXlt instances via the same procedure. +Letusseeinthissimpleschemeanymagic.Itisnothingmorethanwhatwasdescribed +above: a way of determining which method should be called for a particular R object. +It can of course be used as a mechanism to mimic (and certainly it was inspired by) +the idea of inheritance in object-oriented programming languages, but note that the +S3 system does not allow for defining classes in any formal manner. +For example, we cannot say that objects of class POSIXct inherit from POSIXt or each +object of class POSIXct is also an instance of POSIXt. The class attribute can still be set +arbitrarily on an per-object basis: we can create ones whose class is simply POSIXct +(without the POSIXt part) or even c("POSIXt", "POSIXct") (in this very order). +10.2.6 +Operator Overloading +Operators are ordinary functions (Section 9.4.4). Even though what follows can par- +tially be implied by what we have said above, as usual in R, there will be some oddities. +For example, let us overload the index operator for objects of class categorical. Look- +ing at help("["), we see that the default method19 has two arguments: x (the categor- +ical object being sliced) and i (the indexer). Ours will have the same interface then: +`[.categorical` <- function(x, i) +{ +structure( +unclass(x)[i], +# `[`(unclass(x), i) +class="categorical", +levels=attr(x, "levels") +# the same levels as input +(continues on next page) +18 For more details on S3 method lookup; see Section 18.3. +19 Note that the default S3 method, `[.default`, is hardcoded at the C language level. Therefore, we can- +not refer to it directly (but unclass does the trick). Also note that we can also call NextMethod here; see Sec- +tion 18.3. + +204 +II DEEPER +(continued from previous page) +) +} +We can also introduce the replacement version of this operator: +`[<-.categorical` <- function(x, i, value) +{ +levels <- attr(x, "levels") +codes <- match(value, levels) +# integer codes corresponding to levels +x <- unclass(x) +x[i] <- codes +# default method for the replacement version of `[` +structure( +x, +class="categorical", +levels=levels +# same levels as input +) +# # or, equivalently: +# structure( +# +`[<-`(unclass(x), i, value=match(value, attr(x, "levels"))), +# +class="categorical", +# +levels=attr(x, "levels") +# ) +} +Testing: +x <- as.categorical(c(3, 6, 4, NA, 9, 9, 6, NA, 3)) +x[1:4] +## [1] "3" "6" "4" NA +## Categories: 3, 4, 6, 9 +x[1:4] <- c("6", "7") +print(x) +## [1] "6" NA +"6" NA +"9" "9" "6" NA +"3" +## Categories: 3, 4, 6, 9 +Note how we handled the case of non-existing levels and that the recycling rule has +been automagically inherited (amongst other features) from the default index oper- +ator. +Exercise 10.14 Do these two operators preserve the names attribute of x? Is indexing with neg- +ative integers or logical vectors supported as well? Why is that/is that not the case? +Furthermore, let us overload the `==` operator. Assume20 that we would like two cat- +20 There are of course many possible ways to implement the `==` operator. For instance, it may return +eitherasingle TRUEor FALSEdependingiftwoobjectsareidentical(althoughprobablyoverloading all.equal + +10 S3 CLASSES +205 +egorical objects be compared based on the actual labels they encode, in an element- +wise manner: +`==.categorical` <- function(e1, e2) +as.character(e1) == as.character(e2) +We are feeling lucky: by not performing any type checking, we rely on the particular +as.character methods corresponding to the types of e1 and e2. Also, assuming that +as.character always21 returns an object of type character, we dispatch to the default +method for `==` (which handles atomic vectors). +Some examples: +as.categorical(c(1, 3, 5, 1)) == as.categorical(c(1, 3, 1, 1)) +## [1] +TRUE +TRUE FALSE +TRUE +as.categorical(c(1, 3, 5, 1)) == c(1, 3, 1, 1) +## [1] +TRUE +TRUE FALSE +TRUE +c(1, 3, 5, 1) == as.categorical(c(1, 3, 1, 1)) +## [1] +TRUE +TRUE FALSE +TRUE +Important In the case of binary operators, dispatching is done based on the classes +of both arguments. In all three example calls above, we call `==.categorical`, regard- +less of whether the classed object is the first or the second operand. If two operands +are classed and different methods are overloaded for both of them, a warning will be +generated and the default internal method will be called. +`==.A` <- function(e1, e2) "A" +`==.B` <- function(e1, e2) "B" +structure(c(1, 2, 3), class="A") == structure(c(2, NA, 3), class="B") +## Warning: Incompatible methods ("==.A", "==.B") for "==" +## [1] FALSE +NA +TRUE +Note In Section 18.4, we will mention that operators as well as certain groups of func- +tions (including min, sum, and all or abs, log, and round) can be overloaded all at once; +see also help("groupGeneric"). +would be a better idea). We could also be comparing the corresponding underlying integer codes instead of +the labels, etc. +21 Which of course does not have to be the case; it is merely an assumption based on our belief in the +common sense of other developers. + +206 +II DEEPER +10.3 +Common Built-in S3 Classes +Let us discuss some noteworthy built-in classes, including the ones that represent +date/time information and factors (ordered or not). +Classes for representing tabular data will be dealt with in separate parts of this text- +book, owing to their importance and ubiquity. Namely, matrices and other arrays are +covered in Chapter 11, and data frames in Chapter 12. +The inspecting of other22 interesting classes is left as a simple exercise to the kind +reader. +10.3.1 +Date, Time, etc. +The Date class can be used to represent… dates. +(x <- c(Sys.Date(), as.Date(c("1969-12-31", "1970-01-01", "2023-02-29")))) +## [1] "2022-12-27" "1969-12-31" "1970-01-01" NA +class(x) +## [1] "Date" +Complex types are built upon basic ones; underneath, what we deal with is: +typeof(x) +## [1] "double" +unclass(x) +## [1] 19353 +-1 +0 +NA +whichisthenumberofdayssincethesocalledUNIXepoch,1970-01-01T00:00:00+0000 +(midnight GMT/UTC). +The POSIXct (calendar time) class can be used to represent date-time objects: +(x <- Sys.time()) +## [1] "2022-12-27 20:49:37 AEDT" +class(x) +## [1] "POSIXct" "POSIXt" +typeof(x) +## [1] "double" +unclass(x) +## [1] 1672134577 +Underneath, it is the number of seconds since the UNIX epoch. By default, whilst +22 An(incomprehensive)approximationtothelistofavailableclassescanbegeneratedbycalling unique(. +S3_methods_table[, 2]). + +10 S3 CLASSES +207 +printing, the current default timezone is used (see Sys.timezone). However, such ob- +jects can be equipped with the tzone attribute. +structure(1, class=c("POSIXct", "POSIXt")) +# using current default timezone +## [1] "1970-01-01 10:00:01 AEST" +structure(1, class=c("POSIXct", "POSIXt"), tzone="UTC") +## [1] "1970-01-01 00:00:01 UTC" +In both cases, the time is 1 second after the beginning of UNIX epoch. In the former, +it is displayed in the current local timezone, though (on the author’s PC). +Exercise 10.15 UseISOdatetimetoinspecthowmidnightsaredisplayedindifferenttimezones. +There is also the POSIXlt (local time) class, which is represented using a list of atomic +vectors23. +(x <- as.POSIXlt(c(a="1970-01-01 00:00:00", b="2030-12-31 23:59:59"))) +## +a +b +## "1970-01-01 00:00:00 AEST" "2030-12-31 23:59:59 AEDT" +class(x) +## [1] "POSIXlt" "POSIXt" +typeof(x) +## [1] "list" +str(unclass(x)) +# calling str instead of print to make display more compact +## List of 11 +## +$ sec +: num [1:2] 0 59 +## +$ min +: int [1:2] 0 59 +## +$ hour +: int [1:2] 0 23 +## +$ mday +: int [1:2] 1 31 +## +$ mon +: int [1:2] 0 11 +## +$ year +: Named int [1:2] 70 130 +## +..- attr(*, "names")= chr [1:2] "a" "b" +## +$ wday +: int [1:2] 4 2 +## +$ yday +: int [1:2] 0 364 +## +$ isdst : int [1:2] 0 1 +## +$ zone +: chr [1:2] "AEST" "AEDT" +## +$ gmtoff: int [1:2] NA NA +Exercise 10.16 Read about the meaning of each named element, especially mon and year; see +help("DateTimeClasses"). +The manual states that POSIXlt is supposedly closer to human-readable forms than +POSIXct, but it is a matter of taste. Some R functions return the former, and some +yield the latter type. +Exercise 10.17 The two main functions for date formatting and parsing, strftime and strp- +23 Which was inspired by C’s tm structure defined in . + +208 +II DEEPER +time, use special field formatters (similar to those used by sprintf). Read about them in the R +manual. What type of inputs do they accept? What outputs do they produce? +There is a number of methods overloaded for objects of the said classes. In fact, the +first call in this section already involved the use of c.Date. +Exercise 10.18 Play around with the overloaded versions of seq, rep, and as.character. +Note that a specific number of days or seconds can be added to or subtracted from a +date or time, respectively. However, - (see also diff) can also be applied on two date- +time objects, which yields an object of class difftime. +Sys.Date() - (Sys.Date() - 1) +## Time difference of 1 days +Sys.time() - (Sys.time() - 1) +## Time difference of 1 secs +Exercise 10.19 Check out how objects of class difftime are internally represented. +Applying other arithmetic operations on date-time objects yields an error. Also note +that because date-time objects are just numbers, they can be compared to each other +using binary operators24 and methods such as sort and order25. +Exercise 10.20 Check out the stringx package [22] which replaces the base R date-time pro- +cessing functions with their more portable counterparts. +Exercise 10.21 system.time can be used to measure the time to execute a given expression: +system.time({ +sum(runif(1e7)) +# whatever, just testing +}) +## +user +system elapsed +## +0.232 +0.012 +0.245 +The function returns an object of class proc_time. Inspect how it is represented internally. +10.3.2 +Formulae (*) +Formulae (created by means of `~`) are quite advanced language constructs and hence +they will be discussed much further: in Section 15.2. +Some R users refer to them in functions such as lm, aggregate, t.test, boxplot, or +plot to specify models or queries such as “y as a function of x1, x2, and x3” and “y +grouped/split by a combination of x1 and x2” where y, x1, etc. are for example column +names in a data frame or named items in a list. +There is no single standard governing how a function should interpret a formula’s +24 The overloaded group generic Ops prevents us from adding or multiplying two dates and defines the +meaning of the comparison operators; see Section 18.4. +25 See an exercise below on the use of xtfrm. + +10 S3 CLASSES +209 +terms. In fact, each procedure is free to introduce its own meaning (a micro-language +built on top of R). Due to this, yours truly discourages26 their use (especially by begin- +ners). +10.3.3 +Factors +The factor class is often used to represent categorical (qualitative) data, e.g., species, +groups, types. In fact, the example categorical class that we played with above has +been inspired by the built-in factor. +(x <- c("spam", "spam", "bacon", "sausage", "spam", "bacon")) +## [1] "spam" +"spam" +"bacon" +"sausage" "spam" +"bacon" +(f <- factor(x)) +## [1] spam +spam +bacon +sausage spam +bacon +## Levels: bacon sausage spam +Take note of how factors are printed: there are no double quote characters around the +labels and the list of levels is given at the end. +Internally, such objects are represented as integer vectors (Section 6.4.1) with ele- +ments between 1 and k with the special (as in Section 4.4.3) levels attribute being a +character vector of length k27. +class(f) +## [1] "factor" +typeof(f) +## [1] "integer" +unclass(f) +## [1] 3 3 1 2 3 1 +## attr(,"levels") +## [1] "bacon" +"sausage" "spam" +attr(f, "levels") +# also: levels(f) +## [1] "bacon" +"sausage" "spam" +Factors are often used instead of character vectors defined over a small number of +unique labels28, where there is a need to manipulate their levels easily. +attr(f, "levels") <- c("a", "b", "c") +# also levels(f) <- c(....new...) +print(f) +(continues on next page) +26 For example, lm.fit can be used instead of lm. It is slightly more difficult to learn, surely, but has +the added benefit of making sure the user knows that all model variables are not magical (especially the +nonlinear/mixed effect terms). +27 [49] states: Factorsarecurrentlyimplementedusinganintegerarraytospecifytheactuallevelsandasecondarray +of names that are mapped to the integers. Rather unfortunately users often make use of the implementation in order to +makesomecalculationseasier.This, however,is animplementationissueand isnotguaranteedtohold in all implement- +ations of R. Still, fortunately, this has been a de facto standard for factors for a very long time. +28 Recallthatthereisaglobal(internal)stringcache,hencehavingmanyduplicatedstringsisnotanissue, +memory-use-wisely. + +210 +II DEEPER +(continued from previous page) +## [1] c c a b c a +## Levels: a b c +The underlying codes remain the same. +Certain operations on vectors of small integers are relatively easy to implement, es- +pecially those concerning element grouping: splitting, counting, plotting (e.g., Fig- +ure 13.1). It is because the integer codes can naturally be used whilst indexing other +vectors. In Section 5.4, we mentioned a few functions related to this, such as match, +split, findInterval,and tabulate.Specifically,thelattercanbeimplementedlike“for +each i, increase count[factor_codes[i]] by one”. +Exercise 10.22 Study the source code of the factor function. Note the use of as.character, +unique, order, and match. +Exercise 10.23 Implementasimplifiedversionof tablebasedon tabulate.Itshouldworkfor +objects of class factor and return a named numeric vector. +Exercise 10.24 Implement your own version of cut based on findInterval. +Important The as.numeric method has not been overloaded for factors. Therefore, +when we call the generic, the default method is used: it returns the underlying integer +codes as-is. This can surprise the unaware users when they play with factors that fea- +ture levels consisting of strings representing integer numbers: +(g <- factor(c(11, 15, 16, 11, 13, 4, 15))) +# converts numbers to strings +## [1] 11 15 16 11 13 4 +15 +## Levels: 4 11 13 15 16 +as.numeric(g) +# the underlying codes +## [1] 2 4 5 2 3 1 4 +as.numeric(as.character(g)) +# to get the numbers en-coded +## [1] 11 15 16 11 13 +4 15 +Unfortunately, support for factors is often hardcoded at the C language level, which +will make this class behave less predictably (from the R perspective). In particular, the +manual overloading of methods for factor objects might have no effect. +Important If f is a factor, then x[f] does not behave like x[as.character(f)] (index- +ing by labels, using the names attribute). Instead, we get x[as.numeric(f)] (the under- +lying codes will determine the positions). +h <- factor(c("a", "b", "a", "c", "a", "c")) +levels(h)[h] +# the same as c("a", "b", "c")[c(1, 2, 1, 3, 1, 3)] +## [1] "a" "b" "a" "c" "a" "c" +c(b="x", c="y", a="z")[h] +# names are not used whilst indexing +(continues on next page) + +10 S3 CLASSES +211 +(continued from previous page) +## +b +c +b +a +b +a +## "x" "y" "x" "z" "x" "z" +c(b="x", c="y", a="z")[as.character(h)] +# names are used now +## +a +b +a +c +a +c +## "z" "x" "z" "y" "z" "y" +More often than not, indexing by factors will happen “accidentally”, leading to our +being slightly puzzled. In particular, factors look much like character vectors when +they are featured in data frames: +(df <- data.frame(A=c("x", "y", "z"), B=factor(c("x", "y", "z")))) +## +A B +## 1 x x +## 2 y y +## 3 z z +class(df[["A"]]) +## [1] "character" +class(df[["B"]]) +## [1] "factor" +(*)UpuntilR4.0,manyfunctions(including data.frameand read.csv)hadthe string- +sAsFactors option (see help("options")) set to TRUE, which resulted in all character +vectors’ being automatically converted to factors when, e.g., creating data frames +(compare Chapter 12). Luckily, this is no longer the case, but they can still be en- +countered sporadically: for instance, the built-in iris dataset has the fifth column of +class: +class(iris[["Species"]]) +## [1] "factor" +Important Be careful when combining factors and not-factors: +x <- factor(c("A", "B", "A")) +c(x, "C") +## [1] "1" "2" "1" "C" +c(x, factor("C")) +## [1] A B A C +## Levels: A B C +Exercise 10.25 Notethatwhensubsettingafactorobject,theresultwillhavethe levelsattrib- +ute inherited as-is. + +212 +II DEEPER +f[c(1, 2)] +## [1] c c +## Levels: a b c +Implement your own version of the droplevels function which removes the unused attributes. +Exercise 10.26 The replacement version of the index operator does not automatically add new +levels to the modified object: +x <- factor(c("A", "B", "A")) +`[<-`(x, 4, value="C") +# like in x[4] <- "C" +## Warning in `[<-.factor`(x, 4, value = "C"): invalid factor level, NA +## generated +## [1] A +B +A + +## Levels: A B +Implement your own version of `[<-.factor]` which is capable of doing so. +10.3.4 +Ordered Factors +Note that when creating factors, we can enforce a particular ordering and the number +of levels: +x <- c("spam", "spam", "bacon", "sausage", "spam", "bacon") +factor(x, levels=c("eggs", "bacon", "sausage", "spam")) +## [1] spam +spam +bacon +sausage spam +bacon +## Levels: eggs bacon sausage spam +If we want the arrangement of the levels to define a linear ordering relation over set +of the labels, we can call: +(f <- factor(x, levels=c("eggs", "bacon", "sausage", "spam"), ordered=TRUE)) +## [1] spam +spam +bacon +sausage spam +bacon +## Levels: eggs < bacon < sausage < spam +class(f) +## [1] "ordered" "factor" +This yields an ordered factor, which enables comparisons like: +f[f >= "bacon"] +# what's not worse than bacon? +## [1] spam +spam +bacon +sausage spam +bacon +## Levels: eggs < bacon < sausage < spam +How is that possible? Well, based on information provided in this chapter it will come +as no surprise that it is because… someone has implemented a comparison operator +for objects of class ordered. + +10 S3 CLASSES +213 +10.4 +Argument Checking Revisited +Recall that anything can be passed as a function’s input. Here are some additions to +the topic we touched upon in Section 9.2.1. +Despitethatcompoundobjectsareinternallyrepresentedthroughbasictypes(suchas +numeric vectors, lists, or combinations thereof) and attributes, unless we really know +better (which, by the way, this book is all about), we should be relying on the hopefully +well-thought-out methods developed by the class’ designer. +Ideally, when checking arguments passed to a function, determining if an object is of +a desired type should be solely done by means of the generics like is.class. If that is +not the case, a call to as.class should be used to make sure we will be dealing with an +object of the desired type. +If a conversion is not possible, either because a specific method is unavailable or be- +cause its designer decided that this must be the case, whatever the consequences are +is not necessarily our problem anymore. +We should explain to the user that the input type assurance is done via this very mech- +anism and, in case they get any surprising results, they should check/redefine the un- +derlying is.class or as.class themselves. +This is of course not watertight, and there will be users complaining that they get un- +expected or confusing (in their opinion) outputs. With infinitely many potential types, +however, we cannot respond to every possible situation. +Example 10.27 As an illustration, here is a function that counts the number of occurrences of +items in a numerised (digitised?) version of a given object: +numtable <- function(x) +{ +if (!is.numeric(x)) x <- as.numeric(x) +# two generics! +u <- unique(x) +structure( +tabulate(match(x, u)), +names=as.character(u) +) +} +Let us assume that the user has been informed (in the corresponding documentation page) that x +must be a numeric vector (as in is.numeric) or an object coercible to (by means of as.numeric). +The callers will be stress-testing our function in many different ways: +numtable(c(1, 3, 5, 5, 1, 5)) +(continues on next page) + +214 +II DEEPER +(continued from previous page) +## 1 3 5 +## 2 1 3 +This is an intended behaviour. +numtable(c("1", "3", "5", "5", "1", "5")) +## 1 3 5 +## 2 1 3 +This makes sense too, a character vector consisting of number-strings has been fed on input. +numtable(c("a", "e", "z", "z", "a", "z")) +## Warning in numtable(c("a", "e", "z", "z", "a", "z")): NAs introduced by +## coercion +## +## +6 +Does the output make no sense? Of course, it does, they have just passed something not easily +coercible to a numeric vector. Note the warning that suggests there is something wrong. The user +needs to correct their possible mistake by themself. +numtable(list(1, 2, 3:10, 2)) +## Error in numtable(list(1, 2, 3:10, 2)): 'list' object cannot be coerced to type 'double' +Again, makes sense. ‘But I think that this function should apply unlist automatically’ – well, +if you want such a behaviour, why don’t you call numtable(unlist(...)) yourself? It is not so +difficult. +numtable(factor(c(1, 3, 5, 5, 1, 5))) +## 1 2 3 +## 2 1 3 +Is this confusing? No; this is a well-documented behaviour of as.numeric on objects of type +factor(whichwasdesignedbyanotherdeveloper).Ausershouldknow(butwecanremindthem +about it in the documentation) that in this case, as.character should rather be called first. +Of course, sometimes users might discover bugs or unexpected behaviours, especially related to +boundarycaseswehavenotbeenconsiderateenoughtoinspect.Weareofcoursetheonestoblame +for the following: +numtable(numeric(0)) +# bug: this should be corrected +## +## +0 + +10 S3 CLASSES +215 +10.5 +(Over)using the Forward-pipe Operator, `|>` (*) +The object-oriented programming paradigm is useful when we wish to define a new +data type, perhaps even a hierarchy of types. Many development teams find it an effi- +cient tool to organise larger pieces of software. Yet, in the broad data science and nu- +merical computing domains, we are more often consumers of OOP rather than class +designers. +Thanks to the discussed method dispatch mechanism, our language is easily extens- +ible and something that mimics a new data type can easily be introduced. Most im- +portantly, methods can be added or removed during run-time, e.g., when importing +external packages. +However, R is still a functional programming language, where functions not only are +first-class citizens; they are privileged. Of course, there are some inherent limitations +stemming from the ingenious simplicity of S3: method dispatch is usually based only +on the type of the first function argument, classes cannot be defined formally (but see +Section 11.5) and that there is no real encapsulation (we cannot actually hide data from +a user29). However, overall the whole concept has proven quite versatile. +In functional programming, emphasis is on operations (verbs), not data (nouns). This +leadstoaveryreadablesyntax,forexample(assumingthatsquare,x,andyaresensibly +defined), the mean squared error can be written as: +mean(square(x-y)) +This is very user-centric. However, when implementing more complex data pro- +cessing pipelines, a programmer thinks “first, I need to do this, then I need to do that, +end afterwards…”. When they write it down, there can be some pressing of HOME and +END keys on the keyboard involved. This should not be a problem for most program- +mers. +finally(thereafter(then(first(x)))) +However, some people are inherently lazy, always complaining and/or always trying +to “optimise”30 things. +Example 10.28 Base R is of course extremely flexible and we can introduce new vocabulary as +we please. In Chapter 12, we study an example, where we define: +29 Which can be good, right? +30 Do not get yours truly wrong, improving things is generally good, but overall, in the long run, as a +compulsive habit (“this is what (some) stakeholders want”, “we need to be agile and responsive”, etc.), it is +not really sustainable (also for the environment!). Less is better, even though a little harder. By introducing +a new, parallel syntax, we not only duplicate the existing features and cause some divide in the community +(someuserswillbeintroducedtothesystemthroughthenewinterfaceandnotknowtheoldone,otherswill +have to learn the new syntax to be able to communicate with the former group) but also introduce a whole +new set of issues (how to make the new functions interoperable with each other in a seamless manner, etc.). + +216 +II DEEPER +• group_by(afunctionthatsplitsadataframewithrespecttoacombinationoflevelsingiven +named columns and returns a list of data frames with class list_dfs), +• aggregate.list_dfs (which applies a given aggregation function on each column of each +data frame in a given list), and +• mean.list_dfs (a specialised version of the former that calls mean). +Thespecificsdonotreallymatternow,letusjustconsiderthenotationweusewhentheoperations +are chained: +# select a few rows and columns from the `iris` data frame: +iris_subset <- iris[51:150, c("Sepal.Width", "Petal.Length", "Species")] +# compute the averages of all variables grouped by Species: +mean(group_by(iris_subset, "Species")) +## +Species +x +Mean +## 1 versicolor +Sepal.Width 2.770 +## 2 versicolor Petal.Length 4.260 +## 3 +virginica +Sepal.Width 2.974 +## 4 +virginica Petal.Length 5.552 +This is quite readable: we compute the mean in groups defined by Species in a subset of the iris +data frame. All verbs appear on the lefthand side of the expression, with the last (the most im- +portant?) operation being listed first. +By the way, self-explanatory variable names and rich comments are priceless. +In more traditional object-oriented programming languages, either the method list +is sealed inside31 the class’ definition (like in C++), or some peculiar patches must be +applied to inject a method (like in Python)32. There, it is the objects that are told what +to do: they are treated as black boxes. +Some popular languages rely on the message-passing syntax, where operations are +propagated (and written) left-to-right instead of inside-out. For instance, in C++ and +Python(amongstmanyothers),“obj.method1().method2()”means“call method1on obj +and then call method2 on the result. +SinceR4.1.0, thereisa pipeoperator33,`|>`,whichismerelyasyntacticsugarfortrans- +lating between the message-passing and function-centric notion. In a nutshell, writ- +ing: +x |> f() |> g(y) |> h() +(x-y) |> square() |> mean() +is equivalent to: +31 When methods are parts of particular classes, there can be a lot of duplicated code. Functional OOP +can be more developer-friendly as we can implement all methods related to roughly the same functionality +in one spot. +32 See also the concept of extension methods in C# or Kotlin or, to some extent, class inheritance. +33 It was inspired by `|` in Bash and `|>@` in F# and Julia (which are part of the language specification). +Also, there is a `%>%` operator (and related ones) in the R package magrittr. + +10 S3 CLASSES +217 +h(g(f(x), y)) +mean(square(x-y)) +This syntax is developer-centric: it emphasises on the order in which the operations +are executed, something that could always be achieved with the function-centric form +and perhaps a few auxiliary variables. +Example 10.29 In the above example, a pipe operator version of the iris aggregation exercise +would look like: +iris_subset |> group_by("Species") |> mean() +This book is minimalistic by design and there is nothing that cannot be achieved +without the pipe operator, hence we will be refraining34 ourselves from using it. +10.6 +Exercises +Exercise 10.30 Answer the following questions: +• How to display the source code of the default methods for head and tail? +• Can there be, at the same time, one object of class c("A", "B") and another one of class +c("B", "A")? +• If f is a factor, what are the relationships between as.character(f), as.numeric(f), as. +character(as.numeric(f)), and as.numeric(as.character(f))? +• If x is a named vector and f is a factor, is x[f] equivalent to x[as.character(f)] or rather +x[as.numeric(f)]? +Exercise 10.31 A user calls: +plot(x, y, col="red", ylim=c(1, max(x)), log="y") +where x and y are numeric vectors. Consult help("plot") for the meaning of the ylim and log +arguments. Was that straightforward? +Exercise 10.32 Explain why the two following calls yield significantly different results and +present a workaround: +c(Sys.Date(), "1970-01-01") +## [1] "2022-12-27" "1970-01-01" +c("1970-01-01", Sys.Date()) +## [1] "1970-01-01" "19353" +34 Which some readers would name an uncool (old-school) approach, but we do not care. Remember that +the functional syntax is the native one and we have to be able to understand it anyway. + +218 +II DEEPER +Exercise 10.33 Write methods head and tail for our example categorical class. +Exercise 10.34 (*)WriteanRpackagethatdefinesS3classcategoricalandacoupleofmeth- +ods therefor. Note the need for the use of the S3method directive NAMESPACE; see [45]. +Exercise 10.35 Inspect the result of a call to binom.test(79, 100). Find the method respons- +ible for the pretty-printing of such objects. +Exercise 10.36 Inspect the result of a call to rle(c(1, 1, 1, 4, 3, 3, 3, 3, 3,, 2, 2)). +Find the method responsible for the pretty-printing of such objects. +Exercise 10.37 Readmoreabouttheconnectionclass;seetheValuesectioninhelp("connections"). +Exercise 10.38 Readaboutthesubsettingoperatorsoverloadedforthepackage_versionclass; +see help("numeric_version"). +Exercise 10.39 There are xtfrm methods overloaded for classes such as numeric_version, +difftime, Date,and factor.Findouthowtheyworkandwheretheymightbeuseful(especially +in relation to order and sort; see also Section 12.3.1). +Exercise 10.40 Give an example where split(x, list(y1, y2)) (with default arguments) +will fail to generate the correct result. +Exercise 10.41 Write a function that determines the mode, i.e., the most frequently occurring +value in a given object of class factor. If the mode is not unique, return a randomly chosen one +(each with the same probability). +Exercise 10.42 Implement your own version of the gl function. +Exercise 10.43 Check out which built-in date-time functions the stringx package replaces +with more portable ones. + +11 +Matrices and Other Arrays +When we equip an atomic or generic vector with the dim attribute, it automatically +becomes an object of S3 class array. In particular, two-dimensional arrays (primary +S3 class matrix) allow us to represent tabular data where items are aligned into rows +and columns: +structure(1:6, dim=c(2, 3)) +# a matrix with 2 rows and 3 columns +## +[,1] [,2] [,3] +## [1,] +1 +3 +5 +## [2,] +2 +4 +6 +This (combined with the fact that there are many built-in functions overloaded for the +matrix class) opens up a range of new possibilities, which we explore in this chapter. +In particular, we discuss how to perform basic algebraic operations such as matrix +multiplication, transpose, finding eigenvalues, and performing various decomposi- +tions. We also cover data wrangling operations such as array subsetting and column- +and rowwise aggregation. +Important Oftentimes, a numeric matrix with n rows and m will be used to represent +n points (samples) in an m-dimensional (with m features or variables) space, ℝ𝑚. +Furthermore, in the next chapter, we will introduce data frames: matrix-like objects +whose columns can be of any (not necessarily the same) type. +11.1 +Creating Arrays +11.1.1 +matrix and array +A matrix can be conveniently created by means of the matrix function. +(A <- matrix(1:6, byrow=TRUE, nrow=2)) +## +[,1] [,2] [,3] +## [1,] +1 +2 +3 +## [2,] +4 +5 +6 + +220 +II DEEPER +The above converted an atomic vector of length six into a matrix with two rows. The +number of columns was determined automatically (ncol=3 could have been passed to +get the same result). +Important By default, the elements of the input vector are read columnwisely: +matrix(1:6, ncol=3) +# byrow=FALSE +## +[,1] [,2] [,3] +## [1,] +1 +3 +5 +## [2,] +2 +4 +6 +A matrix can be equipped with dimension names, being a list of two character vectors +of appropriate sizes, labelling each row and column, in this order: +matrix(1:6, byrow=TRUE, nrow=2, dimnames=list(c("x", "y"), c("a", "b", "c"))) +## +a b c +## x 1 2 3 +## y 4 5 6 +Alternatively, to create a matrix, we can use the array function, which requires the +number of rows and columns be specified explicitly. +array(1:6, dim=c(2, 3)) +## +[,1] [,2] [,3] +## [1,] +1 +3 +5 +## [2,] +2 +4 +6 +Note that the elements are consumed in a column-major manner. +Arrays of dimensionality other than 2 are also possible. Here is a one-dimensional ar- +ray. When printed, it is indistinguishable from an atomic vector (but still the class +attribute is set to array): +array(1:6, dim=6) +## [1] 1 2 3 4 5 6 +And now for something completely different: a three-dimensional array of size 3-by- +4-by-2 +array(1:24, dim=c(3, 4, 2)) +## , , 1 +## +## +[,1] [,2] [,3] [,4] +## [1,] +1 +4 +7 +10 +## [2,] +2 +5 +8 +11 +## [3,] +3 +6 +9 +12 +(continues on next page) + +11 MATRICES AND OTHER ARRAYS +221 +(continued from previous page) +## +## , , 2 +## +## +[,1] [,2] [,3] [,4] +## [1,] +13 +16 +19 +22 +## [2,] +14 +17 +20 +23 +## [3,] +15 +18 +21 +24 +which can be thought of as two matrices of size 3-by-4 (because how else can we print +out a 3D object on a 2D console?). +The array function can be fed with the dimnames argument too. For instance, the above +three-dimensional hypertable would require a list of three character vectors, of sizes +3, 4, and 2, respectively. +Exercise 11.1 That 10-dimensional arrays are also possible the reader is encouraged to try out +themself. +11.1.2 +Promoting and Stacking Vectors +We can promote an ordinary vector to a column vector, i.e., a matrix with one column +by calling: +as.matrix(1:2) +## +[,1] +## [1,] +1 +## [2,] +2 +cbind(1:2) +## +[,1] +## [1,] +1 +## [2,] +2 +and to a row vector: +t(1:3) +# transpose +## +[,1] [,2] [,3] +## [1,] +1 +2 +3 +rbind(1:3) +## +[,1] [,2] [,3] +## [1,] +1 +2 +3 +Actually, cbind and rbind stand for column- and row-bind; they allow multiple vectors +and matrices be stacked one after/below another: +rbind(1:4, 5:8, 9:10, 11) +# row bind +## +[,1] [,2] [,3] [,4] +(continues on next page) + +222 +II DEEPER +(continued from previous page) +## [1,] +1 +2 +3 +4 +## [2,] +5 +6 +7 +8 +## [3,] +9 +10 +9 +10 +## [4,] +11 +11 +11 +11 +cbind(1:4, 5:8, 9:10, 11) +# column bind +## +[,1] [,2] [,3] [,4] +## [1,] +1 +5 +9 +11 +## [2,] +2 +6 +10 +11 +## [3,] +3 +7 +9 +11 +## [4,] +4 +8 +10 +11 +cbind(1:2, 3:4, rbind(11:13, 21:23)) +# vector, vector, 2x3 matrix +## +[,1] [,2] [,3] [,4] [,5] +## [1,] +1 +3 +11 +12 +13 +## [2,] +2 +4 +21 +22 +23 +and so forth. +Unfortunately, the generalised recycling rule is not implemented in full: +cbind(1:4, 5:8, cbind(9:10, 11)) +# different than cbind(1:4, 5:8, 9:10, 11) +## Warning in cbind(1:4, 5:8, cbind(9:10, 11)): number of rows of result is +## not a multiple of vector length (arg 1) +## +[,1] [,2] [,3] [,4] +## [1,] +1 +5 +9 +11 +## [2,] +2 +6 +10 +11 +Note that the first two arguments are of length four. +11.1.3 +Simplifying Lists +simplify2array is an extension of the unlist function. Given a list of atomic vectors, +each of length one, it will return a flat atomic vector. However, if a list of equisized +vectors of greater lengths is given, these will be converted to a matrix. +simplify2array(list(1, 11, 21)) +# each of length 1 +## [1] +1 11 21 +simplify2array(list(1:3, 11:13, 21:23, 31:33)) +# each of length 3 +## +[,1] [,2] [,3] [,4] +## [1,] +1 +11 +21 +31 +## [2,] +2 +12 +22 +32 +## [3,] +3 +13 +23 +33 +simplify2array(list(1, 11:12, 21:23)) +# no can do +## [[1]] +## [1] 1 +## +(continues on next page) + +11 MATRICES AND OTHER ARRAYS +223 +(continued from previous page) +## [[2]] +## [1] 11 12 +## +## [[3]] +## [1] 21 22 23 +Note that in the second example, each vector becomes a separate column of the res- +ulting matrix1. +See Section 12.3.7 for a few more examples. +Example 11.2 Therearequiteafewfunctionsthatcalltheaboveautomaticallybydefault(com- +pare the simplify or SIMPLIFY (sic!) argument in sapply, tapply, mapply, replicate, etc.). +For instance: +min_mean_max <- function(x) c(Min=min(x), Mean=mean(x), Max=max(x)) +sapply(split(iris[["Sepal.Length"]], iris[["Species"]]), min_mean_max) +## +setosa versicolor virginica +## Min +4.300 +4.900 +4.900 +## Mean +5.006 +5.936 +6.588 +## Max +5.800 +7.000 +7.900 +Take note of what constitutes the columns of the return matrix. +Exercise 11.3 Note the behaviour of as.matrix on list arguments. Write your own version +of simplify2array named as.matrix.list that always returns a matrix. If a list of non- +equisized vectors is given, fill the missing cells with NAs. +Important +Sometimes a call to do.call(cbind, x)) might be a better idea than a +referral to simplify2array. Provided that x is a list of atomic vectors, it always returns +a matrix: shorter vectors are recycled (which might be welcome, but not necessarily). +do.call(cbind, list(a=c(u=1), b=c(v=2, w=3), c=c(i=4, j=5, k=6))) +## Warning in (function (..., deparse.level = 1) : number of rows of result +## is not a multiple of vector length (arg 2) +## +a b c +## i 1 2 4 +## j 1 3 5 +## k 1 2 6 +Example 11.4 Consider a named toy list of numeric vectors: +1 Which can easily be explained by the fact that matrix elements are stored in a columnwise order. + +224 +II DEEPER +x <- list(a=runif(10), b=rnorm(15)) +Compare the results generated by sapply (which calls simplify2array): +sapply(x, function(e) c(Mean=mean(e))) +## +a.Mean +b.Mean +## 0.57825 0.12431 +sapply(x, function(e) c(Min=min(e), Max=max(e))) +## +a +b +## Min 0.045556 -1.9666 +## Max 0.940467 +1.7869 +with its version based on do.call and cbind: +sapply2 <- function(...) +do.call(cbind, lapply(...)) +sapply2(x, function(e) c(Mean=mean(e))) +## +a +b +## Mean 0.57825 0.12431 +sapply2(x, function(e) c(Min=min(e), Max=max(e))) +## +a +b +## Min 0.045556 -1.9666 +## Max 0.940467 +1.7869 +Note that sapply may return an atomic vector with somewhat surprising names. +11.1.4 +Beyond Numeric Arrays +Arrays built upon atomic vectors other than numeric ones are possible too. For in- +stance, later we will stress that comparisons featuring matrices are performed ele- +mentwisely, which results in logical matrices: +A >= 3 +## +[,1] +[,2] [,3] +## [1,] FALSE FALSE TRUE +## [2,] +TRUE +TRUE TRUE +Furthermore, matrices of character strings can be useful too: +matrix(strrep(LETTERS[1:6], 1:6), ncol=3) +## +[,1] [,2] +[,3] +## [1,] "A" +"CCC" +"EEEEE" +## [2,] "BB" "DDDD" "FFFFFF" +And of course complex matrices: + +11 MATRICES AND OTHER ARRAYS +225 +A + 1i +## +[,1] [,2] [,3] +## [1,] 1+1i 2+1i 3+1i +## [2,] 4+1i 5+1i 6+1i +We are not limited to atomic vectors: lists can be a basis for arrays as well: +matrix(list(1, 11:21, "A", list(1, 2, 3)), nrow=2) +## +[,1] +[,2] +## [1,] 1 +"A" +## [2,] integer,11 list,3 +Some elements are not displayed properly, but they are still there. +11.1.5 +Internal Representation +AnobjectofS3class arrayisanatomicvectororalistequippedwiththe dimsattribute, +which is a vector of nonnegative integers. Interestingly, we do not have to set the class +attribute explicitly: the accessor function class will return an implicit2 class anyway +(compare Section 4.4.3). +class(1) +# atomic vector +## [1] "numeric" +class(structure(1, dim=rep(1, 1))) +# 1D array (vector) +## [1] "array" +class(structure(1, dim=rep(1, 2))) +# 2D array (matrix) +## [1] "matrix" "array" +class(structure(1, dim=rep(1, 3))) +# 3D array +## [1] "array" +Note that a 2-dimensional array is additionally of class matrix. +Optional dimension names are represented by means of the dimnames attribute, which +is a list of d character vectors, where d is the array’s dimensionality. +(A <- structure(1:6, dim=c(2, 3), dimnames=list(letters[1:2], LETTERS[1:3]))) +## +A B C +## a 1 3 5 +## b 2 4 6 +dim(A) +# or attr(A, "dim") +## [1] 2 3 +dimnames(A) +# or attr(A, "dimnames") +## [[1]] +## [1] "a" "b" +(continues on next page) +2 Also, note that calling unclass on a matrix has no effect. + +226 +II DEEPER +(continued from previous page) +## +## [[2]] +## [1] "A" "B" "C" +Important +Internally, elements in an array are always stored in the columnwise +(column-major, Fortran) order: +as.numeric(A) +# drop all attributes to reveal the underlying numeric vector +## [1] 1 2 3 4 5 6 +Setting byrow=TRUE in a call to the matrix only affects the order in which this function +reads a given source vector, not the column/row-majorness. +(B <- matrix(1:6, ncol=3, byrow=TRUE)) +## +[,1] [,2] [,3] +## [1,] +1 +2 +3 +## [2,] +4 +5 +6 +as.numeric(B) +## [1] 1 4 2 5 3 6 +The two said special attributes can be modified through the replacement functions +`dim<-` and `dimnames<-` (and of course `attr<-` as well). In particular, changing dim +does not alter the underlying atomic vector; it only affects how other functions, in- +cluding the corresponding print method, interpret their placement on a virtual grid: +`dim<-`(A, c(3, 2)) +# not the same as transpose of A +## +[,1] [,2] +## [1,] +1 +4 +## [2,] +2 +5 +## [3,] +3 +6 +What we have obtained is a different view on the same flat data vector. Also, dimnames +were dropped because its size became incompatible with the newly requested dimen- +sionality. +Exercise 11.5 Study the source code of the nrow, NROW, ncol, NCOL, rownames, row.names, and +colnames functions. +Interestingly, for one-dimensional arrays, the names function returns a sensible value +(based on the dimnames attribute which is a list featuring one character vector), despite +the names attribute’s not being set. +What is more, dimnames itself can be named: + +11 MATRICES AND OTHER ARRAYS +227 +names(dimnames(A)) <- c("ROWS", "COLUMNS") +print(A) +## +COLUMNS +## ROWS A B C +## +a 1 3 5 +## +b 2 4 6 +It is still a numeric matrix, but the presentation thereof is slightly prettified. +Exercise 11.6 outer applies a given (vectorised elementwisely) function on each pair of ele- +ments from two vectors, forming a two-dimensional result grid. Based on two calls to rep, im- +plement your own version thereof. +Some examples: +outer(c(x=1, y=10, z=100), c(a=1, b=2, c=3, d=4), "*") +# multiplication +## +a +b +c +d +## x +1 +2 +3 +4 +## y +10 +20 +30 +40 +## z 100 200 300 400 +outer(c("A", "B"), 1:8, paste, sep="-") +# concatenate strings +## +[,1] +[,2] +[,3] +[,4] +[,5] +[,6] +[,7] +[,8] +## [1,] "A-1" "A-2" "A-3" "A-4" "A-5" "A-6" "A-7" "A-8" +## [2,] "B-1" "B-2" "B-3" "B-4" "B-5" "B-6" "B-7" "B-8" +Exercise 11.7 Showhow match(y, z)canbeimplementedwith outer.Isitstimeandmemory +complexity optimal, though? +Exercise 11.8 tablecreatesacontingencymatrix/arraythatcountsthenumberofuniquepairs +ofcorrespondingelementsfromoneormorevectorsofequallengths.Implementitsone-andtwo- +argument version based on tabulate. +For example: +tips <- read.csv(paste0("https://github.com/gagolews/teaching-data/raw/", +"master/other/tips.csv"), comment.char="#") +# a data.frame (list) +table(tips[["day"]]) +## +## +Fri +Sat +Sun Thur +## +19 +87 +76 +62 +table(tips[["smoker"]], tips[["day"]]) +## +## +Fri Sat Sun Thur +## +No +4 +45 +57 +45 +## +Yes +15 +42 +19 +17 + +228 +II DEEPER +11.2 +Array Indexing +Array subsetting can be performed by means of an overloaded3 `[` method, which we +will usually provide with many indexers – two in the matrix case; see help("["). +In this section, we will be referring to the two following example matrices. +(A <- matrix(1:12, byrow=TRUE, nrow=3)) +## +[,1] [,2] [,3] [,4] +## [1,] +1 +2 +3 +4 +## [2,] +5 +6 +7 +8 +## [3,] +9 +10 +11 +12 +B <- A +dimnames(B) <- list( +c("a", "b", "c"), +# row labels +c("x", "y", "z", "w") # column labels +) +B +## +x +y +z +w +## a 1 +2 +3 +4 +## b 5 +6 +7 +8 +## c 9 10 11 12 +Subsetting higher-dimensional arrays will be covered at the end. +11.2.1 +Arrays Are Built upon Basic Vectors +Firstly, let us note, though, that subsetting based on one indexer (as in Chapter 5) will +refer to the underlying flat vector. +For instance: +A[6] +## [1] 10 +This is the element in the third row, second column: recall that values are stored in a +column-major order. +11.2.2 +Selecting Individual Elements +Mathematically, we say that our example 3-by-4 real matrix 𝐀 ∈ ℝ3×4 is like: +𝐀 = ⎡⎢⎢ +⎣ +𝑎1,1 +𝑎1,2 +𝑎1,3 +𝑎1,4 +𝑎2,1 +𝑎2,2 +𝑎2,3 +𝑎2,4 +𝑎3,1 +𝑎3,2 +𝑎3,3 +𝑎3,4 +⎤⎥⎥ +⎦ += ⎡⎢⎢ +⎣ +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +⎤⎥⎥ +⎦ +. +3 Hidden deeply at the C language level. + +11 MATRICES AND OTHER ARRAYS +229 +Matrix elements are aligned in a two-dimensional grid. They are organised into rows +and columns. Hence, we can pinpoint a cell using two indexes: 𝑎𝑖,𝑗 refers to the i-th +row and the j-th column. +Similarly in R: +A[3, 2] +# 3rd row, 2nd column +## [1] 10 +B["c", "y"] +# using dimnames == B[3, 2] +## [1] 10 +11.2.3 +Selecting Rows and Columns +Some textbooks, and we are fond of this notation here as well, mark with 𝐚𝑖,⋅ a vector +thatconsistsofalltheelementsinthei-throwandwith𝐚⋅,𝑗 allitemsinthej-thcolumn. +In R, these will correspond to one of the indexers being left out. +A[3, ] +# 3rd row +## [1] +9 10 11 12 +A[, 2] +# 2nd column +## [1] +2 +6 10 +B["c", ] +# or B[3, ] +## +x +y +z +w +## +9 10 11 12 +B[, "y"] +# or B[, 2] +## +a +b +c +## +2 +6 10 +Let us stress that A[1], A[1, ], and A[, 1] have all different meanings. Also, we see +that the results’ dimnames are adjusted accordingly; see also unname which can take care +of them once and for all. +Exercise 11.9 Use duplicated to remove repeating rows in a given numeric matrix (see also +unique). +11.2.4 +Dropping Dimensions +Extracting an individual element or a single row/column from a matrix yields an +atomic vector. If the dim attribute consists of 1s only, it will be removed whatsoever. +In order to obtain proper row and column vectors, we can request the preservation +of the dimensionality of the output object (and, more precisely, the length of dim) by +passing drop=FALSE to `[`. +A[1, 2, drop=FALSE] +# 1st row, 2nd columns +## +[,1] +## [1,] +2 +(continues on next page) + +230 +II DEEPER +(continued from previous page) +A[1, +, drop=FALSE] +# 1st row +## +[,1] [,2] [,3] [,4] +## [1,] +1 +2 +3 +4 +A[ , 2, drop=FALSE] +# 2nd column +## +[,1] +## [1,] +2 +## [2,] +6 +## [3,] +10 +Important The drop argument unfortunately defaults to TRUE. Many bugs could be +avoided more easily otherwise, especially when the indexers are generated program- +matically. +See also the drop function which gets rid of the dimensions that have only one level. +Note For list-based matrices, we can also use a multi-argument version of `[[` to +extract the individual elements. +C <- matrix(list(1, 11:12, 21:23, 31:34), nrow=2) +C[1, 2] +# for `[`, input type is the same as the output type, hence a list +## [[1]] +## [1] 21 22 23 +C[1, 2, drop=FALSE] +## +[,1] +## [1,] integer,3 +C[[1, 2]] +# extract +## [1] 21 22 23 +11.2.5 +Selecting Submatrices +Indexing based on two vectors, both of length two or more, extracts a sub-block of a +given matrix: +A[1:2, c(1, 2, 4)] +# rows 1,2 columns 1,2,4 +## +[,1] [,2] [,3] +## [1,] +1 +2 +4 +## [2,] +5 +6 +8 +B[c("a", "b"), -3] +## +x y w +## a 1 2 4 +## b 5 6 8 + +11 MATRICES AND OTHER ARRAYS +231 +Note again that drop=TRUE is the default, which affects the behaviour if one of the in- +dexers is a scalar. +A[c(1, 3), 3] +## [1] +3 11 +A[c(1, 3), 3, drop=FALSE] +## +[,1] +## [1,] +3 +## [2,] +11 +Exercise 11.10 Overload the split function for the matrix class in such a way that, given a +matrix with n rows and an object of class factor of length n (or a list of such objects), a list of n +matrices is returned. For example: +split.matrix <- ...to.do... +A <- matrix(1:12, nrow=3) +# matrix whose rows are to be split +s <- factor(c("a", "b", "a")) +# determines the grouping of rows +split(A, s) +## $a +## +[,1] [,2] [,3] [,4] +## [1,] +1 +4 +7 +10 +## [2,] +3 +6 +9 +12 +## +## $b +## +[,1] [,2] [,3] [,4] +## [1,] +2 +5 +8 +11 +11.2.6 +Selecting Elements Based on Logical Vectors +Logical vectors can also be used as indexers, with consequences that are not hard to +guess: +A[c(TRUE, FALSE, TRUE), -1] +# select 1st and 3rd row, all but 1st column +## +[,1] [,2] [,3] +## [1,] +4 +7 +10 +## [2,] +6 +9 +12 +B[B[, "x"]>1 & B[, "x"]<=9, ] +# all rows where x is in (1, 9] +## +x +y +z +w +## b 5 +6 +7 +8 +## c 9 10 11 12 +A[2, colMeans(A)>6, drop=FALSE] +# 2nd row of the columns with means > 6 +## +[,1] [,2] +## [1,] +8 +11 +Note In Section 11.3, we note that comparisons involving matrices are performed in +an elementwise manner, for example: + +232 +II DEEPER +A>7 +## +[,1] +[,2] +[,3] [,4] +## [1,] FALSE FALSE FALSE TRUE +## [2,] FALSE FALSE +TRUE TRUE +## [3,] FALSE FALSE +TRUE TRUE +Such logical matrices can be used to index other matrices of the same size. This always +yields a (flat) vector in return. +A[A>7] +## [1] +8 +9 10 11 12 +This nothing else than the single-indexer subsetting involving two flat vectors (a nu- +meric and a logical one); the dim attributes are not considered here. +Exercise 11.11 Implement your own versions of max.col, lower.tri, and upper.tri. +11.2.7 +Selecting Based on Two-Column Numeric Matrices +We can also index a matrix A with a two-column matrix of positive integers I, for in- +stance: +(I <- cbind( +c(1, 3, 2, 1, 2), +c(2, 3, 2, 1, 4) +)) +## +[,1] [,2] +## [1,] +1 +2 +## [2,] +3 +3 +## [3,] +2 +2 +## [4,] +1 +1 +## [5,] +2 +4 +Now A[I] gives an easy access to: +• A[I[1, 1], I[1, 2]], +• A[I[2, 1], I[2, 2]], +• A[I[3, 1], I[3, 2]], +• … +and so forth. In other words, each row of I gives the coordinates of the elements to +extract. +A[I] +## [1] +4 +9 +5 +1 11 + +11 MATRICES AND OTHER ARRAYS +233 +This is exactly A[1, 2], A[3, 3], A[2, 2], A[1, 1], A[2, 4]. The result is always a +flat vector. +Note which can also return a list of index matrices: +which(A>7, arr.ind=TRUE) +## +row col +## [1,] +2 +3 +## [2,] +3 +3 +## [3,] +1 +4 +## [4,] +2 +4 +## [5,] +3 +4 +Moreover, arrayInd can be used to convert flat indexes to multidimensional ones. +Exercise 11.12 Implementyourownversionof arrayIndandafunctionperformingtheinverse +operation. +Exercise 11.13 Implement your own version of diag. +11.2.8 +Higher-Dimensional Arrays +For d-dimensional arrays, indexing can involve up to d indexes. +Thisisparticularlyusefulfordim-namedarraysthatrepresentcontingencytablesover +a Cartesian product of multiple factors. The built-in datasets::Titanic object is an +example of this: +str(dimnames(Titanic)) +# for reference (note that dimnames are named) +## List of 4 +## +$ Class +: chr [1:4] "1st" "2nd" "3rd" "Crew" +## +$ Sex +: chr [1:2] "Male" "Female" +## +$ Age +: chr [1:2] "Child" "Adult" +## +$ Survived: chr [1:2] "No" "Yes" +Titanic["Crew", "Male", "Adult", "Yes"] +## [1] 192 +gives the number of adult male members of the crew who survived the accident. Also: +Titanic["Crew", , "Adult", ] +## +Survived +## Sex +No Yes +## +Male +670 192 +## +Female +3 +20 +and so on. + +234 +II DEEPER +Exercise 11.14 Check if the above four-dimensional array can be indexed by means of matrices +with four columns. +11.2.9 +Replacing Elements +Thereisofcoursealsoamultidimensionalversionofthereplacementsubsettingfunc- +tion, `[<-`. +Generally, subsetting drops all attributes except names, dim, and dimnames (unless it +does not make sense otherwise). The replacement variant of the index operator mod- +ifies vector values but generally preserves all the attributes. +This enables transforming matrix elements like: +B[B<10] <- A[B<10]^2 +print(B) +## +x +y +z +w +## a 1 16 49 100 +## b 4 25 64 121 +## c 9 10 11 +12 +B[] <- rep(seq_len(NROW(B)), NCOL(B)) +# NOT the same as B <- ... +print(B) +## +x y z w +## a 1 1 1 1 +## b 2 2 2 2 +## c 3 3 3 3 +Take note of the preservation of dim and dimnames. +Exercise 11.15 Given a character matrix with entities that can be interpreted as numbers like: +(X <- rbind(x=c(a="1", b="2"), y=c("3", "4"))) +## +a +b +## x "1" "2" +## y "3" "4" +convert it to a numeric matrix with a single line of code. +11.3 +Common Operations +11.3.1 +Matrix Transpose +The matrix transpose, mathematically denoted with 𝐀𝑇, is available via a call to t: + +11 MATRICES AND OTHER ARRAYS +235 +(A <- matrix(1:6, byrow=TRUE, nrow=2)) +## +[,1] [,2] [,3] +## [1,] +1 +2 +3 +## [2,] +4 +5 +6 +t(A) +## +[,1] [,2] +## [1,] +1 +4 +## [2,] +2 +5 +## [3,] +3 +6 +Hence, if 𝐁 = 𝐀𝑇, then it is a matrix such that 𝑏𝑖,𝑗 = 𝑎𝑗,𝑖. In other words, in the +transposed matrix, rows become columns and columns become rows. +For higher-dimensional arrays, a generalised transpose can be achieved with aperm +(try permuting the dimensions of Titanic). Also note that the conjugate transpose of +a complex matrix 𝐀 is done via Conj(t(A)). +11.3.2 +Vectorised Mathematical Functions +Vectorised functions such as sqrt, abs, round, log, exp, cos, sin, etc., operate on each +element of a given array4. +A <- matrix(1/(1:6), nrow=2) +round(A, 2) +# rounds every element in A +## +[,1] [,2] [,3] +## [1,] +1.0 0.33 0.20 +## [2,] +0.5 0.25 0.17 +Exercise 11.16 Using a single call to matplot, which accepts the y argument be a matrix, draw +a plot of sin(𝑥), cos(𝑥), | sin(𝑥)|, and | cos(𝑥)| for 𝑥 ∈ [−2𝜋, 6𝜋]. +11.3.3 +Aggregating Rows and Columns +Whenwecallanaggregationfunctiononanarray,itwillreduceallelementstoasingle +number: +(A <- matrix(1:12, byrow=TRUE, nrow=3)) +## +[,1] [,2] [,3] [,4] +## [1,] +1 +2 +3 +4 +## [2,] +5 +6 +7 +8 +## [3,] +9 +10 +11 +12 +mean(A) +## [1] 6.5 +4 They are simply applied on each element of the underlying flat vector. In Section 5.5, we have men- +tioned that unary functions preserve all attributes of their inputs, hence also dim and dimnames. + +236 +II DEEPER +The apply function may be used to summarise individual rows or columns in a matrix: +• apply(A, 1, f) applies a given function f on each row of a matrix A; +• apply(A, 2, f) applies f on each column of A. +For instance: +apply(A, 1, mean) +# synonym: rowMeans(A) +## [1] +2.5 +6.5 10.5 +apply(A, 2, mean) +# synonym: colMeans(A) +## [1] 5 6 7 8 +Note that the function being applied does not have to return a single number: +apply(A, 2, range) +# min and max +## +[,1] [,2] [,3] [,4] +## [1,] +1 +2 +3 +4 +## [2,] +9 +10 +11 +12 +apply(A, 1, function(row) c(Min=min(row), Mean=mean(row), Max=max(row))) +## +[,1] [,2] [,3] +## Min +1.0 +5.0 +9.0 +## Mean +2.5 +6.5 10.5 +## Max +4.0 +8.0 12.0 +Take note of the columnwise order of the output values. +apply works on higher-dimensional arrays too: +apply(Titanic, 1, mean) +# 1st dimension - Class +## +1st +2nd +3rd +Crew +## +40.625 +35.625 +88.250 110.625 +apply(Titanic, c(1, 3), mean) +# w.r.t. Class (1st) and Age (3rd) +## +Age +## Class +Child +Adult +## +1st +1.50 +79.75 +## +2nd +6.00 +65.25 +## +3rd +19.75 156.75 +## +Crew +0.00 221.25 +11.3.4 +Binary Operators +In Section 5.5, we have stated that binary elementwise operations, such as addition +or multiplication, preserve the attributes of the longer input or both (with the first +argument preferred to the second) if they are of equal sizes. +Taking into account that: +• an array is simply a flat vector equipped with the dim attribute, and + +11 MATRICES AND OTHER ARRAYS +237 +• we refer to the respective default methods when applying binary operators +allows us to deduce how `+`, `<=`, `&`, etc. behave in a number of different contexts. +Array-Array. First, let us note what happens when we operate on two arrays of +identical dimensionalities. +(A <- rbind(c(1, 10, 100), c(-1, -10, -100))) +## +[,1] [,2] [,3] +## [1,] +1 +10 +100 +## [2,] +-1 +-10 -100 +(B <- matrix(1:6, byrow=TRUE, nrow=2)) +## +[,1] [,2] [,3] +## [1,] +1 +2 +3 +## [2,] +4 +5 +6 +A + B +# elementwise addition +## +[,1] [,2] [,3] +## [1,] +2 +12 +103 +## [2,] +3 +-5 +-94 +A * B +# elementwise multiplication (not: algebraic matrix multiply) +## +[,1] [,2] [,3] +## [1,] +1 +20 +300 +## [2,] +-4 +-50 -600 +This is simply the addition and multiplication of the corresponding elements of two +given matrices. +Array-Scalar. Second, we can apply scalar-matrix operations: +(-1)*B +## +[,1] [,2] [,3] +## [1,] +-1 +-2 +-3 +## [2,] +-4 +-5 +-6 +A^2 +## +[,1] [,2] +[,3] +## [1,] +1 +100 10000 +## [2,] +1 +100 10000 +These multiplied each element in B by -1 and squared every element in A, respectively. +Also note that the behaviour of comparison operators is similar: +A >= 1 & A <= 100 +## +[,1] +[,2] +[,3] +## [1,] +TRUE +TRUE +TRUE +## [2,] FALSE FALSE FALSE + +238 +II DEEPER +Array-Vector. Next, based on the recycling rule and the fact that elements are ordered +columnwisely, we get that: +B * c(10, 100) +## +[,1] [,2] [,3] +## [1,] +10 +20 +30 +## [2,] +400 +500 +600 +multiplied every element in the first row by 10 and each element in the second row by +100. +Note that if wish to multiply each element in the first, second, …, etc. column by the +first, second, …, etc. value in a vector, we should not call: +B * c(1, 100, 1000) +## +[,1] [,2] [,3] +## [1,] +1 2000 +300 +## [2,] +400 +5 6000 +but rather: +t(t(B) * c(1, 100, 1000)) +## +[,1] [,2] [,3] +## [1,] +1 +200 3000 +## [2,] +4 +500 6000 +or: +t(apply(B, 1, `*`, c(1, 100, 1000))) +## +[,1] [,2] [,3] +## [1,] +1 +200 3000 +## [2,] +4 +500 6000 +Exercise 11.17 Write a function which standardises the values in each column of a given mat- +rix: for each column, from every element, subtract the mean and then divide it by the standard +deviation. Try to do it in a few different ways, including via a call to apply, sweep, scale, or +based solely on arithmetic operators. +Note Some sanity checks are being done on the dim attributes, so not every configur- +ation is possible. Notice the following peculiarities: +getOption("error") +## NULL +A + t(B) +# dim==c(2, 3) vs dim==c(3, 2) +## Error in A + t(B): non-conformable arrays +A * cbind(1, 10, 100) +# this is too good to be true +## Error in A * cbind(1, 10, 100): non-conformable arrays +(continues on next page) + +11 MATRICES AND OTHER ARRAYS +239 +(continued from previous page) +A * rbind(1, 10) +# but A * c(1, 10) works... +## Error in A * rbind(1, 10): non-conformable arrays +A + 1:12 +## Error in eval(expr, envir, enclos): dims [product 6] do not match the length of object [12] +A + 1:5 +# partial recycling is okay +## Warning in A + 1:5: longer object length is not a multiple of shorter +## object length +## +[,1] [,2] [,3] +## [1,] +2 +13 +105 +## [2,] +1 +-6 +-99 +11.4 +Numerical Matrix Algebra (*) +Many data analysis and machine learning algorithms, in their essence, involve quite +simple matrix algebra and numerical mathematics. Suffice to say that anyone serious +about data science and scientific computing should learn the necessary theory; see, +for example, [25] and [26]. +Risaconvenientinterfacetothewell-testedandstablealgorithmsfrom,amongstoth- +ers, LAPACK and BLAS5. Below we mention only a few of them. Note that there are many +third-party packages featuring hundreds of algorithms tackling differential equa- +tions, constrainedand unconstrainedoptimisation, etc.;exploring the relevant CRAN +Task Views6 can give a good overview. +11.4.1 +Matrix Multiplication +`*` performs elementwise multiplication. For what we call (algebraic) matrix multi- +plication, we should use the `%*%` operator. +Refreshing from a basic linear algebra course, matrix multiplication can only be per- +formed on two matrices of compatible sizes: the number of columns in the left matrix +must match the number of rows in the right operand. +Given 𝐀 ∈ ℝ𝑛×𝑝 and 𝐁 ∈ ℝ𝑝×𝑚, their multiply is a matrix 𝐂 = 𝐀𝐁 ∈ ℝ𝑛×𝑚 such +that 𝑐𝑖,𝑗 is the dot product of the i-th row in 𝐀 and the j-th column in 𝐁: +𝑐𝑖,𝑗 = 𝐚𝑖,⋅ ⋅ 𝐛⋅,𝑗 = +𝑝 +∑ +𝑘=1 +𝑎𝑖,𝑘𝑏𝑘,𝑗, +5 (*) Note that we can select the underlying implementation of BLAS at R’s compile time; see Section A.3 +in [47]. Some of them are faster than others. +6 https://cran.r-project.org/web/views/ + +240 +II DEEPER +for 𝑖 = 1, … , 𝑛 and 𝑗 = 1, … , 𝑚. +For instance: +(A <- rbind(c(1, 1, 1), c(-1, 1, 0))) +## +[,1] [,2] [,3] +## [1,] +1 +1 +1 +## [2,] +-1 +1 +0 +(B <- rbind(c(3, -1), c(1, 2), c(6, 1))) +## +[,1] [,2] +## [1,] +3 +-1 +## [2,] +1 +2 +## [3,] +6 +1 +A %*% B +## +[,1] [,2] +## [1,] +10 +2 +## [2,] +-2 +3 +Note When applying `%*%` on one or more flat vectors, their dimensionality will be +promoted automatically to make the operation possible. Note that, however, c(a, b) +%*% c(c, d) gives a scalar 𝑎𝑐 + 𝑏𝑑, and not a 2-by-2 matrix. +Further, crossprod(A, B) yields 𝐀𝑇𝐁 and tcrossprod(A, B) determines 𝐀𝐁𝑇 more +efficiently than relying on `%*%`. Note that we can omit the second argument and get +𝐀𝑇𝐀 and 𝐀𝐀, respectively +crossprod(c(1, 1)) +# Euclidean norm squared +## +[,1] +## [1,] +2 +crossprod(c(1, 1), c(-1, 1)) +# dot product of two vectors +## +[,1] +## [1,] +0 +crossprod(A) +# same as t(A) %*% A, i.e., dot products of all column pairs +## +[,1] [,2] [,3] +## [1,] +2 +0 +1 +## [2,] +0 +2 +1 +## [3,] +1 +1 +1 +Recall that if the dot product of two vectors is equal to 0, we say that they are ortho- +gonal (perpendicular). +Exercise 11.18 (*)Writeyourownversionsof covand cor:functionstocomputethecovariance +andcorrelationmatrices.Makeuseofthefactthattheformercanbedeterminedwith crossprod +based on a centred version of an input matrix. + +11 MATRICES AND OTHER ARRAYS +241 +11.4.2 +Solving Systems of Linear Equations +The solve function can be used to solve m systems of n linear equations of the form +𝐀𝐗 = 𝐁, where 𝐀 ∈ ℝ𝑛×𝑛 and 𝐗, 𝐁 ∈ ℝ𝑛×𝑚 (via the LU decomposition with partial +pivoting and row interchanges). +11.4.3 +Norms and Metrics +Given an n-by-m matrix 𝐀, calling norm(A, "1"), norm(A, "2"), and norm(A, "I"), we +can compute the operator norms: +‖𝐀‖1 += +max𝑗=1,…,𝑚 ∑𝑛 +𝑖=1 |𝑎𝑖,𝑗|, +‖𝐀‖2 += +𝜎1(𝐀) = sup𝟎≠𝐱∈ℝ𝑚 +‖𝐀𝐱‖2 +‖𝐱‖2 +‖𝐀‖𝐼 += +max𝑖=1,…,𝑛 ∑𝑚 +𝑗=1 |𝑎𝑖,𝑗|, +where 𝜎1 gives the largest singular value (see below). +Also, passing "F" as the second argument yields the Frobenius norm, ‖𝐀‖𝐹 += +√∑𝑛 +𝑖=1 ∑𝑚 +𝑗=1 𝑎2 +𝑖,𝑗, and "M" computes the max norm, ‖𝐀‖𝑀 = max 𝑖=1,…,𝑛 +𝑗=1,…,𝑚 |𝑎𝑖,𝑗|. +Notethatif𝐀isacolumnvector,then‖𝐀‖𝐹 and‖𝐀‖2 areequivalentandarereferredto +astheEuclideannorm.Moreover,‖𝐀‖𝑀 = ‖𝐀‖𝐼 givethesupremumnormandoutputs +‖𝐀‖1 the Manhattan (taxicab) one. +Exercise 11.19 Given an n-by-m matrix 𝐀 representing m vectors in ℝ𝑛, normalise each +column so that you obtain m unit vectors, i.e., whose Euclidean norm is 1. +Further, dist determines all pairwise distances between a set of n vectors in ℝ𝑚, writ- +ten as a n by m matrix. +For example, let us consider three vectors in ℝ2: +(X <- rbind(c(1, 1), c(1, -2), c(0, 0))) +## +[,1] [,2] +## [1,] +1 +1 +## [2,] +1 +-2 +## [3,] +0 +0 +as.matrix(dist(X, "euclidean")) +## +1 +2 +3 +## 1 0.0000 3.0000 1.4142 +## 2 3.0000 0.0000 2.2361 +## 3 1.4142 2.2361 0.0000 +From that we see that the distance between the 1st and the 3rd vector is ca. 1.41421. +Euclidean, maximum, Manhattan, and Canberra distances/metrics are available, +amongst others. +Exercise 11.20 dist returns an object of S3 class dist. Inspect how it is represented. +Example 11.21 adist implements a couple of string metrics. For example: + +242 +II DEEPER +x <- c("spam", "bacon", "eggs", "spa", "spams", "legs") +names(x) <- x +(d <- adist(x)) +## +spam bacon eggs spa spams legs +## spam +0 +5 +4 +1 +1 +4 +## bacon +5 +0 +5 +5 +5 +5 +## eggs +4 +5 +0 +4 +4 +2 +## spa +1 +5 +4 +0 +2 +4 +## spams +1 +5 +4 +2 +0 +4 +## legs +4 +5 +2 +4 +4 +0 +gives the Levenshtein distances between each pair of strings. In particular, we need two edit op- +erations (character insertions, deletions, or replacements) to turn "eggs" into "legs" (add l and +remove g). +Example 11.22 Objectsofclassdistcanbeusedtoperformhierarchicalclusteringsofdatasets. +For example: +h <- hclust(as.dist(d), method="average") +# see also: plot(h, labels=x) +cutree(h, 3) +## +spam bacon +eggs +spa spams +legs +## +1 +2 +3 +1 +1 +3 +yields a grouping into 3 clusters determined by the average linkage ("legs" and "eggs" are +grouped together, "spam", "spa", "spams" form another cluster, and "bacon" is a singleton). +11.4.4 +Eigenvalues and Eigenvectors +eigen returns a sequence of eigenvalues (𝜆1, … , 𝜆𝑛) (ordered nondecreasingly w.r.t. +|𝜆𝑖|) and a matrix 𝐕 whose columns define the corresponding eigenvectors (scaled to +unit length) of a given matrix 𝐗. To recall, by definition it holds that 𝐗𝐯⋅,𝑖 = 𝜆𝑖𝐯⋅,𝑖. +Here are the eigenvalues and the corresponding eigenvectors of an example matrix +(defining rotation in 2D by 𝜋/3): +(R <- rbind(c(cos(pi/3), -sin(pi/3)), c(sin(pi/3), cos(pi/3)))) +## +[,1] +[,2] +## [1,] 0.50000 -0.86603 +## [2,] 0.86603 +0.50000 +eigen(R) +## eigen() decomposition +## $values +## [1] 0.5+0.86603i 0.5-0.86603i +## +## $vectors +## +[,1] +[,2] +(continues on next page) + +11 MATRICES AND OTHER ARRAYS +243 +(continued from previous page) +## [1,] 0.70711+0.00000i 0.70711+0.00000i +## [2,] 0.00000-0.70711i 0.00000+0.70711i +Example 11.23 Consider a pseudorandom sample from a bivariate7 normal distribution; see +Figure 11.1. +Z <- matrix(rnorm(2000), ncol=2) +# independent N(0, 1) +Z <- Z %*% rbind(c(1, 0), c(0, sqrt(5))) +# scaling +Z <- Z %*% R +# rotation +Z <- t(c(10, -5) + t(Z)) +# translation +plot(Z, asp=1) +5 +10 +15 +-8 +-6 +-4 +-2 +0 +Z[,1] +Z[,2] +Figure 11.1: Example bivariate normal sample +It is known that eigenvectors of the covariance matrix correspond to the principal components of +the original dataset and the eigenvalues give the variance explained by them: +eigen(cov(Z)) +## eigen() decomposition +## $values +## [1] 5.18609 0.98386 +## +## $vectors +## +[,1] +[,2] +(continues on next page) +7 For drawing random samples from any multivariate distribution, refer to the theory of copulas, e.g., +[37]. There are a few R packages on CRAN that implement the most popular models. + +244 +II DEEPER +(continued from previous page) +## [1,] -0.86715 +0.49804 +## [2,] -0.49804 -0.86715 +this roughly corresponds to the principal directions [sin(𝜋/3), cos(𝜋/3)] and the thereto- +orthogonal [cos(𝜋/3), − sin(𝜋/3)] with variances of 5 and 1, respectively. Still, this method +of performing a PCA is not particularly numerically stable; see below for an alternative. +11.4.5 +QR Decomposition +We say that a real n-by-m matrix 𝐐, 𝑛 ≥ 𝑚, is orthogonal, whenever 𝐐𝑇𝐐 = 𝐈 (iden- +tity matrix) which is equivalent to its columns being orthogonal unit vectors (note that +if 𝐐 is a square matrix, then 𝐐𝑇 = 𝐐−1 if and only if 𝐐𝑇𝐐 = 𝐐𝐐𝑇 = 𝐈). +Let 𝐀 be a real8 n-by-m matrix with 𝑛 ≥ 𝑚. Then 𝐀 = 𝐐𝐑 is its QR decomposition +(in the so-called narrow form), if 𝐐 is an orthogonal n-by-m matrix and 𝐑 is an upper +triangular m-by-m one. +The qr function returns an object of S3 class qr from which we can extract the two +components; see the qr.Q and qr.R functions. +Example 11.24 Let 𝐗 be an n-by-m data matrix, representing n points in ℝ𝑚, and a vector +𝐲 ∈ ℝ𝑛 of the desired outputs corresponding to each input. For fitting a linear model 𝐱𝑇𝜽, +where 𝜽 is a vector of m parameters, we can use the method of least squares, which minimises +ℒ(𝜽) = +𝑛 +∑ +𝑖=1 +(𝐱𝑇 +𝑖,⋅𝜽 − 𝑦𝑖) +2 = ‖𝐗𝜽 − 𝐲‖2 +2 +It might be shown that if 𝐗 = 𝐐𝐑, then 𝜽 = (𝐗𝑇𝐗) +−1 𝐗𝑇𝐲 = 𝐑−1𝐐𝑇𝐲, which can +conveniently be determined via a call to qr.coef. +Inparticular,wecanfita simplelinear regressionmodel 𝑦 = 𝑎𝑥 +𝑏 byconsidering𝐗 = [𝑥, 1] +and 𝜽 = [𝑎, 𝑏], for example (see Figure 11.2): +x <- cars[["speed"]] +y <- cars[["dist"]] +X <- cbind(x, 1) +# the model is theta[1]*x + theta[2]*1 +qrX <- qr(X) +(theta <- solve(qr.R(qrX)) %*% t(qr.Q(qrX)) %*% y) +# or: qr.coef(qrX, y) +## +[,1] +## x +3.9324 +## +-17.5791 +plot(x, y, xlab="speed", ylab="dist") +# scatter plot +abline(theta[2], theta[1], lty=2) +# add the regression line +8 𝐀 can also be a complex matrix, which results in its QR decomposition’s being such that 𝐐 is a unitary +matrix. + +11 MATRICES AND OTHER ARRAYS +245 +5 +10 +15 +20 +25 +0 +20 +40 +60 +80 +100 +120 +speed +dist +Figure 11.2: The built-in cars dataset and the fitted regression line +Note that solve with one argument determines the inverse of a given matrix. The fitted model is +𝑦 = 3.93241𝑥 − 17.5791. +Thesameapproachisusedby lm.fit,whichistheworkhorsebehindthe lmmethodacceptingan +R formula (which some readers might be familiar with; compare Section 15.2). +lm.fit(cbind(x, 1), y)[["coefficients"]] +# also: lm(dist~speed, data=cars) +## +x +## +3.9324 -17.5791 +11.4.6 +SVD Decomposition +Given a real n-by-m matrix 𝐗, its singular value decomposition (SVD) is given by 𝐗 = +𝐔𝐃𝐕𝑇, where 𝐃 is a p-by-p diagonal matrix (featuring the so-called singular values of +𝐗, 𝑑1,1 ≥ … ≥ 𝑑𝑝,𝑝 ≥ 0, 𝑝 = min{𝑛, 𝑚}) and 𝐔, 𝐕 are orthogonal matrices of size +n-by-p and m-by-p, respectively. +svdmaynotonlybeusedtodeterminethesolutiontolinearregression9 butalsotoper- +form the principal component analysis10. Namely, 𝐕 gives the eigenvectors of 𝐗𝑇𝐗. +Assuming that 𝐗 is centred at 0, the latter is precisely its scaled covariance matrix. +Example 11.25 Continuing the PCA example above, we can determine the principal directions +also by calling: +9 As the pseudoinverse 𝐗+ = (𝐗𝑇𝐗) +−1 𝐗𝑇 = 𝐕𝐃+𝐔𝑇 = 𝐑−1𝐐𝑇, with 𝐗+𝐗 = 𝐈. Here 𝐃+ is a +transposed version of 𝐃 featuring the reciprocals of its non-zero elements. +10 See the source code of getS3method("prcomp", "default"). + +246 +II DEEPER +Zc <- apply(Z, 2, function(x) x-mean(x)) +# centred version of Z +svd(Zc)[["v"]] +## +[,1] +[,2] +## [1,] -0.86715 +0.49804 +## [2,] -0.49804 -0.86715 +11.5 +S4 Classes (*) +The concept of the S3-style object oriented programming is based on a brilliantly +simple idea (see Chapter 10): calling a generic f(x) automatically dispatches to a +method f.class_of_x(x) or f.default(x) in the case where the former does not ex- +ist. Naturally, it has some inherent limitations: +• classes cannot be formally defined; the class attribute may be assigned arbitrarily +onto any object11, +• argument dispatch is performed only12 with regard to one data type13. +In most cases, and with appropriate level of mindfulness, this is not a problem at all. +However, it is a typical condition of programmers who come to our world from more +mainstream languages (e.g., C++; yours truly included) until they appreciate the true +beauty of R’s being somewhat different. Before they fully develop such an acquired +taste, though, they grow restless as “R is not a real object oriented system because it +lacks polymorphism, encapsulation, formal inheritance, and so on and so forth, and +something must be done about it”. The truth is that it had not have to, but with high +probability it would have anyway in one way or another. +And so when the fourth version of the S language was introduced in 1998 (see [4]), it +brought a new object oriented system which we are used to referring to as S4. Its R +version has been implemented in the methods package. Below we discuss it briefly; for +more details, see help("Classes_Details") and help("Methods_Details") as well as [5] +and [6]. +Note (*) S4 was loosely inspired by the Common Lisp Object System (with its def- +class, defmethod, etc.; see, e.g., [15]). In the current author’s opinion, the S4 system +is somewhat an afterthought. Due to appendages like this, R seems like a patchwork +11 A partial solution to this could involve defining a method like validate.class_name, that is called fre- +quently and which checks whether a given object enjoys some desired constraints. +12 Although there are functions featuring some workarounds (see, e.g., cbind which dispatches to cbind. +data.frame if one argument is a data frame and the remaining ones are vectors or matrices). Also, binary +operators overloaded via group generics consider the classes of both operands; see Section 18.4. +13 Hypothetically, we can imagine an OOP system relying on methods named like method.class_name1. +class_name2 where dispatching is based on two argument types. + +11 MATRICES AND OTHER ARRAYS +247 +language; suffice it to say that it was not the last attempt to do a somewhat more real +OOP in the overall functional R: the story will continue in Section 16.3. +The main problem with all the OOP approaches is that each of them is parallel to S3 +which never lost its popularity and is still at the very core of our language. We are +thus covering them for the sake of completeness, because that’s what must be done. +After all, with non-zero probability, the reader will sooner or later come across such +objects (e.g., below we explain the meaning of notation like x@slot). Also, yours truly +rebelliously suggests taking statements such as “for new projects, it is recommended +to use the more flexible and robust S4 scheme provided in the methods package” (see +help("UseMethod")) with a pinch of salt. +11.5.1 +Defining S4 Classes +An S4 class can formally be registered by means of a call to setClass. +For instance: +library("methods") +# in the case where it is not auto-loaded +setClass("categorical", slots=c(data="integer", levels="character")) +defines a class named categorical with two slots data and levels being integer and +character vectors, respectively. Note that this notation is already quite peculiar: there +is no assignment which would suggest that we have introduced something novel. +An object of the above class can be instantiated by calling new: +z <- new("categorical", data=c(1L, 2L, 2L, 1L, 1L), levels=c("a", "b")) +print(z) +## An object of class "categorical" +## Slot "data": +## [1] 1 2 2 1 1 +## +## Slot "levels": +## [1] "a" "b" +That z is of the recently-introduced class can be verified as follows: +is(z, "categorical") +## [1] TRUE +class(z) +# also: attr(z, "class") +## [1] "categorical" +## attr(,"package") +## [1] ".GlobalEnv" +Important Some R packages will be importing from the methods only for the sake of + +248 +II DEEPER +being able to access the convenient is function – it does not mean they are defining +new S4 classes. +Note S4 objects are marked as being of the following basic type: +typeof(z) +## [1] "S4" +For technical details on how they are internally represented, see Section 1.12 in [48]. +In particular, in our case, all the slots are simply stored as object attributes: +attributes(z) +## $data +## [1] 1 2 2 1 1 +## +## $levels +## [1] "a" "b" +## +## $class +## [1] "categorical" +## attr(,"package") +## [1] ".GlobalEnv" +11.5.2 +Accessing Slots +Reading or writing slot contents can be done by means of the `@` operator and the slot +function or their replacement versions. +z@data +# or slot(z, "data") +## [1] 1 2 2 1 1 +z@levels <- c("A", "B") +Note +The `@` operator can only be used on S4 objects and some sanity checks are +automatically performed: +z@unknown <- "spam" +## Error in (function (cl, name, valueClass) : 'unknown' is not a slot in class "categorical" +z@data <- "spam" +## Error in (function (cl, name, valueClass) : assignment of an object of class "character" is not valid for @'data' in an object of class "categorical"; is(value, "integer") is not TRUE + +11 MATRICES AND OTHER ARRAYS +249 +11.5.3 +Defining Methods +For the S4 counterparts of the S3 generics (Section 10.2), see help("setGeneric"). +Luckily, there is a good degree of interoperability between the S3 and S4 systems. +Let us start by introducing a new method for the well-known as.character generic. +Instead of defining as.character.categorical, we need to register a new routine with +setMethod. +setMethod( +"as.character", +# name of the generic +"categorical", +# class of 1st arg; or: signature=c(x="categorical") +function(x, ...) +# method definition +x@levels[x@data] +) +Testing: +as.character(z) +## [1] "A" "B" "B" "A" "A" +The S4 counterpart of print is show: +setMethod( +"show", +"categorical", +function(object) { +x_character <- as.character(object) +print(x_character) +# calls `print.default` +cat(sprintf("Categories: %s\n", +paste(object@levels, collapse=", "))) +} +) +Interestingly, it is involved automatically upon a call to print: +print(z) +# calls `show` for `categorical` +## [1] "A" "B" "B" "A" "A" +## Categories: A, B +Methodsthatdispatchonthetypeofmultipleargumentsarepossibletoo,forexample: +setMethod( +"split", +c(x="ANY", f="categorical"), +function (x, f, drop=FALSE, ...) +split(x, as.character(f), drop=drop, ...) +) + +250 +II DEEPER +allows the first argument to be of any type (like a default method), and: +setMethod( +"split", +c(x="matrix", f="categorical"), +function (x, f, drop=FALSE, ...) +lapply( +split(seq_len(NROW(x)), f, drop=drop, ...), +# calls the above +function(i) x[i, , drop=FALSE]) +) +is a version tailored for matrices. Testing: +A <- matrix(1:35, nrow=5) +# whatever +split(A, z) +# matrix,categorical +## $A +## +[,1] [,2] [,3] [,4] [,5] [,6] [,7] +## [1,] +1 +6 +11 +16 +21 +26 +31 +## [2,] +4 +9 +14 +19 +24 +29 +34 +## [3,] +5 +10 +15 +20 +25 +30 +35 +## +## $B +## +[,1] [,2] [,3] [,4] [,5] [,6] [,7] +## [1,] +2 +7 +12 +17 +22 +27 +32 +## [2,] +3 +8 +13 +18 +23 +28 +33 +split(1:5, z) +# ANY,categorical +## $A +## [1] 1 4 5 +## +## $B +## [1] 2 3 +Exercise 11.26 Overload the `[` operator for the categorical class +11.5.4 +Defining Constructors +We can also overload the initialize method which is automatically called by new: +setMethod( +"initialize", +# class name +"categorical", +# method name +function(.Object, x) { +# the method itself +x <- as.character(x) +# see above +xu <- unique(sort(x)) +# drops NAs +.Object@data <- match(x, xu) +(continues on next page) + +11 MATRICES AND OTHER ARRAYS +251 +(continued from previous page) +.Object@levels <- xu +.Object +# return value - a modified object +} +) +This allows for constructing new objects of class categorical based on an object like x +above, for instance: +w <- new("categorical", c("a", "c", "a", "a", "d", "c")) +print(w) +## [1] "a" "c" "a" "a" "d" "c" +## Categories: a, c, d +Note that we have not set the two slots directly. They were automatically taken care of +by initialize. +Exercise 11.27 Set up a validating method for our class; see help("setValidity"). +11.5.5 +Inheritance +New S4 classes can be derived from existing ones, for instance: +setClass("binary", contains="categorical") +is a child class inhering all slots from its parent. We can, for example, overload the +initialisation method for it: +setMethod( +"initialize", +"binary", +function(.Object, x) +{ +x <- as.character(as.integer(as.logical(x))) +xu <- c("0", "1") +.Object@data <- match(x, xu) +.Object@levels <- xu +.Object +} +) +Testing: +new("binary", c(TRUE, FALSE, TRUE, FALSE, NA, TRUE)) +## [1] "1" "0" "1" "0" NA +"1" +## Categories: 0, 1 + +252 +II DEEPER +Note that we are still using the show method of the parent class. +11.5.6 +A Note on the Matrix Package +The Matrix package is perhaps the most widely known showcase of the S4 object- +orientation (and that is the reason why we cover S4 in this very chapter). It defines +classes and methods for dense and sparse matrices, including rectangular, symmet- +ric, triangular, band, and diagonal ones. +For instance, large graph (e.g., in network sciences) or preference (e.g., in recom- +mender systems) data can be represented using sparse matrices (those which feature +many 0s; after all, it is extremely more common for two vertices in a network to not be +joined by an edge than to be connected). +For example: +library("Matrix") +(A <- Diagonal(x=1:5)) +## 5 x 5 diagonal matrix of class "ddiMatrix" +## +[,1] [,2] [,3] [,4] [,5] +## [1,] +1 +. +. +. +. +## [2,] +. +2 +. +. +. +## [3,] +. +. +3 +. +. +## [4,] +. +. +. +4 +. +## [5,] +. +. +. +. +5 +created a real diagonal matrix. Moreover: +B <- as(A, "sparseMatrix") +B[1, 2] <- 7 +B[4, 1] <- 42 +print(B) +## 5 x 5 sparse Matrix of class "dgCMatrix" +## +## [1,] +1 7 . . . +## [2,] +. 2 . . . +## [3,] +. . 3 . . +## [4,] 42 . . 4 . +## [5,] +. . . . 5 +yields a general sparse real matrix in the CRC (compressed, sparse, column-oriented) +format. +For more information on the package, see vignette(package="Matrix"). + +11 MATRICES AND OTHER ARRAYS +253 +11.6 +Exercises +Exercise 11.28 Let X be a matrix with dimnames set, e.g.: +X <- matrix(1:12, byrow=TRUE, nrow=3) +# example matrix +dimnames(X)[[2]] <- c("a", "b", "c", "d") +# set column names +print(X) +## +a +b +c +d +## [1,] 1 +2 +3 +4 +## [2,] 5 +6 +7 +8 +## [3,] 9 10 11 12 +Explain (in your own words) the meaning of the following expressions involving matrix subset- +ting. Note that not each of them is valid. +• X[1, ], +• X[, 3], +• X[, 3, drop=FALSE], +• X[3], +• X[, "a"], +• X[, c("a", "b", "c")], +• X[, -2], +• X[X[,1] > 5, ], +• X[X[,1]>5, c("a", "b", "c")], +• X[X[,1]>=5 & X[,1]<=10, ], +• X[X[,1]>=5 & X[,1]<=10, c("a", "b", "c")], +• X[, c(1, "b", "d")]. +Exercise 11.29 Assuming that X is an array, what are the differences between the following in- +dexing schemes? +• X["1", ] vs X[1, ], +• X[, "a", "b", "c"]vs X["a", "b", "c"]vs X[, c("a", "b", "c")]vs X[c("a", "b", +"c")], +• X[1] vs X[, 1] vs X[1, ], +• X[X>0] vs X[X>0, ] vs X[, X>0], +• X[X[, 1]>0] vs X[X[, 1]>0,] vs X[,X[,1]>0], +• X[X[, 1]>5, X[1, ]<10] vs X[X[1, ]>5, X[, 1]<10]. + +254 +II DEEPER +Exercise 11.30 For a given real n-by-m matrix 𝐗, determine the bounding hyperrectangle of +thusly encoded n input points in an m-dimensional space. Return a 2-by-m matrix 𝐁 with +𝑏1,𝑗 = min𝑖 𝑥𝑖,𝑗 and 𝑏2,𝑗 = max𝑖 𝑥𝑖,𝑗. +Exercise 11.31 Let 𝐭 be vector of n integers in {1, … , 𝑘}. Write a function to one-hot-encode +each 𝑡𝑖: return a0-1 matrix 𝐑 ofsize n-by-k suchthat 𝑟𝑖,𝑗 = 1 ifand onlyif 𝑗 = 𝑡𝑖.Forexample, +if 𝐭 = [1, 2, 3, 2, 4] and 𝑘 = 4, then: +𝐑 = +⎡⎢⎢⎢⎢ +⎣ +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +1 +0 +0 +0 +0 +0 +1 +⎤⎥⎥⎥⎥ +⎦ +. +Onasidenote,sucharepresentationisusefulwhensolving,e.g.,amulticlassclassificationprob- +lem by means of k binary classifiers. +Then, write another function, but this time setting 𝑟𝑖,𝑗 = 1 if and only if 𝑗 ≥ 𝑡𝑖, e.g.: +𝑅 = +⎡⎢⎢⎢⎢ +⎣ +1 +1 +1 +1 +0 +1 +1 +1 +0 +0 +1 +1 +0 +1 +1 +1 +0 +0 +0 +1 +⎤⎥⎥⎥⎥ +⎦ +. +Important Kind reminder: as usual, try to solve all the exercises without the use of +explicit for and while loops (provided that it is possible). +Exercise 11.32 Given an n-by-k real matrix, apply the softmax function on each row, i.e., map +𝑥𝑖,𝑗 to +exp(𝑥𝑖,𝑗) +∑𝑘 +𝑙=1 exp(𝑥𝑖,𝑙). Then, one-hot decode the values in each row, i.e., find the column number +with the greatest value. Return a vector of size n with elements in {1, … , 𝑘}. +Exercise 11.33 Assume that an n-by-d real matrix 𝐗 represents n points in ℝ𝑑. Write a func- +tion(butdonotreferto dist)thatdeterminesthepairwisedistancesbetweenallthenpointsand +a given 𝐲 ∈ ℝ𝑑. Return a vector 𝐝 of length n with 𝑑𝑖 = ‖𝐱𝑖,⋅ − 𝐲‖2. +Exercise 11.34 Let 𝐗 and 𝐘 be two real-valued matrices of sizes n-by-d and m-by-d, respect- +ively, representing two sets of points in ℝ𝑑. Return an integer vector 𝐫 of length m such that 𝑟𝑖 +indicates the index of the point in 𝐗 with the least distance to (the closest to) the i-th point in 𝐘, +i.e., 𝑟𝑖 = arg min𝑗 ‖𝐱𝑗,⋅ − 𝐲𝑖,⋅‖2. +Exercise 11.35 Write your own version of the built-in utils::combn. +Exercise 11.36 Time series are vectors or matrices of class ts equipped with the tsp attribute, +amongst others. Refer to help("ts") for more information about how they are represented and +what S3 methods have been overloaded for them. +Exercise 11.37 (*) Numeric matrices can be stored in a CSV file, amongst others. Usually, we +will be loading them via read.csv, which returns a data frame (see Chapter 12), for example: + +11 MATRICES AND OTHER ARRAYS +255 +X <- as.matrix(read.csv( +paste0( +"https://github.com/gagolews/teaching-data/", +"raw/master/marek/eurxxx-20200101-20200630.csv" +), +comment.char="#", +sep="," +)) +Writeyourownfunction read_numeric_matrix(file_name, comment, sep)whichisinstead +based on a few calls to scan. Use file to establish a file connection to be able to ignore the com- +ment lines and fetch the column names before reading the actual numeric values. +Exercise 11.38 (*) Using readBin, read the t10k-images-idx3-ubyte.gz from the MNIST +database homepage14. The output object should be a three-dimensional, 10000-by-28-by-28 ar- +ray with real elements between 0 and 255. Refer to the File Formats section therein for more de- +tails. +Exercise 11.39 (**) Circular convolution of discrete-valued multidimensional signals can be +performed by means of fft and matrix multiplication, whereas affine transformations require +onlythelatter.Applyvariousimagetransformationssuchassharpening,shearing,androtating +on the MNIST digits and plot the results using the image function. +Exercise 11.40 (*) Using constrOptim, find the minimum of the Constrained Betts Function +𝑓 (𝑥1, 𝑥2) = 0.01𝑥2 +1 +𝑥2 +2 −100withlinearconstraints2 ≤ 𝑥1 ≤ 50,−50 ≤ 𝑥2 ≤ 50,and +10𝑥1 ≥ 10 + 𝑥2. (**) Also, use solve.QP from the quadprog package of find the minimum. +14 https://web.archive.org/web/20211107114045/http://yann.lecun.com/exdb/mnist/ + + +12 +Data Frames +Mostmatricesarebuiltontopofatomicvectorsandhenceallowitemsofthesametype +to be arranged into rows and columns. Data frames (objects of S3 class data.frame, +firstintroducedin[8]),ontheotherhand,arecollectionsofvectorsofidenticallengths +or matrices with identical row counts, hence allowing to represent structured1 data of +possibly heterogeneous types, for instance: +class(iris) +# `iris` is an example built-in data frame +## [1] "data.frame" +iris[c(1, 51, 101), ] +# 3 chosen rows from `iris` +## +Sepal.Length Sepal.Width Petal.Length Petal.Width +Species +## 1 +5.1 +3.5 +1.4 +0.2 +setosa +## 51 +7.0 +3.2 +4.7 +1.4 versicolor +## 101 +6.3 +3.3 +6.0 +2.5 +virginica +is a mix of numeric and factor-type data. +The good news is that not only data frames are built upon named lists (e.g., to extract +a column we can refer to `[[`), but also many functions recognise them to be matrix- +like, (e.g., to select specific rows and columns, two indexes can be passed to `[` like in +the example above). Hence, it will soon turn out that we already know a lot about how +to perform basic data wrangling activities, even if we do not full realise it now. +Important Some of us will approach this chapter biased by what we know from else- +where, including our experience with some popular third-party packages for data +frame processing. The art is to filter out that information as noise (at least, for the +time being). We will show how powerful base R vocabulary is and how much can be +implied from the material covered in the preceding chapters. And yes, this book is like +a good thriller/drama/love story: it is meant to be read from the beginning to end, so +please go back to the start of this comprehensive course if you happened to pop in here +by accident or driven by “but I need to know now”. Good morning. +1 We are already highly skilled in handling unstructured data and turning it to something that is much +more regular: the numerous functions for processing numeric and character vectors as well as lists that we +have covered in the first part of this book allow us to extract meaningful data from text, handle missing +values, engineer features, and so forth. + +258 +II DEEPER +12.1 +Creating Data Frames +12.1.1 +data.frame and as.data.frame +Most frequently, we create data frames based on a series of logical, numeric, or char- +acters vectors of identical lengths. The data.frame function is particularly useful in +such a scenario: +(x <- data.frame( +a=c(TRUE, FALSE), +b=1:6, +c=runif(6), +d=c("spam", "spam", "eggs") +)) +## +a b +c +d +## 1 +TRUE 1 0.77437 spam +## 2 FALSE 2 0.19722 spam +## 3 +TRUE 3 0.97801 eggs +## 4 FALSE 4 0.20133 spam +## 5 +TRUE 5 0.36124 spam +## 6 FALSE 6 0.74261 eggs +Note that shorter vectors were recycled. That the diverse column types were retained +and no coercion has been made, can be verified, e.g., by calling: +str(x) +## 'data.frame': +6 obs. of +4 variables: +## +$ a: logi +TRUE FALSE TRUE FALSE TRUE FALSE +## +$ b: int +1 2 3 4 5 6 +## +$ c: num +0.774 0.197 0.978 0.201 0.361 ... +## +$ d: chr +"spam" "spam" "eggs" "spam" ... +We can also fetch the class of each column directly by calling (compare Section 12.3): +sapply(x, class) +# the same as unlist(Map(class, x)) +## +a +b +c +d +## +"logical" +"integer" +"numeric" "character" +Important For many reasons (see, e.g., Section 12.1.5 and Section 12.1.6), we recom- +mend to have the type of each column always checked, for instance by calling the str +function. + +12 DATA FRAMES +259 +Many objects, such as matrices, can easily be coerced to data frames using particular +as.data.frame methods. +Here is an example matrix: +(A <- matrix(1:6, nrow=3, +dimnames=list( +NULL, +# no row labels +c("u", "v") +# some column labels +))) +## +u v +## [1,] 1 4 +## [2,] 2 5 +## [3,] 3 6 +Let us convert it to a data frame: +as.data.frame(A) +# as.data.frame.matrix +## +u v +## 1 1 4 +## 2 2 5 +## 3 3 6 +Note that a matrix with no row labels is printed slightly differently than a data frame +with (as it will soon turn out) the default row.names. +Named lists are amongst other candidates for a meaningful conversion. Consider an +example list, where each element is a vector of the same length as the other ones: +(l <- Map( +function(x) { +c(Min=min(x), Median=median(x), Mean=mean(x), Max=max(x)) +}, +split(iris[["Sepal.Length"]], iris[["Species"]]) +)) +## $setosa +## +Min Median +Mean +Max +## +4.300 +5.000 +5.006 +5.800 +## +## $versicolor +## +Min Median +Mean +Max +## +4.900 +5.900 +5.936 +7.000 +## +## $virginica +## +Min Median +Mean +Max +## +4.900 +6.500 +6.588 +7.900 + +260 +II DEEPER +Each list element will be turned to a separate column: +as.data.frame(l) +# as.data.frame.list +## +setosa versicolor virginica +## Min +4.300 +4.900 +4.900 +## Median +5.000 +5.900 +6.500 +## Mean +5.006 +5.936 +6.588 +## Max +5.800 +7.000 +7.900 +Sadly, as.data.frame.list is not particularly fond of lists of vectors of incompatible +lengths: +as.data.frame(list(a=1, b=11:12, c=21:23)) +## Error in (function (..., row.names = NULL, check.rows = FALSE, check.names = TRUE, : arguments imply differing number of rows: 1, 2, 3 +The above vectors could have been recycled with a warning, but they were not. +as.data.frame(list(a=1:4, b=11:12, c=21)) +# recycling rule okay +## +a +b +c +## 1 1 11 21 +## 2 2 12 21 +## 3 3 11 21 +## 4 4 12 21 +The method for the S3 class table (mentioned in Chapter 11) can be helpful as well. +Here is an example contingency table together with its unstacked version. +(t <- table(mtcars[["vs"]], mtcars[["cyl"]])) +## +## +4 +6 +8 +## +0 +1 +3 14 +## +1 10 +4 +0 +as.data.frame(t) +# as.data.frame.table; see the stringsAsFactors note below! +## +Var1 Var2 Freq +## 1 +0 +4 +1 +## 2 +1 +4 +10 +## 3 +0 +6 +3 +## 4 +1 +6 +4 +## 5 +0 +8 +14 +## 6 +1 +8 +0 +Actually, as.data.frame.table is so useful that we might want to call it directly on any +array. This way, we can convert it from the so-called wide format to the long one; see +Section 12.3.6 for more details. +Note The above method is based on expand.grid, which determines all combinations +of a given series of vectors. + +12 DATA FRAMES +261 +expand.grid(1:2, c("a", "b", "c")) +# see the stringsAsFactors note below! +## +Var1 Var2 +## 1 +1 +a +## 2 +2 +a +## 3 +1 +b +## 4 +2 +b +## 5 +1 +c +## 6 +2 +c +Overall, many classes of objects can be included2 in a data frame; the popular choices +include Date, POSIXct, and factor. It is worth noting that the data.frame function calls +the corresponding as.data.frame method, and format is used on printing. +Example 12.1 Here are two custom methods for what we would like to call from now on an S3 +class spam: +as.data.frame.spam <- function(x, ...) +structure( +list(x), +class="data.frame", +row.names=seq_along(x) +) +format.spam <- function(x, ...) +paste0("*", x, "*") +Testing data frame creation and printing: +data.frame( +a=structure(c("a", "b", "c"), class="spam"), +b=factor(c("spam", "bacon", "spam")), +c=Sys.Date()+1:3 +) +## +a +b +c +## 1 *a* +spam 2022-12-29 +## 2 *b* bacon 2022-12-30 +## 3 *c* +spam 2022-12-31 +12.1.2 +cbind.data.frame and rbind.data.frame +There are data frame-specific versions of cbind or rbind (which we discussed in +the context of stacking matrices in Section 11.1.2). They are used quite eagerly: +2 Also, the attributes of objects stored as matrix columns will generally be preserved (even if they are not +displayed by print; see str though). + +262 +II DEEPER +help("cbind") states that they will be referred to if at least3 one of its arguments is +a data frame and the other arguments are atomic vectors or lists (possibly with the +dim attribute). +For example: +x <- iris[c(1, 51, 101), c("Sepal.Length", "Species")] +# whatever +cbind(Yummy=c(TRUE, FALSE, TRUE), x) +## +Yummy Sepal.Length +Species +## 1 +TRUE +5.1 +setosa +## 51 +FALSE +7.0 versicolor +## 101 +TRUE +6.3 +virginica +added a new column to a data frame x. Moreover: +rbind(x, list(42, "virginica")) +## +Sepal.Length +Species +## 1 +5.1 +setosa +## 51 +7.0 versicolor +## 101 +6.3 +virginica +## 11 +42.0 +virginica +added a new row. Note that columns are of different types. Hence, the values to row- +bind were provided as a generic vector. The list can also be named. It can consist of +vectors of length greater than one, given in any order: +rbind(x, list( +Species=c("virginica", "setosa"), +Sepal.Length=c(42, 7) +)) +## +Sepal.Length +Species +## 1 +5.1 +setosa +## 51 +7.0 versicolor +## 101 +6.3 +virginica +## 11 +42.0 +virginica +## 2 +7.0 +setosa +Sometimesreferring to these methods directlywill be necessary.Consideran example +list of atomic vectors: +x <- list(a=1:3, b=11:13, c=21:23) +First, we call the generic which dispatches to the default method: +3 This is a clear violation of the rule that an S3 generic dispatches on the type of only one (usually: first) +argument; an exception made for the sake of the questionable user convenience. Also, note that there is no +cbind.default method available: it is hardcoded at the C language level. + +12 DATA FRAMES +263 +do.call(cbind, x) +## +a +b +c +## [1,] 1 11 21 +## [2,] 2 12 22 +## [3,] 3 13 23 +If we want to make sure we garner a data frame in result, we need to write: +do.call(cbind.data.frame, x) +## +a +b +c +## 1 1 11 21 +## 2 2 12 22 +## 3 3 13 23 +This is particularly useful in the context of fetching outputs from Map and its friends, +which are wrapped inside a list. For instance: +l <- unname(Map( +function(x) list( +Sepal.Length=mean(x[["Sepal.Length"]]), +Sepal.Width=mean(x[["Sepal.Width"]]), +Species=x[["Species"]][1] +), +split(iris, iris[["Species"]]) +# split.data.frame; see below +)) +str(l) +## List of 3 +## +$ :List of 3 +## +..$ Sepal.Length: num 5.01 +## +..$ Sepal.Width : num 3.43 +## +..$ Species +: Factor w/ 3 levels "setosa","versicolor",..: 1 +## +$ :List of 3 +## +..$ Sepal.Length: num 5.94 +## +..$ Sepal.Width : num 2.77 +## +..$ Species +: Factor w/ 3 levels "setosa","versicolor",..: 2 +## +$ :List of 3 +## +..$ Sepal.Length: num 6.59 +## +..$ Sepal.Width : num 2.97 +## +..$ Species +: Factor w/ 3 levels "setosa","versicolor",..: 3 +This was nothing more than a fancy way to obtain an illustrative list, which we may +now turn into a data frame by calling: +do.call(rbind.data.frame, l) +## +Sepal.Length Sepal.Width +Species +## 1 +5.006 +3.428 +setosa +(continues on next page) + +264 +II DEEPER +(continued from previous page) +## 2 +5.936 +2.770 versicolor +## 3 +6.588 +2.974 +virginica +On the other hand, do.call(rbind, l) does not return a particularly friendly object +type: +do.call(rbind, l) +## +Sepal.Length Sepal.Width Species +## [1,] 5.006 +3.428 +setosa +## [2,] 5.936 +2.77 +versicolor +## [3,] 6.588 +2.974 +virginica +Despite the pretty face, it is a matrix… over a list: +str(do.call(rbind, l)) +## List of 9 +## +$ : num 5.01 +## +$ : num 5.94 +## +$ : num 6.59 +## +$ : num 3.43 +## +$ : num 2.77 +## +$ : num 2.97 +## +$ : Factor w/ 3 levels "setosa","versicolor",..: 1 +## +$ : Factor w/ 3 levels "setosa","versicolor",..: 2 +## +$ : Factor w/ 3 levels "setosa","versicolor",..: 3 +## +- attr(*, "dim")= int [1:2] 3 3 +## +- attr(*, "dimnames")=List of 2 +## +..$ : NULL +## +..$ : chr [1:3] "Sepal.Length" "Sepal.Width" "Species" +12.1.3 +Reading Data Frames +Structureddatacanbeimportedfromexternalsources,suchasCSV/TSV(comma/tab- +separated values) or HDF5 files, relational databases supporting SQL (see Sec- +tion 12.1.4) web APIs (e.g., through the curl and jsonlite packages), spreadsheets +[46], and so on. +Inparticular, read.csvandthelikefetchdatafromplaintextfilesconsistingofrecords +where fields are separated by commas, semicolons, tabs, etc. +For instance: +x <- data.frame(a=runif(3), b=c(TRUE, FALSE, TRUE)) +# example data frame +f <- tempfile() +# temporary file name +write.csv(x, f, row.names=FALSE) +# export + +12 DATA FRAMES +265 +This created a CSV file which looks like: +cat(readLines(f), sep="\n") +# print file contents +## "a","b" +## 0.287577520124614,TRUE +## 0.788305135443807,FALSE +## 0.4089769218117,TRUE +The above can be read by calling: +read.csv(f) +## +a +b +## 1 0.28758 +TRUE +## 2 0.78831 FALSE +## 3 0.40898 +TRUE +Exercise 12.2 Checkout help("read.table")foralonglistoftunableparameters,especially: +sep, dec, quote, header, comment.char, and row.names. Further, note that reading from com- +pressed files is supported directly. +Important CSV is by far the most portable and user-friendly format for exchanging +matrix-like objects between different programs and computing languages (e.g., Py- +thon, Julia, LibreOffice Calc, etc.). Such files can be opened in any text editor. +Note As mentioned in Section 8.3.5, it is possible to process data frames on a chunk- +by-chunk basis, which is beneficial especially when data do not fit into memory (com- +pare the nrows argument to read.csv). +12.1.4 +Interfacing Relational Databases and Querying with SQL (*) +The DBI package provides a universal interface for particular database management +systemswhosedriversareimplementedinadditionaladd-onssuchas RSQLite, RMari- +aDB, RPostgreSQL, etc., or, more generally, RODBC or odbc. For more details, see Section +4 of [46]. +Example 12.3 Let us play with an in-memory (volatile) instance of an SQLite database. +library("DBI") +con <- dbConnect(RSQLite::SQLite(), ":memory:") +This returns an object representing a database connection which we can refer to in further com- +munication. +An easy way to create a database table is to call: + +266 +II DEEPER +dbWriteTable(con, "mtcars", mtcars) +# `mtcars` is a toy built-in data frame +Alternatively, dbExecute could have been referred to in order to send SQL statements such as +CREATE TABLE ... followed by a series of INSERT INTO .... +Some data retrieval can now follow: +dbGetQuery(con, " +SELECT cyl, vs, AVG(mpg) AS mpg_ave, AVG(hp) AS hp_ave +FROM mtcars +GROUP BY cyl, vs +") +## +cyl vs mpg_ave hp_ave +## 1 +4 +0 +26.000 +91.00 +## 2 +4 +1 +26.730 +81.80 +## 3 +6 +0 +20.567 131.67 +## 4 +6 +1 +19.125 115.25 +## 5 +8 +0 +15.100 209.21 +This gives us an ordinary R data frame which we can process in the same fashion as any other +object of this kind. +At the end, the database connection must be closed. +dbDisconnect(con) +Exercise 12.4 Database passwords should never be stored in plain text files, let alone in R +scripts in version-controlled repositories. Consider a few ways for fetching credentials program- +matically: +• using environment variables (see help("Sys.getenv")), +• using the keyring package, +• callingsystem2(Section7.3.3)toretrieveitfromthesystemkeyring(e.g.,thekeyringpack- +age for Python provides a platform-independent command-line utility). +12.1.5 +Strings as Factors? +The following is so critical that we will devote a separate subsection to discuss it, so +that we always remain vigilant (such is life: maintaining some level of mindfulness is +often a good idea). +Important Some functions related to data frames automatically convert character +vectors to factors. This behaviour is frequently controlled by the stringsAsFactors ar- +gument thereto. + +12 DATA FRAMES +267 +Thisisparticularlyproblematicduetothefactthat,whenprinted,factorandcharacter +columns look identical: +(x <- data.frame(a=factor(c("U", "V")), b=c("U", "V"))) +## +a b +## 1 U U +## 2 V V +We recall from Section 10.3.3 that factors can be nasty. For example, passing factors +as indexers in `[` or converting them with as.numeric might give counterintuitive +(for the uninformed) results. Also, new factor levels must be added manually when +we want to extend them with more diverse data. This can cause some unexpected be- +haviour in contexts such as: +rbind(x, c("W", "W")) +## Warning in `[<-.factor`(`*tmp*`, ri, value = "W"): invalid factor level, +## NA generated +## +a b +## 1 +U U +## 2 +V V +## 3 W +It is therefore a good habit to have the data types always checked, for instance: +str(x) +## 'data.frame': +2 obs. of +2 variables: +## +$ a: Factor w/ 2 levels "U","V": 1 2 +## +$ b: chr +"U" "V" +Before R 4.0, a number of functions, including data.frame and read.csv had the +stringsAsFactors argument defaulting to TRUE. This is no longer the case for many +of them. +However, exceptions to this rule still exist, e.g., including as.data.frame.table and +expand.grid. Besides, some built-in example data frames have factor-typed columns +inherited from the old days, e.g.: +class(iris[["Species"]]) +## [1] "factor" +We observe that the Species column in iris is not of type character. Thence, adding a +new variety might be oblique: +iris2 <- iris[c(1, 51, 101), ] +# example subset +levels(iris2[["Species"]]) <- c(levels(iris2[["Species"]]), "croatica") +rbind(iris2, c(6, 3, 3, 2, "croatica")) +## +Sepal.Length Sepal.Width Petal.Length Petal.Width +Species +(continues on next page) + +268 +II DEEPER +(continued from previous page) +## 1 +5.1 +3.5 +1.4 +0.2 +setosa +## 51 +7 +3.2 +4.7 +1.4 versicolor +## 101 +6.3 +3.3 +6 +2.5 +virginica +## 4 +6 +3 +3 +2 +croatica +Alternatively, we could have simply converted the Species column to character. +12.1.6 +Internal Representation +Objects of S3 class data.frame are built upon lists of vectors of the same length or +matrices with identical row counts, which define consecutive columns thereof. Apart +from class, they must be equipped with the following special attributes: +• names – a character vector (as usual in any named list) labelling the columns or +their groups, +• row.names – a character or integer vector with no duplicates nor missing values, +doing what advertised. +Therefore, a data frame can be created from scratch by calling, for example: +structure( +list(a=11:13, b=21:23), +# sets the `names` attribute already +row.names=1:3, +class="data.frame" +) +## +a +b +## 1 11 21 +## 2 12 22 +## 3 13 23 +Here is a data frame based on a length-5 list, a matrix with five rows, and a length-5 +numeric vector, with some fancy row names on top: +structure( +list( +a=list(1, 1:2, 1:3, numeric(0), -(4:1)), +b=cbind(u=11:15, v=21:25), +c=runif(5) +), +row.names=c("spam", "bacon", "eggs", "ham", "aubergine"), +class="data.frame" +) +## +a b.u b.v +c +## spam +1 +11 +21 0.28758 +## bacon +1, 2 +12 +22 0.78831 +(continues on next page) + +12 DATA FRAMES +269 +(continued from previous page) +## eggs +1, 2, 3 +13 +23 0.40898 +## ham +14 +24 0.88302 +## aubergine -4, -3, -2, -1 +15 +25 0.94047 +In general, the columns of type list can contain anything, e.g., other lists or R func- +tions. Including atomic vectors of varying lengths just like above allows for creating +something à la ragged arrays – a pretty handy scenario. +The issue with matrix entries, on the other hand, is that they appear as if they were +many, but – as it will turn out in the sequel – they are often treated as a single com- +plex column, e.g., by the index operator (see Section 12.2). Therefore, from this per- +spective, the above data frame has three columns, not four. Such objects can be output +by aggregate (see Section 12.3), amongst others. Nevertheless, they can be very useful +too, forming natural column groups which can be easily accessed and batch-processed +in the same way. +Important Unfortunately, data frames with list or matrix columns cannot be nor- +mally created with the data.frame nor cbind functions which might explain why they +are less popular. This behaviour is dictated by the particular underlying as.data.frame +methods which are called by both of them. As a curiosity, see help("I") though. +Exercise 12.5 Verifythatforadataframefeaturingamatrixcolumn,thelatterdoesnotrequire +column names (the second dimnames) set. +The names and row.names attributes are special in the sense of Section 4.4.3. In partic- +ular, they can be accessed or modified by the corresponding functions. +It is worth noting that row.names(df) always returns a character vector, even when +attr(df, "row.names") is an integer vector. Further, setting row.names(df) <- NULL +will re-set4 this attribute to the most commonly desired case of consecutive natural +numbers, for example: +(x <- iris[c(1, 51, 101), ]) +# comes with some sad row names +## +Sepal.Length Sepal.Width Petal.Length Petal.Width +Species +## 1 +5.1 +3.5 +1.4 +0.2 +setosa +## 51 +7.0 +3.2 +4.7 +1.4 versicolor +## 101 +6.3 +3.3 +6.0 +2.5 +virginica +row.names(x) <- NULL +# reset to seq_len(NROW(x)) +print(x) +## +Sepal.Length Sepal.Width Petal.Length Petal.Width +Species +## 1 +5.1 +3.5 +1.4 +0.2 +setosa +(continues on next page) +4 `attr<-`(df, "row.names") does not feature the same sanity checks as `row.names<-`(df) does. For +instance, it is easy to corrupt a data frame by setting a too-short row.names attribute. + +270 +II DEEPER +(continued from previous page) +## 2 +7.0 +3.2 +4.7 +1.4 versicolor +## 3 +6.3 +3.3 +6.0 +2.5 +virginica +Exercise 12.6 What is the name of the replacement version of the row.names method for the +data.frame class? +Exercise 12.7 Implement your own version of expand.grid. +Exercise 12.8 Implement your own version of xtabs, but which does not rely on a formula in- +terface. Allow three parameters: a data frame, the name of the “counts” column and the names of +the cross-classifying variables. Hence, my_xtabs(x, "Freq", c("Var1", "Var2")) should be +equivalent to xtabs(Freq~Var1+Var2, x). +12.2 +Data Frame Subsetting +12.2.1 +Data Frames are Lists +Data frames are named lists, where each element represents an individual column. +Therefore5, length yields the number of columns and names gives their respective la- +bels. +Let us play with the following data frame: +(x <- data.frame( +a=runif(6), +b=rnorm(6), +c=LETTERS[1:6], +d1=c(FALSE, TRUE, FALSE, NA, FALSE, NA), +d2=c(FALSE, TRUE, FALSE, TRUE, FALSE, TRUE) +)) +## +a +b c +d1 +d2 +## 1 0.287578 +0.070508 A FALSE FALSE +## 2 0.788305 +0.129288 B +TRUE +TRUE +## 3 0.408977 +1.715065 C FALSE FALSE +## 4 0.883017 +0.460916 D +NA +TRUE +## 5 0.940467 -1.265061 E FALSE FALSE +## 6 0.045556 -0.686853 F +NA +TRUE +typeof(x) +# each data frame is a list +## [1] "list" +(continues on next page) +5 This is a strong word. This implication relies on an implicit assumption that the primitive functions +length and names have not be contaminated by treating data frames differently than named lists. Luckily, +thatisindeednotthecase.Also,despitethefactthatwehavetheindexoperatorsspeciallyoverloadedforthe +data.frame class, they behave quite reasonably and, as we will see, they allow for a mix of list- and matrix- +like behaviours. + +12 DATA FRAMES +271 +(continued from previous page) +length(x) +# the number of columns +## [1] 5 +names(x) +# column labels +## [1] "a" +"b" +"c" +"d1" "d2" +The one-argument versions of extract and index operators behave as expected. `[[` +fetches (looks inside) the contents of a given column: +x[["a"]] +# or x[[1]] +## [1] 0.287578 0.788305 0.408977 0.883017 0.940467 0.045556 +and `[` returns a data frame (a list with extras) comprised of the specified elements: +x["a"] +# or x[1] +## +a +## 1 0.287578 +## 2 0.788305 +## 3 0.408977 +## 4 0.883017 +## 5 0.940467 +## 6 0.045556 +x[c(TRUE, TRUE, FALSE, TRUE, FALSE)] +## +a +b +d1 +## 1 0.287578 +0.070508 FALSE +## 2 0.788305 +0.129288 +TRUE +## 3 0.408977 +1.715065 FALSE +## 4 0.883017 +0.460916 +NA +## 5 0.940467 -1.265061 FALSE +## 6 0.045556 -0.686853 +NA +Just like with lists, the replacement versions of the said operators can be used to add +new or replace existing columns. +y <- head(x, 1) +# for a more compact display +y[["a"]] <- round(y[["a"]], 1) +# replaces the column with new content +y[["b"]] <- NULL +# removes the column, like, totally +y[["e"]] <- 10*y[["a"]]^2 +# adds a new column at the end +print(y) +## +a c +d1 +d2 +e +## 1 0.3 A FALSE FALSE 0.9 +Example 12.9 Some spam for thought to show how much we already know: some common use +cases of indexing and vectorised functions: + +272 +II DEEPER +y <- head(x, 1) +# for a more compact display +Move column a to the end: +y[unique(c(names(y), "a"), fromLast=TRUE)] +## +b c +d1 +d2 +a +## 1 0.070508 A FALSE FALSE 0.28758 +Remove column a and c: +y[-match(c("a", "c"), names(y))] +## +b +d1 +d2 +## 1 0.070508 FALSE FALSE +All columns between a and c: +y[match("a", names(y)):match("c", names(y))] +## +a +b c +## 1 0.28758 0.070508 A +Names starting with d: +y[grep("^d", names(y))] +## +d1 +d2 +## 1 FALSE FALSE +Change name of column c to z: +names(y)[names(y) == "c"] <- "z" +# in-place +print(y) +## +a +b z +d1 +d2 +## 1 0.28758 0.070508 A FALSE FALSE +Change names: d2 to u and d1 to v: +names(y)[match(c("d2", "d1"), names(y))] <- c("v", "u") +# in-place +print(y) +## +a +b z +u +v +## 1 0.28758 0.070508 A FALSE FALSE +Note Some R users might prefer the `$` operator over `[[`, but we do not. By de- +fault, the former supports partial matching of column names which might be appeal- +ing when R is used interactively. However, it does not work on matrices, nor it allows +for programmatically generated names. It is also trickier to use on non-syntactically +valid labels; compare Section 9.4.1. + +12 DATA FRAMES +273 +Exercise 12.10 Write a function names_replace that changes the name of a data frame +columns based on a translation table given in a from=to fashion, for instance: +names_replace <- function(x, ...) ...to.do... +x <- data.frame(a=1, b=2, c=3) +names_replace(x, c="new_c", a="new_a") +## +new_a b new_c +## 1 +1 2 +3 +12.2.2 +Data Frames are Matrix-like +Data frames can be considered “generalised” matrices. They store data of any kind +(possiblymixed)organisedinatabularfashion.Somefunctionsmentionedinthepre- +viouschapterwillhencebeoverloadedforthedataframecase.Theseinclude: dim(des- +pite the lack of the dim attribute), NROW, NCOL, and dimnames (which is of course based +on row.names and names). +For example: +(x <- data.frame( +a=runif(6), +b=rnorm(6), +c=LETTERS[1:6], +d1=c(FALSE, TRUE, FALSE, NA, FALSE, NA), +d2=c(FALSE, TRUE, FALSE, TRUE, FALSE, TRUE) +)) +## +a +b c +d1 +d2 +## 1 0.287578 +0.070508 A FALSE FALSE +## 2 0.788305 +0.129288 B +TRUE +TRUE +## 3 0.408977 +1.715065 C FALSE FALSE +## 4 0.883017 +0.460916 D +NA +TRUE +## 5 0.940467 -1.265061 E FALSE FALSE +## 6 0.045556 -0.686853 F +NA +TRUE +dim(x) +# the number of rows and columns +## [1] 6 5 +dimnames(x) +# it is not a matrix, but a matrix-like object +## [[1]] +## [1] "1" "2" "3" "4" "5" "6" +## +## [[2]] +## [1] "a" +"b" +"c" +"d1" "d2" +In addition to the list-like behaviour, which only allows for dealing with particular +columns or groups thereof, the `[` operator was also equipped with the ability to take +two indexers: + +274 +II DEEPER +x[1:2, ] +# first two rows +## +a +b c +d1 +d2 +## 1 0.28758 0.070508 A FALSE FALSE +## 2 0.78831 0.129288 B +TRUE +TRUE +x[x[["a"]] >= 0.3 +& +x[["a"]] <= 0.8, -2] +# or use x[, "a"] +## +a c +d1 +d2 +## 2 0.78831 B +TRUE +TRUE +## 3 0.40898 C FALSE FALSE +Recall the drop argument to `[` and its effects on matrix indexing. It the current case, +its behaviour will be similar with regard to the operations on individual columns: +x[, 1] +# synonym: x[[1]], because drop=TRUE +## [1] 0.287578 0.788305 0.408977 0.883017 0.940467 0.045556 +x[, 1, drop=FALSE] +# synonym: x[1] +## +a +## 1 0.287578 +## 2 0.788305 +## 3 0.408977 +## 4 0.883017 +## 5 0.940467 +## 6 0.045556 +Also, note that when we extract a single row and more than one column, drop does not +really apply. It is because columns (unlike in matrices) can potentially be of different +types: +x[1, 1:2] +# two numeric columns but the result is still a numeric +## +a +b +## 1 0.28758 0.070508 +However: +x[1, 1] +## [1] 0.28758 +x[1, 1, drop=FALSE] +## +a +## 1 0.28758 +Note Take note of logical indexing featuring missing values: +x[x[["d1"]], ] +## +a +b +c +d1 +d2 +## 2 +0.78831 0.12929 +B TRUE TRUE +(continues on next page) + +12 DATA FRAMES +275 +(continued from previous page) +## NA +NA +NA +NA +NA +## NA.1 +NA +NA +NA +NA +x[which(x[["d1"]]), ] +# drops missing values +## +a +b c +d1 +d2 +## 2 0.78831 0.12929 B TRUE TRUE +The default behaviour is actually correct as it indicates about something being miss- +ing. However, no warning is emitted and thus this can lead to bugs in our code. +By far, we might have already noted that the index operator adjusts (not: resets) the +row.names attribute. For instance: +(xs <- x[head(order(x[["a"]], decreasing=TRUE), 3), ]) +## +a +b c +d1 +d2 +## 5 0.94047 -1.26506 E FALSE FALSE +## 4 0.88302 +0.46092 D +NA +TRUE +## 2 0.78831 +0.12929 B +TRUE +TRUE +It is a version of x comprised of only top three values in the u column. Indexing by +means of character vectors will refer to row.names and names: +xs["5", c("a", "b")] +## +a +b +## 5 0.94047 -1.2651 +Note that this is not the same as “xs[5, c("a", "b")]”, despite the fact that row.names +is formally an integer vector here. +Note +If a data frame features a matrix, we need to use the index/extract operator +twice in order to access a specific sub-column: +(x <- aggregate(iris[1], iris[5], function(x) c(Min=min(x), Max=max(x)))) +## +Species Sepal.Length.Min Sepal.Length.Max +## 1 +setosa +4.3 +5.8 +## 2 versicolor +4.9 +7.0 +## 3 +virginica +4.9 +7.9 +x[["Sepal.Length"]][, "Min"] +## [1] 4.3 4.9 4.9 +In other words, neither “x[["Sepal.Length.Min"]]” nor “x[, +"Sepal.Length.Min"]” +works. +As far as the replacement version of the index operator is concerned, it is a quite flex- +ible tool, allowing the new content to be a vector, a data frame, a list, or even a matrix. + +276 +II DEEPER +Exercise 12.11 Write two replacement functions6. First, set_row_names which replaces the +row.names of a data frame with the contents of a specific column, for example: +(x <- aggregate(iris[1], iris[5], mean)) +# some data frame +## +Species Sepal.Length +## 1 +setosa +5.006 +## 2 versicolor +5.936 +## 3 +virginica +6.588 +set_row_names(x) <- "Species" +print(x) +## +Sepal.Length +## setosa +5.006 +## versicolor +5.936 +## virginica +6.588 +Second, reset_row_names which converts row.names to a standalone column of a given name, +for instance: +reset_row_names(x) <- "Type" +print(x) +## +Sepal.Length +Type +## 1 +5.006 +setosa +## 2 +5.936 versicolor +## 3 +6.588 +virginica +These two functions may be handy as they allow for writing “x[something, +]” instead of +“x[x[["column"]] %in% something, ]”. +12.3 +Common Operations +Below we review the most commonly applied operations related to data frame +wrangling. We have a few dedicated functions or methods overloaded for the data. +frame class. However, we have already mastered the necessary skills to deal with this +kind of objects through our hard work, in particular involving the solving of the ex- +ercises in the preceding chapters. Let us repeat: data frames are just lists exhibiting +matrix-like behaviour. +12.3.1 +Ordering Rows +Ordering rows in a data frame with respect to different criteria can be easily achieved +by means of the order function and the two-argument version of `[`. +6 (*) Compare pandas.DataFrame.set_index and pandas.DataFrame.reset_index in Python. + +12 DATA FRAMES +277 +For instance, here are the top six cars in terms of the time (in seconds) to complete a +402-metre race: +mtcars6 <- mtcars[order(mtcars[["qsec"]])[1:6], ] +mtcars6[["model"]] <- row.names(mtcars6) +row.names(mtcars6) <- NULL +print(mtcars6) +## +mpg cyl disp +hp drat +wt +qsec vs am gear carb +model +## 1 15.8 +8 +351 264 4.22 3.17 14.50 +0 +1 +5 +4 Ford Pantera L +## 2 15.0 +8 +301 335 3.54 3.57 14.60 +0 +1 +5 +8 +Maserati Bora +## 3 13.3 +8 +350 245 3.73 3.84 15.41 +0 +0 +3 +4 +Camaro Z28 +## 4 19.7 +6 +145 175 3.62 2.77 15.50 +0 +1 +5 +6 +Ferrari Dino +## 5 14.3 +8 +360 245 3.21 3.57 15.84 +0 +0 +3 +4 +Duster 360 +## 6 21.0 +6 +160 110 3.90 2.62 16.46 +0 +1 +4 +4 +Mazda RX4 +order uses a stable sorting algorithm, therefore sorting with respect to a different cri- +terion will not break the relative ordering of qsec in row groups with ties: +mtcars6[order(mtcars6[["cyl"]]), ] +## +mpg cyl disp +hp drat +wt +qsec vs am gear carb +model +## 4 19.7 +6 +145 175 3.62 2.77 15.50 +0 +1 +5 +6 +Ferrari Dino +## 6 21.0 +6 +160 110 3.90 2.62 16.46 +0 +1 +4 +4 +Mazda RX4 +## 1 15.8 +8 +351 264 4.22 3.17 14.50 +0 +1 +5 +4 Ford Pantera L +## 2 15.0 +8 +301 335 3.54 3.57 14.60 +0 +1 +5 +8 +Maserati Bora +## 3 13.3 +8 +350 245 3.73 3.84 15.41 +0 +0 +3 +4 +Camaro Z28 +## 5 14.3 +8 +360 245 3.21 3.57 15.84 +0 +0 +3 +4 +Duster 360 +Example 12.12 Notice the difference between ordering by cyl and gear vs gear and cyl: +mtcars6[order(mtcars6[["cyl"]], mtcars6[["gear"]]), ] +## +mpg cyl disp +hp drat +wt +qsec vs am gear carb +model +## 6 21.0 +6 +160 110 3.90 2.62 16.46 +0 +1 +4 +4 +Mazda RX4 +## 4 19.7 +6 +145 175 3.62 2.77 15.50 +0 +1 +5 +6 +Ferrari Dino +## 3 13.3 +8 +350 245 3.73 3.84 15.41 +0 +0 +3 +4 +Camaro Z28 +## 5 14.3 +8 +360 245 3.21 3.57 15.84 +0 +0 +3 +4 +Duster 360 +## 1 15.8 +8 +351 264 4.22 3.17 14.50 +0 +1 +5 +4 Ford Pantera L +## 2 15.0 +8 +301 335 3.54 3.57 14.60 +0 +1 +5 +8 +Maserati Bora +mtcars6[order(mtcars6[["gear"]], mtcars6[["cyl"]]), ] +## +mpg cyl disp +hp drat +wt +qsec vs am gear carb +model +## 3 13.3 +8 +350 245 3.73 3.84 15.41 +0 +0 +3 +4 +Camaro Z28 +## 5 14.3 +8 +360 245 3.21 3.57 15.84 +0 +0 +3 +4 +Duster 360 +## 6 21.0 +6 +160 110 3.90 2.62 16.46 +0 +1 +4 +4 +Mazda RX4 +## 4 19.7 +6 +145 175 3.62 2.77 15.50 +0 +1 +5 +6 +Ferrari Dino +## 1 15.8 +8 +351 264 4.22 3.17 14.50 +0 +1 +5 +4 Ford Pantera L +## 2 15.0 +8 +301 335 3.54 3.57 14.60 +0 +1 +5 +8 +Maserati Bora + +278 +II DEEPER +Note Mixing a increasing and decreasing ordering is tricky as the decreasing argu- +ment to order currently does not accept multiple flags in all the contexts. Perhaps the +easiest way to change the ordering direction is to use the unary minus operator on the +column(s) to be sorted decreasingly. +mtcars6[order(mtcars6[["gear"]], -mtcars6[["cyl"]]), ] +## +mpg cyl disp +hp drat +wt +qsec vs am gear carb +model +## 3 13.3 +8 +350 245 3.73 3.84 15.41 +0 +0 +3 +4 +Camaro Z28 +## 5 14.3 +8 +360 245 3.21 3.57 15.84 +0 +0 +3 +4 +Duster 360 +## 6 21.0 +6 +160 110 3.90 2.62 16.46 +0 +1 +4 +4 +Mazda RX4 +## 1 15.8 +8 +351 264 4.22 3.17 14.50 +0 +1 +5 +4 Ford Pantera L +## 2 15.0 +8 +301 335 3.54 3.57 14.60 +0 +1 +5 +8 +Maserati Bora +## 4 19.7 +6 +145 175 3.62 2.77 15.50 +0 +1 +5 +6 +Ferrari Dino +For factor and character columns, xtfrm can be used to convert them to sort keys first. +mtcars6[order(mtcars6[["cyl"]], -xtfrm(mtcars6[["model"]])), ] +## +mpg cyl disp +hp drat +wt +qsec vs am gear carb +model +## 6 21.0 +6 +160 110 3.90 2.62 16.46 +0 +1 +4 +4 +Mazda RX4 +## 4 19.7 +6 +145 175 3.62 2.77 15.50 +0 +1 +5 +6 +Ferrari Dino +## 2 15.0 +8 +301 335 3.54 3.57 14.60 +0 +1 +5 +8 +Maserati Bora +## 1 15.8 +8 +351 264 4.22 3.17 14.50 +0 +1 +5 +4 Ford Pantera L +## 5 14.3 +8 +360 245 3.21 3.57 15.84 +0 +0 +3 +4 +Duster 360 +## 3 13.3 +8 +350 245 3.73 3.84 15.41 +0 +0 +3 +4 +Camaro Z28 +Both of the above behave like decreasing=c(FALSE, TRUE). +Exercise 12.13 Write a method sort.data.frame that orders a data frame with respect to a +given set of columns. +sort.data.frame <- function(x, decreasing=FALSE, cols) ...to.do... +sort(mtcars6, cols=c("cyl", "model")) +## +mpg cyl disp +hp drat +wt +qsec vs am gear carb +model +## 4 19.7 +6 +145 175 3.62 2.77 15.50 +0 +1 +5 +6 +Ferrari Dino +## 6 21.0 +6 +160 110 3.90 2.62 16.46 +0 +1 +4 +4 +Mazda RX4 +## 3 13.3 +8 +350 245 3.73 3.84 15.41 +0 +0 +3 +4 +Camaro Z28 +## 5 14.3 +8 +360 245 3.21 3.57 15.84 +0 +0 +3 +4 +Duster 360 +## 1 15.8 +8 +351 264 4.22 3.17 14.50 +0 +1 +5 +4 Ford Pantera L +## 2 15.0 +8 +301 335 3.54 3.57 14.60 +0 +1 +5 +8 +Maserati Bora +Unfortunately, that decreasing must be of length one and be placed as the second method argu- +ment is imposed by the sort S3 generic. + +12 DATA FRAMES +279 +12.3.2 +Handling Duplicated Rows +duplicated, anyDuplicated, and unique have methods overloaded for the data.frame +class. They can be used to indicate, get rid of, or replace the repeating rows. +sum(duplicated(iris)) +# how many duplicated rows are there? +## [1] 1 +iris[duplicated(iris), ] +# show the duplicated rows +## +Sepal.Length Sepal.Width Petal.Length Petal.Width +Species +## 143 +5.8 +2.7 +5.1 +1.9 virginica +12.3.3 +Joining (Merging) Data Frames +The merge function can perform the JOIN operation that some readers might know +from SQL7. It matches the items in the columns that two given data frames somewhat +share, and then returns their combination. +Example 12.14 Twocallstomergecouldbeusedtomatchdataonprogrammers(eachidentified +by developer_id and giving such details as their name, location, main skill, etc.) with the in- +formationabouttheopen-sourceprojects(eachidentifiedbyproject_idandinformingusabout +its title, scope, web site, and so forth) they are engaged in (based on a third data frame featuring +developer_id and project_id pairs). +As an simple illustration, consider the two following objects: +A <- data.frame( +u=c("b0", "b1", "b2", "b3"), +v=c("a0", "a1", "a2", "a3") +) +B <- data.frame( +v=c("a0", "a2", "a2", "a4"), +w=c("c0", "c1", "c2", "c3") +) +The two common columns, i.e., storing data of similar nature (a-something strings), +are both named v. +First, the inner (natural) join, where we list only the matching pairs: +merge(A, B) +# x=A, y=B, by="v", all.x=FALSE, all.y=FALSE +## +v +u +w +## 1 a0 b0 c0 +(continues on next page) +7 JOIN is the reverse operation to data normalisation known from theory of relational databases, which +itself reduces data redundancy and increases their integrity. What data scientists need for succeeding with +their daily activities (analysis, visualisation, processing) is thus the opposite of what the art of data man- +agement focuses on (efficient collection and storage). Readers are encouraged to learn about various nor- +malisation forms from, e.g., [11] or any other course covering this topic. + +280 +II DEEPER +(continued from previous page) +## 2 a2 b2 c1 +## 3 a2 b2 c2 +Note that the common column (or, more generally, columns) is included only once in +the result. +The left join guarantees that all elements in the first data frame will be included in the +result: +merge(A, B, all.x=TRUE) +# by="v", all.y=FALSE +## +v +u +w +## 1 a0 b0 +c0 +## 2 a1 b1 +## 3 a2 b2 +c1 +## 4 a2 b2 +c2 +## 5 a3 b3 +The right join includes all records in the second argument: +merge(A, B, all.y=TRUE) +# by="v", all.x=FALSE +## +v +u +w +## 1 a0 +b0 c0 +## 2 a2 +b2 c1 +## 3 a2 +b2 c2 +## 4 a4 c3 +And the full outer join is their set-theoretic union: +merge(A, B, all.x=TRUE, all.y=TRUE) +# by="v" +## +v +u +w +## 1 a0 +b0 +c0 +## 2 a1 +b1 +## 3 a2 +b2 +c1 +## 4 a2 +b2 +c2 +## 5 a3 +b3 +## 6 a4 +c3 +Exercise 12.15 Show how match (Section 5.4.1) can be used to implement a very basic version +of merge. +12.3.4 +Aggregating and Transforming Columns +Let us discuss how to perform data aggregation or engineer features. Despite the fact +that we already know how to access individual columns with `[` and process them +using the many vectorised functions, we still have something interesting to add about +the said matter. + +12 DATA FRAMES +281 +It would be tempting to try implementing such operations with apply. Unfortunately, +currently this function coerces its argument to a matrix. Hence, we should refrain +from applying it on data frames whose columns are of mixed types8. +However, taking into account that data frames are special lists, we can always call Map +and its relatives. +Example 12.16 Given an example data frame: +(iris_sample <- iris[sample(NROW(iris), 6), ]) +## +Sepal.Length Sepal.Width Petal.Length Petal.Width +Species +## 28 +5.2 +3.5 +1.5 +0.2 +setosa +## 80 +5.7 +2.6 +3.5 +1.0 versicolor +## 101 +6.3 +3.3 +6.0 +2.5 +virginica +## 111 +6.5 +3.2 +5.1 +2.0 +virginica +## 137 +6.3 +3.4 +5.6 +2.4 +virginica +## 133 +6.4 +2.8 +5.6 +2.2 +virginica +To get the class of each column, we can call: +sapply(iris_sample, class) +# or unlist(Map(class, iris)) +## Sepal.Length +Sepal.Width Petal.Length +Petal.Width +Species +## +"numeric" +"numeric" +"numeric" +"numeric" +"factor" +Next, here is a way to compute some aggregates of the numeric columns: +unlist(Map(mean, Filter(is.numeric, iris_sample))) +## Sepal.Length +Sepal.Width Petal.Length +Petal.Width +## +6.0667 +3.1333 +4.5500 +1.7167 +or: +sapply(iris_sample[sapply(iris_sample, is.numeric)], mean) +## Sepal.Length +Sepal.Width Petal.Length +Petal.Width +## +6.0667 +3.1333 +4.5500 +1.7167 +We can also fetch more than a single summary of each column: +as.data.frame(Map( +function(x) c(Min=min(x), Max=max(x)), +Filter(is.numeric, iris_sample) +)) +## +Sepal.Length Sepal.Width Petal.Length Petal.Width +## Min +5.2 +2.6 +1.5 +0.2 +## Max +6.5 +3.5 +6.0 +2.5 +or: +8 Due to this, storing data as matrix columns inside data frames is not such a bad idea. + +282 +II DEEPER +sapply(iris_sample[sapply(iris_sample, is.numeric)], quantile, c(0, 1)) +## +Sepal.Length Sepal.Width Petal.Length Petal.Width +## 0% +5.2 +2.6 +1.5 +0.2 +## 100% +6.5 +3.5 +6.0 +2.5 +Note that the latter called simplify2array automatically, thus the result is a matrix. +On the other hand, standardisation of all the numeric features can be performed, e.g., via a call: +iris_sample[] <- Map(function(x) { +if (!is.numeric(x)) x else (x-mean(x))/sd(x) +}, iris_sample) +print(iris_sample) +## +Sepal.Length Sepal.Width Petal.Length Petal.Width +Species +## 28 +-1.70405 +1.03024 +-1.76004 +-1.65318 +setosa +## 80 +-0.72094 +-1.49854 +-0.60591 +-0.78117 versicolor +## 101 +0.45878 +0.46829 +0.83674 +0.85384 +virginica +## 111 +0.85202 +0.18732 +0.31738 +0.30884 +virginica +## 137 +0.45878 +0.74927 +0.60591 +0.74484 +virginica +## 133 +0.65540 +-0.93659 +0.60591 +0.52684 +virginica +12.3.5 +Handling Missing Values +The is.na method for objects of class data.frame returns a logical matrix of the same +dimensionality9 indicating whether the corresponding items are missing or not. Of +course, this function can still be called on individual columns as well. +Further, na.omit can be used to get rid of rows with missing values. +Exercise 12.17 Given a data frame, use is.na and other functions such as apply, approx, etc., +to: +1. remove all rows that feature at least one missing value, +2. remove all rows that only consist of missing values, +3. remove all columns that feature at least one missing value, +4. for each column, replace all missing values with the column averages, +5. for each column, replace all missing values with values that linearly interpolate between the +precedingandsucceedingwell-definedobservations(whichisusefulontimeseries),e.g.,the +blanks in c(0.60, 0.62, NA, 0.64, NA, NA, 0.58) should be filled so as to obtain +c(0.60, 0.62, 0.63, 0.64, 0.62, 0.60, 0.58). +12.3.6 +Reshaping Data Frames +Consider an example matrix: +9 Provided that a data frame does not feature a matrix column. + +12 DATA FRAMES +283 +A <- matrix(round(runif(6), 2), nrow=3, +dimnames=list( +c("X", "Y", "Z"), +# row labels +c("u", "v") +# column labels +)) +names(dimnames(A)) <- c("Row", "Col") +print(A) +## +Col +## Row +u +v +## +X 0.29 0.88 +## +Y 0.79 0.94 +## +Z 0.41 0.05 +The as.data.frame method for the table class can be called directly on any array: +as.data.frame.table(A, responseName="Val") +## +Row Col +Val +## 1 +X +u 0.29 +## 2 +Y +u 0.79 +## 3 +Z +u 0.41 +## 4 +X +v 0.88 +## 5 +Y +v 0.94 +## 6 +Z +v 0.05 +This is an instance of reshaping an array, and more precisely, stacking: converting from +a wide (okay, in this example, not so wide, as we have only two columns) to a long +format. +This can be also achieved by means of the reshape function which is more flexible and +operates directly on data frames (but is harder to use): +(df <- `names<-`( +data.frame(row.names(A), A, row.names=NULL), +c("Row", "Col.u", "Col.v"))) +## +Row Col.u Col.v +## 1 +X +0.29 +0.88 +## 2 +Y +0.79 +0.94 +## 3 +Z +0.41 +0.05 +(stacked <- reshape(df, varying=2:3, direction="long")) +## +Row time +Col id +## 1.u +X +u 0.29 +1 +## 2.u +Y +u 0.79 +2 +## 3.u +Z +u 0.41 +3 +## 1.v +X +v 0.88 +1 +## 2.v +Y +v 0.94 +2 +## 3.v +Z +v 0.05 +3 + +284 +II DEEPER +Maybe the default column names are not superb, but we can always adjust them +manually afterwards. +The reverse operation is called unstacking: +reshape(stacked, idvar="Row", timevar="time", drop="id", direction="wide") +## +Row Col.u Col.v +## 1.u +X +0.29 +0.88 +## 2.u +Y +0.79 +0.94 +## 3.u +Z +0.41 +0.05 +Exercise 12.18 Given a named numeric vector, convert it to a data frame with two columns, for +instance: +covert <- function(x) ...to.do... +x <- c(spam=42, eggs=7, bacon=3) +convert(x) +## +key value +## 1 +spam +42 +## 2 +eggs +7 +## 3 bacon +3 +Exercise 12.19 Reshape (stack) the built-in WorldPhones dataset. Then, reshape (unstack) the +stacked WorldPhones dataset. Further, unstack the stacked set but first remove10 five random +rows from it, and then randomly permute all the remaining rows. Fill the missing entries with +NAs. +Exercise 12.20 Implement a basic version of as.data.frame.table manually (using rep +etc.). Also, write a function as.table.data.frame that implements its reverse. Make sure both +functions are compatible with each other. +Exercise 12.21 The built-in Titanic is a four-dimensional array. Convert it to a long data +frame. +Exercise 12.22 Perform what follows on the data frame defined below: +1. convert the second column from character to a list of character vectors (split at ","); +2. extract first elements from each of the vectors; +3. extract last elements; +4. (*) unstack the data frame; +5. (*) stack it back to a data frame featuring a list; +6. convert the list back to a character column (concatenate with "," as separator). +10 The original dataset can be thought of as representing a fully crossed design experiment (all combina- +tions of two grouping variables are present). Its truncated version is like an incomplete cross design. + +12 DATA FRAMES +285 +(x <- data.frame( +name=c("Kat", "Ron", "Jo", "Mary"), +food=c("buckwheat", "spam,bacon,spam", "", "eggs,spam,spam,lollipops") +)) +## +name +food +## 1 +Kat +buckwheat +## 2 +Ron +spam,bacon,spam +## 3 +Jo +## 4 Mary eggs,spam,spam,lollipops +Exercise 12.23 Write a function that converts all matrix-based columns in a given data frame +to separate, atomic columns. Also, write a function to that does the opposite: one that groups all +columns with similar prefixes and turns them into matrices. +12.3.7 +Aggregating Data in Groups +We can straightforwardly apply various transforms on data groups determined by a +factor-like variable or a combination thereof thanks to the split.data.frame method, +which returns a list of data frames. +For example: +x <- data.frame( +a=c( +10, +20, +30, +40, +50), +u=c("spam", "spam", "eggs", "spam", "eggs"), +v=c( +1, +2, +1, +1, +1) +) +split(x, x["u"]) +# i.e., split.data.frame(x, x["u"]) or x[["u"]] +## $eggs +## +a +u v +## 3 30 eggs 1 +## 5 50 eggs 1 +## +## $spam +## +a +u v +## 1 10 spam 1 +## 2 20 spam 2 +## 4 40 spam 1 +This split x with respect to the u column serving as the grouping variable. On the other +hand: +split(x, x[c("u", "v")]) +# sep="." +## $eggs.1 +## +a +u v +## 3 30 eggs 1 +(continues on next page) + +286 +II DEEPER +(continued from previous page) +## 5 50 eggs 1 +## +## $spam.1 +## +a +u v +## 1 10 spam 1 +## 4 40 spam 1 +## +## $eggs.2 +## [1] a u v +## <0 rows> (or 0-length row.names) +## +## $spam.2 +## +a +u v +## 2 20 spam 2 +partitioned with respect to a combination of two factor-like sequences. Note that a +non-existing level pair (eggs, 2) results in an empty data frame. +Exercise 12.24 split.data.frame (when called explicitly) can also be used to break a matrix +into a list of matrices (rowwisely). Given a matrix, perform its train-test split: allocate, say, 70% +of the rows at random into one matrix and the remaining 30% into another one. +If the aggregation of grouped data in numeric columns is needed, sapply is quite con- +venient. To recall, it is a combination of lapply (one-vector version of Map) and sim- +plify2array (Section 11.1.3). +sapply(split(iris[1:2], iris[5]), sapply, mean) +## +setosa versicolor virginica +## Sepal.Length +5.006 +5.936 +6.588 +## Sepal.Width +3.428 +2.770 +2.974 +If the function being to apply returns more than a single value, sapply will not return +a too-informative result by default: the list of matrices converted to a matrix will not +have the row.names argument set. As a workaround, we either call simplify2array ex- +plicitly or pass simplify="array" to sapply: +(res <- sapply( +split(iris[1:2], iris[5]), +sapply, +function(x) c(Min=min(x), Max=max(x)), +simplify="array" +)) +# or simplify2array(lapply or Map etc.) +## , , setosa +## +## +Sepal.Length Sepal.Width +## Min +4.3 +2.3 +(continues on next page) + +12 DATA FRAMES +287 +(continued from previous page) +## Max +5.8 +4.4 +## +## , , versicolor +## +## +Sepal.Length Sepal.Width +## Min +4.9 +2.0 +## Max +7.0 +3.4 +## +## , , virginica +## +## +Sepal.Length Sepal.Width +## Min +4.9 +2.2 +## Max +7.9 +3.8 +This yields a three-dimensional array which is particularly handy if we now would like +to access specific results by name: +res[, "Sepal.Length", "setosa"] +## Min Max +## 4.3 5.8 +Also, the previously mentioned as.data.frame.table method works like a charm on it +(up to the column names): +as.data.frame.table(res) +## +Var1 +Var2 +Var3 Freq +## 1 +Min Sepal.Length +setosa +4.3 +## 2 +Max Sepal.Length +setosa +5.8 +## 3 +Min +Sepal.Width +setosa +2.3 +## 4 +Max +Sepal.Width +setosa +4.4 +## 5 +Min Sepal.Length versicolor +4.9 +## 6 +Max Sepal.Length versicolor +7.0 +## 7 +Min +Sepal.Width versicolor +2.0 +## 8 +Max +Sepal.Width versicolor +3.4 +## 9 +Min Sepal.Length +virginica +4.9 +## 10 +Max Sepal.Length +virginica +7.9 +## 11 +Min +Sepal.Width +virginica +2.2 +## 12 +Max +Sepal.Width +virginica +3.8 +Note If the grouping (by) variable is a list of two or more factors, the combined levels +will be concatenated to a single string: +as.data.frame.table(as.array(sapply( +split(ToothGrowth["len"], ToothGrowth[c("supp", "dose")]), +(continues on next page) + +288 +II DEEPER +(continued from previous page) +sapply, +mean +))) +## +Var1 +Freq +## 1 OJ.0.5.len 13.23 +## 2 VC.0.5.len +7.98 +## 3 +OJ.1.len 22.70 +## 4 +VC.1.len 16.77 +## 5 +OJ.2.len 26.06 +## 6 +VC.2.len 26.14 +Also,thenameoftheaggregatedcolumn(len)hasbeenincluded.Thisbehaviouryields +a result that may be deemed convenient in some contexts, but not necessarily so in +other ones. +Exercise 12.25 Many aggregation functions are idempotent, which means that when they are +fed with a vector with all the elements being identical, the result is exactly that unique element: +min, mean, median, and max behave exactly this way. +Overload the mean and median methods for character vectors and factors so that they return NA +when they are fed with a sequence of not all elements being the same and the unique value other- +wise. +mean.character <- function(x, na.rm=FALSE, ...) ...to.do... +mean.factor <- function(x, na.rm=FALSE, ...) ...to.do... +This way, we can also aggregate the grouping variables in a convenient way: +do.call(rbind.data.frame, +lapply(split(ToothGrowth, ToothGrowth[c("supp", "dose")]), lapply, mean)) +## +len supp dose +## OJ.0.5 13.23 +OJ +0.5 +## VC.0.5 +7.98 +VC +0.5 +## OJ.1 +22.70 +OJ +1.0 +## VC.1 +16.77 +VC +1.0 +## OJ.2 +26.06 +OJ +2.0 +## VC.2 +26.14 +VC +2.0 +The built-in aggregate method can assist us in a situation where a single function is to +be applied on all columns in a data frame. +aggregate(iris[-5], iris[5], mean) +# not: ...[[5]] +## +Species Sepal.Length Sepal.Width Petal.Length Petal.Width +## 1 +setosa +5.006 +3.428 +1.462 +0.246 +(continues on next page) + +12 DATA FRAMES +289 +(continued from previous page) +## 2 versicolor +5.936 +2.770 +4.260 +1.326 +## 3 +virginica +6.588 +2.974 +5.552 +2.026 +aggregate(ToothGrowth["len"], ToothGrowth[c("supp", "dose")], mean) +## +supp dose +len +## 1 +OJ +0.5 13.23 +## 2 +VC +0.5 +7.98 +## 3 +OJ +1.0 22.70 +## 4 +VC +1.0 16.77 +## 5 +OJ +2.0 26.06 +## 6 +VC +2.0 26.14 +Note that the second argument, by, must be list-like (therefore also a data frame is +accepted), not a factor nor an atomic vector. Also, if the function being applied returns +many values, they will be wrapped into a matrix column: +(x <- aggregate(iris[2], iris[5], function(x) c(Min=min(x), Max=max(x)))) +## +Species Sepal.Width.Min Sepal.Width.Max +## 1 +setosa +2.3 +4.4 +## 2 versicolor +2.0 +3.4 +## 3 +virginica +2.2 +3.8 +class(x[["Sepal.Width"]]) +## [1] "matrix" "array" +x[["Sepal.Width"]] +# not: Sepal.Width.Max, etc. +## +Min Max +## [1,] 2.3 4.4 +## [2,] 2.0 3.4 +## [3,] 2.2 3.8 +It is actually handy, because by referring to x[["Sepal.Width"]] we have access to all +the stats for this column. Further, if many columns are being aggregated at the same +time, we can process all the summaries in the same way. +Exercise 12.26 Check out the built-in by function which supports some basic split-apply-bind +use cases. Note the particularly peculiar behaviour of the print method for the by class. +The most flexible scenario involves applying a custom function returning any set of +aggregatesintheformofalistandthenrow-bindingtheresultstoobtainadataframe. +Example 12.27 The following implements an R version of what we would express in SQL as: +SELECT supp, dose, AVG(len) AS ave_len, COUNT(*) AS count +FROM ToothGrowth +GROUP BY supp, dose +Ad rem: + +290 +II DEEPER +do.call(rbind.data.frame, lapply( +split(ToothGrowth, ToothGrowth[c("supp", "dose")]), +function(df) list( +supp=df[1, "supp"], +dose=df[1, "dose"], +ave_len=mean(df[["len"]]), +count=NROW(df) +) +)) +## +supp dose ave_len count +## OJ.0.5 +OJ +0.5 +13.23 +10 +## VC.0.5 +VC +0.5 +7.98 +10 +## OJ.1 +OJ +1.0 +22.70 +10 +## VC.1 +VC +1.0 +16.77 +10 +## OJ.2 +OJ +2.0 +26.06 +10 +## VC.2 +VC +2.0 +26.14 +10 +Example 12.28 As an exercise, let us study a function that takes a named list x (can be a data +frame) and a sequence of col=f pairs and applies the function f (or each function from a list of +functions f) on the named element col in x: +napply <- function(x, ...) +{ +fs <- list(...) +stopifnot(is.list(x), !is.null(names(x))) +stopifnot(all(names(fs) %in% names(x))) +do.call( +c, +# concatenates lists +lapply( +structure(seq_along(fs), names=names(fs)), +function(i) { +# always returns a list +y <- x[[ names(fs)[i] ]] +if (is.function(fs[[i]])) +list(fs[[i]](y)) +else +lapply(fs[[i]], function(f) f(y)) +} +) +) +} +For example: +first <- function(x, ...) head(x, n=1L, ...) +# we use it below +napply(ToothGrowth, +(continues on next page) + +12 DATA FRAMES +291 +(continued from previous page) +supp=first, dose=first, len=list(ave=mean, count=length) +) +## $supp +## [1] VC +## Levels: OJ VC +## +## $dose +## [1] 0.5 +## +## $len.ave +## [1] 18.813 +## +## $len.count +## [1] 60 +applies first on both ToothGrowth[["supp"]] and ToothGrowth[["dose"]] as well as mean +and length on ToothGrowth[["len"]]. List names are there for some dramatic effects. +And now: +do.call( +rbind.data.frame, +lapply( +split(ToothGrowth, ToothGrowth[c("supp", "dose")]), +napply, +supp=first, dose=first, len=list(ave=mean, count=length) +) +) +## +supp dose len.ave len.count +## OJ.0.5 +OJ +0.5 +13.23 +10 +## VC.0.5 +VC +0.5 +7.98 +10 +## OJ.1 +OJ +1.0 +22.70 +10 +## VC.1 +VC +1.0 +16.77 +10 +## OJ.2 +OJ +2.0 +26.06 +10 +## VC.2 +VC +2.0 +26.14 +10 +or even: +aaaggg <- function(x, by, ...) +do.call(rbind.data.frame, lapply(split(x, x[by]), napply, ...)) +so that: +aaaggg(iris, "Species", Species=first, Sepal.Length=mean) +## +Species Sepal.Length +(continues on next page) + +292 +II DEEPER +(continued from previous page) +## setosa +setosa +5.006 +## versicolor versicolor +5.936 +## virginica +virginica +6.588 +This brings fun back to R programming in the sad times when many things are given to us on a +plate. And by the way, the above has not been tested thoroughly, it is a proof of concept; as usual, +testing, debugging, and extending is left as an exercise to the reader. +Example 12.29 In Section 10.5, we have considered an example where we have used our own +group_by function and an aggregation method overloaded for the object’s class it returns. +Hereisthefunctionthatsplitsadataframeintoalistofdataframeswithrespecttoacombination +of levels in given named columns: +group_by <- function(df, by) +{ +stopifnot(is.character(by), is.data.frame(df)) +df <- droplevels(df) +# in case there are factors with empty levels +structure( +split(df, df[names(df) %in% by]), +class="list_dfs", +by=by +) +} +The next function applies a set of aggregates on every column of each data frame in a given list +(two nested lapplys plus some cosmetic additions): +aggregate.list_dfs <- function(x, FUN, ...) +{ +aggregates <- lapply(x, function(df) { +is_by <- names(df) %in% attr(x, "by") +res <- lapply(df[!is_by], FUN, ...) +res_mat <- do.call(rbind, res) +if (is.null(dimnames(res_mat)[[2]])) +dimnames(res_mat)[[2]] <- paste0("f", seq_len(NCOL(res_mat))) +cbind( +`row.names<-`(df[1, is_by, drop=FALSE], NULL), +x=row.names(res_mat), +`row.names<-`(res_mat, NULL) +) +}) +combined_aggregates <- do.call(rbind.data.frame, aggregates) +`row.names<-`(combined_aggregates, NULL) +} +aggregate(group_by(ToothGrowth, c("supp", "dose")), range) +(continues on next page) + +12 DATA FRAMES +293 +(continued from previous page) +## +supp dose +x +f1 +f2 +## 1 +OJ +0.5 len +8.2 21.5 +## 2 +VC +0.5 len +4.2 11.5 +## 3 +OJ +1.0 len 14.5 27.3 +## 4 +VC +1.0 len 13.6 22.5 +## 5 +OJ +2.0 len 22.4 30.9 +## 6 +VC +2.0 len 18.5 33.9 +We really want our API be bloated, hence let us introduce a convenience function being a spe- +cialised version of the above: +mean.list_dfs <- function(x, ...) +aggregate.list_dfs(x, function(y) c(Mean=mean(y, ...))) +mean(group_by(iris[51:150, c(2, 3, 5)], "Species")) +## +Species +x +Mean +## 1 versicolor +Sepal.Width 2.770 +## 2 versicolor Petal.Length 4.260 +## 3 +virginica +Sepal.Width 2.974 +## 4 +virginica Petal.Length 5.552 +12.3.8 +Transforming Data in Groups +Somevariableswillsometimesneedtobetransformedrelativetowhatishappeningin +subsets of a dataset. This is the case, e.g., where we decide that missing values should +be replaced by the corresponding within-group averages, or want to compute the rel- +ative ranks or z-scores. +If the losing of the original ordering of rows is not an issue, the standard split-apply- +bind will suffice. +An example data frame: +(x <- data.frame( +a=c( 10, +1, +NA, +NA, +NA, +4), +b=c( -1, +10, +40, +30, +1, +20), +c=runif(6), +d=c("v", "u", "u", "u", "v", "u") +)) +## +a +b +c d +## 1 10 -1 0.52811 v +## 2 +1 10 0.89242 u +## 3 NA 40 0.55144 u +## 4 NA 30 0.45661 u +## 5 NA +1 0.95683 v +## 6 +4 20 0.45333 u + +294 +II DEEPER +Some operations: +fill_na <- function(x) `[<-`(x, is.na(x), value=mean(x[!is.na(x)])) +standardise <- function(x) (x-mean(x))/sd(x) +And now: +do.call(rbind.data.frame, lapply( +split(x, x["d"]), +function(df) { +df[["a"]] <- fill_na(df[["a"]]) +df[["b"]] <- rank(df[["b"]]) +df[["c"]] <- standardise(df[["c"]]) +df +} +)) +## +a b +c d +## u.2 +1.0 1 +1.46357 u +## u.3 +2.5 4 -0.17823 u +## u.4 +2.5 3 -0.63478 u +## u.6 +4.0 2 -0.65057 u +## v.1 10.0 1 -0.70711 v +## v.5 10.0 2 +0.70711 v +Note that only the relative ordering of rows within groups has been retained. Overall, +the rows are in a different order. +If this is an issue, we can use the unsplit function: +unsplit( +lapply( +split(x, x["d"]), +function(df) { +df[["a"]] <- fill_na(df[["a"]]) +df[["b"]] <- rank(df[["b"]]) +df[["c"]] <- standardise(df[["c"]]) +df +} +), +x["d"] +) +## +a b +c d +## 1 10.0 1 -0.70711 v +## 2 +1.0 1 +1.46357 u +## 3 +2.5 4 -0.17823 u +## 4 +2.5 3 -0.63478 u +(continues on next page) + +12 DATA FRAMES +295 +(continued from previous page) +## 5 10.0 2 +0.70711 v +## 6 +4.0 2 -0.65057 u +Exercise 12.30 Show how we can do the above also via the replacement version of split. +Example 12.31 Reverting to the previous ordering can be done manually too. It is because the +split operation behaves as if we first ordered the data frame with respect to the grouping vari- +able(s) (using a stable sorting algorithm). +Here is some transformation of a sample data frame split by a combination of two factors: +(x <- `row.names<-`(ToothGrowth[sample(NROW(ToothGrowth), 10), ], NULL)) +## +len supp dose +## 1 +23.0 +OJ +2.0 +## 2 +23.3 +OJ +1.0 +## 3 +29.4 +OJ +2.0 +## 4 +14.5 +OJ +1.0 +## 5 +11.2 +VC +0.5 +## 6 +20.0 +OJ +1.0 +## 7 +24.5 +OJ +2.0 +## 8 +10.0 +OJ +0.5 +## 9 +9.4 +OJ +0.5 +## 10 +7.0 +VC +0.5 +(y <- do.call(rbind.data.frame, lapply( +split(x, x[c("dose", "supp")]), +# two grouping variables +function(df) { +df[["len"]] <- df[["len"]] * 100^df[["dose"]] * +# whatever +ifelse(df[["supp"]] == "OJ", -1, 1) +# do not overthink it +df +} +))) +## +len supp dose +## 0.5.OJ.8 +-100 +OJ +0.5 +## 0.5.OJ.9 +-94 +OJ +0.5 +## 1.OJ.2 +-2330 +OJ +1.0 +## 1.OJ.4 +-1450 +OJ +1.0 +## 1.OJ.6 +-2000 +OJ +1.0 +## 2.OJ.1 +-230000 +OJ +2.0 +## 2.OJ.3 +-294000 +OJ +2.0 +## 2.OJ.7 +-245000 +OJ +2.0 +## 0.5.VC.5 +112 +VC +0.5 +## 0.5.VC.10 +70 +VC +0.5 +In Section 5.4.4, we have mentioned that by calling order, we ca determine the inverse of a given +permutation. Hence, we can call: + +296 +II DEEPER +y[order(order(x[["supp"]], x[["dose"]])), ] +# not: dose, supp +## +len supp dose +## 2.OJ.1 +-230000 +OJ +2.0 +## 1.OJ.2 +-2330 +OJ +1.0 +## 2.OJ.3 +-294000 +OJ +2.0 +## 1.OJ.4 +-1450 +OJ +1.0 +## 0.5.VC.5 +112 +VC +0.5 +## 1.OJ.6 +-2000 +OJ +1.0 +## 2.OJ.7 +-245000 +OJ +2.0 +## 0.5.OJ.8 +-100 +OJ +0.5 +## 0.5.OJ.9 +-94 +OJ +0.5 +## 0.5.VC.10 +70 +VC +0.5 +Additionally, we can manually restore the original row.names, et voilà. +12.3.9 +Metaprogramming-Based Techniques (*) +In Section 9.5.7, we have mentioned that due to R’s being equipped with the ability +to write programs that manipulate unevaluated R expressions, some functions can +provide us with quite weird interfaces to a few common operations. These include +transform, subset, with, and basically every procedure accepting a formula. Also, the +popular data.table and dplyr packages that we briefly mention in Section 12.3.10 fall +into this class. +In some contexts, they all may be found convenient11. +However, overall, each of these methods must be carefully studied separately. This +is because they can arbitrarily interpret the form of the arguments passed thereto, +without taking into account their real meaning. +We try to avoid12 them in this course, as we can do perfectly without them. However, +they are not only interesting on their own, but also quite popular in other users’ code, +hence the honourable mention. Learning them in more detail is left to the kind reader +as an optional exercise. In Chapter 18, we will return to these functions as they will +serve as a very interesting illustration of how to implement our own procedures that +rely on metaprogramming techniques. +Example 12.32 For instance, let us consider an example call to the subset function: +11 And, in the case of third-party packages, sometimes faster and more memory efficient (on larger data- +sets), as it is usually the case with more specialised tools. However, in many daily programming contexts, +speed of the data wrangling operations is not that often an issue. Remember that we always have SQL- +supporting relational databases at our disposal too. +12 We are not alone in our calling to refrain from using them. help("subset") warns (and +help("transform") quite similarly): This is a convenience function intended for use interactively. For programming, +it is better to use the standard subsetting functions like `[`, and in particular the non-standard evaluation of argument +subset can have unanticipated consequences. The same in help("with"): For interactive use, this is very effective +and nice to read. For programming however, i.e., in one’s functions, more care is needed, and typically one should refrain +from using with, as, e.g., variables in data may accidentally override local variables. + +12 DATA FRAMES +297 +subset(iris, Sepal.Length>7.5, -(Sepal.Width:Petal.Width)) +## +Sepal.Length +Species +## 106 +7.6 virginica +## 118 +7.7 virginica +## 119 +7.7 virginica +## 123 +7.7 virginica +## 132 +7.9 virginica +## 136 +7.7 virginica +NeitherSepal.Length>7.5nor -(Sepal.Width:Petal.Width)makesenseasstandaloneRex- +pressions, because we have not defined the named variables used therein: +Sepal.Length>7.5 +# utter nonsense +## Error in eval(expr, envir, enclos): object 'Sepal.Length' not found +-(Sepal.Width:Petal.Width) +# gibberish +## Error in eval(expr, envir, enclos): object 'Sepal.Width' not found +Only from help("subset"), we can learn that this tool generously decides that the second ex- +pressionplaystheroleofarowselectorandthethirdoneremovesallthecolumnsbetweenthetwo +given ones. +In our course, we pay attention to developing transferable skills. Assuming that R is not the only +language we are going to learn during of our long and happy lives, it is much more likely that in +the next environment, we will rather be writing something more of the more basic form: +between <- function(x, from, to) (which(from == x):which(to == x)) +iris[iris[["Sepal.Length"]]>7.5, +-between(names(iris), "Sepal.Width", "Petal.Width")] +## +Sepal.Length +Species +## 106 +7.6 virginica +## 118 +7.7 virginica +## 119 +7.7 virginica +## 123 +7.7 virginica +## 132 +7.9 virginica +## 136 +7.7 virginica +Let us stress again that this is a book on how to become a great chef who proudly uses produce +from sustainable sources, and not how to order ultra-processed food from DeliverNoodles.com. +Example 12.33 transform can be used to add, modify, and remove columns in a data frame +with the possibility of referring to existing features as if they were ordinary variables: +head(transform(mtcars, log_hp=log(hp), am=2*am-1, hp=NULL)) +## +mpg cyl disp drat +wt +qsec vs am gear carb log_hp +## Mazda RX4 +21.0 +6 +160 3.90 2.620 16.46 +0 +1 +4 +4 4.7005 +## Mazda RX4 Wag +21.0 +6 +160 3.90 2.875 17.02 +0 +1 +4 +4 4.7005 +## Datsun 710 +22.8 +4 +108 3.85 2.320 18.61 +1 +1 +4 +1 4.5326 +(continues on next page) + +298 +II DEEPER +(continued from previous page) +## Hornet 4 Drive +21.4 +6 +258 3.08 3.215 19.44 +1 -1 +3 +1 4.7005 +## Hornet Sportabout 18.7 +8 +360 3.15 3.440 17.02 +0 -1 +3 +2 5.1648 +## Valiant +18.1 +6 +225 2.76 3.460 20.22 +1 -1 +3 +1 4.6540 +Similarly,attachaddsanynamedlisttothesearchpath(seeChapter16)sothatthecolumnscan +be accessed by name. This, however, does not allow any alterations thereof to be performed. As an +alternative, with and within may be referred to if writing df[["..."]] each time is so difficult +to us (it should not be): +within(head(mtcars), { +log_hp <- log(hp) +fuel_economy <- 235/mpg +am <- factor(am, levels=c(0, 1), labels=c("no", "yes")) +rm(list=c("mpg", "hp", "vs", "qsec")) +}) +## +cyl disp drat +wt +am gear carb fuel_economy log_hp +## Mazda RX4 +6 +160 3.90 2.620 yes +4 +4 +11.190 4.7005 +## Mazda RX4 Wag +6 +160 3.90 2.875 yes +4 +4 +11.190 4.7005 +## Datsun 710 +4 +108 3.85 2.320 yes +4 +1 +10.307 4.5326 +## Hornet 4 Drive +6 +258 3.08 3.215 +no +3 +1 +10.981 4.7005 +## Hornet Sportabout +8 +360 3.15 3.440 +no +3 +2 +12.567 5.1648 +## Valiant +6 +225 2.76 3.460 +no +3 +1 +12.983 4.6540 +Example 12.34 As mentioned in Section 10.3.2, see Section 15.2 for more details, formulae are +special objects that consist of two unevaluated expressions separated by a tilde (`~`). +Functionscansupportformulaeanddowhattheypleasewiththem,butapopularapproachisto +allow them to express “something grouped by something else” or “one thing as a function of other +things”. +do.call(rbind.data.frame, lapply(split(ToothGrowth, ~supp+dose), head, 1)) +## +len supp dose +## OJ.0.5 15.2 +OJ +0.5 +## VC.0.5 +4.2 +VC +0.5 +## OJ.1 +19.7 +OJ +1.0 +## VC.1 +16.5 +VC +1.0 +## OJ.2 +25.5 +OJ +2.0 +## VC.2 +23.6 +VC +2.0 +aggregate(cbind(mpg, log_hp=log(hp))~am:cyl, mtcars, mean) +## +am cyl +mpg log_hp +## 1 +0 +4 22.900 4.4186 +## 2 +1 +4 28.075 4.3709 +## 3 +0 +6 19.125 4.7447 +## 4 +1 +6 20.567 4.8552 +## 5 +0 +8 15.050 5.2553 +(continues on next page) + +12 DATA FRAMES +299 +(continued from previous page) +## 6 +1 +8 15.400 5.6950 +head(model.frame(mpg+hp~log(hp)+I(1/qsec), mtcars)) +## +mpg + hp log(hp) +I(1/qsec) +## Mazda RX4 +131.0 +4.7005 0.060753.... +## Mazda RX4 Wag +131.0 +4.7005 0.058754.... +## Datsun 710 +115.8 +4.5326 0.053734.... +## Hornet 4 Drive +131.4 +4.7005 0.051440.... +## Hornet Sportabout +193.7 +5.1648 0.058754.... +## Valiant +123.1 +4.6540 0.049455.... +If these seem esoteric, it is because that is exactly the case. We need to consult the corresponding +functions’ manuals to be able to understand what they do. And, as we do not recommend their +use, we are not going to explain them here. +Exercise 12.35 In the last example, the peculiar printing of the last column is due to which +method being overloaded? +12.3.10 +A Note on the dplyr (tidyverse) and data.table Packages (*) +The popular third-party packages data.table and dplyr implement the most common +data frame wrangling procedures. Moreover, some of the operations may be much +faster for larger data sets. +TheybothintroduceacompletelynewAPIfortheoperationswealreadyknowwellhow +to perform. Furthermore, they are heavily based on metaprogramming (nonstandard +evaluation). A good way to learn them is by solving some of the exercises listed below. +Note that dplyr is part of a huge system of interdependent packages called tidyverse +which tend to do things their own way and which became quite invasive over the last +years. Importantly, of course, R programmers should remember that they are able to +do without them – and they need to when processing other important data structures +is needed, e.g., fancy lists and matrices. Base R always comes first as the more funda- +mental layer. +Important Some functions we may find useful will (annoyingly to base R users) re- +turn objects of class tibble (tbl_df) (e.g., haven::read.xpt that reads SAS data files). +However, those are in fact data.frame subclasses and we can always use as.data.frame +to get our favourite objects back. +Also, we cannot stress enough that it is SQL that we recommend to learn as perhaps +the most powerful interface to more considerable amounts of data, and also one that +gives skills which can be used at a later time in other programming environments. +We should remember that base R has already proven long time ago to be a versatile +tool for rapid prototyping, calling specialised procedures written in C or Java, and +wrangling data that fit into memory. For larger problems, techniques for working with + +300 +II DEEPER +batches of data, sampling methods, or aggregating data stored elsewhere is often the +way to go, especially when building machine learning models or visualisation13 is re- +quired. Usually, the most recent data will be stored in normalised databases and you +will need to join a few tables in order to fetch something of interest in the current +analysis context. +12.4 +Exercises +Exercise 12.36 Answer the following questions: +• What attributes a data frame must be equipped with? +• If row.names is an integer vector, how to access rows labelled 1, 7, and 42? +• Howtocreateadataframethatfeaturesacolumnthatisalistofcharactervectorsofdifferent +lengths? +• How to create a data frame that includes a matrix column? +• How to convert all numeric columns in a data frame to a numeric matrix? +• Assuming that x is an atomic vector, what is the difference between “as.data.frame(x)” vs +“as.data.frame(as.list(x))”vs“as.data.frame(list(a=x))”vs“data.frame(a=x)”? +Exercise 12.37 Assuming that x is a data frame, what is the meaning of/difference between the +following: +• “x["u"]” vs “x[["u"]]” vs “x[, "u"]”? +• “x["u"][1]” vs “x[["u"]][1]” vs “x[1, "u"]” vs “x[1, "u", drop=FALSE]”? +• “x[which(x[[1]] > 0), ]” vs “x[x[[1]] > 0, ]”? +• “x[grep("^foo", names(x))]”? +Exercise 12.38 Assume we have a data frame with columns named like: ID (character), +checked (logical, possibly with missing values), category (factor), x0, … x9 (ten separate nu- +meric columns), y0, … y9 (ten separate numeric columns), coords (numeric matrix with two +columns named lat and long), and features (list of character vectors of different lengths). +• How to extract the rows where checked is TRUE? +• How to extract a subset comprised only of ID and x-something columns? +• How to extract the rows for which ID is like 3 letters and then 5 digits (e.g., XYZ12345)? +• How to select all the numeric columns in one go? +13 For example, drawing a scatter plot of one billion points barely makes sense and may result in unread- +able images of large file sizes. They need to be sampled or summarised (e.g., binned) somehow first. + +12 DATA FRAMES +301 +• Assuming that the IDsarelikethreelettersand then fivedigits,howtoadd twocolumns: ID3 +(the letters) and ID5 (the five digits). +• How to get rid of all the columns between x3 and y7? +• How to check where both lat and long in coords are positive? +• How to add the row indicating the number of features? +• How to extract the rows where "spam" is amongst the features? +• How to convert it to a long data frame with two columns: ID and feature (individual +strings)? +• How to change the name of the ID column to id? +• How to make the y-foo columns appear before the x-bar ones? +• How to order the rows with respect to checked (FALSE first, then TRUE) and IDs (decreas- +ingly)? +• How to remove rows with duplicate IDs? +• How to determine how many entries correspond to each category? +• How to compute the average lat and long in each category? +• How to compute the average lat and long for each category and checked combined? +Exercise 12.39 Consider the flights14 dataset. Give some ways to select all rows between +March and October (regardless of the year). +Exercise 12.40 In this task, you will be working with a version of a dataset on 70k+ Melbourne +trees (urban_forest15). Before proceeding any further, read the dataset’s description available +here16. +1. Load the downloaded dataset by calling the read.csv function. +2. FetchtheIDs(CoM.ID)andtrunkdiameters(Diameter.Breast.Height)offivehorsechest- +nutswiththesmallestdiametersatbreastheight.Theoutputdataframemustbesortedwith +respect to Diameter.Breast.Height, decreasingly. +3. Create a new data frame that gives the number of trees planted in each year. +4. Computetheaverageage(inyears,basedon Year.Planted;using aggregate)ofthetreesof +genera (each genus separately): Eucalyptus, Platanus, Ficus, Acer, and Quercus. Depict the +sorted data with barplot. +Exercise 12.41 (*) Consider the historic data dumps of https://travel.stackexchange.com/ +availableathttps://github.com/gagolews/teaching-data/tree/master/travel_stackexchange_com_ +2017. +14 https://github.com/gagolews/teaching-data/blob/master/other/flights.csv +15 https://github.com/gagolews/teaching-data/raw/master/marek/urban_forest.csv.gz +16 https://data.melbourne.vic.gov.au/Environment/Trees-with-species-and-dimensions-Urban-Forest-/ +fp38-wiyy + +302 +II DEEPER +Export the CSV files located therein to an SQLite database. Then, write some R code that corres- +pond to the following SQL queries (use dbGetQuery to verify your results): +--- 1) +SELECT +Users.DisplayName, +Users.Age, +Users.Location, +SUM(Posts.FavoriteCount) AS FavoriteTotal, +Posts.Title AS MostFavoriteQuestion, +MAX(Posts.FavoriteCount) AS MostFavoriteQuestionLikes +FROM Posts +JOIN Users ON Users.Id=Posts.OwnerUserId +WHERE Posts.PostTypeId=1 +GROUP BY OwnerUserId +ORDER BY FavoriteTotal DESC +LIMIT 10 +--- 2) +SELECT +Posts.ID, +Posts.Title, +Posts2.PositiveAnswerCount +FROM Posts +JOIN ( +SELECT +Posts.ParentID, +COUNT(*) AS PositiveAnswerCount +FROM Posts +WHERE Posts.PostTypeID=2 AND Posts.Score>0 +GROUP BY Posts.ParentID +) AS Posts2 +ON Posts.ID=Posts2.ParentID +ORDER BY Posts2.PositiveAnswerCount DESC +LIMIT 10 +--- 3) +SELECT +Posts.Title, +UpVotesPerYear.Year, +MAX(UpVotesPerYear.Count) AS Count +FROM ( +SELECT +PostId, +COUNT(*) AS Count, +STRFTIME('%Y', Votes.CreationDate) AS Year +FROM Votes +WHERE VoteTypeId=2 +(continues on next page) + +12 DATA FRAMES +303 +(continued from previous page) +GROUP BY PostId, Year +) AS UpVotesPerYear +JOIN Posts ON Posts.Id=UpVotesPerYear.PostId +WHERE Posts.PostTypeId=1 +GROUP BY Year +--- 4) +SELECT +Questions.Id, +Questions.Title, +BestAnswers.MaxScore, +Posts.Score AS AcceptedScore, +BestAnswers.MaxScore-Posts.Score AS Difference +FROM ( +SELECT Id, ParentId, MAX(Score) AS MaxScore +FROM Posts +WHERE PostTypeId==2 +GROUP BY ParentId +) AS BestAnswers +JOIN ( +SELECT * FROM Posts +WHERE PostTypeId==1 +) AS Questions +ON Questions.Id=BestAnswers.ParentId +JOIN Posts ON Questions.AcceptedAnswerId=Posts.Id +WHERE Difference>50 +ORDER BY Difference DESC +--- 5) +SELECT +Posts.Title, +CmtTotScr.CommentsTotalScore +FROM ( +SELECT +PostID, +UserID, +SUM(Score) AS CommentsTotalScore +FROM Comments +GROUP BY PostID, UserID +) AS CmtTotScr +JOIN Posts ON Posts.ID=CmtTotScr.PostID +AND Posts.OwnerUserId=CmtTotScr.UserID +WHERE Posts.PostTypeId=1 +ORDER BY CmtTotScr.CommentsTotalScore DESC +LIMIT 10 +--- 6) +(continues on next page) + +304 +II DEEPER +(continued from previous page) +SELECT DISTINCT +Users.Id, +Users.DisplayName, +Users.Reputation, +Users.Age, +Users.Location +FROM ( +SELECT +Name, UserID +FROM Badges +WHERE Name IN ( +SELECT +Name +FROM Badges +WHERE Class=1 +GROUP BY Name +HAVING COUNT(*) BETWEEN 2 AND 10 +) +AND Class=1 +) AS ValuableBadges +JOIN Users ON ValuableBadges.UserId=Users.Id +--- 7) +SELECT +Posts.Title, +VotesByAge2.OldVotes +FROM Posts +JOIN ( +SELECT +PostId, +MAX(CASE WHEN VoteDate = 'new' THEN Total ELSE 0 END) NewVotes, +MAX(CASE WHEN VoteDate = 'old' THEN Total ELSE 0 END) OldVotes, +SUM(Total) AS Votes +FROM ( +SELECT +PostId, +CASE STRFTIME('%Y', CreationDate) +WHEN '2017' THEN 'new' +WHEN '2016' THEN 'new' +ELSE 'old' +END VoteDate, +COUNT(*) AS Total +FROM Votes +WHERE VoteTypeId=2 +GROUP BY PostId, VoteDate +(continues on next page) + +12 DATA FRAMES +305 +(continued from previous page) +) AS VotesByAge +GROUP BY VotesByAge.PostId +HAVING NewVotes=0 +) AS VotesByAge2 ON VotesByAge2.PostId=Posts.ID +WHERE Posts.PostTypeId=1 +ORDER BY VotesByAge2.OldVotes DESC +LIMIT 10 +Exercise 12.42 (*)GenerateaCSVfilefeaturingsomerandomdataarrangedinafewcolumns +of the size at least two times larger than your available RAM. Then, export the CSV file to an +SQLite database. Use file connections (Section 8.3.5) and the nrow argument to read.table to +be able to process it on a chunk-by-chunk basis. +Determine whether setting colClasses in read.table speeds up the reading of large CSV files +significantly or not. +Exercise 12.43 (*) Export the whole XML data dump of StackOverflow17 published at https: +//archive.org/details/stackexchange (see also https://data.stackexchange.com/) to an SQLite +database. +17 https://stackoverflow.com + + +13 +￿ Graphics +The R Project homepage1 advertises our free software as an environment for statistical +computing and graphics. Hence, our course would not be complete if we have not dealt +with the latter use case. +R is equipped with two independent systems for graphics generation. +1. The (historically) newer one, grid, is quite complicated. Some readers might have +comeacrossthe latticeand ggplot2packagesbefore:theyarebuiltontopof grid. +2. On the other hand, its traditional (S-style) counterpart, graphics, is much easier +to master. Still, it gives their users full control over the drawing process. Its being +both simple, fast, and low-level makes it very attractive from the perspective of +this course’s philosophy. +This is why, in this chapter, we will only cover the second one. Note that all figures +in this book were generated using graphics and its dependants. They are sufficiently +aesthetic, aren’t they? +￿ This chapter is under construction. Please come back later. +13.1 +￿ Placeholders for Plots Referred to Elsewhere +￿ Plotting and factors; see Figure 13.1. +plot(iris[["Sepal.Length"]], +# x (it is a vector) +iris[["Petal.Width"]], +# y (it is a vector) +col=as.numeric(iris[["Species"]]), +# colours +pch=as.numeric(iris[["Species"]]) +) +1 https://www.r-project.org/ + +308 +II DEEPER +4.5 +5.0 +5.5 +6.0 +6.5 +7.0 +7.5 +8.0 +0.5 +1.0 +1.5 +2.0 +2.5 +iris[["Sepal.Length"]] +iris[["Petal.Width"]] +Figure 13.1: as.numeric on factors can be used to create different plotting styles + +Part III +Deepest + + +14 +￿ Interfacing Compiled Code +R is a nice glue language: it is perfect for implementing data wrangling pipelines, visu- +alisation, and developing prototypes of data analysis algorithms. In other words, it +makes connecting larger buildingblocks very easy. Still, the more computing-intensive +tasks should be done at the C/C++/Fortran level. +￿ This chapter is under construction. Please come back later. +14.1 +￿ R/C API +14.2 +￿ External Pointers +14.3 +￿ RCpp + + +15 +￿ Expressions +￿ This chapter is under construction. Please come back later. +15.1 +￿ The Dollar Operator, `$` +15.2 +￿ Formulae, `~` + + +16 +￿ Environments +￿ This chapter is under construction. Please come back later. +16.1 +￿ Classification of R Data Types (Revisited) +Recall our first approximation to the classification of R Data Types that we presented +in the Preface. +16.2 +￿ Copying on Demand +16.3 +￿ A Note on Reference Classes (*) + +316 +III DEEPEST +RDataTypes +Basic +Atomic +NULL +logical +raw +numeric +integer +double +complex +character +Recursive +list +pairlist +function +closure +primitive: special/builtin +environment +LanguageObjects +symbol (name) +call +expression +Internal +promise +externalptr +S4 +... +Compound +factor +matrix +array +data.frame +formula +Date +kmeans +... +Figure 16.1: R data types + +17 +￿ Evaluating Expressions +￿ This chapter is under construction. Please come back later. + + +18 +￿ Evaluating Functions +￿ This chapter is under construction. Please come back later. +18.1 +￿ Evaluation of Default Arguments +18.2 +￿ Ellipsis Revisited +18.3 +￿ S3 Method Lookup by UseMethod +18.4 +￿ Overloading S3 Group Generics +18.5 +￿ Package Namespaces + + +Part IV +Appendix + + +A +Changelog +Important This book is still a work in progress. The first 12 chapters are already quite +readable, but there will be more. Stay tuned. +Any bug/typos reports/fixes1 are appreciated. +Below is the list of the most noteworthy changes. +• 2022-12-29 (v0.1.12): +– First public release at https://deepr.gagolewski.com. +– Beta (complete) versions of Chapters 1–12 (basic and compound types, func- +tions, etc.) published. +– Preface drafted (alpha version). +– ISBN 978-0-6455719-2-9 reserved. +– Cover. +1 https://github.com/gagolews/deepr/issues + + +References +[1] +Abelson, H., Sussman, G.J., Sussman, J. (1996). StructureandInterpretationofCom- +puter Programs. MIT Press. +[2] +Abramowitz, M., Stegun, I.A. (1972). Handbook of Mathematical Functions with For- +mulas, Graphs, and Mathematical Tables. Dover. URL: https://people.math.sfu.ca/ +~cbm/aands/. +[3] +Becker, R.A., Chambers, J.M., Wilks, A.R. (1988). The New S Language. Chapman +& Hall. +[4] +Chambers,J.M.(1998).ProgrammingwithData.AGuidetotheSLanguage.Springer- +Verlag. +[5] +Chambers, J.M. (2008). Software for Data Analysis. Programming with R. Springer. +[6] +Chambers, J.M. (2016). Extending R. Chapman & Hall. +[7] +Chambers, J.M. (2020). S, R, and data science. The R Journal, 12(1):462–476. DOI: +10.32614/RJ-2020-028. +[8] +Chambers, J.M., Hastie, T.J. (1991). Statistical Models in S. Chapman & Hall. +[9] +Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C. (2009). Introduction to Al- +gorithms. MIT Press and McGraw-Hill. +[10] Crawley, M.J. (2007). The R Book. John Wiley & Sons. +[11] Date, C.J. (2003). An Introduction to Database Systems. Pearson. +[12] Davis, M., Whistler, K. (2021). Unicode standard annex #15: Unicode normaliza- +tion forms. URL: http://www.unicode.org/reports/tr15/. +[13] Davis, M., Whistler, K., Scherer, M. (2021). Unicode technical standard #10: Uni- +code collation algorithm. URL: http://www.unicode.org/reports/tr10/. +[14] Deisenroth,M.P.,Faisal,A.A.,Ong,C.S.(2020).MathematicsforMachineLearning. +Cambridge University Press. URL: https://mml-book.com/. +[15] DeMichiel, L.G., Gabriel, R.P. (1987). The Common Lisp Object System: An over- +view. ECOOP. URL: https://www.dreamsongs.com/Files/ECOOP.pdf. +[16] Devroye, L. (1986). Non-UniformRandomVariateGeneration. Springer-Verlag. URL: +http://luc.devroye.org/rnbookindex.html. +[17] Forbes, C., Evans, M., Hastings, N., Peacock, B. (2010). Statistical Distributions. +Wiley. + +326 +REFERENCES +[18] Friedl, J.E.F. (2006). Mastering Regular Expressions. O'Reilly. +[19] Gagolewski, M. (2016). Programowanie w języku R. 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URL: http: +//www.R-project.org. + diff --git a/uNAzT4oBgHgl3EQfPvtT/content/tmp_files/load_file.txt b/uNAzT4oBgHgl3EQfPvtT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..25932627485904e7b926280c00614ba34116d063 --- /dev/null +++ b/uNAzT4oBgHgl3EQfPvtT/content/tmp_files/load_file.txt @@ -0,0 +1,14877 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf,len=14876 +page_content='DeepR Programming Marek Gagolewski v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='12 (draft) Dr habil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Marek Gagolewski Deakin University, Australia 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names mentioned herein may be the trademarks of their respective owners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Rather than use a trademark symbol with every occurrence of a trademarked name, the names are used in an editorial fashion to the benefit of the trademark owner, with no intention of infringement of the trademark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Weird is the world we live in, but the following had to be written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Everyefforthasbeenmadeinthepreparationofthisbooktoensuretheaccuracyofthe information presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, the information contained in this book is provided withoutwarranty,eitherexpressorimplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Theauthorwillofcoursenotbeheldliable for any damages caused or alleged to be caused directly or indirectly by this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Anyway, any bug reports/corrections/feature requests are welcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' To make this text- book even better, please file them at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews/deepr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Typeset with XeLATEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Please be understanding: it was an algorithmic process, hence the results are ∈ [good enough, perfect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Homepage: https://deepr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/ Datasets: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews/teaching-data Release: v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='12 (draft) (2022-12-29T10:59:45+1100) ISBN: 978-0-6455719-2-9 (reserved) (vX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Melbourne: Marek Gagolewski) Contents Preface xi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 To R, or not to R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' xi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 R as a Language and an Environment .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' xi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Aims, Scope, and Design Philosophy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' xii 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 \uffff Classification of R Data Types and Book Structure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' xiv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 About the Author .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 26 IV CONTENTS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Special Functions (*) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 44 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Operator Precedence Revisited .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Aggregating with all, any, and sum .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 46 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Simplifying Predicates .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 58 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 NULL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Object Attributes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Developing Perceptual Indifference to Most Attributes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 But There Are Some Use Cases .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 61 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Special Attributes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 95 6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 98 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Formatting Objects .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 100 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Partial Matching .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 168 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Assuring Quality Code .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 169 Managing Changes and Working Collaboratively .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 169 CONTENTS VII Test-driven Development and Continuous Integration .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 170 Debugging .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 170 Profiling .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 171 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Special Functions: Syntactic Sugar .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 172 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Operators are Functions Too .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 174 Creating Own Replacement Functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 174 Substituting Parts of Vectors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 176 Compositions of Replacement Functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 221 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Simplifying Lists .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 222 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Beyond Numeric Arrays .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 264 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Interfacing Relational Databases and Querying with SQL (*) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 265 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Strings as Factors?' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 266 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Internal Representation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 268 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Data Frame Subsetting .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 270 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Data Frames are Lists .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 273 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Common Operations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': 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Interfacing Compiled Code 311 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 \uffff R/C API .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 319 IV Appendix 321 A Changelog 323 References 325 X CONTENTS DeepRProgramming is a comprehensive course on one of the most popular languages in data science (statistical computing, graphics, machine learning, data wrangling and analytics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It introduces the base language in-depth and is aimed at ambitious students,practitioners,and researcherswho wouldliketobecome independentusers of this powerful environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This early draft is distributed in the hope that it will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For many students around the world, educational resources are hardly affordable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, I have decided that this book should remain an independent, non-profit, open-access project (available both in PDF1 and HTML2 forms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Whilst, for some people, the presence of a “designer tag” from a major publisher might still be a proxy forquality,itismyhopethatthispublicationwillproveusefultothosewhoseekknow- ledge for knowledge’s sake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Please spread the news about it by sharing the above URLs with your mates, peers, or students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thank you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, check out my other book, Minimalist Data Wrangling with Python3 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Anybug/typosreports/fixes4 areappreciated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Althoughavailableonline,thisisawhole course, and should be read from the beginning to the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Please refer to the Preface for general introductory remarks and design philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Consider citing this book as: Gagolewski M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (2023), DeepRProgramming, Zenodo, Mel- bourne, ISBN: 978-0-6455719-2-9, URL: https://deepr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1 https://deepr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/deepr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='pdf 2 https://deepr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/ 3 https://datawranglingpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/ 4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews/deepr/issues 0 Preface 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 To R, or not to R R [50] has been named the eleventh most dreaded programming language in the 2022 StackOverflow Developer Survey5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, it is a free app, so there must be something wrong with it, right?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' But whatever, R is deprecated anyway;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' the “modern” way is to use tidyverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Or we should all just switch to Python6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Well, not really7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 R as a Language and an Environment Let us get one thing straight: R is not just a statistical package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is a general-purpose, high-level programming language, that just happens to be very powerful for any kind of numerical, data-intense computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It offers extensive support for statistical, ma- chine learning, data analysis, data wrangling, and data visualisation applications, but there is a lot more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Initially, R was written “for statisticians by statisticians”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, it may be thought ofasafreeyetmorecapablealternativetoStata,SAS,SPSS,Statistica,Minitab,Weka, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Unlike some of them, however, a spreadsheet-like GUI is not the main gateway for performing computations on data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In R, a user must write code to get things actually done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Despite the learning curve’s being a little steeper for non-programmers, in the long run, it empowers their users because they are not limited only to the most com- mon scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If some functionality is missing or does not suit their needs, they can easily implement everything themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Itisthusveryconvenientforrapidprototyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Ithelpsturnourideasintooperational code that can be tested, extended, polished, run in production, and otherwise enjoyed overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' As an interpreted language, it can be run not only in an interactive read-eval- 5 https://survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='stackoverflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='co/2022/ 6 https://datawranglingpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/ 7 Or, as Aussies would say, yeah, nah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' XII PREFACE print loop (command–result, question–answer, …), but also in batch mode (running whole, standalone scripts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thus, we would rather position R amongst such tools/languages for numerical or sci- entific computing as Python with the NumPy ecosystem, Julia, GNU Octave, Scilab, MATLAB, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, it is more specialised in data science applications than all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, it provides a much smoother experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is why, over the years, R has become the de facto standard in statistics and many other related fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important R is a whole ecosystem (environment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It not only consists of the R lan- guage interpreter, but also features advanced: graphical capabilities (see Chapter 13), a help system (Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4), ways for convenient interfacing with compiled code (Chapter 14), apackagesystemandcentralisedpackagerepositories(suchasCRANandBiocon- ductor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1), a lively community of users and developers – curious and passionate people, just like you and me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note R’s predecessor is the popular S system designed in the 1980s by John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Cham- bersandhiscolleaguesatBellLabsS:[3,4,8,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='RiscalledGNUS,afree,open-source version of its commercial counterpart developed in the mid-1990s8by Robert Gentle- man and Ross Ihaka of the Statistics Department, University of Auckland, and a large number of contributors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see [7, 31] for some historical notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' R has a C language-like syntax that involves the use of {curly braces}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Still, in principle, it is a beautiful, functional programming language: its design was heavily inspired by Scheme (see [1] and Chapter 17 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is also somewhat object-oriented (Chapter 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Aims, Scope, and Design Philosophy Many users have been introduced to R by means of some very advanced operations involving data frames, formulas, and functions that rely on nonstandard evaluation (metaprogramming), like: 8 R version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='49 released in April 1997 (the first for which source code?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' is available on CRAN), was already quitefeature-rich(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=',implementedS3methods,formulae,anddataframesintroducedinthe1991version of S [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8 https://cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='r-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/src/base/R-0/ PREFACE XIII lm( Ozone~Solar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R+Temp, data=subset(airquality, Temp>60, select=-(Month:Day)) ) |> summary() This is horrible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Another group has been isolated from the base R through a thick layer of third-party packagesthatfeatureanoverwhelmingnumberoffunctions(everyoperation,regard- less of its complexity, has a different name), often duplicating the core functionality, and sometimes being quite incompatible with our traditional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Both families should be fine — as long as they limit themselves to solving only the simplest and most common data processing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' But we yearn for more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We do not want hundreds of prefabricated recipes for popular dishes that we can mindlessly apply without much understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Our aim is to learn baseR, which is supposedtobe the common language (lingua franca) to all R users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We want to be able to write code that everybody should be able to under- stand, and which will be likely to work without modifications ten years from now (no slang!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We want to be able to tackle any data-intense problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Furthermore, we want to de- velop skills that are transferable, so that learning new tools such as Julia or Python with NumPy and Pandas will be much easier later (because R is not the only notable envir- onment out there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Anyway, enough preaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This graduate9-level textbook is for independent readers who do not mind a slightly steeper learning curve at the beginning, but who are able to appreciate a more cohesively and comprehensively10 organised material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some will benefit from it as a first introduction to R (but without all the pampering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For others11, this will be a good course from intermediate to advanced (do not skip the first chapters, though).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Either way, do not forget to solve all the prescribed exercises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Good luck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9 The author taught similar courses for his wonderfully ambitious undergraduate data/computer sci- ence and maths students at Warsaw University of Technology, where such an approach has proven not dif- ficult at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It requires a more independent, curious, and motivated mindset, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' And that’s the way to go, in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10 Yours truly is neither a historian, a stenographer, nor a grammarian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We allow ourselves to make a few noninvasive idealisations for didactic purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Languages evolve over time, R now is different than it used to be, and we can shape it (slowly, because we value its stable API) to become something even better in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 11 It might also happen that for some, this will not be a good course at all, either at this stage of their career (come back later) or in general (no dramas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is a non-profit, open-access project, but it does not mean it is ideal for everyone – in such a case, give other sources a try, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', [5, 10, 35, 41, 42, 43, 49], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some of them are also freely available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' XIV PREFACE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 \uffff Classification of R Data Types and Book Structure RDataTypes Basic Atomic NULL logical numeric character list function .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Compound factor matrix array data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='frame formula Date kmeans .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Figure 1: An overview of the most prevalent R data types (see Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 for a more comprehensive list) The most commonly used R data types can be classified as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see also Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Basic types – which we discuss in the first part of this book – internal or built-in types, upon which more complex ones are hinged: atomic vectors – represent whole sequences of values, where every element is of the same type: – logical (Chapter 3) – includes items that are TRUE (“yes”, “present”), FALSE (“no”, “absent”), or NA (“not available”, “missing”);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – numeric(Chapter2)–featuresrealnumbers,suchas1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='14,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000001, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – character (Chapter 6) – contains strings of characters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', "groß", "123", or “Добрий день”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' function (Chapter 7) – used to group a series of expressions (code lines) so that they can be applied on different kinds of input data to generate the (hopefully) desired outcomes, for instance, cat, print, plot, sample, and sum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' list (Chapter 4) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' a generic vector – can store elements of mixed types;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The above will be complemented with a discussion on vector indexing (Chapter 5) and ways to control the program flow (Chapter 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' PREFACE XV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Compound types – discussed in the second part – wrappers around objects of basic types that might behave differently from the underlying primitives thanks to the dedicated operations overloaded for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They are factor (Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3) – a vector-like object that represents qualitative data (on a nominal or an ordered scale);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' matrix (Chapter 11) – stores tabular data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', arranged into rows and columns, where each cell is usually of the same type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='frame (Chapter 12) – also used for depositing tabular data, but this time such that each column can be of different type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' and many more, which we or third-parties can define arbitrarily using, amongst others, the principles of S3-style object orientated-programming (Chapter 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In this part of the book, we also discuss the principles of sustainable coding (Chapter 9) as well as introduce the basic ways to prepare publication-quality graphics (Chapter 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' \uffff Some more advanced material that, in most cases, we can easily do without, but which is still essential to gain a full understanding of and control over the environ- ment, is discussed in the first part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This includes, amongst others, the following data types: externalptr (sec:to-do);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' environment (sec:to-do);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' symbol (name), call, expression (sec:to-do);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' formula (Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2) – used by some functions to specify supervised learn- ing models or define operations to be performed within data subgroups, amongst others;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' \uffff Also, we will discuss other things, but this is an early draft of this book, so right now, we only provide a placeholder therefor (sec:to-do).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Please come back later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note The above classification is just a first approximation to the complete type clas- sification that we will discuss in Section 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see also Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, we should not be surprised that above we do not see any of the data types defined by a few very popular12 third-party packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We will later see that we can most often do without them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If that is not the case, we will become skilled enough to learn them easily ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 12 Which does not automatically mean good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, sugar, salt, and some drugs are very popular, but it does not make them healthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' XVI PREFACE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 About the Author I, Marek Gagolewski13 (pronounced like Ma’rek Gong-olive-ski), am currently a Senior Lecturer in Applied AI at Deakin University in Melbourne, VIC, Australia and an Associate Professor in Data Science at the Systems Research Institute of the Polish Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' My research interests are related to data science, in particular: modelling complex phenomena, developing usable, general-purpose algorithms, studying their analyt- ical properties, and finding out how people use, misuse, understand, and misunder- stand methods of data analysis in research, commercial, and decision-making set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' I’m an author of 90+ publications, including journal papers in outlets such as Proceedings of the National Academy of Sciences (PNAS), Information Fusion, International Journal of Forecasting, Statistical Modelling, Journal of Statistical Software, Information Sci- ences, Knowledge-Based Systems, IEEE Transactions on Fuzzy Systems, and Journal of Infor- metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In my “spare” time, I write books for my students (also check out my Minimalist Data Wrangling with Python14 [20]) and develop open-source (libre) data analysis software, such as stringi15 (one of the most often downloaded R packages), genieclust16 (a fast and robust clustering algorithm in both Python and R), and many others17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Acknowledgements DeepRProgramming is based on my experience as an author of a quite successful Polish textbook ProgramowaniewjęzykuR (see [19]) which was published by PWN (1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2014, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The current book is an entirely different work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, its predecessor served as an excellent testbed for many ideas conveyed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Theteachingstyleexercisedinthisbookhasprovensuccessfulinmanysimilarcourses that yours truly has been responsible for, including at Warsaw University of Techno- logy, Data Science Retreat (Berlin), and Deakin University (Melbourne).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' I thank all my students and colleagues for the feedback given over the last 10+ years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We describe R version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Patched (2022-11-10 r83330).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, we expect 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9% of material covered here to be valid in future releases (consider filing a bug report if you discover that this is not the case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 13 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com 14 https://datawranglingpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/ 15 https://stringi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com 16 https://genieclust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com 17 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews PREFACE XVII This book was prepared in a Markdown superset called MyST18, Sphinx19, and TeX (XeLaTeX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Code chunks were processed with the R package knitr [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' All fig- ures were plotted with the low-level graphics package using the author’s own style template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' A little help from Makefiles, custom shell scripts, and Sphinx plugins (sphinxcontrib-bibtex20, sphinxcontrib-proof21) dotted the j’s and crossed the f ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The Ubuntu Mono22 font is used for the display of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Typesetting of the main text relies upon the Alegreya23 and Lato24 typefaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This book received no funding, administrative, technical, or editorial support from Deakin University, Warsaw University of Technology, Polish Academy of Sciences, or any other source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 18 https://myst-parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='io/en/latest/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='html 19 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='sphinx-doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/ 20 https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/project/sphinxcontrib-bibtex/ 21 https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/project/sphinxcontrib-proof/ 22 https://design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='ubuntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/font/ 23 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='huertatipografica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/en 24 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='latofonts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/ Part I Deep 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Hello, World!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Traditionally, every programming journey starts with the printing of a “Hello, World”- like greeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let’s then get it over with asap: cat("My hovercraft is full of eels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='") ## My hovercraft is full of eels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' By calling the cat function, we printed out a given character string that we enclosed in double quote characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Documenting code is a good development practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is thus worth knowing that any text followed by a hash sign (that is not part of a string) is a comment, ignored by the interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' # This is a comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' # This is another comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' cat("I cannot wait", "till lunchtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='") # two arguments (another comment) ## I cannot wait till lunchtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' cat("# I will not buy this record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='\\n# It is scratched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='") # `\\n` == newline ## # I will not buy this record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ## # It is scratched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' By convention, in this book, the textual outputs generated by R itself are always pre- ceded by two hashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This makes copy-pasting all code chunks easier in the case where the kind reader would like to experiment with them by themself (which is always highly encouraged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Whenever a call to some function is to be made, the round brackets are oblig- atory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' All objects within the parentheses (they are separated by commas) con- stitute the input data to be consumed by the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thus, the syntax is: some_function_to_be_called(argument1, argument2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 4 I DEEP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Setting up the Development Environment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Installing R Itisquitenaturaltopinefortheabilitytoexecutetheabovecodeourselves–wecannot learn programming without getting our hands dirty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The official precompiled binary distributions of R can be downloaded from https:// cran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='r-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For serious programming work1, we recommend, sooner rather than later, switching to2 one of the Unix-like operating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This includes the free, open-source (== good) variants of GNU/Linux, amongst others, or the proprietary (== very far from good) m**OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The users thereof might employ their favourite package manager (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', apt, dnf, pacman, or Homebrew) to install R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Users of other operating systems (such as Wi***ws) might consider installing Anaconda or Miniconda if they require some level of interoperability with the Py- thon environment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', they would like to work with the Jupyter environment (Sec- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Below we review several ways in which we can write and execute R code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is up to the benign reader to research, setup, and learn the development environment that suits their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' As usual in real life, there is no single universal approach that always works best in all the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Interactive Mode R’s read-eval-print loop (REPL) can give us instant gratification whenever we would like to compute something quickly, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', determine basic aggregates of a few numbers entered by hand or evaluate a mathematical expression like “2+2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' How to start the R console varies from system to system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', users of Unix-like boxes can simply execute R from the terminal (shell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Wi***ws folks can fire up the RGui from the Start menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important When working interactively, the default3 command prompt, “>”, means: I am awaiting an order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, “+” denotes: Please continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In such a case, we should either complete the unfinished expression, or cancel the operation by pressing ESC or CTRL+C (depends on the operating system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' > cat("And now (continues on next page) 1 For instance, when an easy interoperability with other programming languages/environments is re- quired or when we think about scheduling jobs on Linux-based computing/container clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 Or at least trying out – by installing a copy of GNU/Linux on a virtual machine (VM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3 It can be changed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see help("options").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1 INTRODUCTION 5 (continued from previous page) + for something + completely different + + + it is an unfinished expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' + awaiting another double quote character and then the closing bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' + + press ESC or CTRL+C to abort input > For readability, we never print out the command prompt characters in this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Batch Mode: Working with R Scripts (**) The interactive mode of operation is unsuitable for more complicated tasks, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The users of Unix-like operating systems will be interested in another extreme, which involves writing standalone R scripts that can be executed one by line, without any user intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' To do so, in the terminal (command line, shell), we can invoke: Rscript file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R where file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R is the path to some source file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 (**) In your favourite text editor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', Notepad++, Kate, vi, Emacs, RStudio, or VSCodium), create a file named test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Write a few calls to the cat function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Then, execute this script from the terminal by invoking the Rscript program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Weaving: Automatic Report Generation (**) Reproducibledataanalysis4 requiresustokeeptheresults(text,tables,plots,auxiliary files) synchronised with their generating code and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' utils::Sweave (the Sweave function from the utils package) and knitr [44] are two example template processors that evaluate R code chunks within documents written in LaTeX, HTML, or other markup languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The chunks are replaced by the outputs they yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This book is a showcase of such an approach – all the results, including Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 and the above “Hello, World”, were generated programmatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thanks to its being writ- teninthehighlyuniversalMarkdown5 language,itcouldbeeasilyconvertedtoasingle 4 The idea dates back to Knuth’s literate programming concept;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 5 https://daringfireball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='net/projects/markdown/ 6 I DEEP PDF document6 as well as the whole website7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Tools like pandoc and docutils facilitate such operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 (**) Install the knitr package by calling install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages("knitr") from within an R session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Then, create a text file named test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Rmd with the following content: # Hello, Markdown!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is my first automatically generated report, where I print stuff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ```{r} print("G\'day!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='") print(2+2) ``` Thank you for your attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Assuming that the file is located in the current working directory (compare Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3), call knitr::knit("test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Rmd") from within the R console or run the following in the terminal: Rscript -e \'knitr::knit("test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Rmd")\' Then, inspect the generated Markdown file, test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Furthermore, if you have the pandoc tool installed, to generate a standalone HTML file, execute in the terminal: pandoc test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='md --standalone -o test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='html Alternatively, for ways to call external programs from R, see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Semi-Interactive Modes (Jupyter Notebooks, Sending Code to an As- sociated R Console, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') The nature of the most frequent use cases of R encourages a semi-interactive work- flow, where we progress with prototyping fast by trial-and-error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In this mode, we write a series of short code fragments inside a standalone R script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Each fragment implements a simple, well-defined task, such as the loading of data files, data cleansing, feature visualisation, computations of some information ag- gregates, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Importantly, any code chunk can be sent to the associated R console and executed therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This way, we can inspect the results it generates at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If we are not happy with the outcome, we can apply any corrections that are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6 https://deepr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/deepr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='pdf 7 https://deepr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gagolewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com 1 INTRODUCTION 7 There are quite a few integrated development environments (IDEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' sometimes re- quiring additional plugins) that enable such a workflow, including JupyterLab, Emacs, RStudio, and VSCodium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Executing an individual code line or a whole text selection is usually done by pressing a (configurable) keyboard shortcut such as Ctrl+Enter or Shift+Enter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 (*) JupyterLab8 isadevelopmentenvironmentthatrunsinawebbrowser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Itwas programmed in Python, but supports many programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thanks to IRkernel9, we can use it with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Install JupyterLab and IRkernel (for instance, if you use Anaconda, run conda install c r r-essentials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' From the File menu, select Create a new R source file and save it as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' From the File menu, select Create a new console for editor running the R kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Type some print “Hello, World”-like calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Press Shift+Enter (whilst working in the editor) to send different code fragments onto the console and execute them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Inspect the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1: JupyterLab: A source file editor and the associated R console, where we can run arbitrary code fragments Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 (*) The Jupyter project, whose JupyterLab is part of, also supports the handling of dedicated Notebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' There, editable and executable code chunks and results they generate can 8 https://jupyterlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='io/en/stable/ 9 https://irkernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='io/ File Edit View Kernel Settings Help Run Tabs C 三 test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R ×+ OPEN TABS Close All test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R 1 # # 2 test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R 3 #Press shift+Enter to executecurrent lineor selection 4 # in the associated console below KERNELS Shut Down All 5 test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R plot(rnorm(1000), rnorm(1000), main="G\'day!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='") 5 > TERMINALS Shut Down All test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R X [1]: plot(rnorm(1000)， rnorm(1000)， main="G\'day!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='") G\'day!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3 O C 2 08 000 1000) O norm(1 O 098 88 :[]8 I DEEP be kept together in a single .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='ipynb (JSON) file;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 for an illustration and Chapter 1 of [20] for a quick introduction (from the Python language kernel perspective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This environment is quite convenient for live coding (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', for teachers) or performing explorat- ory data analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, for more serious programming work, the code can get quite messy (luckily, there is always an option to export a notebook to an executable, plain text R script).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2: An example Jupyter Notebook, where we can keep the code and the results together 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Atomic Vectors at a Glance After the printing of the “Hello, World” message, a typical programming course would normally proceed with the discussion on basic data types for storing individual nu- meric or logical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Next, we would be introduced to arithmetic and comparison operations on such scalars, followed by the definition of whole arrays or other collec- tions of such values, complemented by the methods to iterate over them, one element after another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In R, no separate types representing individual values have been defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Instead, what seems to be a single datum, is already a vector (sequence, array) of length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='71828 # input a number (here: the same as print(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='71828)) ## [1] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7183 length(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='71828) # it is a vector featuring one element ## [1] 1 Jupyter Welcome (unsaved changes) R Logout File Edit View Insert Cell Kernel Widgets Help Trusted RO Example Jupyter Notebook In [1l:plot(rnorm(1000), rnorm(1000), main="G\'day!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='") G\'day!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 00 8 8 00 2 8 80 00 morm(1000) O 8 00 QQ Q 2 0% 8 8 O ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='- 4 4 2 0 2 morm(1000)1 INTRODUCTION 9 To create a vector of any length, we can call the c function, which combines given ar- guments into a single sequence: c(1, 2, 3) # three vectors of length 1 > one vector of length 3 ## [1] 1 2 3 length(c(1, 2, 3)) ## [1] 3 In Chapter 2, Chapter 3, and Chapter 6, we will discuss the most prevalent types of atomic vectors: numeric, logical, and character ones, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' c(0, 1, -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='14159, 12345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6) # four numbers ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1416 12345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6000 c(TRUE, FALSE) # two logical values ## [1] TRUE FALSE c("spam", "spam", "bacon and spam") # three character strings ## [1] "spam" "spam" "bacon and spam" We call them atomic, because they can only group together values of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Lists, which we will discuss in Chapter 4, are, on the other hand, referred to as generic vectors – they can be used for storing items of mixed types – other lists as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note Not having separate scalar types greatly simplifies the programming of numer- ical computing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Vectors are prevalent in our main areas of interest – statistics, simulations, data science, machine learning, and all other data-oriented computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example, columns and rows in tables (values of different features describing cli- ents,ratingsofitemsgivenbyusers)ortimeseries(stockmarketprices,readingsfrom temperature sensors) are all best represented by means of such sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, the fact that vectors are the core part of the R language makes their use very natural – as opposed to the languages that require special add-ons for vector processing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', numpy for Python [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' By learning different ways to process them as a whole, instead of one element at a time, we will assure that our ideas can quickly be turnedintoworkingcode(rapidprototyping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Forinstance,computingsummarystat- isticssuchas,say,themeanabsolutedeviationofsomesequence x,willbeaseffortless as writing mean(abs(x-mean(x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Such a code is not only easy to read and maintain, but it is also fast to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Getting Help Our aim is to become independent, advanced R programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Independent, however, does not mean omniscient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The R help system is the authorit- 10 I DEEP ative source of knowledge about specific functions or more general topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' To open a help page, we call: help("topic") # equivalently: ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "topic" Exercise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Sight (without going into detail) the manual on the length function by calling help("length").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that most help pages are structured as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Header: “package:base” means that the function is a base one (see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 for more details on the R package system);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Title;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Description: a short description of what the function does;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Usage: the list of formal arguments (parameters) to the function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Arguments: the meaning of each formal argument explained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Details: technical information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Value: return value explained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' References: further reading;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' See Also: links to other help pages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Examples: R code that is worth to run and study by yourself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We can also search within all the installed help pages by calling: help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='search("vague topic") # equivalently: ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "vague topic" Oftentimes, this way we will be able to find answers to our questions more reliably than when asking DuckDuckGo or G**gle (which commonly feature many low qual- ity/irrelevant/distracting results that can make us lose the sacred code writer’s flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important All code chunks, including code comments and textual outputs, form an integral part of this book’s text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They should not be skipped by the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' On the con- trary, they should become objects of our intense reflection and thorough investiga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, whenever we introduce a few function, it may be a good idea to look it up in the help system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, playing with the presented code (running, modify- ing, experimenting, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') is also very beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We should develop the habit of asking ourselves questions like “what would happen if…”, and then finding the answers on our own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Wearenowreadytodiscussthemostimportantoperationsonnumericvectors,which constitute the main theme of the next chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' See you there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1 INTRODUCTION 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Exercises Exercise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 What are the three most important types of atomic vectors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 According to the classification of the R data types we introduced in the previous chapter, are atomic vectors basic or compound types?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 Numeric Vectors In this chapter, we discuss the uttermost common operations on numeric vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They are so fundamental that we will also find them in other scientific computing en- vironments, including Python with NumPy or TensorFlow, Julia, MATLAB, GNU Octave, or Scilab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' At first blush, the number of functions we are going to explore may seem quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Still, the reader is kindly asked to place some trust (a rare thing these days) in yours truly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is because our selection is comprised only of the most representative and edu- cational amongst the plethora of possible choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' More complex building blocks can either be reduced to a creative combination of the former or be easily found – should the need arise – in a number additional packages or libraries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', the GNU GSL [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' A solid understanding of base R programming is necessary for the effective dealing with the popular packages (such as data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='table, dplyr, or caret).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Most importantly, base R’s API is stable, hence the code we write today will most likely work the same way in 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is often not the case when we rely on third-party add-ons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In the sequel, we will be advocating a minimalistic, keep-it-simple approach to the art of programming of data processing pipelines, one that is a good balance between “do- ing it all by oneself”, “minimising the information overload”, “being lazy”, and “stand- ing on the shoulders of giants”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note The exercises that we suggest below are all self-contained, unless explicitly stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The use of language constructs that are yet to be formally intro- duced (in particular, if, for, and while which we will explain in Chapter 8) is not only unnecessary, but discouraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, we recommend against taking shortcuts by looking up partial solutions on the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Rather, to get the most out of this course, the reader should be seeking relevant information within the current and preceding chapters as well as the R help system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Creating Numeric Vectors 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Numeric Constants The simplest numeric vectors are those of length one: 14 I DEEP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='14 ## [1] -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='23e-4 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='000123 The latter is in what we call the scientificnotation which is convenient means of entering numbers of very large or small order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Here, “e” stands for “… times 10 to the power of…”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='23e-4 is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='23×10−4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='000123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In other words, given 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='23, we move the decimal separator by 4 digits towards the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In real life, some information items may be inherently or temporarily missing, un- known, or Not Available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' R is data processing-oriented, hence it is equipped with a special indicator: NA_real_ # numeric NA (missing value) ## [1] NA This is similar to the Null marker in database query languages such as SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that NA_real_ is displayed simply as “NA”, chiefly for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, Inf denotes the infinity (∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' a value that is larger than the largest represent- abledoubleprecision–64bit–floatingpointnumber)and NaNstandsfornot-a-number (it is returned as the result of some illegal operations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', 0/0 or ∞ − ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Concatenating Vectors with c Letusprovidesomewaystocreatenumericvectorswithpossiblymorethan1element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' First, the c function we introduced in the previous chapter, can be used to combine (concatenate) many numeric vectors, each of any length, so as to form a single object: c(1, 2, 3) # 3 vectors of length 1 > 1 vector of length 3 ## [1] 1 2 3 c(1, c(2, NA_real_, 4), 5, c(6, c(7, Inf))) ## [1] 1 2 NA 4 5 6 7 Inf Note Running help("c"), we will see that its usage is like “c(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In the current context, this means that the c function takes an arbitrary number of arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 we will study the dot-dot-dot (ellipsis) parameter in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Repeating Entries with rep Second, rep replicates the elements in a given vector a given number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 NUMERIC VECTORS 15 rep(1, 5) ## [1] 1 1 1 1 1 rep(c(1, 2, 3), 4) ## [1] 1 2 3 1 2 3 1 2 3 1 2 3 In the second case, the whole vector (1, 2, 3) has been recycled (tiled) four times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Inter- estingly, if the second argument was a vector of the same length as the first one, the behaviour would be quite different: rep(c(1, 2, 3), c(2, 1, 4)) ## [1] 1 1 2 3 3 3 3 rep(c(1, 2, 3), c(4, 4, 4)) ## [1] 1 1 1 1 2 2 2 2 3 3 3 3 Here, each element is repeated the corresponding number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If we call help("rep"), we will come across the notion like “rep(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=')” in the Usage section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Unfortunately, it is rather peculiar, but reading further we discover the dot- dot-dot stands for one of the following further parameters (see the Arguments section): times, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out, each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' So far, we have been playing with times, which is listed second in the parameter list (after x – the vector whose elements are to be repeated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important It turns out that the following function calls are all equivalent: rep(c(1, 2, 3), 4) # positional matching of arguments: `x`, then `times` rep(c(1, 2, 3), times=4) # `times` is the second argument rep(x=c(1, 2, 3), times=4) # keyword arguments of the form name=value rep(times=4, x=c(1, 2, 3)) # keyword arguments can be given in any order rep(times=4, c(1, 2, 3)) # mixed positional and keyword arguments Wecanalsopasseachorlength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out(adothasnospecialmeaninginR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='seeSection2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2), but their names should be mentioned explicitly: rep(c(1, 2, 3), length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=7) ## [1] 1 2 3 1 2 3 1 rep(c(1, 2, 3), each=3) ## [1] 1 1 1 2 2 2 3 3 3 rep(c(1, 2, 3), length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=7, each=3) ## [1] 1 1 1 2 2 2 3 16 I DEEP Note Whether it was a good programming practice to actually implement a range of variedbehavioursinsideasinglefunctionisamatteroftaste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Ontheonehand,inallof the examples above, we do repeat the input elements somehow, so remembering just one function name is really convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Nevertheless, a drastic change in the repeti- tion pattern depending, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', on the length of the times argument can be bug-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Anyway, we have been warned1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Zero-length vectors are also possible: rep(c(1, 2, 3), 0) ## numeric(0) Even though their handling might be a little tricky (compare Chapter 9), we will see later that they are useful in contexts like “create an empty data frame with a specific column structure”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Generating Arithmetic Progressions with seq and `:` Third, we can call the seq function to create a sequence of equally-spaced numbers (on a linear scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', an arithmetic progression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' seq(1, 15, 2) ## [1] 1 3 5 7 9 11 13 15 Reading the function’s help page, we note that it has the following parameters: from, to, by, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out, amongst others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thus, the above call is equivalent to: seq(from=1, to=15, by=2) ## [1] 1 3 5 7 9 11 13 15 Note that to actually means “up to”: seq(from=1, to=16, by=2) ## [1] 1 3 5 7 9 11 13 15 We can also pass length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out instead of by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In such a case, the increments or decre- ments will be computed via the formula ((to - from)/(length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out - 1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' this default value is reported in the Usage section in help("seq").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1 Some“caring”Rusersmightbetemptedtointroducetwonewfunctionsnow,oneforgenerating(1,2,3, 1, 2, 3, …) only and the other outputting patterns like (1, 1, 1, 2, 2, 2, …).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They would most likely wrap them in a new package and announce that on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' But this is nothing else than a multiplication of entities without actual necessity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' we would end up with three functions: the original one, rep, which everyone should know anyway because it is so basic and has been and will be used everywhere by almost everybody so far, and the two redundant ones, whose user-friendliness is only illusory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' See also Chapter 9 for discussion on the design of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 NUMERIC VECTORS 17 seq(1, 0, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=5) ## [1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 Also, this: seq(length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=5) # default `from` is 1 ## [1] 1 2 3 4 5 Arithmetic progressions with step equal to 1 or -1 can also be generated via the `:` operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1:10 # seq(1, 10) or seq(1, 10, 1) ## [1] 1 2 3 4 5 6 7 8 9 10 1:10 # seq(-1, 10) or seq(-1, 10, 1) ## [1] -1 0 1 2 3 4 5 6 7 8 9 10 1:-10 # seq(-1, -10) or seq(-1, -10, -1) ## [1] 1 2 3 4 5 6 7 8 9 -10 Note the order of precedence of this operator: “-1:10” means “(-1):10” and not “-(1:10)”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' compare Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Takealookatthemanualpageofseq_alongandseq_lenanddeterminewhether they can easily be done without, having seq2 at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Generating Pseudorandom Numbers Wecanalsogeneratesequencesdrawnindependentlyfromarangeofunivariateprob- ability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' runif(7) # uniform U(0, 1) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='287578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='788305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='408977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='883017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='940467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='045556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='528105 rnorm(7) # normal N(0, 1) ## [1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='23950 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='10897 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='11724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='18308 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='28055 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='72727 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='69018 These correspond to seven pseudorandom deviates following the uniform distribu- tion on the unit interval (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', (0, 1)) and the standard normal distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', with expectation 0 and standard deviation 1), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' compare Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For more named distribution classes (frequently occurring in various real-world stat- istical modelling exercises), see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Another useful function samples a number of values from a given vector, either with or without replacement: 2 Also note that certain configurations of seq and its variants might return vectors of type integer in- stead of double, some of them in a compact (ALTREP) form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 18 I DEEP sample(1:10, 20, replace=TRUE) # 20 with replacement (allow repetitions) ## [1] 3 3 10 2 6 5 4 6 9 10 5 3 9 9 9 3 8 10 7 10 sample(1:10, 5, replace=FALSE) # 5 without replacement (do not repeat) ## [1] 9 3 4 6 1 Thedistributionofthesampledvaluesdoesnotneedtobeuniform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='the probargument may be fed with a vector of the corresponding probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example, here are 20 independent realisations of the random variable 𝑋 such that Pr(𝑋 = 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 (the probability that we obtain 0 is equal to 90%) and Pr(𝑋 = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1: sample(0:1, 20, replace=TRUE, prob=c(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1)) ## [1] 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 Note If n is a single number (a numeric vector of length 1), then sample(n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') is equivalent to sample(1:n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Similarly, seq(n) is a synonym for seq(1, n) or seq(1, length(n)), depending on the length of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is a dangerous behaviour which can occasionally backfire and lead to bugs (check what happens when n is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Non- etheless, we have been warned and from now on are going to be extra careful (but are we really?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Read more at help("sample") and help("seq").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us stress that the numbers we obtain are merely pseudorandom, because they are generated algorithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' R uses the Mersenne-Twister MT19937 method [36] by de- fault;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see help("RNG") and [16, 24, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' By setting the seed of the random number gener- ator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', re-setting its state to a given one, we can obtain results that are reproducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='seed(12345) # seeds are specified with integers sample(1:10, 5, replace=TRUE) # a,b,c,d,e ## [1] 3 10 8 10 8 sample(1:10, 5, replace=TRUE) # f,g,h,i,j ## [1] 2 6 6 7 10 Setting the seed to the one used previously gives: set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='seed(12345) sample(1:10, 5, replace=TRUE) # a,b,c,d,e ## [1] 3 10 8 10 8 We did not(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') expect that!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' And now for something completely different: set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='seed(12345) sample(1:10, 10, replace=TRUE) # a,b,c,d,e,f,g,h,i,j ## [1] 3 10 8 10 8 2 6 6 7 10 Reproducibilityisacrucialfeatureofeachtrulyscientificexperiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Thesameinitial condition (here: the same seed), leads to exactly the same outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 NUMERIC VECTORS 19 Note Some claim that the only unsuspicious seed is 42, but each programmer can have their own picks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Yours truly, for example, uses 123, 1234, and 12345 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' When performing many runs of Monte Carlo experiments, it may be a good idea to call set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' seed(i) in the i-th iteration of a simulation we are trying to program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Anyhow, we should make sure that our seed settings are applied consistently across all ourscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Otherwise,wemightbeaccusedoftamperingwithevidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Forinstance, here is the ultimate proof that we are very lucky today: set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='seed(1679619) # totally unsuspicious, right?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' sample(0:1, 20, replace=TRUE) # so random ## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is exactly why reproducible scripts and auxiliary data should be published along- side all research reports or papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Only open, transparent science can be fully trust- worthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='seed is not called explicitly, and therandomstate is not restoredfromthe previ- ously saved R session (see Chapter 16), then the random generator is initialised based on the current wall time and the identifier of the running R instance (PID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This may give the impression that the numbers we generate are surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In order to understand the “pseudo” part of the said randomness better, in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3, we will build a very simple random generator ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Reading Data with scan The example text file named euraud-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv3 gives the EUR to AUD exchange rates (how many Australian Dollars can one buy for 1 Euro) from 1 January to 30 June 2020 (remember COVID-19?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us preview the first couple of lines: # EUR/AUD Exchange Rates # Source: Statistical Data Warehouse of the European Central Bank System # https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='ecb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='eu/stats/policy_and_exchange_rates/ # (provided free of charge) NA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6006 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6031 NA The four first lines that begin with “#” merely serve as comments for us, humans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' they should be ignored by the interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The first “real” value, NA corresponds to 1 January (Wednesday;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' New Years Day;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Forex markets were closed, hence a missing observa- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews/teaching-data/raw/master/marek/euraud-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv 20 I DEEP The scan function can be used to read all the inputs and convert them to a single nu- meric vector: scan(paste0("https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews/teaching-data/raw/", "master/marek/euraud-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv"), comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='char="#") ## [1] NA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6006 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6110 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6188 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6115 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6122 NA NA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6154 ## [21] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6177 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6184 1.' 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reached getOption("max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='print") -- omitted 83 entries ] We used the paste0 function to concatenate two long (too long to fit a single line of code) strings to form a single URL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We can also read the files located on our computer, for example: scan("~/teaching-data/marek/euraud-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv", comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='char="#") uses an absolute file path that starts at the user’s home directory, denoted “~”: yours truly’s case is /home/gagolews/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note For portability reasons, we should use slashes, “/”, as path separators (but see help("file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='path") and help(".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Platform")).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' These are not only recognised by all Unix- like boxes but also other popular operating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that URLs (such as https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='r-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/) feature slashes too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Paths can also be relative to the current working directory, denoted “.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It can be read viaacallto getwd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Usually,itisthedirectoryfromwheretheRsessionhasbeenstarted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, if the current working directory was /home/gagolews/teaching-data/ marek,wecouldhavewrittenthefilepathequivalentlyas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='/euraud-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' csv or even euraud-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' On as side note, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='./ would denote the parent directory of the current work- ing directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='./r/iris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv would be equivalent to /home/gagolews/ teaching-data/r/iris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Read the help page about scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Take note of the following formal arguments and their meaning: dec, sep, what, comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='char, and na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Later we will discuss the read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='table and read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv, which are wrappers around scan 2 NUMERIC VECTORS 21 that can be used to read tabular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that write can be used to export an atomic vector’s contents to a text file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Figure2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1showsthegraphoftheaforementionedexchangerates,whichwasgen- erated by calling: plot(scan(paste0("https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews/teaching-data/raw/", "master/marek/euraud-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv"), comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='char="#"), xlab="Day", ylab="EUR/AUD") 0 50 100 150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='85 Day EUR/AUD Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1: EUR/AUD exchange rates from 2020-01-01 (day 1) to 2020-06-30 (day 182) Somewhat misleadingly (and for the reasons that will become apparent later), the document- ation of plot can be accessed by calling help("plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Read about, and experiment with,differentvaluesofthemain,xlab,ylab,type,col,pch,cex,lty,andlwdarguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='More plotting routines will be discussed in Chapter 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Creating Named Objects Often,theobjectswebringforthwillneedtobememorisedsothattheycanbereferred to in further computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The assignment operator, `<-`, can be used for this very purpose: x <- 1:3 # creates a numeric vector and binds the name `x` to it The now-named object can be recalled and dealt with as we please: 22 I DEEP print(x) # or just `x` in the R console ## [1] 1 2 3 sum(x) # example operation: compute the sum of all elements in `x` ## [1] 6 Important In R, all names are case-sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, x and X can coexist peacefully: when set, they refer to two different objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, if we tried to call Print(x) above, we would get an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Typically, we will be using what we refer to as syntactic names (see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 for an exception though).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In the R help system (see help("make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='names") and also help("Quotes")), we read: A syntactically valid name consists of letters, numbers and the dot or underline characters and starts with a letter or the dot not followed by a number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Names such as .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2way are not valid, and neither are the reserved words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For the list of the latter, see help("Reserved").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' A good name is self-explanatory and thus reader-friendly: patients, mean, and aver- age_scoresarewaybetter(iftheyreallyarewhattheyclaimtheyare)than xyz123, crap, or spam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, it might not be such a bad idea to get used to denoting: vectors with x, y, z, matrices (and matrix-like objects) with A, B, …, X, Y, Z, integer indexes with letters i, j, k, l, object sizes with n, m, d, p or nx, ny, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', especially when they are only of temporary nature (for storing some auxiliary results, iterating over collections of objects, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' There are numerous naming conventions that we can adopt, but most often they are a matter of taste;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' snake_case, lowerCamelCase, UpperCamelCase, flatcase, or dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='case are equally good as long as they are used coherently (for instance, some use snake_case for vectors and UpperCamelCase for functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It may even be the case that we have little choice but to adhere to the naming conventions agreed upon in the project we are about to contribute to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note Let us stress that a dot, “.”, has no special meaning (however, see Chapter 10 and Chapter 16 for some asterisks);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='omit is as good a name as na_omit, naOmit, NA- OMIT, naomit, and NaOmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Users coming from some other (C, C++, Java, Python, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') programming languages will need to habituate themselves to this convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' R, as a dynamic language, allows for introducing new variables at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, existing names can be re-bound to new values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance: 2 NUMERIC VECTORS 23 (y <- c(1, 10, 100)) # bracketed expression - printing not suppressed ## [1] 1 10 100 x <- y print(x) ## [1] 1 10 100 Now x refers to a verbatim copy of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note Objects are automatically destroyed when there are no more names bound with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular, by now the garbage collector should have got rid of the 1:3 vector begotten above (to which the name x was bound previously).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' See sec:to-do for more details on memory management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Vectorised Mathematical Functions Mathematically, we will be denoting a given vector 𝒙 of length n as (𝑥1, 𝑥2, … , 𝑥𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In other words, its i-th element is equal to 𝑥𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us review some ubiquitous operations in numerical computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 abs and sqrt R implements vectorised versions of the most popular mathematical functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', abs (absolute value, |𝑥|) and sqrt (square root, √𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' abs(c(2, -1, 0, -3, NA_real_)) ## [1] 2 1 0 3 NA Here, vectorised means that instead of being defined to act on a single numeric value, the function of interest is applied on each element in a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The i-th resulting item is a transformed version of the i-th input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If an input is a missing value, the corres- ponding output will be marked as “don’t know” as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Another example: x <- c(4, 2, -1) (y <- sqrt(x)) ## Warning in sqrt(x): NaNs produced ## [1] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4142 NaN To attract our attention to the fact that computing the square root of a negative value yields a not-a-number, R generated an informative warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' A warning is not an error though: the result is being reckoned as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 24 I DEEP Also the fact that the irrational √2 is displayed as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4142 does not mean that it is such a crudeapproximationto1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='41421356237309504880168872420969807856967187537694 …;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' it is only rounded when printing, for aesthetic reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In fact, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 we will point out that the computer’s floating-point arithmetic allows for roughly 16 decimal digits precision (but we shall see that the devil is in the detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' print(y, digits=16) # display more significant figures ## [1] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='000000000000000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='414213562373095 NaN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Rounding The following functions get rid of all or portions of fractional parts of numbers: floor(x) (rounds down to the nearest integer, denoted ⌊𝑥⌋), ceiling(x) (rounds up, denoted ⌈𝑥⌉), trunc(x) (rounds towards zero), and round(x, digits=0) (rounds to the nearest number with digits decimal digits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance: x <- c(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0001, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9999, -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3149, -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='19999, 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4567, -765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4321, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) floor(x) ## [1] 7 6 5 6 123 -766 0 1 2 ceiling(x) ## [1] 8 7 4 5 124 -765 1 2 3 trunc(x) ## [1] 7 6 4 5 123 -765 0 1 2 Note If we call help("round"), we will read that its usage is like round(x, digits=0), which means that the digits parameter is equipped with the defaultvalue of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In other words, if rounding to 0 decimal digits is what we need, the second argument can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' round(x) # the same as round(x, 0) ## [1] 7 7 4 5 123 -765 0 2 2 round(x, 1) ## [1] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 -765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 round(x, -2) ## [1] 0 0 0 0 100 -800 0 0 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Natural Exponential Function and Logarithm Moreover: 2 NUMERIC VECTORS 25 exp(x)outputsthenaturalexponentialfunction,𝑒𝑥,wheretheEuler’snumber𝑒 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='718, log(x, base=exp(1)) computes, by default, the natural logarithm of 𝑥, log𝑒 𝑥 (which is most often denoted simply as log 𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Recall that if 𝑥 = 𝑒𝑦, then log𝑒 𝑥 = 𝑦, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', one is the inverse of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' log(c(0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7183, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3891, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0855)) # grows slowly ## [1] -Inf 0 1 2 3 exp(c(0, 1, 2, 3)) # grows fast ## [1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7183 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3891 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0855 Note These functions enjoy a number of very useful identities and inequalities, in- cluding: log(𝑥 ⋅ 𝑦) = log 𝑥 + log 𝑦, log(𝑥𝑦) = 𝑦 log 𝑥, 𝑒𝑥+𝑦 = 𝑒𝑥 ⋅ 𝑒𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For more properties like these, take a glance at Chapter 4 of the freely available hand- book [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For the logarithm to a different base, say log10 𝑥, we can call: log(c(0, 1, 10, 100, 1000, 1e10), 10) # or log(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', base=10) ## [1] -Inf 0 1 2 3 10 Note that if log𝑏 𝑥 = 𝑦, then 𝑥 = 𝑏𝑦, for any 1 ≠ 𝑏 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note Commonly, a logarithmic scale is used for variables that grow rapidly when expressed as functions of each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- seq(0, 10, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=1001) par(mfrow=c(1, 2)) # two plots in one figure (1 row, 2 columns) plot(x, exp(x), type="l") plot(x, exp(x), type="l", log="y") # log-scale on the y-axis 26 I DEEP 0 2 4 6 8 10 0 5000 10000 15000 20000 x exp(x) 0 2 4 6 8 10 1 10 100 1000 10000 x exp(x) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2: Linear- vs log-scale on the y-axis Note that 𝑒𝑥 on the log-scale is just a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, keep in mind that such a trans- formation of the axes can only be applied in the case of values strictly greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Probability Distributions (*) It should come as no surprise that R offers an extensive support for many univariate probability distribution families,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' including: continuousdistributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='whichtakevaluesbeingarbitraryrealnumbers(overthe whole possible range or in some interval): – *unif (uniform),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – *norm (normal),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – *exp (exponential),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – *gamma (gamma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – *beta (beta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – *lnorm (log-normal),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – *t (Student),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – *cauchy (Cauchy–Lorentz),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – *chisq (chi-squared,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 𝜒2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – *f (Snedecor–Fisher),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 NUMERIC VECTORS 27 – *weibull (Weibull);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' with the prefix “*” being one of: – “d” (probability density function, PDF), – “p” (cumulative distribution function, CDF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' or survival function, SF), – “q” (quantile function, being the inverse of the CDF), – “r” (generation of random deviates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' already mentioned);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' discrete distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', those whose possible outcomes can be easily enumer- ated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', some integers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' – *binom (binomial), – *geom (geometric), – *pois (Poisson), – *hyper (hypergeometric), – *nbinom (negative binomial);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' here, prefixes “p” and “r” have the same meaning as above, however: – “d” now gives the probability mass function (PMF), – “q”yieldsthequantilefunction,butonethatisdefinedasageneralisedinverse of the CDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Each distribution is characterised by a set of underlying parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, a normal distribution N(𝜇, 𝜎) can be pinpointed by setting its expected value 𝜇 ∈ ℝ and standard deviation 𝜎 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In R, these two have been named mean and sd, respect- ively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see help("dnorm").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note The parametrisations assumed in R can be subtly different from what we know from statistical textbooks or probability courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example, the normal distribu- tion can be parameterised based on either standard deviation or variance, and the ex- ponential distribution can be defined via its expected value or the reciprocal thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We thus advise the reader to study carefully the documentation of help("dnorm"), help("dunif"), help("dexp"), help("dbinom"), and the like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is also worth to know the typical use cases of each of the distribution listed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', a Poisson distribution can describe the probability of observing the number of in- dependent events in a fixed time interval (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', the number of users downloading a copy of R from CRAN per hour), and an exponential distribution can model the time between such events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' compare [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Acalltohist(x)drawsahistogram,whichcanserveasanestimatoroftheunder- lyingcontinuousprobabilitydensityfunctionofagivensample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='seeFigure2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3foranillustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 28 I DEEP par(mfrow=c(1, 2)) # 2 plots in 1 figure # Uniform U(0, 1) hist(runif(10000, 0, 1), col="white", probability=TRUE, main="") x <- seq(0, 1, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=101) lines(x, dunif(x, 0, 1), lwd=2) # draw the true density function (PDF) # Normal N(0, 1) hist(rnorm(10000, 0, 1), col="white", probability=TRUE, main="") x <- seq(-4, 4, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=101) lines(x, dnorm(x, 0, 1), lwd=2) # draw the PDF runif(10000, 0, 1) Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 rnorm(10000, 0, 1) Density 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3: Example histograms of some pseudorandom samples and the true under- lying probability density functions: the uniform distribution on the unit interval (left) and the standard normal distribution (right) Draw a histogram of some random samples of different sizes n from the following distributions: rnorm(n, µ, σ) — normal N(𝜇, 𝜎) with expected values 𝜇 ∈ {−1, 0, 5} (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', 𝜇 being equal to either −1, 0, or 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' read “∈” as “belongs to the given set” or “in”) and standard devi- ations 𝜎 ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, 1, 5};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' runif(n, a, b) — uniform U(𝑎, 𝑏) on the interval (𝑎, 𝑏) with 𝑎 = 0 and 𝑏 = 1 as well as 𝑎 = −1 and 𝑏 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' rbeta(n, α, β) — beta B(𝛼, 𝛽) with 𝛼, 𝛽 ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, 1, 2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' rexp(n, λ) — exponential E(𝜆) with rates 𝜆 ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, 1, 10};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover,readaboutandplaywiththe breaks, main, xlab, ylab, xlim, ylim,and colparamet- ers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see help("hist").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Werollasix-sideddice12times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Let𝐶bearandomvariabledenotingthenumber 2 NUMERIC VECTORS 29 of cases wherethe “1” face is thrown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 𝐶 follows a binomial distribution Bin(𝑛, 𝑝) with paramet- ers 𝑛 = 12 (the number of Bernoulli trials) and 𝑝 = 1/6 (the probability of success in a single roll).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Theprobabilitiesthatthenumberof“1”srolledwillbeequalto0,1,…,4,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=',𝑃(𝐶 = 0),𝑃(𝐶 = 1),…,𝑃(𝐶 = 4),respectively,canbecomputedbasedontheprobabilitymassfunction(dbinom): dbinom(0:4, 12, 1/6) # probability mass function at 5 different points ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='112157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='269176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='296094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='197396 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='088828 On the other hand, the probability that we throw more than three “1”s, 𝑃(𝐶 > 3) = 1 − 𝑃(𝐶 ≤ 3), can be determined by means of the cumulative distribution function (pbinom) or survival function (pbinom(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='tail=FALSE)): 1-pbinom(3, 12, 1/6) # pbinom(3, 12, 1/6, lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='tail=FALSE) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='12518 The smallest 𝑐 such that 𝑃(𝐶 ≤ 𝑐) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='95 can be computed based on the quantile function: qbinom(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, 12, 1/6) ## [1] 2 pbinom(3:4, 12, 1/6) # for comparison - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='95 is in-between ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='87482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='96365 In other words, at least 95% of the time we will be observing no more than 4 successes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also here are some pseudorandom realisations of 𝐶 – the number of “1”s in 30 simulations of 12 independent dice rolls each: rbinom(30, 12, 1/6) ## [1] 1 3 2 4 4 0 2 4 2 2 4 2 3 2 0 4 1 0 1 4 4 3 2 6 2 3 2 2 1 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Special Functions (*) Within mathematical formulae and across assorted application areas, certain func- tions appear more frequently than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, for the sake of notational brevity and computational precision, many of them have been assigned special names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, the following may be mentioned in the definitions related to some of the probability distributions listed above: gamma(x) for 𝑥 > 0 computes Γ(𝑥) = ∫ ∞ 0 𝑡𝑥−1𝑒−𝑡 𝑑𝑡, beta(a, b) for 𝑎, 𝑏 > 0 yields 𝐵(𝑎, 𝑏) = Γ(𝑎)Γ(𝑏) Γ(𝑎+𝑏) = ∫ 1 0 𝑡𝑎−1(1 − 𝑡)𝑏−1 𝑑𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Whydo wehave beta if itis merely amixof gammas?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Aspecific,tailoredfunctionshould be faster and more precise than its DIY version;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' its underlying implementation does not have to involve any calls to gamma at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 30 I DEEP beta(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25, 250) # okay ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='91213 gamma(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25)*gamma(250)/gamma(250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25) # not okay ## [1] NaN The Γ function grows so rapidly that already gamma(172) yields Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is due to the fact that a computer’s arithmetic is not infinitely precise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' compare Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Special functions are plentiful;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see the open-access [38] for one of the most definitive references (and also [2] for its predecessor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' R package gsl [28] provides a vectorised interface to the famous GNU GSL [23] library, which implements many of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 The Pochhammer symbol, (𝑎)𝑥 = Γ(𝑎 + 𝑥)/Γ(𝑎), can be computed via a call to gsl::poch(a, x) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', the poch function from the gsl package;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1): # call install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages("gsl") first library("gsl") # load the package poch(10, 3:6) # calls gsl_sf_poch() from GNU GSL ## [1] 1320 17160 240240 3603600 Read the documentation of the corresponding gsl_sf_poch function in the GNU GSL manual available here4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' And since you are there, do not hesitate to go through the list of all the other functions, including those related to statistics, permutations, combinations, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Manyfunctionsalsohavetheirlogarithm-ofversions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='see,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', lgammaand lbeta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Also, for instance, dnorm and dbeta has the log parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Its classical use case is the (nu- merical) maximum likelihood estimation, which involves the sums of the logarithms of densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Arithmetic Operations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Vectorised Arithmetic Operators R features the following arithmetic operators: `+` (addition) and `-` (subtraction), `*` (multiplication) and `/` (division), `%/%` (integer division) and `%%` (modulo, division remainder), `^` (exponentiation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' synonym: `**`).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They are all vectorised: they take two vectors on input and yield another vector in result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 4 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/software/gsl/doc/html/ 2 NUMERIC VECTORS 31 c(1, 2, 3) * c(10, 100, 1000) ## [1] 10 200 3000 We note that the multiplication was performed in an elementwise fashion: the 1st ele- ment in the left vector was multiplied by the corresponding element in the right vector and the result has been stored in the 1st element of the output, then the 2nd element in the left… all right, we get the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Other operators are vectorised in the same manner: 0:10 + seq(0, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 0:7 / rep(3, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=8) # division by 3 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='33333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='66667 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='33333 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='66667 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='33333 0:7 %/% rep(3, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=8) # integer division ## [1] 0 0 0 1 1 1 2 2 0:7 %% rep(3, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=8) # division remainder ## [1] 0 1 2 0 1 2 0 1 Note that operations involving missing values also yield NAs: c(1, NA_real_, 3, NA_real_) + c(NA_real_, 2, 3, NA_real_) ## [1] NA NA 6 NA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Recycling Rule Some of the above statements can be written more concisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' When the operands are of different lengths, the shorter one is recycled (think: rep(y, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=length(x))) as many times as necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 0:7 / 3 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='33333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='66667 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='33333 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='66667 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='33333 1:10 * c(-1, 1) ## [1] -1 2 -3 4 -5 6 -7 8 -9 10 2 ^ (0:10) ## [1] 1 2 4 8 16 32 64 128 256 512 1024 We call this the recycling rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Ifanoperandcannotberecycledinitsentirety,awarning5 isgenerated,buttheoutput is still available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 5 A few built-in functions do not warn at all when incomplete recycling is performed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', paste) or can even give an error (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='list).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Consider this inconsistency an annoying bug and hope it will be fixed in the next decade or so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 32 I DEEP c(1, 10, 100) * 1:8 ## Warning in c(1, 10, 100) * 1:8: longer object length is not a multiple of ## shorter object length ## [1] 1 20 300 4 50 600 7 80 Note Some functions are also deeply vectorised, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', with respect to multiple argu- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example, runif(3, c(10, 20, 30), c(11, 22, 33)) ## [1] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='288 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='577 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='227 generates three random numbers uniformly distributed over the intervals (10, 11), (20, 22), and (30, 33), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, pmin and pmax return the parallel minimum and maximum of the corresponding elements of the input vectors: pmin(c(1, 2, 3, 4), c(4, 2, 3, 1)) ## [1] 1 2 3 1 pmin(3, 1:5) ## [1] 1 2 3 3 3 pmax(0, pmin(1, c(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25, -2, 5, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='99))) # clipping to [0, 1] ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='99 Note Vectorisationand the recyclingrule areperhapsmost useful whenapplying bin- aryoperatorsonsequencesofidenticallengthsorwhenperformingvector-scalar(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', a sequence vs a single value) operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, there is much more: schemes like “every k-th element” appear in Taylor series expansions (multiply by c(-1, 1)), k-fold cross validation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see also Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 for use cases in matrix/tensor processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Operator Precedence Apart from the seven binary arithmetic operators, other noteworthy, already men- tioned ones include: the unary `-` (change of sign), `:` (sequence generation), and `<-` (assignment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Expressions involving multiple operations need a set of rules governing the order of computations (unless we enforce it using round brackets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We have said that “-1:10” means “(-1):10” rather than “-(1:10)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' But what about, say, “1+1+1+1+1*0” or “3*2^0:5+10”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us list the aforementioned operators in their order of precedence, from the least to the most binding (see also help("Syntax")): 2 NUMERIC VECTORS 33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `<-` (right-to-left), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `+` and `-`, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `*` and `/`, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `%%` and `%/%`, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `:`, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `+` and `-` (unary), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `^` (right-to-left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, “-2^2/3+3*4” means “((-(2^2))/3)+(3*4)” and not, for example, -((2^(2/ (3+3)))*4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that `+` and `-`, `*` and `/`, as well as `%%` and `%/%` have the same priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Expressions involving a series of operations in the same group, are evaluated left-to- right, with the exception of `^` and `<-`, which are performed from right to left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore: “2*3/4*5” is equivalent to “((2*3)/4)*5”, “2^3^4” is the same as “2^(3^4)” (which, mathematically, we would write as 234 = 281), “x <- y <- 4*3%%8/2” binds both y and x with 6 and not x with the previous value of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' And let us remember: when in doubt, we can always bracket a subexpression to make sure it is executed in the intended order (which can also increase readability of the code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Accumulating The `+` and `*` operators as well as the pmin and pmax functions implement element- wise operations that are applied on the corresponding elements taken from two given vectors: ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 𝑥1 𝑥2 𝑥3 ⋮ 𝑥𝑛 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟ ⎠ + ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 𝑦1 𝑦2 𝑦3 ⋮ 𝑦𝑛 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟ ⎠ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 𝑥1 + 𝑦1 𝑥2 + 𝑦2 𝑥3 + 𝑦3 ⋮ 𝑥𝑛 + 𝑦𝑛 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟ ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, we can also scan through all the values in a single vector and combine the successive elements that we inspect using the corresponding operation: cumsum(x) gives the cumulative sum of the elements in a vector, cumprod(x) computes the cumulative product, cummin(x) yields the cumulative minimum, 34 I DEEP cummax(x) generates the cumulative maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The i-th element in the output vector will consist of the sum/product/min/max of the first i inputs: cumsum ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 𝑥1 𝑥2 𝑥3 ⋮ 𝑥𝑛 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟ ⎠ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝ 𝑥1 𝑥1 + 𝑥2 𝑥1 + 𝑥2 + 𝑥3 ⋮ ⋱ 𝑥1 + 𝑥2 + 𝑥3 + ⋯ + 𝑥𝑛 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟ ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example: cumsum(1:8) ## [1] 1 3 6 10 15 21 28 36 cumprod(1:8) ## [1] 1 2 6 24 120 720 5040 40320 cummin(c(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 0)) ## [1] 3 2 2 2 1 1 0 cummax(c(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 0)) ## [1] 3 3 4 5 5 6 6 If we are interested only in the last cumulant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' summarising all the inputs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' we have the following functions at our disposal: sum(x) computes the sum of elements in a vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ∑𝑛 𝑖=1 𝑥𝑖 = 𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' prod(x) outputs the product of all elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ∏𝑛 𝑖=1 𝑥𝑖 = 𝑥1𝑥2 ⋯ 𝑥𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' min(x) computes the minimum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' max(x) reckons the greatest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example: sum(1:8) ## [1] 36 prod(1:8) ## [1] 40320 min(c(3, 2, 4, 5, 1, 6, 0)) ## [1] 0 max(c(3, 2, 4, 5, 1, 6, 0)) ## [1] 6 Note In Chapter 7, we will discuss the Reduce function, which generalises the above by allowing any binary operation to be propagated over a given vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 diff can be considered an inverse to cumsum: it computes the iterative difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 NUMERIC VECTORS 35 Namely, it subtracts the first two elements, then the 2nd from the 3rd one, the 3rd from the 4th, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In other words, diff(x) gives 𝒚 such that 𝑦𝑖 = 𝑥𝑖+1 − 𝑥𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- c(-2, 3, 6, 2, 15) diff(x) ## [1] 5 3 -4 13 cumsum(diff(x)) ## [1] 5 8 4 17 cumsum(c(-2, diff(x))) # recreates x ## [1] -2 3 6 2 15 Thanks to diff, we can compute the daily changes to the EUR/AUD forex rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' aud <- scan(paste0("https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews/teaching-data/raw/", "master/marek/euraud-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv"), comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='char="#") aud_all <- na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='omit(aud) # remove all missing values plot(diff(aud_all), type="s", ylab="Daily change [EUR/AUD]") abline(h=0, lty="dotted") # draw a horizontal line at y=0 0 20 40 60 80 100 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='04 Index Daily change [EUR/AUD] Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4: Iterative differences of the exchange rates (non-missing values only) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Aggregating The above functions form the basis for some popular summary statistics6 (sample ag- gregates), such as: mean(x) gives the arithmetic mean, sum(x)/length(x), 6 Actually, var and median, amongst others, are defined by the stats package, but this one is automatic- ally loaded by default, so let us not make a fuss about it now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 36 I DEEP var(x) yields the (unbiased) sample variance, sum((x-mean(x))^2)/(length(x)-1), sd(x) is the standard deviation, sqrt(var(x)), median(x) computes the sample median, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', the middle value in the sorted ver- sion of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance7: x <- runif(1000) c(min(x), mean(x), median(x), max(x), sd(x)) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00046535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='49727780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='48995025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='99940453 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='28748391 Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 Let 𝒙 be any vector of length 𝑛 with positive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Compute its geometric and harmonic mean, which are given by, respectively, 𝑛 √√√ ⎷ 𝑛 ∏ 𝑖=1 𝑥𝑖 = 𝑒 1 𝑛 ∑𝑛 𝑖=1 log 𝑥𝑖 and 𝑛 ∑𝑛 𝑖=1 1 𝑥𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Whensolvingexerciseslikethisone,itdoesnotreallymatterwhatdatayouapplythesefunctions on (see, however, Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 for discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We are being abstract in the sense that the 𝒙 vec- tor can be anything: from the one that features very accurate financial predictions that will help minimiseinequityandmakethisworldlessmiserable,throughthedatayouhavebeencollecting for the last the years in relation to your definitely-super-important PhD research, whatever your company asked you to crunch today, to something related to your hobby project that you enjoy doing after hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, just test the above on something like “x <- runif(10)”, and move on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' All the aforementioned functions return a missing value if any of the input elements is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Luckily, they are equipped with the na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm parameter on behalf of which we can request the removal of NAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' aud <- scan(paste0("https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews/teaching-data/raw/", "master/marek/euraud-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv"), comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='char="#") c(min(aud), mean(aud), max(aud)) ## [1] NA NA NA c(min(aud, na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=TRUE), mean(aud, na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=TRUE), max(aud, na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=TRUE)) ## [1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6006 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6775 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8635 Note In the documentation, we read that the usage of some of the aforementioned functions is like sum(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=FALSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' prod, min, and max are defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They acceptanynumberofinputvectors,eachofthemcanbeofarbitrarylength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Therefore, min(1, 2, 3), min(c(1,2,3)) as well as min(c(1,2),3) all return the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, we can also read that we have mean(x, trim=0, na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=FALSE, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This 7 Notethat min, median,and maxisaspecialcaseof quantile,whichwewilldiscussmuchfurther,namely, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is because it returns a named vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 NUMERIC VECTORS 37 time, only one vector can be aggregated and any further arguments (except trim and na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm) are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The extra flexibility (which we do not have to rely upon, ever) of the former group is due their being associative operations: it holds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', (2+3)+4 = 2+(3+4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, the operations can be performed in any order, in any groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also note that they are more primitive operations: it is mean that is based on sum, not vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Exercises Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 Answer the following questions: What is the meaning of the dot-dot-dot parameter in the definition of the c function?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We say that the round function is vectorised: what does that mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What do we mean by saying that multiplication operates element-by-element?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' How does the recycling rule work when applying `+`?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' How to (and why) set the seed of the pseudorandom number generator?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What is the difference between NA_real_ and NaN?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' How are default arguments specified in the manual of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', the round function?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Is a call to rep(times=4, x=1:5)” equivalent to rep(4, 1:5)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' List a few ways to generate a sequence like (-1, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='75, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, …, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='75, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Is“-3:5”thesameas "-(3:5)"?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Whatabouttheprecedenceofoperatorsinexpressionssuch as “2^3/4*5^6”, “5*6+4/17%%8”, and “1+-2^3:4”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If x is a numeric vector of length 𝑛 (for some 𝑛 ≥ 0), how many values will sample(x) output?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Does scan support reading directly from compressed archives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gz files?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' When in doubt, refer back to the material discussed in this chapter and/or the R manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='10 The following code generates an example graph of arcsine and arccosine, whose preparation – thanks to vectorisation – is quite straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- seq(-1, 1, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=11) # increase length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out for a smoother curve plot(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' asin(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' # asin() computed for 11 points type="l",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' # lines ylim=c(-pi/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' pi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' # y axis limits like c(y_min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' y_max) ylab="asin(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' acos(x)") # y axis label (continues on next page) 38 I DEEP (continued from previous page) lines(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' acos(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' col="red",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' lty="dashed") # adds to the current plot legend("topright",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' c("asin(x)",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "acos(x)"),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' lty=c("solid",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "dashed"),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' col=c("black",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "red"),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' bg="white") Inspired by the above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' plot the following functions: | sin 𝑥2|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' |sin |𝑥||,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' √⌊𝑥⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' and 1/(1 + 𝑒−𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Recall that the documentation of plot can be viewed by calling help("plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='11 It can be shown that: 4 𝑛 ∑ 𝑖=1 (−1)𝑖+1 2𝑖 − 1 = 4 (1 1 − 1 3 + 1 5 − 1 7 + ⋯) slowly converges to 𝜋 as 𝑛 approaches ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Compute the above for 𝑛 = 1,000,000 and 𝑛 = 1,000,000,000 using the vectorised functions and operators discussed in this chapter, making use of the recycling rule as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='12 Let x and y be two vectors of identical lengths 𝑛, say: x <- rnorm(100) y <- 2*x+10+rnorm(100, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) Compute the Pearson linear correlation coefficient given by: 𝑟 = ∑𝑛 𝑖=1 (𝑥𝑖 − 1 𝑛 ∑𝑛 𝑗=1 𝑥𝑗) (𝑦𝑖 − 1 𝑛 ∑𝑛 𝑗=1 𝑦𝑗) √∑𝑛 𝑖=1 (𝑥𝑖 − 1 𝑛 ∑𝑛 𝑗=1 𝑥𝑗) 2 √∑𝑛 𝑖=1 (𝑦𝑖 − 1 𝑛 ∑𝑛 𝑗=1 𝑦𝑗) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' To make sure you have come up with a correct implementation, compare your result to a call to the built-in cor(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='13 (*) Look up on the internet an R package that features functions to compute the 5-day moving (rolling) average and median of a given vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Apply them on the EUR/AUD cur- rency exchange data and plot thus obtained smoothened versions of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='14 (**)Computethe𝑘-movingaverageusingacalltoconvolve(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', type="filter").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In the next chapter we will study operations that involve logical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3 Logical Vectors There are three logical constants in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Wait… how many?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Creating Logical Vectors R defines three logical constants: TRUE, FALSE, and NA – meant to represent “yes”, “no”, and “?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Each of them, when instantiated, is an atomic vector of length one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some of the functions we introduced in the previous chapter can be used to generate logical vectors as well: c(TRUE, FALSE, FALSE, NA, TRUE, FALSE) ## [1] TRUE FALSE FALSE NA TRUE FALSE rep(c(TRUE, FALSE, NA), each=2) ## [1] TRUE TRUE FALSE FALSE NA NA sample(c(TRUE, FALSE), 10, replace=TRUE, prob=c(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2)) ## [1] TRUE TRUE TRUE FALSE FALSE TRUE TRUE FALSE TRUE TRUE Note “T” is a synonym for TRUE and “F” stands for FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, these are not re- servedkeywordsandcanbere-assignedanyothervalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Therefore,weadviseagainst relying on them and hence we will never use them throughout the course of this course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also note that the logical missing value is spelled simply as “NA” and not “NA_logical_”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The fact that both the logical “NA” and the numeric "NA_real_" are, for the sake of our mental well-being, both printed as "NA" on the R console, does not mean they are identical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 for discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 40 I DEEP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Comparing Elements 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Vectorised Comparison Operators Logical vectors frequently come into being as results of various testing activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular, the binary operators: `<` (less than), `<=` (less than or equal), `>` (greater than), `>=` (greater than or equal) `==` (equal), `!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='=` (not equal), comparethecorrespondingelementsoftwonumericvectorsandoutputalogicalvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1 < 3 ## [1] TRUE c(1, 2, 3, 4) == c(2, 2, 3, 8) ## [1] FALSE TRUE TRUE FALSE 1:10 <= 10:1 ## [1] TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE Thus, they operate in an elementwise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, the recycling rule is applied if necessary: 3 < 1:5 # c(3, 3, 3, 3, 3) < c(1, 2, 3, 4, 5) ## [1] FALSE FALSE FALSE TRUE TRUE c(1, 4) == 1:4 # c(1, 4, 1, 4) == c(1, 2, 3, 4) ## [1] TRUE FALSE FALSE TRUE Therefore, we can say that they are vectorised in the same manner as the arithmetic operators `+`, `*`, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' compare Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Testing for NA, NaN, and Inf Comparisons against missing values and not-numbers yield NAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, instead of the incorrect x == NA_reals_ or x == NaN, testing for missingness should rather be performed via a call to the vectorised is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='na function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='na(c(NA_real_, Inf, -Inf, NaN, -1, 0, 1)) ## [1] TRUE FALSE FALSE TRUE FALSE FALSE FALSE is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='nan(c(NA_real_, Inf, -Inf, NaN, -1, 0, 1)) (continues on next page) 3 LOGICAL VECTORS 41 (continued from previous page) ## [1] FALSE FALSE FALSE TRUE FALSE FALSE FALSE is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='na(c(TRUE, FALSE, NA, TRUE)) # works for logical vectors too ## [1] FALSE FALSE TRUE FALSE Moreover, is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='finite is noteworthy, because it returns FALSE on Infs, NA_real_s and NaNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='finite(c(NA_real_, Inf, -Inf, NaN, -1, 0, 1)) ## [1] FALSE FALSE FALSE FALSE TRUE TRUE TRUE See also the more specific is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='nan and is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Dealing with Floating Point Round-Off Errors (*) In mathematics, real numbers are merely an idealisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In practice, however, it is impossibletostorethemwithinfiniteprecision(think𝜋 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1415926535897932384626433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='): computer memory is limited and our time is precious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, a widely agreed upon consensus had to be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In R, we rely on the so- called double-precision floating point format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Floating point means that the numbers can be both small (close to zero) and large: ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='23 × 10−308 and ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='79 × 10308 are both acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='23e-308 == 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000223 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='79e308 == 17900000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 These two are quite distant from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Everynumericvaluetakes8bytes(orequivalently64bits)ofmemory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Weare,however, able to store only about 15-17 decimal digits: 42 I DEEP print(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='12345678901234567890123456789012345678901234, digits=22) # 22 is max ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1234567890123456773699 whichlimitstheprecisionofourcomputations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Theabout partis–unfortunately–due to the numbers’ being written in the computer-friendly binary, not human-aligned decimal, base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This can lead to some unexpected outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 cannot be represented exactly, because it cannot be written as a finite series of reciprocals of powers of 2 (it holds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 = 2−4 + 2−5 + 2−8 + 2−9 + …).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This leads to surprising results such as: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 == 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 ## [1] FALSE Despite the fact that what follows does not show anything suspicious: c(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Printing involves rounding, hence, in the above context, is misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Above, we have something more like: print(c(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3), digits=22) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1000000000000000055511 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3000000000000000444089 ## [3] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2999999999999999888978 All integers between −253 and 253 all stored exactly – this is good news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, the next integer is beyond the representable range: 2^53 + 1 == 2^53 ## [1] TRUE The above suggests that, more generally, the order of operations may matter, in particular, the associativity property may be violated when dealing with numbers of different orders of magnitude: 2^53 + 2^-53 - 2^53 - 2^-53 # should be == 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 ## [1] -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1102e-16 Some numbers may just be just too large, too small, or too close to zero to be rep- resented exactly: c(sum(2^((1023-52):1023)), sum(2^((1023-53):1023))) ## [1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7977e+308 Inf c(2^(-1022-52), 2^(-1022-53)) ## [1] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9407e-324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000e+00 3 LOGICAL VECTORS 43 Important The double-precision floating point format (IEEE 754) is not specific to R: it is used by most other computing environments, including Python and C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For discussion, see [27, 30, 33] ([26] can be of particular interest to the general statist- ical/data analysis audience).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Can we do anything about these issues?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' First, when dealing with integers of reasonable order of magnitude (a frequent case wherewearedealingvariousresourceorcaseIDsinourdatasets),restassuredthatwe aresafe:theircomparison,addition,subtraction,andmultiplicationisalwaysprecise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In all other cases (including applying other operations on integers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', division or sqrt), we need to be very careful with comparisons, especially involving testing for equality, `==`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The sole fact that sin 𝜋 = 0, mathematically speaking, does not mean that we should expect that: sin(pi) == 0 ## [1] FALSE Instead, they are so close to each other that we can treat the difference between them as negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thus, in practice, instead of testing if 𝑥 = 𝑦, we will be considering: |𝑥 − 𝑦| (absolute error) or |𝑥−𝑦| |𝑦| (relative error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' which takes the order of magnitude of the numbers into ac- count but obviously cannot be applied if 𝑦 is very close of 0), and determining if these are less than some assumed error margin, 𝜀 > 0, say, 10−8 or 2−26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example: abs(sin(pi) - 0) < 2^-26 ## [1] TRUE Note Note that rounding can sometimes have a similar effect as testing for almost- equality in terms of the absolute error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' round(sin(pi), 8) == 0 ## [1] TRUE Important Our recommendations are valid for the most popular applications of R, 44 I DEEP i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', statistical and, more generally, scientific computing1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The datasets we handle on a daily basis do not represent accurate measurements themselves, bah, the World it- self is far from ideal, therefore we do not have to lose sleep over our not being able to precisely pinpoint the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Logical Operations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Vectorised Logical Operators The comparison operators such as `==` and `>` accept only two arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Their chaining is forbidden;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' a test which we would mathematically write as 0 ≤ 𝑥 ≤ 1 (or 𝑥 ∈ [0, 1]) cannot be expressed as “0<=x<=1” in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, we need a way to combine two logical conditions so as to be able to state that “𝑥 ≥ 0 and, at the same time, 𝑥 ≤ 1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In such situations, the following logical operators and functions come in handy: `!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='` (not, negation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' unary), `&` (and, conjunction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' are both predicates true?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ), `|` (or, alternation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' is at least one true?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ), xor (exclusive-or, exclusive disjunction, either-or;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' is one and only one of the pre- dicates true?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They again act elementwisely and implement the recycling rule if necessary (and ap- plicable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- c(-10, -1, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, 1, 5, 100) (x >= 0) & (x <= 1) ## [1] FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE (x < 0) | (x > 1) ## [1] TRUE TRUE TRUE FALSE FALSE FALSE TRUE TRUE !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ((x < 0) | (x > 1)) ## [1] FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE xor(x >= -1, x <= 1) ## [1] TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE 1 However,infinancialapplications,weshouldratherrelyonbase-10numbers(comparethe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1problem above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also note that there exist some libraries implementing higher precision floating-point numbers or even interval arithmetic that keeps track of error propagation operation chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3 LOGICAL VECTORS 45 Important The vectorised `&` and `|` operators should not be confused with their scalar, short-circuit counterparts, `&&` and `||`, which we discuss in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Operator Precedence Revisited The operators introduced in this chapter have lower precedence than the arithmetic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular, the binary `+` and `-`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Calling help("Syntax") reveals that we can extend our listing from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `<-` (right-to-left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' least binding), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `|`, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `&`, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='` (unary), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `<`, `>`, `<=`, `>=`, `==`, and `!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='=`, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `+` and `-`, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `*` and `/`, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' … 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Dealing with Missingness Operations involving missing values follow the principles of the Łukasiewicz’s three- valued logic, which is based on common sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, “NA | TRUE” is TRUE, be- cause or needs at least one argument to be TRUE to generate such a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' On the other hand, “NA | FALSE” is NA, because the result would be different depending on what we substituted NA for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us take a moment to contemplate the operations’ truth tables for all the possible combinations of inputs: u <- c(TRUE, FALSE, NA, TRUE, FALSE, NA, TRUE, FALSE, NA) v <- c(TRUE, TRUE, TRUE, FALSE, FALSE, FALSE, NA, NA, NA) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='u ## [1] FALSE TRUE NA FALSE TRUE NA FALSE TRUE NA u & v ## [1] TRUE FALSE NA FALSE FALSE FALSE NA FALSE NA u | v ## [1] TRUE TRUE TRUE TRUE FALSE NA TRUE NA NA xor(u, v) ## [1] FALSE TRUE NA TRUE FALSE NA NA NA NA 46 I DEEP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Aggregating with all, any, and sum Just like in the case of numeric vectors, we can summarise the contents of logical se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' all tests whether every element in a logical vector is equal to TRUE and any determines if there exists an element that is TRUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- runif(10000) all(x <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2) # are all values in x <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ## [1] FALSE any(x <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2) # is there at least one element in x that is <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ## [1] TRUE Note The all function will frequently be used in conjunction with “==”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is because the latter, as we have said above, is itself vectorised: it does not test whether a vector as a whole is equal to another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' z <- c(1, 2, 3) z == 1:3 # elementwise equal ## [1] TRUE TRUE TRUE all(z == 1:3) # elementwise equal summarised ## [1] TRUE However, let us keep in mind the warning about the testing for exact equality of floating-point numbers stated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Sometimes, considering absolute or relative errors might be more appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' z <- sin((0:10)*pi) # sin(0), sin(pi), sin(2*pi), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', sin(10*pi) all(z == 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0) # danger zone!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=" please don't." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ## [1] FALSE all(abs(z - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0) < 1e-9) # are the absolute errors negligible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ## [1] TRUE We can also call sum on a logical vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Taken into account that it interprets TRUE as numeric 1 and FALSE as 0 (more on this in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1), it will give us the number of elements equal to TRUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' sum(x <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2) # how many elements in x are <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ## [1] 1998 Also, by computing sum(x)/length(x), we can obtain the proportion (fraction) of val- ues equal to TRUE in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Equivalently: mean(x <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2) # proportion of elements <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1998 3 LOGICAL VECTORS 47 Naturally, we expect mean(runif(n) <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2)” to be equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 (20%), but with ran- domness we can never be sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Simplifying Predicates Eachaspiringprogrammerneeds to becomefluentwiththerulesgoverningthetrans- formations of logical conditions, for example, that the negation of “(x >= 0) & (x < 1)” is equivalent to “(x < 0) | (x >= 1)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Each such rule is called a tautology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Here are some of them: !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='p) is equivalent to p (double negation), !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (p & q) holds if and only if !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='p | !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='q (De Morgan’s law), !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (p | q) is !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='p & !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='q (another De Morgan’s law), all(p) is equivalent to !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='any(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Various combinations thereof are of course possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some further simplifications are enabled by other properties of the binary operations: commutativity (symmetry), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', 𝑎 + 𝑏 = 𝑏 + 𝑎, 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎, associativity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐), max(max(𝑎, 𝑏), 𝑐) = max(𝑎, max(𝑏, 𝑐)), distributivity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', 𝑎 ∗ 𝑏 + 𝑎 ∗ 𝑐 = 𝑎 ∗ (𝑏 + 𝑐), min(max(𝑎, 𝑏), max(𝑎, 𝑐)) = max(𝑎, min(𝑏, 𝑐)), and relations, including: transitivity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', if 𝑎 ≤ 𝑏 and 𝑏 ≤ 𝑐 then surely 𝑎 ≤ 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Assuming that a, b, and c are numeric vectors, simplify the following expressions: !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (b>a & b *See the comment lines within the files themselves for" ## [4] "> a detailed description of each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' *" ## [5] "" ## [6] "*Good* datasets are actually hard to find!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='" writeLines is its counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' There is also an option to read or write parts of files at a time, which me mention in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, cat(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', append=TRUE) can be used to create a text file incrementally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Pattern Searching 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Comparing Whole Strings Wehavealready revieweda coupleof ways tocomparestringsasa whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Forinstance, the `==` operator implements elementwise testing: c("spam", "spam", "bacon", "eggs") == c("spam", "eggs") # recycling rule ## [1] TRUE FALSE FALSE TRUE Moreover, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1, we have introduced the match function and its derivative, the `%in%` operator, which are vectorised in a different way: match(c("spam", "spam", "bacon", "eggs"), c("spam", "eggs")) ## [1] 1 1 NA 2 c("spam", "spam", "bacon", "eggs") %in% c("spam", "eggs") ## [1] TRUE TRUE FALSE TRUE Note We should stress that these are simple, bytewise comparisons of the cor- responding code points and as such they might not be valid in, for example, nat- ural language processing activities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' compare [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular, German word groß is not deemed equal to gross, although we expect that should be the case, at least in a German language setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, in the rare situations where we read Unicode- unnormalised data (say, not in the NFC form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see [12]), canonically equivalent strings may be considered as different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Partial Matching When only a consideration of the initial part of each string is required, we can call: 6 CHARACTER VECTORS 101 startsWith(c("s", "spam", "spamtastic", "spontaneous", "spoon"), "spam") ## [1] FALSE TRUE TRUE FALSE FALSE Both the above and endsWith are applied elementwisely in case of many search pre- fixes/suffixes, just like in `==`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Partial matching of strings can be performed with charmatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is a each-vs-all ver- sion of startsWith: charmatch(c("s", "sp", "spam", "spams", "eggs", "bacon"), c("spam", "eggs")) ## [1] 1 1 1 NA 2 NA charmatch(c("s", "sp", "spam", "spoo", "spoof"), c("spam", "spoon")) ## [1] 0 0 1 2 NA Note that 0 designates that there was an ambiguity in the matching of a string to a given table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note (*) In Chapter 18, we discuss the more-advanced match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='arg which is (unfortu- nately)frequentlycalledfromwithinotherRfunctions,andinChapter15,wemention the (discouraged) partial matching of argument names in function calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Matching Anywhere Within a String Fixedpatternscanbealsosearchedforanywherewithin characterstringsusing grepl: x <- c("spam", "y spammite spam", "yummy SPAM", "sram") grepl("spam", x, fixed=TRUE) # fixed patterns, as opposed to regexes below ## [1] TRUE TRUE FALSE FALSE Important Note that the order of arguments is like grepl(needle, haystack), not the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, this function is not vectorised with respect to the first argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Determine how can a call to grep(y, x, value=FALSE) and grep(y, x, value=TRUE) be implemented based on grepl and other operations that we are already famil- iar with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note As a curiosity, agrepl performs approximate matching based on Levenshtein’s edit distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This can account for a small number of “typos”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' agrepl("spam", x) ## [1] TRUE TRUE FALSE TRUE (continues on next page) 102 I DEEP (continued from previous page) agrepl("ham", x, ignore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='case=TRUE) ## [1] TRUE TRUE TRUE TRUE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Using Regular Expressions (*) Setting perl=TRUE allows for identifying occurrences of patterns specified by the PCRE2 regular expressions (regexes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' grepl("^spam", x, perl=TRUE) # strings that begin with `spam` ## [1] TRUE FALSE FALSE FALSE grepl("(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='i)^spam|spam$", x, perl=TRUE) # begin or end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' case ignored ## [1] TRUE TRUE TRUE FALSE Note For more details on regular expressions in general, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The ultimate reference for PCRE2 pattern syntax is the man7 page pcre2pattern(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' R also gives ac- cess to ERE-like TRE library (see help("regex")), which is the default one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, we discourage its use, because it is feature-poorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 The list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='files function generates the list of file names in a given directory that matchagivenregularexpression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Forinstance,thefollowinggivesallCSVfilesinsomedirectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='files(".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='./.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='./Projects/teaching-data/r/", r"(\\.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv$)") # or "\\\\.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv$" ## [1] "air_quality_1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv" "anscombe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv" "iris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv" ## [4] "titanic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv" "tooth_growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv" "trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv" ## [7] "world_phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv" Writeasingleregularexpressionthatmatchesfilenamesendingwith“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv”or“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gz”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Also, write a regex that matches CSV files whose names do not begin with “eurusd”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Locating Pattern Occurrences regexpr finds the first occurrence of a pattern in each string: regexpr("spam", x, fixed=TRUE) ## [1] 1 3 -1 -1 ## attr(,"match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='length") ## [1] 4 4 -1 -1 ## attr(,"index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='type") ## [1] "chars" (continues on next page) 7 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='pcre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/current/doc/html/pcre2pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='html 6 CHARACTER VECTORS 103 (continued from previous page) ## attr(,"useBytes") ## [1] TRUE In particular, there is a pattern occurrence starting at the 3th code point of the 2nd string in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, there is no pattern match in the last string (denoted with -1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='length attribute is generally more worthwhile when searching with regular expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' To locate all the matches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', globally, we use gregexpr: # `spam` followed by 0 or more letters, case insensitively gregexpr("(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='i)spam\\\\p{L}*", x, perl=TRUE) ## [[1]] ## [1] 1 ## attr(,"match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='length") ## [1] 4 ## attr(,"index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='type") ## [1] "chars" ## attr(,"useBytes") ## [1] TRUE ## ## [[2]] ## [1] 3 12 ## attr(,"match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='length") ## [1] 8 4 ## attr(,"index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='type") ## [1] "chars" ## attr(,"useBytes") ## [1] TRUE ## ## [[3]] ## [1] 7 ## attr(,"match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='length") ## [1] 4 ## attr(,"index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='type") ## [1] "chars" ## attr(,"useBytes") ## [1] TRUE ## ## [[4]] ## [1] -1 ## attr(,"match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='length") ## [1] -1 ## attr(,"index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='type") (continues on next page) 104 I DEEP (continued from previous page) ## [1] "chars" ## attr(,"useBytes") ## [1] TRUE As we have noted in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2, wrapping the results in a list was a clever choice as the number of matches can obviously vary between strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2, we will take a look at the Map function, which, along with substring introduced below, can aid in getting the most out of such data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Meanwhile, let us just mention that regmatches extracts the matching substrings: regmatches(x, gregexpr("(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='i)spam\\\\p{L}*", x, perl=TRUE)) ## [[1]] ## [1] "spam" ## ## [[2]] ## [1] "spammite" "spam" ## ## [[3]] ## [1] "SPAM" ## ## [[4]] ## character(0) Note (*) Let us consider what happens when a regular expression contains parenthes- ised subexpressions (capture groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' r <- "(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='[^.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ]+)\\\\.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='[^ ]*)" The above regex consists of two such parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The first one is labelled “basename” and is comprised of a number of arbitrary characters except for the space and the dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The second group, named “extension” is a substring of anything but the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' These two are separated by a dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Such a pattern can be used for unpacking space-delimited lists of file names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' z <- "dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gz something_else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='txt spam" regexpr(r, z, perl=TRUE) ## [1] 1 ## attr(,"match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='length") ## [1] 14 ## attr(,"index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='type") ## [1] "chars" ## attr(,"useBytes") ## [1] TRUE (continues on next page) 6 CHARACTER VECTORS 105 (continued from previous page) ## attr(,"capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='start") ## basename extension ## [1,] 1 9 ## attr(,"capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='length") ## basename extension ## [1,] 7 6 ## attr(,"capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='names") ## [1] "basename" "extension" gregexpr(r, z, perl=TRUE) ## [[1]] ## [1] 1 16 ## attr(,"match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='length") ## [1] 14 18 ## attr(,"index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='type") ## [1] "chars" ## attr(,"useBytes") ## [1] TRUE ## attr(,"capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='start") ## basename extension ## [1,] 1 9 ## [2,] 16 31 ## attr(,"capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='length") ## basename extension ## [1,] 7 6 ## [2,] 14 3 ## attr(,"capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='names") ## [1] "basename" "extension" The capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' * attributes give us access to the matches to the individual capture groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', the basename and the extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 (*) Check out the difference between the results generated by regexec and reg- expr as well as gregexec and gregexpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Replacing Pattern Occurrences sub and gsub can replace first and all,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' matches to a pattern: x <- c("spam",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "y spammite spam",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "yummy SPAM",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "sram") sub("spam",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "ham",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' fixed=TRUE) ## [1] "ham" "y hammite spam" "yummy SPAM" "sram" gsub("spam",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "ham",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' fixed=TRUE) ## [1] "ham" "y hammite ham" "yummy SPAM" "sram" 106 I DEEP Note (*) If a regex features some capture groups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' matches thereto can be mentioned not only in the pattern itself,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' but also in the replacement string: gsub("(\\\\p{L})\\\\p{L}\\\\1",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "\\\\1",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "aha egg gag NaN spam",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' perl=TRUE) ## [1] "a egg g N spam" The above matches a letter (it is a capture group),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' another letter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' and the former letter again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Each such palindrome of length 3 is replaced with just the repeated letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 (*)Displaythesourcecodeof glob2rxbycalling print(glob2rx)andstudyhow this function converts wildcards such as file?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='??' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='. * or *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv to regular expressions that can be passed to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 Splitting Strings into Tokens strsplit divides each string in a character vector into chunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This time, though, the search pattern, specifying the token delimiter, is given as the second argument: strsplit(c("spam;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='spam;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='eggs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='bacon", "spam"), ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='", fixed=TRUE) ## [[1]] ## [1] "spam" "spam" "eggs" "" "bacon" ## ## [[2]] ## [1] "spam" 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Other String Operations 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Extracting Substrings substring extracts parts of strings between given character position ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' substring("spammity spam", 1, 4) # from 1st to 4th character ## [1] "spam" substring("spammity spam", 10) # from 10th to end ## [1] "spam" substring("spammity spam", c(1, 10), c(4, 14)) # vectorisation ## [1] "spam" "spam" substring(c("spammity spam", "bacon and eggs"), 1, c(4, 5)) ## [1] "spam" "bacon" Note There is also a replacement (compare Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) version of the above: 6 CHARACTER VECTORS 107 x <- "spam, spam, bacon, and spam" substring(x, 7, 11) <- "eggs" print(x) ## [1] "spam, eggs, bacon, and spam" Unfortunately, the number of characters in the replacement string should not exceed the length of the part being substituted (try “chickpeas” instead of “eggs”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, substring replacement can be written as a composition of substring extraction and concatenation: paste(substring(x, 1, 6), "chickpeas", substring(x, 11), sep="") ## [1] "spam, chickpeas, bacon, and spam" Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Taketheoutputgeneratedbyregexprandapplysubstringtoextractthepattern occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If there is no match in some string, the corresponding output should be NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Translating Characters tolower and toupper can be used to convert between lower and upper case: toupper("spam") ## [1] "SPAM" Note Just like many other string operations in base R, these functions perform very simple character substitutions and they might not be valid in natural language pro- cessing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, groß is not converted to GROSS, which is the correct case folding in German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, chartr translates individual characters: chartr("\\\\", "/", "c:\\\\windows\\\\system\\\\cmd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='exe") # chartr(old, new, x) ## [1] "c:/windows/system/cmd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='exe" chartr("([S", ")]*", ":( :S :[") ## [1] ":) :* :]" In the first line, we replace each backslash with slash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The second example replaces “(”, “[”, and “S” with “)”, “]”, and “*”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 108 I DEEP 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Ordering Strings We have previously mentioned that operators such as `<` and `>=` as well as func- tions like sort, order, rank, but also xtfrm8 are based on the lexicographic ordering of strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' sort(c("chłodny", "hardy", "chladný", "hladný")) ## [1] "chladný" "chłodny" "hardy" "hladný" It is worth noting that the ordering is dependent on the currently selected locale, see Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='getlocale("LC_COLLATE").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, in the Slovak language setting, we would obtain "hardy" < "hladný" < "chladný" < "chłodny".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note Many “structured” data items can be displayed or transmitted as human- readablestrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Inparticular,weknowthat as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numericcanbeusedtoconvertastring to a number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 we will discuss date-time objects such as "1970-01-01 00:00:00 GMT".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We will be processing them with specialised functions such as strptime and strftime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important (*) As we have mentioned, many string operations in base R are not neces- sarily portable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The stringx package [22] defines drop-in, “fixed” replacements there- for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='TheyarebasedontheInternationalComponentsforUnicode(ICU9)library,which is a de facto standard for the processing of Unicode text, and the R package stringi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' # call install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages("stringx") first suppressPackageStartupMessages(library("stringx")) # load the package sort(c("chłodny", "hardy", "chladný", "hladný"), locale="sk_SK") ## [1] "hardy" "hladný" "chladný" "chłodny" toupper("gro\\u00DF") # compare base::toupper("gro\\u00DF") ## [1] "GROSS" detach("package:stringx") # unload the package 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Other Atomic Vector Types (*) We have discussed four vector types: logical, double, character, and list (the lat- ter being a generic-recursive vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' To get the complete picture of the sequence-like 8 See Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 for a use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9 http://site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='icu-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/ 6 CHARACTER VECTORS 109 types in R, let us briefly mention integer, complex, and raw atomic types, so that we are not surprised when we encounter them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Integer Vectors (*) Integer scalars can be input manually by using the L suffix: (x <- c(1L, 2L, -1L, NA_integer_)) # looks like numeric ## [1] 1 2 -1 NA typeof(x) # but is integer ## [1] "integer" Some functions return them in certain contexts10: typeof(1:10) # seq(1, 10) as well, but not seq(1, 10, 1) ## [1] "integer" as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='integer(c(-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1)) # truncate/round towards 0 ## [1] -1 0 1 2 In the vast majority of expressions, integer vectors behave like numeric ones, and are silently coerced to double if need be, so there is no real practical reason to distinguish between them (they are of internal interest, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', when writing C/C++ extensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Chapter 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example: 1L/2L # like 1/2 == 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Note (*) R integers are 32-bit signed types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The double type can store more integers than them (with the maximal contiguously representable integer being 253 vs 231 − 1 in the former case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3): as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='integer(2^31-1) + 1L # 32-bit integer overflow ## Warning in as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='integer(2^31 - 1) + 1L: NAs produced by integer overflow ## [1] NA as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='integer(2^31-1) + 1 == 2^31 # integer+double == double – OK ## [1] TRUE (2^53 - 1) + 1 == 2^53 # OK ## [1] TRUE (2^53 + 1) - 1 == 2^53 # lost due to FP rounding, left result is 2^53 - 1 ## [1] FALSE 10 Actually, 1:10returnsanintegervectorinacompact(ALTREP,see[39])form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='comparetheresultsofthe call to “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Internal(inspect(1:10))” and “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Internal(inspect(seq(1, 10, 1)))”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This way, the whole vector does not have to be allocated which saves memory and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' At the R level, though, it behaves as any other integer (numeric) sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 110 I DEEP Note Since R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0, there is support for vectors longer than 231 − 1 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' As there areno64-bitintegersinR,theseareindexedbydoublesanyway(aswehavebeendoing all this time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Interestingly, x[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9] is the same as x[1] and x[-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9] means x[-1] (trun- cation of the fractional part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is why the notation like x[length(x)*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2] works re- gardless of whether the length of x is a multiple of 5 or not, which is neat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Raw Vectors (*) Vectors of type raw can store bytes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', unsigned 8-bit integers, whose range is 0-255 (there are no raw NAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='raw(c(-1, 0, 1, 2, 0xc0, 254, 255, 256, NA)) ## Warning: out-of-range values treated as 0 in coercion to raw ## [1] 00 00 01 02 c0 fe ff 00 00 They are displayed as two-digit hexadecimal (base-16) numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also note that we may enter such numbers using the “0x” prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' There are only few functions that deal with such vectors: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', readBin, charToRaw, and rawToChar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Complex Vectors (*) We can also play with vectors of type complex, with “1i” representing the imaginary unit, √−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Complex numbers appear in quite a few engineering or scientific applic- ations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', in physics, electronics, or signal processing and are (at least: should be) part of many university-level subjects or textbooks in mathematics11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' c(0, 1i, pi+pi*1i, NA_complex_) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000i 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1416+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1416i NA Apart from the basic operators, mathematical and aggregation functions, procedures like fft, solve, qr, or svd can be fed with or produce such data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For more details, see help("complex") and some matrix examples in Chapter 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Exercises Exercises marked with (*) might require tinkering with regular expressions or third- party R packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 11 Even the statistics/machine learning oriented ones, because of their heavy use of numerical comput- ing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', [14, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6 CHARACTER VECTORS 111 Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Answer the following questions: How many characters are there in the string "ab\\n\\\\\\t\\\\\\\\\\""?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What about "-{ab\\n\\\\\\ t\\\\\\\\\\"-)}-"?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What is the result of calling “paste(NA, 1:5, collapse="")”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Whatisthemeaningofthefollowing sprintfformatstrings: "%s", "%20s", "%-20s", "%f", "%g", "%e", "%5f", "%5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2f%%", "%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2f", "%0+5f", and "[%+-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2f]"?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What is the difference between regexpr and gregexpr?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What does “g” in the latter name stand for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What is the result of a call to “grepl(c("spam", "spammity spam", "aubergines"), "spam")”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Is it always the case that “"Aaron" < "Zorro"”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If x is a character vector, is “x == x” always equal to TRUE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If x and y are character vectors of lengths n and m, respectively, what is the length of the output of “match(x, y)”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If x is a named vector, why there is a difference between “x[NA]” and “x[NA_character_]”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What is the difference between “x == y” and “x %in% y”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 Let x, y, and z be atomic vectors and a and b be single strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Generate the same results as “pastena(x, collapse=b)”, “pastena(x, y, sep=a)”, “pastena(x, y, sep=a, collapse=b)”, “pastena(x, y, z, sep=a)”, “pastena(x, y, z, sep=a, collapse=b)”, assuming that pastena is a version of paste (which we do not have) that handles missing data in a way consistent with most other functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 Based on list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='files and glob2rx, generate the list of all PDFs on your com- puter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Then, using file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='size filter out the files smaller than 10 MiB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 Read a text file that stores a long paragraph of some banal prose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Concatenate all the lines to form a single, long string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Using strwrap and cat, output the paragraph on the console, nicely formatted to fit an aesthetic width, say, 60 text columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='10 (*) Implement your own, simplified version of basename and dirname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='11 (*) Implement an operation similar to trimws using the functions introduced in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='12 (*) Write a regex that extracts all words from each string in a given character vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='13 (*) Write a regex that extracts, from each string in a character vector, all: integers numbers (signed or unsigned), floating-point numbers, numbers of any kind (including those in scientific notation), #hashtags, 112 I DEEP email@addresses, hyperlinks of the form http://… and https://….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='14 (*) What does 42i, 42L, and 0x42 stand for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='15 (*) Check out stri_sort in the stringi package (or sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character in stringx) for a way to obtain an ordering like "a1" < "a2" < "a10" < "a11" < "a100".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='16 (*) In sprintf, the formatter "%20s" means that if a string is less than 20 bytes long,theremainingbyteswillbereplacedwithspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='OnlyforASCIIcharacters(Englishletters, digits, some punctuation marks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') it is true that one character is represented by 1 byte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Other Unicode code points can take up between 2 and 4 bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' cat(sprintf(".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='.%6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='.", c("abc", "1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='<", "aßc", "ąß©")), sep="\\n") # aligned?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ## .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='. abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='. ## .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='. 1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='. ## .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='. aßc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='. ## .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='.ąß©.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='. Use the stri_pad function from the stringi package to align the strings aesthetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Altern- atively, check out sprintf from stringx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='17 (*) Implement an operation similar to stri_pad from stringi using the func- tions introduced in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7 Functions R is a functional language, where functions play first fiddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Each action we perform reduces itself to a call to some function, or a combination thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' So far we have been tinkering with dozens of available functions which arepart of base R, with only few exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They constitute the essential vocabulary that everyone must be able to speak fluently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Any operation, be it sum, sqrt, or paste, when fed with a number of arguments, gen- erates some (hopefully useful) return value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' sum(1:10) # invoking `sum` on a specific argument ## [1] 55 From a user’s perspective, each function is merely a tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' To achieve a goal at hand, we do not really have to care about what is going on under its hood, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', how the inputs are actually being transformed so that, after a couple of nanoseconds or hours, we can enjoy what has been yielded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is very convenient: all we need to know is the function’s specification which can be stated, for example, informally, in plain Polish or Malay, in its help page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In this chapter, we will learn how to write our own functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The use of this skill is a good development practice when we expect that some operations are to be executed many times but perhaps on different data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, some R functions are meant to invoke other functions, for instance on every ele- ment in a list or every section of a data frame grouped by a qualitative variable, so it is good to learn know how we can specify a custom operation to be propagated there- over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Given some objects (whatever): x1 <- runif(16) x2 <- runif(32) x3 <- runif(64) when we want to apply the same action on different data, say, compute the root mean square, instead of re-typing almost identical expressions (or a bunch of them) over and over again: sqrt(mean(x1^2)) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6545 (continues on next page) 114 I DEEP (continued from previous page) sqrt(mean(x2^2)) # the same second time - borderline okay ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='56203 sqrt(mean(x3^2)) # tedious, barbarous, and error-prone ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='57206 we can generalise the operation to any object like x: rms <- # bound what follows to name `rms` function(x) # a function that takes one parameter, `x` sqrt(mean(x^2)) # expression to transform the input to yield output and then re-use it on different concrete data instances: rms(x1) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6545 rms(x2) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='56203 rms(x3) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='57206 or even combine it with other function calls: rms(sqrt(c(x1, x2, x3)))^2 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='50824 Important Does writing your own functions equal reinventing the wheel?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Can everything be found on the internet these days (including on Stack Overflow, GitHub, or CRAN)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Luckily, this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Otherwise, data analysts’, researchers’, and developers’ lives could be considered monotonous, dreary, and uninspiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Plus, sometimes it is much quicker to write a function from scratch than to get through the whole garbage dump from where, only occasionally, we can dig out some pearls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Not to mention the self-educative side: we become better programmers by crunching those exercises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We are advocating for minimalism here, remember?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This and many more other important issues in function design will be reflected upon in Chapter 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7 FUNCTIONS 115 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Creating and Invoking Functions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Anonymous Functions Functions are usually created by means of the following notation: function(args) body First, args is a (possibly empty) list of comma-separated parameter names which are supposed to act as input variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Second, body is a single R expression which will be evaluated when the function is called.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The value that this expression yields will constitute the function’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example, here is a definition of a function which takes no inputs and generates a constant output: function() 1 ## function() 1 Wethuscreatedafunctionobject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='However,ithasdisappearedimmediatelythereafter, as we have not used it at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Any function, say, f can be invoked, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', evaluated on concrete data, by using the nota- tion f(arg1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', argn), where “arg1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', argn” are the arguments to be passed to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (function() 1)() # invoking f like f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' here, no arguments are expected ## [1] 1 Only now we have obtained a return value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note (*) Calling typeof on a function object will report "closure" (for user-defined functions), "builtin", or "primitive" (for some built-in, base ones), for reasons that we explain in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' typeof(function() 1) ## [1] "closure" 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Named Functions Function objects can be bound with names so that they can be referred to multiple times: 116 I DEEP one <- function() 1 # one <- (function() 1) We created an object named one (we use bold font to indicate that it is of type function, because functions are so important in R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We are very familiar with such a notation, as not since yesterday we are used to writing “x <- 1” etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Invoking one, which can be done by writing one(), will yield a return value: one() # (function() 1)() ## [1] 1 This output can be used in further computations, for instance: 0:2 - one() # 0:2 - (function() 1)(), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', 0:2 - 1 ## [1] -1 0 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Passing Arguments To Functions Functions with no arguments are kind of boring, thus let us distil a more serious op- eration: concat <- function(x, y) paste(x, y, sep="") Here we have created a mapping whose aim is to concatenate two objects by means of a specialised call to paste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Yours faithfully pleads guilty to multiplying entities need- lessly, because it should not be a problem for anyone to write paste(x, y, sep="") each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Yet, ‘tis merely an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The concat function has two parameters, “x” and “y”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, calling it will require the provision of two arguments, which we put within round brackets and separate from each other by commas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' u <- 1:5 concat("spam", u) # i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', concat(x="spam", y=1:5) ## [1] "spam1" "spam2" "spam3" "spam4" "spam5" Important Notice the distinction: parameters (also called formal arguments) are ab- stract, general, or symbolic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' “something, anything that will be put in place of x when the function is invoked”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' By contrast, arguments (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' actual parameters) are con- crete, specific, and real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' During the above call, x in the function’s body is precisely "spam", and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, the u object from the caller’s environment is seen under the name y there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Most of the time (however, see Chapter 18), it is best to think of the function as being fed not with u per se, but the value that u is bound to, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', “1:5”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7 FUNCTIONS 117 Also: x <- 1:5 y <- "spam" concat(y, x) # concat(x="spam", y=1:5) ## [1] "spam1" "spam2" "spam3" "spam4" "spam5" This is still a call to equivalent to concat(x=y, y=x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The argument x is being assigned with the value of y from the calling environment, "spam".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Yes, one x is not the same as the other x, and which is which is unambiguously defined by the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Under- standing and being able to manipulate such abstractions is basic logic and common sense that everyone should master.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Write a function called standardise that takes a numeric vector x as argument and returns its standardised version, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', from each element in x subtract the sample arithmetic mean and then divide it by the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note Recall from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 that, syntactically speaking, the following are perfectly valid alternatives to the positionally-matched call concat("spam", u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' concat(x="spam", y=u) concat(y=u, x="spam") concat("spam", y=u) concat(u, x="spam") concat(x="spam", u) concat(y=u, "spam") However,thelasttwoshouldparticularlybeavoided,forthesakeofthereaders’sanity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is best to provide positionally-matched arguments before the keyword-based ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, we introduce the (overused) forward-pipe operator, `|>`, which enables the above to be written as “"spam" |> concat(u)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Grouping Expressions with Curly Braces, `{` We have been informed that a function’s body is a single R expression whose evalu- ated value is passed to the user as its output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This may sound restrictive and contrast with what we have experienced so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Rarely are we faced with such simple comput- ing tasks and we have already seen R functions performing quite sophisticated oper- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It turns out that, grammatically, a single R expression can be arbitrarily complex (Chapter 15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' we can use curly braces to group many calls that are to be evaluated one after another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance: 118 I DEEP { cat("first expression\\n") cat("second expression\\n") # .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' cat("last expression\\n") } ## first expression ## second expression ## last expression Notethatweusedfourspacestovisuallyindenttheconstituentsforgreaterreadability (somedevelopersprefertabsoverspaces,othersfindtwoorthreespacesmoreurbane, but we do not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This single (compound) expression can now play a role of a function’s body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important The last expression evaluated in a curly-braces delimited block will be con- sidered its the output value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- { 1 2 3 # <--- last expression: will be taken as the output value } print(x) ## [1] 3 Note (*)Theabovecodeblockcanalsobewrittenmoreconciselybyreplacingnewlines with semicolons, although with perhaps some loss in readability: {1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3} ## [1] 3 In Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4, we will give a few more details about `{`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Here is a version of the above concat function which takes care of a more Chapter 2-style missing values’ propagation: concat <- function(a, b) { z <- paste(a, b, sep="") z[is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='na(a) | is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='na(b)] <- NA_character_ z # last expression in the block – return value } 7 FUNCTIONS 119 Example calls: concat("a", 1:3) ## [1] "a1" "a2" "a3" concat(NA_character_, 1:3) ## [1] NA NA NA concat(1:6, c("a", NA_character_, "c")) ## [1] "1a" NA "3c" "4a" NA "6c" Letusappreciatethefactthatwecouldkeepthecodebriefthanksto pasteand`|`implementing the recycling rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Write a function called normalise that takes a numeric vector x and returns its version shifted and scaled to the [0, 1] interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' To do so, from each element subtract the sample minimumandthendivideitbytherange,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=',thedifferencebetweenthemaximumandthemin- imum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Avoid computing min(x) twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Write a function that applies the robust standardisation of a numeric vector: sub- tract the median and divide it by the median absolute deviation, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4826 times the median of the absolute differences between the values and their median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note R is an open-source (free, libre) project – users are not only encouraged to run the software for whatever the purpose, but also study and modify its source code without any restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This applies both to functions that we have authored ourselves: print(concat) ## function(a, b) ## { ## z <- paste(a, b, sep="") ## z[is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='na(a) | is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='na(b)] <- NA_character_ ## z # last expression in the block – return value ## } ## and to the routines that are part of base R or any other extension packages: print(union) ## function (x, y) ## { ## u <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='vector(x) ## v <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='vector(y) ## unique(c(u, v)) ## } ## ## Nevertheless, some functionality might be implemented in a compiled programming 120 I DEEP language such as C, C++, or Fortran;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' notice a call to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Internal in the source code of paste, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Primitive in list, or .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Call in runif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, we will sometimes have to dig a little bit deeper to access the underlying source code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Chapter 14 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Functional Programming R is a functional programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' As such, it shares a number of common fea- tures with other languages that emphasise on the role of function manipulation in software development (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', Common Lisp, Scheme, OCaml, Haskell, Clojure, F#).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us explore them now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Functions are Objects R functions were given the right to a fair go;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' they are what we refer to as first-class cit- izens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In other words, our interaction with them is not limited to their invocation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' we treat them as any other language objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Namely, they can be: stored inside list objects: list(identity, nrow, sum) # a list with three elements of type function ## [[1]] ## function (x) ## x ## ## ## ## [[2]] ## function (x) ## dim(x)[1L] ## ## ## ## [[3]] ## function (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm = FALSE) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Primitive("sum") This is possible owing to the fact that lists, as we recall, can embrace R objects of any kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' created and then called inside another function’s body: euclidean_distance <- function(x, y) { square <- function(z) z^2 # auxiliary/internal/helper function (continues on next page) 7 FUNCTIONS 121 (continued from previous page) sqrt(sum(square(x-y))) # square root of the sum of squares } euclidean_distance(c(0, 1), c(1, 0)) # example call ## [1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4142 This is why we tend to classify functions as representatives of recursive types (com- pare is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='recursive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' passed as arguments to other operations: # Replaces missing values with a given aggregate # of all non-missing elements: fill_na <- function(x, filler_fun) { missing_ones <- is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content="na(x) # otherwise, we'd call is." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='na twice replacement_value <- filler_fun(x[!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='missing_ones]) x[missing_ones] <- replacement_value x } fill_na(c(0, NA_real_, NA_real_, 2, 3, 7, NA_real_), mean) ## [1] 0 3 3 2 3 7 3 fill_na(c(0, NA_real_, NA_real_, 2, 3, 7, NA_real_), median) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 We call these higher-order functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note More advanced techniques, which we will discuss later (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', closures, lazy eval- uation, metaprogramming, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ), will let the functions be: returned as other function’s outputs (sec:to-do), equipped auxiliary data (sec:to-do), generated programmatically on the fly (sec:to-do), and modified at runtime (sec:to-do).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Below we review some noteworthy higher-order functions, in particular: do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='call and Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Many other ones will be introduced in due course or are left as an educative exer- cise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 122 I DEEP 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Calling on Precomputed Arguments with do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='call The notation like f(arg1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', argn) has no monopoly over how we are supposed to call a function on a specific sequence of comma-delimited arguments: the latter do not have to be hardcoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Here is an alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We can first prepare a number of objects to be passed as f’s inputs, wrap them in a list l, and then invoke do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='call(f, l) to get the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' words <- list( c("spam", "bacon", "eggs"), c("buckwheat", "quinoa", "barley"), c("ham", "spam", "spam") ) do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='call(paste, words) # paste(words[[1]], words[[2]], words[[3]]) ## [1] "spam buckwheat ham" "bacon quinoa spam" "eggs barley spam" do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='call(cbind, words) # column-bind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' returns a matrix (explained later) ## [,1] [,2] [,3] ## [1,] "spam" "buckwheat" "ham" ## [2,] "bacon" "quinoa" "spam" ## [3,] "eggs" "barley" "spam" do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='call(rbind, words) # row-bind (explained later) ## [,1] [,2] [,3] ## [1,] "spam" "bacon" "eggs" ## [2,] "buckwheat" "quinoa" "barley" ## [3,] "ham" "spam" "spam" Note that the length and content of the list passed as the 2nd argument of do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='call can be arbitrary (possibly unknown at the time of writing the code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' See Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 for more use cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', ways to concatenate a list of data frames (perhaps produced by some complex chain of commands) into a single data frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If elements of the list are named, they will be matched to the corresponding keyword arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- 2^(seq(-2, 2, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=101)) plot_opts <- list(col="red", lty="dashed", type="l") do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='call(plot, c(list(x, log2(x), xlab="x", ylab="log2(x)"), plot_opts)) ## (the displaying of the plot has been suppressed) Note that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', plot_opts can now be reused in further calls to graphical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is very convenient as it avoids repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Common Higher-Order Functions There is an important class of higher-order functions that allow us to apply custom operations on consecutive elements of sequences without relying on loop-like state- ments, at least explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They can be found in all functional programming languages 7 FUNCTIONS 123 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', Lisp, Haskell, Scala) and have been ported to various add-on libraries (functools in Python, more recent versions of the C++ Standard Library, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') or frameworks (Apache Spark and the like).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Their presence reflects the obvious truth that some kinds of operations occur more frequently than other ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular: Map calls a function on each element of a sequence in order to transform: – their individual components (just like sqrt, round, or the unary `!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='` operator in R), or – the corresponding elements of many sequences so as to vectorise a given op- eration elementwisely (compare the binary `+` or paste), Reduce (also called accumulate) applies a binary operation to combine consecutive elementsinasequence,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=',togeneratetheaggregates,like,totally(compare sum, prod, all, max) or cumulatively (compare cumsum, cummmin), Filter creates a subset of a sequence that is comprised of elements that enjoy a given property (which we typically achieve in R by means of the `[` operator), Find locates the first element that fulfils some logical condition (compare which), and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Below we will only focus on the Map function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The inspection of the remaining ones is left as an exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is because, oftentimes, we can be better-off with their more R-ish versions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', using the subsetting operator, `[`).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Vectorising Functions with Map In data-centric computing, we are frequently faced with tasks that involve processing eachandeveryelementinasequenceindependently,oneafteranother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Suchusecases can benefit from vectorised operations like those discussed in Chapter 2, Chapter 3, and Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Most of the functions that we have introduced in the preceding parts, unfortunately, cannot be applied on lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, if we try calling sqrt on a list, we will get an error, even if it is a list of numeric vectors only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' One way to compute the square root of all elements would be to invoke sqrt(unlist(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=')).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is a go-to approach if we wish to treat all the list’s elements as one sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' But this comes at a price of losing the list’s structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Wehavealsodiscussedsomeoperationsthatarenotvectorisedwithrespecttoalltheir arguments, even though they could have been designed this way, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', grepl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The Map function1 applies an operation on each element in a vector or the correspond- ing elements in a number of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In many situations, it may be used as a more elegant alternative to for loops that we will introduce in the next chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1 Yes, the author is aware that Map was implemented using the slightly more primitive mapply, but we are not fond of the latter’s having the SIMPLIFY argument set to TRUE by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 124 I DEEP First2, a call to Map(f, x) yields a list whose i-th element is equal to f(x[[i]]) (recall that `[[` works on atomic vectors too).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example: x <- list( # an example named list x1=1:3, x2=seq(0, 1, by=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25), x3=c(1, 0, NA_real_, 0, 0, 1, NA_real_) ) Map(sqrt, x) # x is named, hence the result will be named too ## $x1 ## [1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4142 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7321 ## ## $x2 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='50000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='70711 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='86603 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00000 ## ## $x3 ## [1] 1 0 NA 0 0 1 NA Map(length, x) ## $x1 ## [1] 3 ## ## $x2 ## [1] 5 ## ## $x3 ## [1] 7 unlist(Map(mean, x)) # compute three aggregates, convert to an atomic vector ## x1 x2 x3 ## 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 NA Map(function(n) round(runif(n, -1, 1), 1), c(2, 4, 6)) # x is atomic now ## [[1]] ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 ## ## [[2]] ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 ## ## [[3]] ## [1] -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 Next, we can vectorise a given function over a number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' A call to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', Map(f, x, y, z) results in a list whose i-th element is equal to f(x[[i]], y[[i]], z[[i]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Just like in case of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', paste, recycling rule will be applied if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 This use case scenario can also be programmed using lapply;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' lapply(x, f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') is equivalent to Map(f, x, MoreArgs=list(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=')).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7 FUNCTIONS 125 For example, the following generates list(seq(1, 6), seq(11, 13), seq(21, 29)): Map(seq, c(1, 11, 21), c(6, 13, 29)) ## [[1]] ## [1] 1 2 3 4 5 6 ## ## [[2]] ## [1] 11 12 13 ## ## [[3]] ## [1] 21 22 23 24 25 26 27 28 29 Moreover, we can get list(seq(1, 40, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=10), seq(11, 40, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=5), seq(21, 40, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=10), seq(31, 40, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=5)) by calling: Map(seq, c(1, 11, 21, 31), 40, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=c(10, 5)) ## [[1]] ## [1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3333 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6667 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3333 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6667 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3333 ## [9] 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6667 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 ## ## [[2]] ## [1] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='50 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='75 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 ## ## [[3]] ## [1] 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='000 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='111 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='222 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='333 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='444 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='556 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='667 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='778 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='889 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='000 ## ## [[4]] ## [1] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='50 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='75 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 Note If we have some additional arguments to be passed to the function applied (which the function does not have to be vectorised over), we can wrap them inside a separate list and toss it via the MoreArgs argument (à la do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='call).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' unlist(Map(mean, x, MoreArgs=list(na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=TRUE))) # mean(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=TRUE) ## x1 x2 x3 ## 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Alternatively, we can always construct a custom anonymous function: unlist(Map(function(xi) mean(xi, na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=TRUE), x)) ## x1 x2 x3 ## 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 126 I DEEP Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Hereisanexamplelistoffiles(seeourteachingdatarepository3)withdailyForex rates: file_names <- c( "euraud-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv", "eurgbp-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv", "eurusd-20200101-20200630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv" ) Call Map to read each dataset with scan and determine the minimal, mean, and maximal value in each series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 Implement your own version of the Filter function based on a call to Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Accessing Third-Party Functions When we indulge in the writing of a software piece, a few questions naturally arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Is the problem we are facing fairly complex?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Has it already been successfully addressed in its entirety?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If not, can it, or its parts, be split into manageable chunks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Can it be constructed based on some readily available nontrivial components?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' A smart developer is independent, but knows when to stand on the shoulders to cry on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us explore some ways in which we can reuse the existing function libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Using R Packages Most contributed R extensions come in the form of the so-called add-on packages, which can include: reusable code (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', new functions), data (which we can exercise on), documentation (manuals, vignettes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 for some more and [45] for all the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Most packages are published in the moderated repository that is part of the Compre- hensive R Archive Network (CRAN4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, there are also other popular sources such as Bioconductor5 which specialises in bioinformatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' To fetch a package pkg from a repository (CRAN by default;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see, however, the repos argument), we call install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages("pkg").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews/teaching-data/tree/master/marek 4 https://cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='r-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/ 5 https://bioconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/ 7 FUNCTIONS 127 A call to library("pkg") loads an indicated package and makes its exported objects available to the user (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', attaches it on the search list;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see sec:to-do).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, in one of the previous chapters, we have mentioned the gsl package: # call install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages("gsl") first library("gsl") # load the package poch(10, 3:6) # calls gsl_sf_poch() from GNU GSL ## [1] 1320 17160 240240 3603600 Here, poch is an object exported by package gsl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If we did not call library("gsl"), trying to access the former would result in an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We could also have accessed the above function without attaching it onto the object search list by using the pkg::object syntax, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', gsl::poch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 Use the find function to determine which packages define the following objects: mean, var, find, and Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Recall from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 where such information can be found in these objects’ manual pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note For more information about any R extension, call help(package="pkg").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, it is a good idea to visit the package’s CRAN entry at an address like https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/package=pkg to access some additional information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', vignettes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see also vignette(package="pkg")).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Why waste our time and energy by querying a web search engine that will lead us to some (usually low-quality) middleman when you can acquire authoritative knowledge directly from the source?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, it is worth exploring various CRAN Task Views6 that group the packages into topics such as Genetics, Graphics, and Optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' These are edited by experts in their relevant fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important Frequently, R packages are written in their respective authors’ free time, many of whom are volunteers/public servants/enthusiasts who are neither paid for doing this nor it is part of the so-called their job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' You can show appreciation for their generosity by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', spreading the word about their software by citing them in public- ations (see citation(package="pkg")), talking about them during lunch time, or men- tioning them in (un)social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' You can also help them improve the existing code base by reporting bugs, polishing documentation, proposing new features, or clean- ing up the redundant fragments of their APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some readers will become one of them someday (when they will come up with something useful for our community).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6 https://cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='r-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/web/views/ 128 I DEEP Default Packages Note that the always-on package base is a must-have that provides us with the most crucial functions (vector addition, c, Map, library).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Certain other packages are also loaded by default: getOption("defaultPackages") ## [1] "datasets" "utils" "grDevices" "graphics" "stats" ## [6] "methods" Although this list can – technically speaking – be changed, in this book we assume that the above are always attached, because it is reasonable to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is why in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, there was no need to call, for example, library("stats") before refer- ring to the var and sd functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' On a side note, grDevices and graphics will be discussed in sec:to-do and methods will be mentioned in sec:to-do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' datasets brings a few example R objects that we can exercise our skills on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Functions from utils, graphics, and stats, on the other hand, already appeared here and there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Source vs Binary Packages (*) R is a free and open project, therefore its packages are published primarily in the source form – so that anyone can study how they work and improve them or reuse parts thereof in different projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If we call install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages("path", repos=NULL, type="source"), we should be able toinstallapackagefromsources: pathcaneitherbepinpointingadirectoryorasource tarball (see help("untar"), most often as a compressed pkg_version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='tar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='gz file).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Notethattype="source"isthedefaultunlessoneisonW****wsorsomem**OSboxes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see getOption("pkgType").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is because these two might require additional build tools to be present in the system, especially if a package features C, C++, or Fortran code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Chapter 14 and Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 of [47]: Rtools7 on W****ws, Xcode Command Line Tools8 on m**OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Because of these systems’ being less developer-oriented, as a courtesy to their users, CRAN also distributes the platform-specific binary versions of the packages (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='zip or .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='tgz files).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages will try to fetch them by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 It is very easy to fetch a package’s source directly from GitLab or GitHub, which arequitepopularhostingplatformsthesedays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Atthetimeofwritingthis,therelevantlinkswere, respectively: https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/user/repo/-/archive/branch/repo-branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='zip https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/user/repo/archive/branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='zip 7 https://cran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='r-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/bin/windows/Rtools/ 8 https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/xcode/resources/ 7 FUNCTIONS 129 For example, to download the contents of the master branch in the repository rpackagedemo owned by gagolews, we can call: f <- tempfile() # temporary file name - download destination download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='file("https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews/rpackagedemo/archive/master.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='zip", destfile=f) Next, the contents can be extracted with unzip: t <- tempdir() # temporary directory to extract the files to (d <- unzip(f, exdir=t)) # returns extracted file paths The path where the files were extracted can be passed to install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages: install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages(dirname(d)[1], repos=NULL, type="source") file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='remove(c(f, d)) # clean up Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='10 Use the git2r package to clone the git repository located at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/ gagolews/rpackagedemo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='git and install the package published therein from within the current R session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Managing Dependencies (*) The currently-installed add-on packages may be upgraded to their most recent ver- sions available on CRAN (or other indicated repository) by calling update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' As a general rule, the more experienced developers we become, the less excited we get aboutthenew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Sure,bugfixesandsomewell-thoughtofadditionalfeaturesareusually welcome, but just we wait until an updated package API for the n-th time, 𝑛 ≥ 2, breaks our program that used to work flawlessly for so long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, when designing software projects (see Chapter 9 for more details), it is essen- tial that we ask ourselves the ultimate question: do we really need to import that pack- age with lots of dependencies from which we will just use only about 3–5 functions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Wouldn’t it be better to write our own version of some functionality (and learn some- thing new, exercise our brain, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') or call a mature terminal-based tool?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Otherwise, as all the historical versions of all the packages are archived on CRAN9, some software dependency management can easily be conducted by storing differ- ent version of packages in different directories (only one version of a package can be loaded at a time though).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This way, wecan createsome sort of an isolated environment for the add-ons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' To fetch the locations where packages are sought (in this very order), call: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='libPaths() ## [1] "/home/gagolews/R/x86_64-pc-linux-gnu-library/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2" (continues on next page) 9 https://cran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='r-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/src/contrib/Archive/ 130 I DEEP (continued from previous page) ## [2] "/usr/local/lib/R/site-library" ## [3] "/usr/lib/R/site-library" ## [4] "/usr/lib/R/library" Thesamefunctioncanbeusedtoaddnewfolderstothesearchpath;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='seealsotheenvir- onment variable R_LIBS_USER (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', help("Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='setenv")).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages func- tion will honour them as target directories, see its lib parameter for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover,thepackagesmaydepositsomeauxiliarydataontheuser’smachine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='There- fore, it might be a good idea to set the following directories (via the corresponding environment variables) as relative to the current project: tools::R_user_dir("pkg", "data") # R_USER_DATA_DIR ## [1] "/home/gagolews/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='local/share/R/pkg" tools::R_user_dir("pkg", "config") # R_USER_CONFIG_DIR ## [1] "/home/gagolews/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='config/R/pkg" tools::R_user_dir("pkg", "cache") # R_USER_CACHE_DIR ## [1] "/home/gagolews/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='cache/R/pkg" 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Calling External Programs Many tasks can naturally be accomplished by calling external programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Such an ap- proachisparticularlynaturalonUnix-likesystems,whichclassicallyfollowamodular, minimalistic design patterns: there are many tools at a developer’s hand and each tool is specialised at solving a single, well-defined problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Apart from the many standard Unix commands10, we can consider, for example: pandoc11 converts documents between markup formats, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', Markdown, reStruc- turedText, LaTeX, LibreOffice Writer, EPUB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' pdflatex, xelatex, and lualatex compile LaTeX documents to PDF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' convert (from ImageMagick12) applies various operations on bitmap graphics (scaling, cropping, conversion between formats);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' graphviz13 and PlantUML14 can be used to create various graphs and diagrams;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' jupyter-nbconvert converts Jupyter15 notebooks (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) to other formats such as LaTeX, HTML, Markdown, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' python,{command}perl,…canbecalledtoperformtasksthatcanbeexpressedmore easily in languages other than R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10 https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/wiki/List_of_Unix_commands 11 https://pandoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/ 12 https://imagemagick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/ 13 https://graphviz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/ 14 https://plantuml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/ 15 https://jupyter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/ 7 FUNCTIONS 131 and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Good news is that R not only can be called from the shell (in an interactive or batch mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2), but also it can serve well as a glue language itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The system2 function can be used to invoke any system command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Communication between such programs can be done by means of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', intermediate text, JSON, CSV, XML, or any other files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The stdin, stdout, and stderr arguments can be used to con- trol the redirection of the standard I/O streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' system2("pandoc", "-s input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='md -o output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='html") system2("bash", "-c \'for i in `seq 1 2 10`;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' do echo $i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' done\'", stdout=TRUE) ## [1] "1" "3" "5" "7" "9" system2("python3", "-", stdout=TRUE, input=c( "import numpy as np", "print(repr(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='arange(5)))" )) ## [1] "array([0, 1, 2, 3, 4])" Note that the current working directory can be read and changed by means of a call to getwd and setwd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is the directory from where the current R session was started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important Relying on system2 assumes that the commands referred to are available onthetargetplatform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Hence,itmightnotbeportable,unlessadditionalassumptions are made (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', that a user runs some Unix system, that certain libraries are installed therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We strongly recommend GNU/Linux or FreeBSD for both software devel- opment and production use, as they are free, open, developer-friendly, user-loving, reliable, ethical, and sustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 A Note on Interfacing C, C++, Python, Java, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (*) Most stand-alone data processing algorithms are implemented in compiled, slightly lower-level programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This usually makes them faster and more re- usable in other environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, it is often the case that an industry- standard library is written in very portable C, C++, or Fortran and has some bindings availableforeasieraccess fromwithin R,Python, Julia,etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='This isthe casewith FFTW, LIBSVM, mlpack, OpenBLAS, ICU, and GNU GSL, amongst many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For basic ways to interact with such compiled code, see Chapter 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, the rJava package can be used to dynamically create JVM objects and access their fields and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Similarly, reticulate can be used to access Python objects, in- cluding numpyarraysand pandasdataframes(butseealsothe rpy2packageforPython).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important We should not feel obliged to use R in all the parts of a data pro- 132 I DEEP cessing pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some activities can be expressed more naturally in other lan- guages/environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', parse raw data and create an SQL database in Python, but visualise it in R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We can use other tools as the glue language (including R, Python, or Bash) that will steer the data flow in the right direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Exercises Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='11 Answer the following questions: What is the result of “x <- 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- function(x) x^2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x(x)”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' How to write a function that returns two objects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What is a higher-order function?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What are the use cases of do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='call?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Why a call to Map is not necessary in the expression “Map(paste, x, y, z)”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What is the difference between Map(mean, x, na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=TRUE) and Map(mean, x, More- Args=list(na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=TRUE))?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What do we mean when we write stringx::sprintf?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' How to get access to the vignettes (tutorials, FAQs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='table and dplyr pack- ages?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Why perhaps 95% of R users would just googleit and what is sub-optimal about this strategy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What is the difference between a source and a binary package?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' How to update the base package?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' How to assure that we will always run an R session with only specific versions of a set of packages?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='12 Write a function that computes the Gini index of a vector of positive integers x, which, assuming 𝑥1 ≤ 𝑥2 ≤ … ≤ 𝑥𝑛, is equal to: 𝐺(𝑥1, … , 𝑥𝑛) = ∑𝑛 𝑖=1(𝑛 − 2𝑖 + 1)𝑥𝑖 (𝑛 − 1) ∑𝑛 𝑖=1 𝑥𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='13 Implement a function between(x, a, b) that verifies whether each element in x is in the [a, b] interval or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Return a logical vector of the same length as x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Make sure the function is correctly vectorised with respect to all the arguments and handles missing data correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='14 Write your own version of the strrep function called dup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7 FUNCTIONS 133 dup <- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' dup(c("a", "b", "c"), c(1, 3, 5)) ## [1] "a" "bbb" "ccccc" dup("a", 1:3) ## [1] "a" "aa" "aaa" dup(c("a", "b", "c"), 4) ## [1] "aaaa" "bbbb" "cccc" Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='15 Given a list x, generate its sublist with all the elements equal to NULL removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='16 Implement your own version of the built-in sequence function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='17 Using Map, how can we generate window indexes like: ## [[1]] ## [1] 1 2 3 ## ## [[2]] ## [1] 2 3 4 ## ## [[3]] ## [1] 3 4 5 ## ## [[4]] ## [1] 4 5 6 Writeafunction windows(k, n)thatyieldskindexwindowswithelementsbetween1andn(the above example is for k=3 and k=6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='18 Implement a function movstat(f, x, k) that computes, using Map, a given ag- gregate f of each k consecutive elements in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance: movstat <- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- c(1, 3, 5, 10, 25, -25) # example data movstat(mean, x, 3) # 3-moving mean ## [1] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3333 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3333 movstat(median, x, 3) # 3-moving median ## [1] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0000 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3333 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3333 Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='19 Write a function to extract all q-grams, q ≥ 1, from a given character vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Return a list of character vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For examples, 2-grams (bigrams) in "abcd" are: "ab", "bc", “cd”`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='20 Recodeacharactervectorwithasmallnumberofdistinctvaluestoavectorwhere each unique code is assigned a positive integer from 1 to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example calls and the corresponding expected results: 134 I DEEP recode <- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' recode(c("a", "a", "a", "b", "b")) ## [1] 1 1 1 2 2 recode(c("x", "z", "y", "x", "y", "x")) ## [1] 1 3 2 1 2 1 Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='21 Implement a function that returns the number of occurrences of each unique ele- mentinagivenatomicvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Thereturnvalueshouldbeanumericvectorequippedwitha names attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' count <- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' count(c(5, 5, 5, 5, 42, 42, 954)) ## 5 42 954 ## 4 2 1 count(c("x", "z", "y", "x", "y", "x", "w", "x", "x", "y", NA_character_)) ## w x y z ## 1 5 3 1 1 Hint: use match and tabulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='22 Implement a function that extends upon the built-in duplicated, indicating which occurrence (starting from the beginning of the vector) of a repeated value a given value constitutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' duplicatedn <- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' duplicatedn(c("a", "a", "a", "b", "b")) ## [1] 1 2 3 1 2 duplicatedn(c("x", "z", "y", "x", "y", "x", "w", "x", "x", "y", "z")) ## [1] 1 1 1 2 2 3 1 4 5 3 2 Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='23 Based on a call to Map, implement a function my_split such that, given a vec- tor x and an atomic vector y of the same length as x, my_split(x, y) yields the same result as split(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='24 Extend my_split to handle the second argument being a list of the form list(y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') that represents the product of many levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If the ys are of different lengths, apply the recycling rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25 Implement my_unsplit being your own version of the built-in unsplit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Make sure it holds my_unsplit(split(x, g), g) == x for x and g of the same lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='26 Write a function that takes as arguments: (a) an integer n, (b) a numeric vector x of length k and no duplicated elements, (c) a vector of probabilities p of length k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' verify that 𝑝𝑖 ≥ 0 for all 𝑖 and ∑𝑘 𝑖=1 𝑝𝑖 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Based on a random number generator from the uniform distribution on the unit interval, generate n independent realisations of a random variable 𝑋 such that Pr(𝑋 = 𝑥𝑖) = 𝑝𝑖 for 𝑖 = 1, … , 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hint: to obtain a single value: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' generate 𝑢 ∈ [0, 1], 7 FUNCTIONS 135 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' find 𝑚 ∈ {1, … , 𝑘} such that 𝑢 ∈ (∑𝑚−1 𝑗=1 𝑝𝑗, ∑𝑚 𝑗=1 𝑝𝑗], 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' the result is then 𝑥𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='27 Write a function that takes as arguments: (a) an increasingly sorted vector x of length n, (b) any vector y of length n, (c) a vector z of length k and elements in [𝑥1, 𝑥𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let 𝑓 be the piecewise linear spline that interpolates the points (𝑥1, 𝑦1), … , (𝑥𝑛, 𝑦𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Return a vector w of length k such that 𝑤𝑖 = 𝑓 (𝑧𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='28 (*) Write functions dpareto, ppareto, qpareto, and rpareto that implement the basic functions related to the Pareto distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' compare Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8 Flow of Execution The ifelse and Map functions are very powerful, but they allow us to process only the consecutive elements in a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thus, let us (finally!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') discuss different ways to alter a program’s control flow manually, based on some criterion, and to evaluate the same expression a number of times, but perhapsondifferentdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Beforeproceedinganyfurther,letus,however,contemplate on the fact that we have managed to do without them for such a long time – and the data processing exercises we learnt to solve were far from trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Conditional Evaluation Life is full of surprises, so we would be nice if we were able to adapt to whatever the circumstances are going to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The following evaluates a given expression if and only if a logical condition is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' if (condition) expression When performing some other_expression is preferred rather than doing nothing in the case of the condition’s being false, we can write: if (condition) expression else other_expression For instance: (x <- runif(1)) # to spice things up ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='28758 if (x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) cat("head") else cat("tail") ## tail Many expressions can of course be grouped with curly braces, “{” if (x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) { cat("head") x <- 1 (continues on next page) 138 I DEEP (continued from previous page) } else { cat("tail") x <- 0 } ## tail print(x) ## [1] 0 Important Atthetoplevel,weshouldnotputanewlinebefore else,otherwisewewill get an error like Error: unexpected \'else\' in "else".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is because the interpreter enthusiastically executes the statements been read line by line as soon as it regards them as stand-alone expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In this case, we first get an if without else, and then, separately, a dangling else without the preceding if.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This does not happen when a conditional statement is part of an expression group, because the latter is read in its entirety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' function (x) { # opening bracket – start if (x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) cat("head") else # not dandling, because {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='} is read as a whole cat("tail") } # closing bracket – expression ends As an exercise, try removing the curly braces and see what happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Return Value `if` is a function (compare Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4), hence has a return value – the result of eval- uating the conditional expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (x <- runif(1)) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='28758 y <- if (x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) "head" # no else print(y) ## NULL y <- if (x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) "head" else "tail" print(y) ## [1] "tail" This is particularly useful when a call to `if` is the last expression in the code block constituting a function’s body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8 FLOW OF EXECUTION 139 mint <- function(x) { if (x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) # the last expression (actually, the only one) "head" # this can be the return value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' else "tail" # or this one, depending on the condition } mint(x) ## [1] "tail" unlist(Map(mint, runif(5))) ## [1] "tail" "head" "tail" "head" "head" Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Add-on packages can also be loaded using requireNamespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Contrary to lib- rary, the former does not fail when a package is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, it does not attach it on the search list;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see sec:to-do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Instead, it returns a logical value indicating if the package is available for use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This can be use- ful inside other functions where the availability of some additional features depends on the user environment’s configuration: process_data <- function(x) { if (requireNamespace("some_extension_package", quietly=TRUE)) some_extension_package::very_fast_method(x) else normal_method(x) } 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Nested ifs If more than two test cases are possible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', when we need to go beyond either con- dition or !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='condition, then we can use the following construction: if (a) { expression_a } else if (b) { expression_b } else if (c) { expression_c } else { expression_else } This evaluates all conditions a, b, … (in this order) until the first positive case is found, and then executes the corresponding expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 140 I DEEP Note that the above is nothing else than a series of nested if statements: if (a) { expression_a } else { if (b) { expression_b } else { if (c) { expression_c } else { expression_else } } } but written in a less readable1 manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Write a function named sign that determines if a given numeric value is "pos- itive", "negative", or "zero".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Condition: Either True of False if expects a condition that is a single, well-defined logical value, either TRUE or FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thence, problems may arise when this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If the condition is of length not equal to one,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' we get an error: if (c(TRUE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' FALSE)) cat("spam") ## Error in if (c(TRUE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' FALSE)) cat("spam"): the condition has length > 1 if (logical(0)) cat("bacon") ## Error in if (logical(0)) cat("bacon"): argument is of length zero We cannot pass a missing value either: if (NA) cat("ham") ## Error in if (NA) cat("ham"): missing value where TRUE/FALSE needed Important If we think that we are absolutely immune to the writing of code violating the above constraints,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' just we wait until the condition becomes a function of data for which there is no sanity-checking in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' mint <- function(x) if (x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) "H" else "T" (continues on next page) 1 Somewhat related is the switch function which we study in sec:to-do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It relies on lazy evaluation of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Still, it can always be replaced by a series of ifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8 FLOW OF EXECUTION 141 (continued from previous page) mint(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25) ## [1] "T" mint(runif(5)) ## Error in if (x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) "H" else "T": the condition has length > 1 mint(log(rnorm(1))) # not obvious, only triggered sometimes ## Warning in log(rnorm(1)): NaNs produced ## Error in if (x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) "H" else "T": missing value where TRUE/FALSE needed In Chapter 9, we will be particularly interested in ways to assure input data integrity, so that situations such as above will either fail gracefully or succeed bombastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Here, we should probably make sure that x is a single finite numeric value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Alternat- ively, we had rather test whether all(x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=TRUE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Interestingly, objects other that logical are accepted: they will be coerced if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- 1:5 if (length(x)) # i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', length(x) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='= 0, but way less readable cat("length is not zero") ## length is not zero Recall that coercion of numeric to logical yields FALSE if and only if the original value is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Short-Circuit Evaluation Specially for formulating logical conditions in if and while (see below), we have the scalar `||` (alternative) and `&&` (conjunction) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' FALSE || TRUE ## [1] TRUE NA || TRUE ## [1] TRUE Contrary to their vectorised counterparts (`|` and `&`), the scalar operators are lazy (Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) in the sense that they evaluate the first operand and then determine if the computing of the second one is necessary (because, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', FALSE && whatever is always FALSE anyway).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, if (a && b) expression is equivalent to: 142 I DEEP if (a) { if (b) { # compute b only if a is TRUE expression } } and: if (a || b) expression corresponds to: if (a) { expression } else if (b) { # compute b only if a is FALSE expression } For instance, “is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='vector(x) && length(x) > 0 && x[[1]] > 0” is a safe test that takes into account that “x[[1]]” has only the desired meaning for objects that are not non-empty vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some other examples (recall that the expressions within the curly braces are evaluated one after another and that the result is determined by the last value in the series): {cat("spam");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' FALSE} || {cat("ham");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' TRUE} || {cat("cherries");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' FALSE} ## spamham ## [1] TRUE {cat("spam");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' TRUE} && {cat("ham");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' FALSE} && {cat("cherries");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' TRUE} ## spamham ## [1] FALSE Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Study the source code of isTRUE and isFALSE and determine if these functions could be useful in formulating the conditions within the if expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Exception Handling Exceptionsareexceptional,buttheymayhappenandbreakthings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Forinstance,when we try to download a file and the internet connection drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Or an optimisation al- gorithm fails to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Or we just have a bug in our code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Or: 8 FLOW OF EXECUTION 143 read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='csv("/path/to/a/file/that/does/not/exist") ## Warning in file(file, "rt"): cannot open file \'/path/to/a/file/that/does/ ## not/exist\': No such file or directory ## Error in file(file, "rt"): cannot open the connection Three types of conditions are frequently observed: errors – they stop the flow of execution, warnings – non critical, but can be turned into errors (see warn in option), messages – they transmit some diagnostic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' These can be manually triggered by means of stop, warning, and message functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Errors (but warnings too) can be handled by means of the tryCatch function, amongst others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' tryCatch({ # block of expressions to execute, until an error occurs cat("a\\n") stop("b") # error – breaks the linear control flow cat("c\\n") }, error = function(e) { # executed immediately upon an error cat(sprintf("error: %s\\n", e[["message"]])) }, finally = { # always executed at the end, regardless of error occurrence cat("finally!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='\\n") } ) ## a ## error: b ## finally!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The two other conditions can be ignored by calling suppressWarnings and suppress- Messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=" log(-1) ## Warning in log(-1): NaNs produced ## [1] NaN suppressWarnings(log(-1)) # yeah, yeah, we know what we're doing ## [1] NaN Exercise 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Atthetimeofwritingofthisbook,the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='tablepackageemitsamessageupon attachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Call suppressMessages to silence it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that consecutive calls to library do not reload an already loaded package, therefore the message will only be seen once per R session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Related functions include stopifnot discussed in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 and on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='exit mentioned in sec:to-do;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see also Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 for some code debugging tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 144 I DEEP 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Repeated Evaluation And now for something completely different… time for the elephant in the room!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We have been able to do without loops so far (and will be quite all right in the second part of the book too), because many data processing tasks can be written in terms of vectorised operations such as `+`, sqrt, sum, `[`, Map, and Reduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Oftentimes, com- pared to their loop-based counterparts, they are not only much more readable but also more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We will explore this in the exercises below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, at times, using an explicit while or for loop might be the only natural way of solving a problem, for instance, when processing chunks of data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, an explicitly “looped” algorithm may occasionally have better2 time or memory complex- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 while if considers a given logical condition and thus determines whether to execute a given statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' On the other hand, while (condition) # single TRUE or FALSE, as in `if` expression evaluates a given expression as long as the logical condition is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, it is ad- visable to make the condition dependent upon some variable that can be modified by the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' i <- 1 while (i <= 3) { cat(sprintf("%d, ", i)) i <- i + 1 } ## 1, 2, 3, Nested loops are of course possible too: i <- 1 while (i <= 2) { j <- 1 while (j <= 3) { cat(sprintf("%d %d, ", i, j)) j <- j + 1 } (continues on next page) 2 But in such cases it will often benefit from a rewrite in C or C++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Chapter 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8 FLOW OF EXECUTION 145 (continued from previous page) cat("\\n") i <- i + 1 } ## 1 1, 1 2, 1 3, ## 2 1, 2 2, 2 3, Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Implement a simple linear congruential pseudorandom number generator that, given some seed 𝑋0 ∈ [0, 𝑚), outputs a sequence (𝑋1, 𝑋2, … ) defined by: 𝑋𝑖 = (𝑎𝑋𝑖−1 + 𝑐) mod 𝑚, with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', 𝑎 = 75, 𝑐 = 74, and 𝑚 = 216 + 1 (here, mod is the division reminder, `%%`).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note thatthisgeneratorhaspoorstatisticalpropertiesandshouldnotbeusedinpractice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Inparticular, after some number of operations 𝑘, we will find a cycle such that 𝑋𝑘 = 𝑋1, 𝑋𝑘+1 = 𝑋2, ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 for The for-each loop: for (name in vector) expression takes each element, from the beginning to the end, in a given vector, assigns it some name, and evaluates the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example: fridge <- c("spam",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "spam",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "bacon",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "eggs") for (food in fridge) cat(sprintf("%s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' food)) ## spam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' spam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' bacon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' eggs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' One more: for (i in 1:length(fridge)) # better: seq_along(fridge),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see below cat(sprintf("%s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' fridge[i])) ## spam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' spam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' bacon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' eggs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Just one more,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' promise: for (i in 1:2) { for (j in 1:3) cat(sprintf("%d %d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' j)) cat("\\n") } ## 1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ## 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 146 I DEEP Note that the iterator still exists after the loop’s watch has ended: print(i) ## [1] 2 print(j) ## [1] 3 Important Writing: for (i in 1:length(x)) print(x[i]) is not necessarily safe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' because if x is an empty vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' then: x <- logical(0) for (i in 1:length(x)) print(x[i]) ## [1] NA ## logical(0) Recall from Chapter 5 that x[1] tries to access an out of bounds element here and x[0] returns nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Wegenerallysuggestreplacing1:length(x)withseq_along(x)orseq_len(length(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' wherever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note The model for loop above is roughly equivalent to: name <- NULL tmp_vector <- vector tmp_iter <- 1 while (tmp_iter <= length(tmp_vector)) { name <- tmp_vector[[tmp_iter]] expression tmp_iter <- tmp_iter + 1 } Note that tmp_vector is determined before the loop itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, any changes to vec- tor will not influence the execution flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also note that due to the use of `[[`, the loop can be applied on lists as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Let x be a list and f be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The following code generates the same result as Map(f, x): n <- length(x) (continues on next page) 8 FLOW OF EXECUTION 147 (continued from previous page) ret <- vector("list", n) # a new list of length `n` for (i in seq_len(n)) ret[[i]] <- f(x[[i]]) Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 Letxandybetwolistsandfbeafunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='HereisthemostbasicversionofMap(f, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that x and y might be of different lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' nx <- length(x) ny <- length(y) n <- max(nx, ny) ret <- vector("list", n) for (i in seq_len(n)) ret[[i]] <- f(x[[((i-1)%%nx)+1]], y[[((i-1)%%ny)+1]]) Feelfreetoupgradetheabovebyaddingawarninglikethe longer argument is not a multiple of the length of the shorter one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Also,rewriteitwithouttheuseofthemodulooperators,`%%`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 break and next breakcanbeusedtoescapethecurrentloop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' nextskipstheremainingexpressionsand advances to the next iteration (to where the testing of the logical condition occurs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Here is a rather random example: x <- runif(1000) s <- 0 for (e in x) { if (e > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1) next print(e) if (e < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='01) break s <- s + e } ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='045556 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='04206 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='024614 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='045831 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='094841 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00062477 print(s) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2529 148 I DEEP Computes the sum of the elements in x that are less than or equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 from the be- ginning, stopping at the first element less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that we have used the frequently occurring design pattern: for (e in x) { if (condition) next many_statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' } which is equivalent to: for (e in x) { if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='condition) { many_statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' } } but avoids introducing a nested block of expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note (*) There is a third loop type, repeat expression which is a shorthand for while (TRUE) expression i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', it is a possibly infinite loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Such loops are useful when implementing situations such as do-stuff-until-a-thing-happens, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', when we want to execute a command at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' i <- 1 repeat { # while (TRUE) # simulate dice casting until we throw "1" if (runif(1) < 1/6) break # not an infinite loop after all i <- i+1 } print(i) ## [1] 6 Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 What is wrong with the following code?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8 FLOW OF EXECUTION 149 j <- 1 while (j <= 10) { if (j %% 2 == 0) next print(j) j <- j + 1 } Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 What about this one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' j <- 1 while (j <= 10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' j <- j + 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 return return, when called from within a function, immediately yields a specified value and goes back to the caller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example, here is a simple recursive function that flattens a given list: my_unlist <- function(x) { if (is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='atomic(x)) return(x) # so if we are here, x is definitely not atomic out <- NULL for (e in x) out <- c(out, my_unlist(e)) out # or return(out);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=" it's the last expression anyway, so not necessary } my_unlist(list(list(list(1, 2), 3), list(4, list(5, list(6, 7:10))))) ## [1] 1 2 3 4 5 6 7 8 9 10 Note that return is a function: the round brackets are obligatory, 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 A Note on Time and Space Complexity of Algorithms (*) Analysis of algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', [9, 34]), can give us a rough estimate of their run times or memory consumption as a function of the input data size, especially for big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In scientific computing and data science, we most often deal with vectors (sequences) or matrices/data frames (tabular data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, we might be interested in determ- ining how many primitive operations need to be performed as a function of their length n or the number of rows n and columns p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 150 I DEEP The O (Big-Oh) notation, for instance, can express the upper bounds for time/resource consumption in asymptotic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, we say that the time complexity is 𝑂(𝑛2), if for large 𝑛, the number of operations to perform will be proportional to at most the square of the vector size (more precisely, there exists 𝑚 and 𝐶 > 0 such that for all 𝑛 > 𝑚, the number of operations is ≤ 𝐶𝑛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, if we have two algorithms that solve the same task, one that has 𝑂(𝑛2) time complexity, and other of 𝑂(𝑛3), it is better to choose the former, because for large problem sizes we expect it to be faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, whether time grows proportionally to log 𝑛, √𝑛, 𝑛, 𝑛 log 𝑛, 𝑛2, 𝑛3, or 2𝑛, can be useful in predicting how big the data can be if we have a fixed deadline or not too much space left on the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='10 The hclust function determines a hierarchical clustering of a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is fed withanobjectthatstoresthedistancebetweenallthepairsofinputpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Thereare𝑛(𝑛−1)/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', 𝑂(𝑛2)) unique point pairs for any given n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' One numeric scalar (double type) takes 8 bytes of storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If you have 16 GB or RAM, what is the largest dataset that you can cluster on your machine using this function?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Oftentimes, we can learn about the time or memory complexity of the functions we use from their documentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', help("findInterval").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='11 Acourseindatastructuresinalgorithms,whichthisoneisnot,willgiveusplenty of opportunities to implement many algorithms ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This way, we can gain a lot of insights and intuitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, this is a 𝑂(𝑛)-time algorithm: for (i in seq_len(n)) expression and this is one runs in 𝑂(𝑛2)-time: for (i in seq_len(n)) for (j in seq_len(n)) expression as long as, of course, the expression is rather primitive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', operations on scalar variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' R is a very expressive language and hence quite complex and lengthy operations can look pretty innocent (it is a glue language for rapid prototyping, after all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example: for (i in seq_len(n)) for (j in seq_len(n)) z <- z + x[[i]] + y[[j]] can be seen as 𝑂(𝑛3) if each element in the lists x and y as well as z itself are atomic vectors of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8 FLOW OF EXECUTION 151 Similarly, Map(mean, x) is 𝑂(𝑛2) if x is a list of n atomic vectors each of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note A quite common statistical scenario involves the generation of a data buffer of a fixed size: ret <- c() for (i in n) ret[[i]] <- generate_data(i) # here: ret[[length(ret)+1]] <- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thisnotation, however,involvesthe growingofthe retarrayineachiteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Luckily, sinceRversion3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0,eachsuchsizeextensionhasamortised𝑂(1)timeduetothefact that some more memory is internally reserved for its prospective growth (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', Chapter 17 of [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, it would still be better to pre-allocate the output vector and grant it the de- sired, final size already upon creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that we can construct vectors of specific lengths and types in an efficient way (more efficient than with rep) by calling: numeric(3) ## [1] 0 0 0 numeric(0) ## numeric(0) logical(5) ## [1] FALSE FALSE FALSE FALSE FALSE character(2) ## [1] "" "" vector("numeric", 8) ## [1] 0 0 0 0 0 0 0 0 vector("list", 2) ## [[1]] ## NULL ## ## [[2]] ## NULL Note Not all data fit into memory, but it does not mean that we should start installing Apache Hadoop and Spark immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some datasets can be processed on a chunk- by-chunk basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 152 I DEEP R enables data stream handling (some can be of infinite length) through file connec- tions, for example: f <- file(paste0("https://raw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='githubusercontent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/gagolews/teaching-data/", "master/README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='md"), open="r") # a big file, the biggest file ever i <- 0 while (TRUE) { few_lines <- readLines(f, n=4) # read only four lines at a time if (length(few_lines) == 0) break i <- i + length(few_lines) } close(f) print(i) # the number of lines ## [1] 93 Many functions support reading from/writing to already established connections of different types, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', file, gzfile, textConnection, batch-by-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' A frequent scenario involves reading a very large CSV, JSON, or XML file only thou- sands of lines/records at a time, parsing and cleansing them, and exporting to SQL databases (which we will exercise in Chapter 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also note that the always-open text connections stdout and stderr (for writing), and stdin (for reading) are by default mapped to the “terminal/console” and “keyboard”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Call scan, cat, and stop to interact with such sources/targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Exercises Note that, from now on, we should stay alert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Many, if not all, of the following tasks can still be implemented without the explicit use of the R loops, but based only on the operations covered in the previous chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If this is the case, try implementing both the looped and loop-free version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Use microbenchmark::microbenchmark or proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='time to compare the run-times3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='12 Answer the following questions: Let x be a numeric vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' When does if(x > 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' yield a warning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' When does it give an error?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' How to prevent this?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What is the dangling else?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What happens if you put if as the last expression in a curly braces block within a function’s body?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3 It might be the case that a for-based solution is faster (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', for larger objects) because of the use of a more efficient algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Such cases will especially benefit from a rewrite in C or C++ (Chapter 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8 FLOW OF EXECUTION 153 Why do we say that `&&` and `||` are lazy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What are their use cases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What is the difference between `&&` and `&`?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Can while always be replaced with for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What about the other way around?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='13 Verify which of the following can be safely used as logical conditions in if state- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If that is not the case for all x, y, …, determine the additional conditions that should be imposed in order to make them valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x == 0, x[y] > 0, any(x>0), match(x, y), any(x %in% y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='14 Whatcango wrongin thefollowingcodechunk,dependingonthetypeandform of x?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Consider as many scenarios as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' count <- 0 for (i in 1:length(x)) if (x[i] > 0) count <- count + 1 Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='15 Implement shift_left(x, n) and shift_right(x, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The former function getsridofthefirst nobservationsin xandaddsnmissingvaluesattheendoftheresultingvector, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', shift_left(c(1, 2, 3, 4, 5), 2) is c(3, 4, 5, NA, NA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' On the other hand, shift_right(c(1, 2, 3, 4, 5), 2) is c(NA, NA, 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='16 Implement your own version of diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='17 Writea functionthat determinesthelongestincreasingtrendinagivennumeric vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', the length of the longest subsequence of consecutive elements that are increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example, the input c(1, 2, 3, 2, 1, 2, 3, 4, 3) should yield 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='18 Implement the functions that round down and round up, to a number of decimal digits, each element in a numeric vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This concludes the first part of this magnificent book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Part II Deeper 9 Designing Functions In Chapter 7, we learnt how to write our own functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This skill is key to enforcing the good development practice of avoiding the repetition of code: running the same command sequence on different data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This chapter is devoted to the designing of such reusable modules so that they are easier to use, test, and maintain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We also provide some more technical details which were not of the highest importance upon our first exposure to this topic, but which our crucial to our better understanding of how R works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Principles of Sustainable Design Good design is more art than science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' As usual in real life, we will need to make many compromises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is because improving things with regard to one criterion some- times makes them worse with respect to other aspects1 (also which we are not aware of).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, not everything that counts can and will be counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Below are some obser- vations, ideas, and food for thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 To Write or to Abstain Functions that we write ourselves can oftentimes be considered merely creative com- binations of the building blocks available in base R or a few high-quality add-on pack- ages2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some are simpler than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thus, there is a question if a new operation should be introduced at all: whether we are faced with the case of multiplying entities without necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Ontheonehand,theDRY(don’trepeatyourself)principletellsusthatmostfrequently used (say, at least 3 times) code chunks should be generalised in the form of a new function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is definitely a correct approach with regard to non-trivial operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' On the other hand, not every generalisation is necessarily welcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us say that we are lazy and tired of writing g(f(x)) for the n-th time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Why don’t we therefore intro- duce h defined as a combination of g and f?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This might seem like a good idea, but let 1 Compare the notion of Pareto efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 If some non-trivial operation is missing, we can always implement it at the C language level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Chapter 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 158 II DEEPER us not take it for granted: being tired might be an indication of our body and mind needing a rest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' being lazy can be a call for more self-discipline (not an overly popular word these days, but still, an important trait).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 paste0 is a specialised version of paste, but having the sep argument hardcoded to an empty string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Even if this might be the most often applied use case, is the introduction of a new function justifiable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Is it so hard to write paste="" each time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Would changing paste’s default argument be better?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' That of course would harm backward compatibility, but what strategies could we apply to make the transition as smooth as pos- sible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Would it be better to introduce a new version of paste with sep defaulting to "", informing the users that the old version is deprecated and to be removed in, say, two years?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Or maybe one year is better?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Or five?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 In R 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0, deparse1 has been introduced: it is merely a combination of deparse (see below) and paste: print(deparse1) ## function (expr, collapse = " ", width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='cutoff = 500L, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') ## paste(deparse(expr, width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='cutoff, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='), collapse = collapse) ## ## Letussaythiscovers90%ofusecases:wasintroducingitajustifiedideathen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Whatifthatnum- ber was 99%?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Overall, more functions contribute to the information overload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We do not want our users to be overwhelmed by too many choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Luckily, nothing is cemented once and for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Had we done bad design choices resulting in our API’s being bloated, we can always clean up those that no longer spark joy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 To Pamper or to Challenge Think about the kind of audience we would like to serve: is it our team only, students, professionals, certain client groups, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Do they have mathematical, programming, engineering, or scientific background?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Not everything that is appropriate for one co- hort, will be valuable for another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' And not everything that is good for some now, will be beneficial for them in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' People (their skills, attitudes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Assumewearewritingafriendlyandinclusivepackagefornoviceswhowouldlike 9 DESIGNING FUNCTIONS 159 to grasp the basics of data analysis as quickly3 as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Without much effort, it would enable them to solve 80–95% of the most common, easy problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Think of introducing the students to a function that returns five largest observations in a given vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us call it nlargest: so pleasant to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It makes the students feel empowered quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Still,whenfacedwiththeremaining5–20%tasks,theywillhavetolearnanother,moreadvanced, generic,andpowerfultoolanyway(inourcase,thebaseRitself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Aretheydeterminedandskilled enough to do that?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Time will tell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The least we can do is to be explicit about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Recall that it took us some time to arrive at order and subsetting via `[`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Assuming that we read this book from the beginning to the end and solve all the exercises, which we should, we are now able to implement the said nlargest (and lots of other functions) ourselves, using a single line of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This will also pay off in many scenarios that we will be facing in the future, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', when we consider matrices and data frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Yes, everyone will be reinventing their own nlargest this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' But this constitutes a great exer- cise: by our being too nice, some might have lost an opportunity to learn a new, more universal skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Although most of the users would really love to minimise the effort put into all their activities, ultimately, they sometimes need to learn new things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us thus not be afraid to teach them stuff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Furthermore, we do not want to discourage experts (or experts to-be) by presenting themwithoverlysimplifiedsolutionsthatkeeptheirhandstiedwhensomethingmore ambitious needs to be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 To Build or to Reuse In the short term, the failfast philosophy encourages us to build our applications using prefabricatedcomponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Thisisfantasticattheearlystageofitslifecycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Ifwebuild somethingreallysimpleorwhosepurposeismerelytoillustrateanidea,show-offhow “awesome” we are, or to educate, let us be explicit about it so that other users do not feel obliged to treat our product (exercise) seriously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In the (not so likely, probabilistically speaking) event of its becoming successful, we should start thinking about the project’s long-term stability and sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' After all, relying on third-party functions, packages, or programs makes our software pro- jects less… independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This may be problematic, because: the dependencies might not be available on every platform or may behave differ- ently across various system configurations, 3 We will leave the reflection upon whether this is at all feasible for another time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that this strategy is employed by many companies (and drug dealers): make the introductory exper- ience super-smooth and fun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' At the same time, do not allow your users to become independent too easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Instead, make them rely on your product lines/proprietary solutions/payable services etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The free software movement with its do-it-yourself approach stresses on users’ becoming autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This does not contradict the user-friendliness (but that many open-source projects could benefit from be- coming less exclusive is a different story, and this book tries to make a change in this area too).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 160 II DEEPER they may be huge (and can depend on other external software too), their APIs may change which could result in our project’s not working anymore, their functionality can change which can lead to some unexpected behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, it might be a good idea to rewrite some parts from scratch on our own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Identify some R packages on CRAN with many dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Seewhat functions do they import from other packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' How often it is just a few lines of code?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' TheUnixphilosophyemphasises uponthebuildingandusingofminimalisticyetnon- trivial, single-purpose, high quality pieces of software that can work as parts of larger, custom pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' R serves as a glue language quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In the long run, some of our software projects might converge to such a tool – it might be a good idea to standardise our API (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', make it available from the command-line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2) so that the users of other languages can benefit from our work too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important If ourprojectis merelyamodified interface/front-endtoa largerprogram developed by others, we should be humble about it and make sure it is not us who get all the credit for other people’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also,weshouldstateveryclearlyhowcantheoriginaltoolsbeusedtoachievethesame goals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', when working from the command line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Managing Data Flow A function, most of the time, can and should be treated as a black box: its callers do not have to care what it hides inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' After all, they are supposed to use it: given some in- puts,theyexpectawell-defined(read:explainedinverydetailinthefunction’smanual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3) outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Checking Input Data Integrity and Argument Handling A function takes R objects of any kind as arguments, but it does not mean that feeding it with every- or any-thing is healthy for its guts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' When designing functions, it is best to handle the inputs in a manner similar to base R’s behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This will make our contributions easier to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Unfortunately, base R functions frequently do not handle arguments of similar kind 100% consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Such variability might be due to many reasons and, in essence, is not necessarily bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Usually, there might be many different possible behaviours and choosingoneoveranotherwillmakeafewusersunhappyanyway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Somechoicesmight not be optimal, but they are for historical compatibility (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', with S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Of course, it 9 DESIGNING FUNCTIONS 161 might also happen (but the probability is low) that there is a bug or something is not at all well designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thisiswhyitisbetterto keepthevocabularyquiterestricted(andweadvocateforsuch minimalism in this book): even if there are exceptions to the general rules, with fewer functions, they are simply easier to remember.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Consider the following case study, illustrating that even the extremely simple scenario where we deal with a single positive integer, is not necessarily straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 In mathematical notation, we usually denote the number of objects in a collection with the famous “n”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is implicitly assumed that such n is a single natural number (although whether this includes 0 or not should be specified at some point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The functions runif, sample, seq, rep, strrep, and class::knn take it arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, nothing prevents their users from trying to challenge them by passing: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, -1, 0, 1-1e-16 (non-positive numbers, non-integers);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' NA_real_, Inf (not finite);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1:5 (not of length 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' after all, there are no scalars in R) numeric(0) (an empty vector);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' TRUE, NA, c(TRUE, FALSE, NA), "1", c("1", "2", "3") (non-numeric, but coercible to);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' list(1), list(1, 2, 3), list(1:3, 4) (non-atomic);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "spam" (utter nonsense);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='matrix(1), factor(7), factor(c(3, 4, 2, 3)), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (compound types;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Chapter 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Read the aforementioned functions’ reference manuals and call them on different inputs, noting how differently they handle such atypical arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Sometimes we will rely on other functions to handle the data integrity checking for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Let us consider the following function that generates n pseudorandom numbers from the unit interval rounded to d decimal digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We strongly believe or hope (good faith and high competence assumption) that its authors knew what they were doing when they wrote: round_rand <- function(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' d) { x <- runif(n) # runif will check if `n` makes sense round(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' d) # round will determine the appropriateness of `d` } What constitutes correct n and d and how the function behaves when not provided with positive integers is determined by the two underlying functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' runif and round: 162 II DEEPER round_rand(4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 1) # the expected use case ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 round_rand(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9) # 4, 2 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='89 round_rand(4, NA) ## [1] NA NA NA NA round_rand(0, 1) ## numeric(0) Ifwellthought-outandproperlydocumented,manysuchdesignchoicescanbedefen- ded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some programmers will opt for high uniformity/compatibility across numerous tools, but there are cases where some exceptions/diversity do more good than harm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Yet, we should keep in mind that the functions we write might be part of a more com- plicated data flow pipeline, where some other function generates a value that we did not expect (because of a bug therein or because we did not study its manual) and this value is used as input to our function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In our case, this would correspond to the said n or d being determined programmatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 Continuing the previous example, the following might be somewhat challenging with regard to our being flexible and open minded: round_rand(c(100, 42, 63, 30), 1) # length(c(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=')), 1) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 round_rand("4", 1) # as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numeric(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='), 1 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 Sure, it is quite convenient, but might lead to problems that are hard to diagnose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also note the not-really informative error messages in cases like: round_rand(NA, 1) ## Error in runif(n): invalid arguments round_rand(4, "1") ## Error in round(x, d): non-numeric argument to mathematical function Hence, some defensive design mechanisms are not a bad idea, especially if they lead to generating an informative error message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important stopifnot gives aconvenientmeanstoassert theenjoymentofourexpect- ations with regard to a function’s arguments (or some intermediate values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' A call to stopifnot(cond1, cond2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') is more or less equivalent to: if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='logical(cond1) && !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='any(is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='na(cond1)) && all(cond1))) stop("`cond1` are not all TRUE") if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='logical(cond2) && !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='any(is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='na(cond2)) && all(cond2))) (continues on next page) 9 DESIGNING FUNCTIONS 163 (continued from previous page) stop("`cond2` are not all TRUE") .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thus, if all the elements in the given logical vectors are TRUE, nothing happens and we can safely move on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 We can rewrite the above function as follows: round_rand2 <- function(n, d) { stopifnot( is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numeric(n), length(n) == 1, is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='finite(n), n > 0, n == floor(n), is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numeric(d), length(d) == 1, is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='finite(d), d > 0, d == floor(d) ) x <- runif(n) # runif will check if n makes sense round(x, d) # round will determine the appropriateness of d } round_rand2(5, 1) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 round_rand2(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4, 1) ## Error in round_rand2(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4, 1): n == floor(n) is not TRUE round_rand2(5, "1") ## Error in round_rand2(5, "1"): is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numeric(d) is not TRUE Thisimplementsthestrictesttestfor“asinglepositiveinteger”possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Inthecaseofanyviolation of the underlying condition, we get a very informative error message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 At other times, we might be interested in argument checking like: if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numeric(n)) n <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numeric(n) if (length(n) > 1) { warning("only the first element will be used") n <- n[1] } n <- floor(n) stopifnot(is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='finite(n), n > 0) This way, "4" and c(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9, 100) will all be accepted as 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We see that there is always a tension between being generous/flexible and pre- 4 Note that here we rely on S3 generics is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numeric and as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numeric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 164 II DEEPER cise/restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, for some functions, it will be better to behave differently than the others, because of their particular use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Too much uniformity is as bad as chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Overall, we should rely on common sense, but add some lightweight foolproof mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It is our duty to be explicit about all the assumptions we make or exceptions we allow (by writing good documentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We will revisit this topic in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note Example exercises related to the improving of the consistency of base R’s hand- ling of arguments in different domains include the vctrs and stringx packages5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Can these contributions be justified?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='10 Reflect on how you would handle the following scenarios (and how base R and other packages or languages you know deals with them): a vectorised mathematical function (empty vectors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' non-numeric inputs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' what if it is equipped with the names attribute?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' what if it has other ones?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' an aggregation function (what about missing values?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' empty vectors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' a function vectorised with regard to two arguments (elementwise vectorisation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' recycling rule?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' only scalar vs vector or vector vs vector of the same length allowed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' what if one argu- ment is a row vector and the other is a column vector);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' a function vectorised with regard to all arguments (really all?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' maybe some exceptions are necessary?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' afunctionvectorisedwithrespecttothefirstargumentbutnotthesecond(whysucharestric- tion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' when?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Find a few functions that match each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Putting Outputs into Context The functions we write do not exist in a vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We should put them into a much wider context: how are they going to be used when combined with other tools?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' As a general rule, our functions should generate outputs of predictable kind, so that when we write and read the code chunks that utilise them, we can easily deduce what is going to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='11 Some base R functions do not adhere to this rule for the sake of (questionable) users’ convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We will meet a few of them in Chapter 11 and Chapter 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular, sap- ply and the underlying simplify2array, can return a list, an atomic vector, or a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 5 Yours truly is the author of the latter and thus is guilty of multiplying entities beyond necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9 DESIGNING FUNCTIONS 165 simplify2array(list(1, 3:4)) # list ## [[1]] ## [1] 1 ## ## [[2]] ## [1] 3 4 simplify2array(list(1, 3)) # vector ## [1] 1 3 simplify2array(list(1:2, 3:4)) # matrix ## [,1] [,2] ## [1,] 1 3 ## [2,] 2 4 Further, the index operator with drop=TRUE, which is the default, may output an atomic vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' But it may as well yield a matrix or a data frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (A <- matrix(1:6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' nrow=3)) # an example matrix ## [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1] [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2] ## [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='] 1 4 ## [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='] 2 5 ## [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='] 3 6 A[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ] # vector ## [1] 1 4 A[1:2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ] # matrix ## [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1] [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2] ## [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='] 1 4 ## [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='] 2 5 A[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' drop=FALSE] # matrix with 1 row ## [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1] [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2] ## [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='] 1 4 We proclaim that the default functions’ behaviour should be to return the object of the most generic kind possible (if there are other options),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' and then to either have a further argument which must be explicitly set if we really wish to simplify the output,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' or we should ask the user to call a simplifier explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In the latter case, the simplifier should probably fail issuing an error if it is unable to neaten the object or at least apply some brute force solution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', or “fill the gaps” somehow itself, possibly with a warning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='12 For instance: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numeric(A[1:2, ]) # always returns a vector ## [1] 1 2 4 5 stringi::stri_list2matrix(list(1, 3:4)) # fills the gaps with NAs ## [,1] [,2] (continues on next page) 166 II DEEPER (continued from previous page) ## [1,] "1" "3" ## [2,] NA "4" Ideally, a function should perform one (and only one) well-defined task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If a function tends to generate objects of different kinds, depending on the arguments provided, maybe it is better to write two functions instead?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='13 Functionssuchas rep, seq,and sampledonotperformasingletask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Ordothey?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note (*) In a purely functional programming language, we can assume the so-called referential transparency: a call to a pure function can always safely be replaced with the value it is supposed to generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If this is true, then for the same set of argument val- ues, the output is always the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Furthermore, there are no side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In R, it is not really the case: a call can introduce/modify/delete the variables in other environments (see sec:to-do), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', the state of the random number generator, metaprogramming and lazy evaluation techniques lead to the functions’ being free to interpret the argument forms (not only: values) freely (see sec:to-do), printing, plotting, file reading, database access have obvious consequences with regard to the state of some external resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important Each function must return some value, but there are several instances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', plotting, printing), where this does not make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Insuchacase,weshouldconsiderreturning invisible(NULL),a NULLwhosefirst print- ing will be suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Compare the following: (function() NULL)() # anonymous function, called instantly ## NULL (function() invisible(NULL))() # printing suppressed x <- (function() invisible(NULL))() print(x) # no longer invisible ## NULL Take a look at the return value of the built-on cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9 DESIGNING FUNCTIONS 167 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Organising and Maintaining Functions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Function Libraries Definitions of frequently-used functions or datasets can be emplaced in separate source files (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R extension) for further reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Such libraries can be executed by calling: source("path_to_file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R") Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='14 Create a source file (script) named mylib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R, where you define a function called nlargest which returns a few largest elements in a given atomic vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' From within another script, call source("mylib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R") (note that relative paths to refer to the cur- rent working director;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (compare Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6) and then write a few lines of code where you test nlargest on some example inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Writing R Packages When a function library grows substantially, or when there is a need for equipping it with the relevant manual pages6 (Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3) or compiled code (Chapter 14), turning it into an own R package (Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1) might be a good idea (even if it is only for our own or small team’s purpose).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' A source package is merely a directory containing some special files and subdirectories: DESCRIPTION – a text file that gives the name of the package, its version, authors, dependencies upon other packages, license, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 of [45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' NAMESPACE – a text file containing directives stating which objects are to be expor- ted so that they are available to the package users, and which names are to be im- ported from other packages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' R – a directory with R scripts (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='R files), which define, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', functions, example datasets, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' man – a directory with R documentation files (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Rd), describing at least all the ex- ported objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' src – optional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' compiled code, see Chapter 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' tests – optional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' tests to run on the package check, see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 of Writing R Extensions [45] for more details and other options: there is no need for us to repeat the information from the official manual as everyone can read it themself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6 This should read: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 168 II DEEPER Important A source package can be built and installed from within an R session by calling: install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages("pkg_directory", repos=NULL, type="source") ThenitcanbeusedasanyotherRpackage(Section9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Inparticular,itcanbeloaded and attached via a call to: library("pkg") This makes all the objects listed in its NAMESPACE file available to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='15 Create your own package mypkg featuring some of the solutions to the exercises you have solved whilst studying the material in the previous chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' When in doubt, refer to the official manual [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important Note that you do not have to publish your package on CRAN7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Many users are tempted to submit whatever they have been tinkering around with for a while.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Have mercy on the busy CRAN maintainers and do not contribute to the information overload, unless you have come up with something potentially useful for other R users (make it less about you, and more about the community;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' thank you in advance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' R packages can always be hosted on and installed from, for instance, GitLab or GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note (*) The building and installing of packages also be done from the command line: R CMD build pkg_directory R CMD INSTALL --build pkg_directory Also, some users could potentially benefit from creating own Makefiles that help auto- mate the processes of building, testing, checking, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Documenting R Packages Documenting functions and commenting code thoroughly is very important, even if we just write for ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Most programmers sooner or later will note that they find it hard to determine what a piece of code is doing after they took a break from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In some sense, we always write for external audience, which incudes our future self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The help system is one of the stronger assets of the R environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' By far we should have interacted with many man pages and got a good idea of what constitutes an in- formative documentation piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7 And always consult the CRAN Repository Policy at https://cran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='r-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='org/web/packages/policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9 DESIGNING FUNCTIONS 169 From the technical side, R Documentation (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Rd) files should be emplaced in the man subdirectory of a source package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' All exported objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', functions) should be de- scribed clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Additional topics can be covered too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' During the package install, the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Rd files are converted to various output formats, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', HTML or plain text, and displayed upon a call to the well-known help function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Documentation files use a LaTeX-like syntax, which looks quite obscure to an un- trained eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The relevant commands are explained in very detail in Section 2 of Writing R Extensions [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note The process of writing .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Rd files by hand might be quite tedious, especially keep- ing track of the changes to the \\usage and \\arguments commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Rarely do we re- commend the use of third-party packages, because base R facilities are usually good enough, but roxygen2 might be worth a try, because it really makes the developers’ lives easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Most importantly, it allows for documentation to be specified alongside the functions’ definitions, which is much more natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='16 Add a few manual pages to your example R package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Assuring Quality Code Below we mention some good development practices related to maintaining quality code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is an important topic, but writing about them is tedious to the same ex- tent that reading about them is boring, because it is the less scientific part of software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' After all, these are some heuristics that are learnt best by observing and mimicking what the others are doing (and hence the exercises below will encourage to do so).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Managing Changes and Working Collaboratively It is a good idea to employ some source code version control system such as git to keep track of the changes made to the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note It is worth investing some time and effort to learn how to use git from the com- mand line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see https://git-scm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='com/doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' There are a few hosting providers for git repositories, with GitLab and GitHub being a popular choice amongst open-source software developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Notonlydotheysupportworkingcollaborativelyontheprojects,butalsoareequipped with additional tools for reporting bugs, suggesting feature requests, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='17 Find where the source code of some of your most favourite R packages or other open-source projects are hosted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Explore the corresponding repositories, feature trackers, wi- kis, discussion boards, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that each community is different and is governed by different guidelines: after all, we are from all over the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 170 II DEEPER Test-driven Development and Continuous Integration It is often hygienic to include some principles of test-driven development when writ- ing own functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='18 Assume that, for some reasons, we were asked to write a function to compute the root mean square (quadratic mean) of a given numeric vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Before implementing the actual routine, it is a good idea to reflect upon what we want to achieve, especially how we want our function to behave in certain boundary cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' stopifnot gives simple means to assure a given assertion is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If that is the case, it will move forward quietly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us say we have come up with the following set of expectations: stopifnot(all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='equal(rms(1), 1)) stopifnot(all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='equal(rms(1:100), 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='16786054171151931769)) stopifnot(all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='equal(rms(rep(pi, 10)), pi)) stopifnot(all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='equal(rms(numeric(0)), 0)) Write a function rms that fulfils the above assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='19 Implement your own version of the sample function (assuming replace=TRUE), using calls to runif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, start by writing a few unit tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' There are also a couple of R packages that support writing and executing of unit tests, including testthat, tinytest (which is a lighter-weight version of the former), RUnit, or realtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, in the most typical use cases, relying on stopifnot is powerful enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='20 (*) Consult the Writing R Extensions manual [45] about where and how to include unit tests in your example package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note (*) R includes a built-in mechanism to check a couple of code quality areas: running R CMD check pkg_directory from the command line (preferably using the most recent version of R) can suggest a number of improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, it is possible to use various continuous integration techniques that are automat- ically triggered when pushing changes to our software repositories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see GitLab CI or GitHub Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, it is possible to run a package build, install, and check process upon every git commit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, CRAN features some continuous integration services, including checking the package on a range of different platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Debugging For all his life, the current author has been debugging all his programs mostly by manually printing the state of suspected variables (printf and the like) in different areas of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' No shame in that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9 DESIGNING FUNCTIONS 171 For an interactive debugger, see the browser function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, refer to Section 9 of [49] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some IDEs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', RStudio) support this feature too;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see their corresponding docu- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Profiling Typically,aprogramspendsrelativelylongtimeexecutingonlyasmallportionofcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The Rprof functioncanbe ahelpful tool toidentifywhich chunksmightneed arewrite, for instance using a compiled language (Chapter 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Please remember, though, that not only implementations of algorithms that have hight computational complexity can form a bottleneck, but also data input and out- put (such as reading files from disk, printing messages, on the console, querying Web APIs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Special Functions: Syntactic Sugar Some functions, such as `*`, are somewhat special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They can be referred to using an alternativesyntaxwhichforsomereasonmostofusacceptedasthedefaultone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Below we will reveal, amongst others, that “5 * 9” reduces in fact to an ordinary function call: `*`(5, 9) # a call to `*` with 2 arguments, equivalent to 5 * 9 ## [1] 45 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 A Note on Backticks In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2, we have mentioned that we can assign (as in `<-`) syntactically valid names to our objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Most identifiers comprised of letters, digits, dots, and under- scores can be used directly in R code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, it is possible to name our objects whichever we like: non-syntactically valid (nonstandard) names just need to be enclosed in backticks (back quotes, grave ac- cents): `42 a quite peculiar name :O lollolll` <- (-5):5 mean(1/(1+exp(-`42 a quite peculiar name :O lollolll`))) ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Of course, they are less convenient, but still: backticks lets us access them in any con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example: x <- list(`1`="a", `2`="b") # structure(list("a", "b"), names=c("1", "2")) (continues on next page) 172 II DEEPER (continued from previous page) x$`1` # x[["1"]] is still okay (and we prefer this syntax) ## [1] "a" 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Curly Braces, `{` A block of statements grouped with curly braces, `{`, corresponds to a function call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' When we write: { print(TRUE) cat("two") 3 } ## [1] TRUE ## two ## [1] 3 The parser translates it to a call to: `{`(print(TRUE), cat("two"), 3) ## [1] TRUE ## two ## [1] 3 When the above is executed, every argument, one by one, is evaluated and then the last value is returned in result of that call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 `if` if is a function too;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' as mentioned in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1, it returns the value corresponding to the expression evaluated conditionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, we may write: if (runif(1) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) "head" else "tail" ## [1] "head" but also: `if`(runif(1) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, "head", "tail") ## [1] "head" Note A call like `if`(test, what_if_true, what_if_false) can only work properly because of the lazy evaluation of function arguments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9 DESIGNING FUNCTIONS 173 On a side note, while, for, repeat can also be called that way, but they return invis- ible(NULL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Operators are Functions Too Calling Built-in Operators as Functions Every arithmetic, logical, and comparison operator is translated to a call to the cor- responding function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance: `<`(`+`(`*`(`-`(3), 4)), 5) # 2+(-3)*4 < 5 ## [1] TRUE Also, x[i] is equivalent to `[`(x, i) and x[[i]] maps to `[[`(x, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Knowing this will not only enable us to manipulate unevaluated R code (Chapter 15) or access the corresponding manual pages (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', help("[")), but also write some expressions in a more concise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, x <- list(1:5, 11:17, 21:23) unlist(Map(`[`, x, 1)) # 1 is a further argument passed to `[` ## [1] 1 11 21 is equivalent to a call to Map(function(e) e[1], x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note Unsurprisingly, the assignment operator, `<-`, is a function too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It returns the assigned value, invisibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Knowingthat`<-`bindsrighttoleft(compare help("Syntax")),thisiswhytheexpres- sion “a <- b <- 1” results in both a and b being assigned 1: it is equivalent to “`<-`("a", `<-`("b", 1))” and “`<-`("b", 1)” returns 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Owing to the pass-by-value semantics (Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1) we can also expect that we will alwaysbe(withtheexceptionofenvironments,Chapter16)assigningacopyofthevalue on the righthand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- 1:6 y <- x # makes a copy (but delayed, on demand, for performance reasons) y[c(TRUE, FALSE)] <- NA_real_ # modify every 2nd element print(y) ## [1] NA 2 NA 4 NA 6 print(x) # state of x has not changed — x and y are different objects ## [1] 1 2 3 4 5 6 This is especially worth pointing out to Python (amongst others) programmers, where the above assignment would mean that x and y both refer to the same (shared) object in the computer’s memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 174 II DEEPER However, with no harm done to semantics, the actual copying of x is postponed until absolutely necessary (Section 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is efficient both time- and memory-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Creating Own Binary Operators We can also introduce our own binary operators named like `%myopname%`: `%:)%` <- function(e1, e2) (e1+e2)/2 5 %:)% 1:10 ## [1] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Recall that `%%` and `%/%` are built-in operators denoting division remainder and in- teger division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Rarely will we be defining our own operators, but when we encounter a similar one next time, we will no longer be surprised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, in Chapter 11 we will learn about `%*%` which implements matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note In Chapter 10, we will note that most existing operators can be overloaded for objects of different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Replacement Functions Functions generally do not change the state of their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, there is some syntactic sugar that allows us to replace objects or parts thereof with new con- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We call them replacement functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' three of the following calls replace the input x with its modified version: x <- 1:5 # example input x[3] <- 0 # replace the 3rd element with 0 length(x) <- 7 # "replace" length names(x) <- LETTERS[seq_along(x)] # replace the names attribute print(x) # x is different than before ## A B C D E F G ## 1 2 0 4 5 NA NA Creating Own Replacement Functions A replacement function is a mapping named like `name<-` with at least two paramet- ers: x (the object to be modified),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (possible further arguments), value (as the last parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' the object on the righthand side of the `<-` operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Most often, we will be interacting with existing replacement functions, not creating 9 DESIGNING FUNCTIONS 175 our own ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, knowing how to do the latter is key to understanding this language feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example: `add<-` <- function(x, where=TRUE, value) { x[where] <- x[where] + value x # the modified object that will replace the original one } The above aims to add some value to a subset of the input vector x (by default, to each element therein) and return its altered version that will replace the object it has been called upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' y <- 1:5 # example vector add(y) <- 10 # calls `add<-`(y, value=10) print(y) # y has changed ## [1] 11 12 13 14 15 add(y, 3) <- 1000 # calls `add<-`(y, 3, value=1000) print(y) # y has changed again ## [1] 11 12 1013 14 15 We see that calling “add(y, w) <- v” works as if we have called “y <- `add<-`(y, w, value=v)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note (*) According to [49], a call “add(y, 3) <- 1000” is a syntactic sugar precisely for: `*tmp*` <- y # temporary substitution y <- `add<-`(`*tmp*`, 3, value=1000) rm("*tmp*") # remove the named object from the current scope This has at least twoimplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' First,in the unlikelyeventthat a variable`*tmp*` ex- istedbeforethecalltothereplacementfunction,itwillbenomore,itwillceasetobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='It will be an ex-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Second, the temporary substitution guarantees that y must ex- ist before the call (a function’s body does not have to refer to all the arguments passed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' because of lazy evaluation, see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, we could get away with it otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Substituting Parts of Vectors The replacement versions of the subsetting operators are named as follows: `[<-` is used in substitutions like “x[i] <- value”, `[[<-` is called when we perform “x[[i]] <- value”, `$<-` is used whilst calling “x$i <- value”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Here is a use case: 176 II DEEPER x <- 1:5 `[<-`(x, c(3, 5), NA_real_) # returns a new object ## [1] 1 2 NA 4 NA print(x) # does not change the original input ## [1] 1 2 3 4 5 On a side note, `length<-` can be used to expand or shorten a given vector by calling “length(x) <- new_length”, see also Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- 1:5 x[7] <- 7 length(x) <- 10 print(x) ## [1] 1 2 3 4 5 NA 7 NA NA NA length(x) <- 3 print(x) ## [1] 1 2 3 Despite the fact that, semantically speaking, calling `[<-` results in the creation of a new vector (a modified version of the original one), we may luckily expect some per- formance optimisations happening behind our back (reference counting, modifica- tion in-place;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see sec:to-do).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='21 Write a function `extend<-` which pushes new elements at the end of a given vector, modifying it in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `extend<-` <- function(x, value) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example use: x <- 1 extend(x) <- 2 # push 2 at the back extend(x) <- 3:10 # add 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', 10 print(x) ## [1] 1 2 3 4 5 6 7 8 9 10 Replacing Attributes Many replacement functions deal with the re-setting of objects’ attributes (Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular, for each special attribute, there is also its replacement version, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', `names<-`, `class<-`, `dim<-`, `levels<-`, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- 1:3 names(x) <- c("a", "b", "c") # change the `names` attribute print(x) # x has been altered (continues on next page) 9 DESIGNING FUNCTIONS 177 (continued from previous page) ## a b c ## 1 2 3 Individual (arbitrary, including non-special ones) attributes can be set using `attr<-` and all of them can be established by means of a single call to `attributes<-`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- "spam" attributes(x) <- list(shape="oval",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' smell="meaty") attributes(x) <- c(attributes(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' taste="umami") attr(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "colour") <- "rose" print(x) ## [1] "spam" ## attr(,"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='shape") ## [1] "oval" ## attr(,"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='smell") ## [1] "meaty" ## attr(,"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='taste") ## [1] "umami" ## attr(,"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='colour") ## [1] "rose" Also note that setting an attribute to NULL results,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' by convention,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' in its removal: attr(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "taste") <- NULL # this is tasteless now print(x) ## [1] "spam" ## attr(,"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='shape") ## [1] "oval" ## attr(,"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='smell") ## [1] "meaty" ## attr(,"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='colour") ## [1] "rose" attributes(x) <- NULL # remove all print(x) ## [1] "spam" Which can be useful in contexts such as: x <- structure(c(a=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' b=2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' c=3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' some_attrib="value") y <- `attributes<-`(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' NULL) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x retains its attributes and y is a version of x with metadata removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Compositions of Replacement Functions Updating only selected names like: 178 II DEEPER x <- c(a=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' b=2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' c=3) names(x)[2] <- "spam" print(x) ## a spam c ## 1 2 3 is possible due to the fact that “names(x)[i] <- v” is equivalent to: old_names <- names(x) new_names <- `[<-`(old_names,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' value=v) x <- `names<-`(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' value=new_names) Important More generally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' a composition of replacement calls “g(f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' b) <- y” yields a result equivalent to “x <- `f<-`(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' value=`g<-`(f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' value=y))”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that both f and `f<-` need to be defined, but having g is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='22 (*) What is “h(g(f(x, a), b), c) <- y” equivalent to?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='23 Write a (actually very useful!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') function `recode<-` which replaces specific ele- ments in a character vector with some other ones, allowing the following interface: `recode<-` <- function(x, value) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- c("spam", "bacon", "eggs", "spam", "eggs") recode(x) <- c(eggs="best spam", bacon="yummy spam") print(x) ## [1] "spam" "yummy spam" "best spam" "spam" "best spam" We see that the named character vector gives a few from="to" pairs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', all eggs are to be re- placed by best spam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Now, determine which calls are equivalent to the following: x <- c(a=1, b=2, c=3) recode(names(x)) <- c(c="z", b="y") # or equivalently = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' print(x) ## a y z ## 1 2 3 y <- list(c("spam", "bacon", "spam"), c("spam", "eggs", "cauliflower")) recode(y[[2]]) <- c(cauliflower="broccoli") # or = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' print(y) ## [[1]] ## [1] "spam" "bacon" "spam" ## ## [[2]] ## [1] "spam" "eggs" "broccoli" 9 DESIGNING FUNCTIONS 179 Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='24 (*) Consider the `recode<-` function from the previous exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hereisanexamplematrixwiththedimnamesattributewhosenamesattributeisset(moredetails in Chapter 11): (x <- Titanic["Crew", "Male", , ]) ## Survived ## Age No Yes ## Child 0 0 ## Adult 670 192 recode(names(dimnames(x))) <- c(Age="age", Survived="survived") print(x) ## survived ## age No Yes ## Child 0 0 ## Adult 670 192 This changes the x object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For each of the following subtasks, write a single call which alters names(dimnames(x)) without modifying x in-place but returning a recoded copy of: names(dimnames(x)), dimnames(x)), x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25 (*) Consider the `recode<-` function once again but now let an example object be a data frame featuring a column of class factor: x <- iris[c(1, 2, 51, 101), ] recode(levels(x[["Species"]])) <- c( setosa="SET", versicolor="VER", virginica="VIR" ) print(x) ## Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width Petal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length Petal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width Species ## 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 SET ## 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 SET ## 51 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 VER ## 101 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 VIR Without modifying x in-place, how to change levels(x[["Species"]]) and return an altered copy of: levels(x[["Species"]]), x[["Species"]], x?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 180 II DEEPER 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Arguments and Local Variables 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Pass by “Value” As a general rule, functions cannot change the state of their arguments8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We can think of them as being passed by value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', as if their copy was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' test_change <- function(y) { y[1] <- 7 y } x <- 1:5 test_change(x) ## [1] 7 2 3 4 5 print(x) # same ## [1] 1 2 3 4 5 If the above was not the case, the state of x would have been changed after the call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Variable Scope Function arguments as well as any other variables we create inside a function’s body are relative to each call to that function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=" test_change <- function(x) { x <- x+1 z <- -x z } x <- 1:5 test_change(x*10) ## [1] -11 -21 -31 -41 -51 print(x) # x in the function's body was a different x ## [1] 1 2 3 4 5 print(z) # z was local ## Error in print(z): object 'z' not found Both x and z are local variables and live only whilst our function is being executed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8 With the exception of objects of type environment, which are passed by reference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Chapter 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, the fact that we have access to unevaluated R expressions can cause further deviations to this rule (see be- low).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9 DESIGNING FUNCTIONS 181 The former temporarily “overshadows”9 the object of the same name from the caller’s context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important It is a good development practice to refrain from referring to objects not created within the current function, especially to “global” variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We can always pass an object as an argument explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note It is a function call as such, not curly braces per se that form a local scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Writing “x <- { y <- 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' y + 1 }”, y is not an auxiliary variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' it is an ordinary named object created alongside x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' On the other hand, in “x <- (function() { z <- 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' z + 1 })()”, z will not be available thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Closures (*) Most user-defined functions are in fact representatives of the so-called closures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Chapter 18 and [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They not only consist of an R expression to evaluate, but also can carry some auxiliary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, given two equal-length numeric vectors x and y, a call to approxfun(x, y) returns a function that linearly interpolates between the consecutive points (𝑥1, 𝑦1), (𝑥2, 𝑦2), and so forth, so that a corresponding 𝑦 can be determined for any 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- seq(0, 1, length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='out=11) f1 <- approxfun(x, x^2) f2 <- approxfun(x, x^3) f1(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='75) # check that it is quite close to the true 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='75^2 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='565 f2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='75) # compare with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='75^3 ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4275 Inspecting, however, the source codes of the above functions: print(f1) ## function (v) ## .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='approxfun(x, y, v, method, yleft, yright, f, na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm) ## ## print(f2) (continues on next page) 9 In Chapter 18, we will discuss this topic in-depth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' objects are bound to their names within environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Moreover, R uses lexical (static) scoping, which is not necessarily intuitive, especially taking into account that a function’s environment can always be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 182 II DEEPER (continued from previous page) ## function (v) ## .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='approxfun(x, y, v, method, yleft, yright, f, na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm) ## ## we might wonder how can they produce different results — it is evident that they are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It turns out, however, that they internally store some additional data that is referred to upon their calls: environment(f1)[["y"]] ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='00 environment(f2)[["y"]] ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='729 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='000 This and many more we will explore in great detail in the third part of this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Default Arguments We have already mentioned above that, when designing functions that perform com- plex tasks, we will sometimes be faced with a design problem: how to find a sweet spot betweenbeinggenerous/mindfulofthediverseneedsofourusersandmakingtheAPI neither overwhelming nor oversimplistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We know that it is best if a function performs one, well-specified task, but also allows its behaviour be tuned-up if one wishes to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This principle can be facilitated by the use of default arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, log computes logarithms, by default the natural ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' log(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='718) # the same as log(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='78, base=exp(1)) — default base ## [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9999 log(4, base=2) # different base ## [1] 2 Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='26 Study the documentation of the following functions and note the default values that they define: round, hist, grep, and download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We can easily define our own functions equipped with such recommended settings: test_default <- function(x=1) x test_default() # use default ## [1] 1 test_default(2) # use something else ## [1] 2 Most often, default arguments are just constants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, they can be any R 9 DESIGNING FUNCTIONS 183 expressions, also including a reference to other arguments passed to the same func- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see more in Section 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note that default arguments will most often appear and the end of the parameter list, but see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 (on replacement functions) for a well-justified exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Lazy Evaluation In some languages, function arguments are always evaluated prior to a call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In R, though, they are only computed when actually needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We call it lazy or delayed evalu- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Recall that in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 we introduced the short-circuit evaluation operators `||` (or) and `&&` (and).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They are able to do their job precisely thanks to this mechan- ism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='27 In the following example, we do not use the function’s argument at all: lazy_test1 <- function(x) 1 # x not used at all lazy_test1({cat("and now for something completely different!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7}) ## [1] 1 Otherwise, we would see a message being printed out on the console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='28 Next, let us use x amidst other expressions in the body: lazy_test2 <- function(x) { cat("it\'s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ") y <- x+x # using x twice cat(" a man with two noses") y } lazy_test2({cat("and now for something completely different!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=" 7}) ## it's." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' and now for something completely different!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' a man with two noses ## [1] 14 Note that an argument is evaluated once and its value is stored for further reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If that was not the case, we would see two messages like and now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='. 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Ellipsis, `.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='` Let us start with an exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='29 Note the presence of `.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='` in the parameter list of c, list, structure, cbind, rbind, cat, Map (and the underlying mapply), lapply (a specialised version of Map), optimise, optim, uniroot, integrate, outer, aggregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What purpose does it serve, according to these functions manual pages?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 184 II DEEPER We can create a variadic function by placing a dot-dot-dot (ellipsis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see help("dots")), `.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='`, somewhere in its parameter list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The ellipsis serves as placeholder for all objects passed to the function but not matched by any formal (named) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The easiest way to process arguments passed via `.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='` programmatically (see also Sec- tion 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2) is by redirecting them to list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' test_dots <- function(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') list(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') test_dots(1, a=2) ## [[1]] ## [1] 1 ## ## $a ## [1] 2 Such a list can be processed just like… any other R list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What we can do with these arguments is only limited by our creativity (in particular, recall from Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 the very powerful do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='call function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Still,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' there are two major use cases of the ellipsis10: create a new object by combining an arbitrary number of other objects: c(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3) # 3 arguments ## [1] 1 2 3 c(1:5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6:7) # 2 arguments ## [1] 1 2 3 4 5 6 7 structure("spam") # 0 additional arguments ## [1] "spam" structure("spam",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' color="rose",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' taste="umami") # 2 further arguments ## [1] "spam" ## attr(,"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='color") ## [1] "rose" ## attr(,"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='taste") ## [1] "umami" cbind(1:2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3:4) ## [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1] [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2] ## [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='] 1 3 ## [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='] 2 4 cbind(1:2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3:4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 5:6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7:8) ## [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1] [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2] [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3] [,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4] ## [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='] 1 3 5 7 ## [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='] 2 4 6 8 sum(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 42) ## [1] 108 10 Which is somewhat similar to Python’s *args and **kwargs in a function’s parameter list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9 DESIGNING FUNCTIONS 185 pass further arguments (as-is) to other methods : lapply(list(c(1, NA, 3), 4:9), mean, na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=TRUE) # mean(x, na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='rm=TRUE) ## [[1]] ## [1] 2 ## ## [[2]] ## [1] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 integrate(dbeta, 0, 1, shape1=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, shape2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) # dbeta(x, shape1=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5, shape2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5) ## 1 with absolute error < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2e-05 Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='30 The documentation of lapply (let us call help("lapply") now) states that this function is defined as lapply(X, FUN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Here, the ellipsis is a placeholder for a number of optional arguments that can be passed to FUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, if we denote the i-th element of a vector X by X[[i]], calling lapply(X, FUN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') will return a list whose i-th element will be equal to FUN(X[[i]], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='31 Usingasinglecallto lapply,generatealistwiththreenumericvectorsoflengths 3, 9, and 7, respectively, drawn from the uniform distribution on the unit interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Then, upgrade your code to get numbers sampled form the interval [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 Metaprogramming (*) Under the hood, lazy evaluation is a quite complicated mechanism that relies upon the storing of unevaluated R expressions and special promises to instantiate them11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It turns out that we have access to such expressions programmatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular, a call to the composition of deparse and substitute can convert them to a character vector: test_deparse_substitute <- function(x) deparse(substitute(x)) test_deparse_substitute(testing+1+2+3) ## [1] "testing + 1 + 2 + 3" test_deparse_substitute(spam & spam^2 & bacon | grilled(spam)) ## [1] "spam & spam^2 & bacon | grilled(spam)" Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='32 Check out the y-axis label generated by plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default((1:100)^2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Inspect its source code and note a call to the two aforementioned functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Similarly, call shapiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='test(log(rlnorm(100))) and take note of the data: field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' A function is free to do with such an expression whatever it likes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, it can manipulate it and evaluate it in a different context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thanks to such a language feature, 11 Such an evaluation model has been heavily inspired by Scheme [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It will be explained in more detail in sec:to-do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 186 II DEEPER certain operations can be designed so that their users can express them much more compactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is certainly (in theory) a very powerful tool but from practice we know many instances where it has been over/misused and made the use of R confusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='33 (*) The built-in subset and transform use metaprogramming techniques to specify basic data frame transformation techniques (see Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance: transform( subset( iris, Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length>=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 & Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width >= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0, select=c(Species, Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length:Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width) ), Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='mm=Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length/10 ) ## Species Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='mm ## 118 virginica 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='77 ## 132 virginica 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='79 ## 136 virginica 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='77 Notethatnoneofthearguments–except iris–makessenseoutsideofthefunctioncallcontexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular, neither Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length nor Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width variables exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The two functions took the liberty to interpret the arguments passed as they felt like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They have created their own virtual reality within our well-defined world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The reader must refer to their documentation to discover the meaning of the special syntax used therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note (*) Some functions have rather peculiar default arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, in the manualpageof prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='test,wereadthatthe alternativeparameterdefaultsto c("two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' sided", "less", "greater") but that "two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='sided" is actually the default one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If we call print(prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='test), we will find the code line responsible for this behaviour: “alternative <- match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='arg(alternative)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Consider the following example: test_match_arg <- function(x=c("a", "b", "c")) match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='arg(x) test_match_arg() # missing argument — choose 1st ## [1] "a" test_match_arg("c") # one of the predefined options ## [1] "c" test_match_arg("d") # unexpected setting ## Error in match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='arg(x): \'arg\' should be one of "a", "b", "c" In this setting, match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='arg allows only an actual parameter amongst a given set of choices, but selects the first option if the argument is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Unfortunately,wehavetolearnthisbehaviourbyheart,becauseactuallylookingatthe above source code gives us no clue about this being possible whatsoever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If such an ex- 9 DESIGNING FUNCTIONS 187 pression was normally evaluated, we would either be passing the default argument or whatever the user passed as x (but then the function would not know about the range of possible choices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' A call to “match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='arg(x, c("a", "b", "c"))” could guarantee the desired functionality and would be much more readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Instead, metaprogramming techniques allowed match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='arg to access the enclosing function’s default argument list without our explicitly referring to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' One may ask “why is it so” and the only sensible answer to this will be “because its programmer decided it must be this way”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us contemplate this for a while.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In cases like this, we are dealing not with some base R language design choice that we might like or dislike, but which we should normally just accept as an inherent feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Rather, we are struggling intellectually because of some R programmer’s (mis)use (in good faith…) of R’s flexibility itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They have introduced a slang/dialect on top of our mother tongue, whose meaning is valid only within this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Blame the middle- man, not the environment, please.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We generally advocate for avoiding metaprogramming wherever possible (and will elaborate on this later on, including formulas (`~`), built-in functions like subset or transform, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Exercises Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='34 Answer the following questions: Will “stopifnot(1)” stop?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What about “stopifnot(NA)”, “stopifnot(TRUE, FALSE)”, and “stopifnot(c(TRUE, TRUE, NA))”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What does the `if` function return?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Does `attributes<-`(x, NULL) modify x?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' When can we be interested in calling `[` and `[<-` as functions (and not as operators) dir- ectly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' How to define our own binary operator?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Can it have some default arguments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What are the main use cases of `.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='`?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' What is wrong with transform, subset, and match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='arg?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' When a call like “f(-1, do_something_that_takes_a_million_years())” does not ne- cessarily have to be a bad idea?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='35 What is the return value of a call to “f(list(1, 2, 3))”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' f <- function(x) { (continues on next page) 188 II DEEPER (continued from previous page) for (e in x) { print(e) } } Is it NULL, invisible(NULL), x[[length(x)]], or invisible(x[[length(x)]])?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='36 The split function also has its replacement version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Study its documentation to learn how it works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='37 A call to ls(envir=baseenv()) returns all objects defined in package base (see Chapter 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' List the names corresponding to some replacement functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important Apply the principle of test-driven development when solving the remain- ing exercises (or those which you have skipped intentionally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='38 Implement your own version of the Position and Find functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Evaluation should stop as soon as the first element fulfilling a given predicate has been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='39 Implement your own version of the Reduce function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='40 Write a function slide(f, x, k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') which returns a list y of size length(x)-k+1 such that y[[i]] = f(x[i:(i+k-1)], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') unlist(slide(sum, 1:5, 1)) ## [1] 1 2 3 4 5 unlist(slide(sum, 1:5, 3)) ## [1] 6 9 12 unlist(slide(sum, 1:5, 5)) ## [1] 15 Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='41 Using slide defined above, write another function that counts how many in- creasing pairs of numbers are featured in a given numeric vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, in c(0,2,1,1, 0,1,6,0) there are three such pairs: (0,2), (0,1), (1,6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='42 (*) Write your own version of tools::package_dependencies with re- verse=TRUE based on information extracted by calling utils::available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10 S3 Classes Let x be a randomly generated matrix with 1,000,000 rows and 1,000 columns, y be a data frame with results from the latest survey indicating that things are not the way most people (no matter the side of the many political spectra) think they are, and and z be another matrix, this time with many zeroes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Human brain is not capable of handling too much information which is too specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is why we have a natural tendency to group different entities based on their sim- ilarities so as to form some more abstract classes thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, many of us are inherently lazy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thus, oftentimes we will take shortcuts to min- imise energy (at a price to be paid later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Printing out a matrix, a data frame, and a time series are all still instances of the dis- playing of things, although they surely differ in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Now that ad probably forgot- ten which objects are hidden behind x, y, and z, being able to simply call “print(y)” without having to recall that, yes, y is a data frame, might seem quite appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This chapter introduces the so-called S3 classes [8], which provide a lightweight object oriented programming (OOP) approach for automated dispatching of calls to generics of the type “print(y)” to concrete methods such as “print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='frame(y)”, based on the class of the object they are invoked upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' S3 classes in their essence are beautifully simple1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They are inspired2 by the well- thought-through concepts present in other functional programming languages (such as the Common Lisp Object System;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Ultimately, those generics and methods are ordinary R functions (Chapter 7) and classes are merely additional object attributes (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Of course this does not mean that wrapping our heads around them will be effortless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, unlike other “class systems”3, S3 is ubiquitous in R programming: suffice it 1 However, some classes, even the built-in ones that we describe here, can be poorly designed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g, some crucial methods might be missing, they can be not-well-interoperable with other classes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Do not blame this messenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Remember that the R environment is still very reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, there are cases where changing the current behaviour in one place could lead to undesirable consequences elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2 They were built on top of the ordinary (“old S”) R, hence have certain limitations what we discuss in the sequel: classes cannot be formally defined (often we will use named lists for representing objects, and we know we cannot be any more flexible than this), and the dispatching can only be based on the class of one (usually the first, but, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', binary operators take both types into account) of the arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3 Other class systems may give an impression that they are alien implants that were forcefully added to our language to solve a specific, rather narrow class of problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', S4 (Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5), Reference Classes (Section 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3), and other ones proposed by third-party packages 190 II DEEPER to say that factors, matrices, and data frames discussed in the next chapters are quite straightforward, S3-based extensions of the concepts we have introduced so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Object Type vs Class Recall that typeof (introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1) returns the internal type of any R object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Even though there are only few admissible cases thereof4, they open the world of end- less possibilities5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Thebasictypeswecoveredsofar(mostlyatomicandgenericvectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='compareFigure1) provide a basis for more complex data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is thanks to the fact that they can be equipped with arbitrary attributes (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' typeof(NULL) ## [1] "NULL" typeof(c(TRUE, FALSE, NA)) ## [1] "logical" typeof(c(1, 2, 3, NA_real_)) ## [1] "double" typeof(c("a", "b", NA_character_)) ## [1] "character" typeof(list(list(1, 2, 3), LETTERS)) ## [1] "list" typeof(function(x) x) ## [1] "closure" The interesting fact is that most compound types, whose most prevalent instances are constructed using the mechanisms discussed in this chapter6, only pretend they are something different than what they actually are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' They are often quite good at doing their job, though, and hence might be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' By knowing what is under their hood we will demystify them and become able to manipulate their state outside of the pre- scribed use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important Setting the class attribute might make some objects behave differently in certain scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Let us consider two identical objects equipped with different class attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 4 TheirlistishardcodedattheClanguagelevel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='comparethelistof SEXPTYPEsin[48]andseealsoChapter 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 5 In particular, later we mention externalptrs which are simply pointers to memory allocated on the heap, so these might be any instances of C structs or C++ classes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This makes R a very extensible lan- guage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 6 But of course there is more;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see the S4 and other systems discussed in Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10 S3 CLASSES 191 xt <- structure(123, class="POSIXct") # POSIX calendar time xd <- structure(123, class="Date") Despite that both objects are being represented using numeric vectors: c(typeof(xt), typeof(xd)) ## [1] "double" "double" When printed, they are decoded quite differently: print(xt) ## [1] "1970-01-01 10:02:03 AEST" print(xd) ## [1] "1970-05-04" In the former case, 123 is treated as the number of seconds since the so-called UNIX epoch, 1970- 01-01T00:00:00+0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The latter is deciphered as the number of days since the said (quite widely used in computer systems by the way) timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Wemayhencesuspect,andweareabsolutelyright,thatthereexistssomeunderlyingmechanism that actually calls a version of print that is dependent on an object’s virtual class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' That this only depends on the class attribute, which might be set, unset, or reset quite freely, is emphasised below: attr(xt, "class") <- "Date" # change class from POSIXct to Date print(xt) # same 123, but now interpreted as Date ## [1] "1970-05-04" as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numeric(xt) # drops all attributes ## [1] 123 unclass(xd) # drops the class attribute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `attr<-`(xd, "class", NULL) ## [1] 123 We are having so much fun that one more illustration can only add to joy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Consider an example data frame: x <- iris[1:3, 1:2] # a subset of a built-in example data frame print(x) ## Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width ## 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 ## 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 ## 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 This is an object of the following class (an object whose class attribute is set to): attr(x, "class") ## [1] "data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='frame" 192 II DEEPER Some may say, and they are absolutely right, that we have not covered data frames yet: this is the topic of Chapter 12, which is still ahead of us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, from the current perspective, we are interestedinthefactthatanRdataframeismerelyalist(Chapter4)ofvectorsofthesamelengths equipped with names and row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='names attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' typeof(x) ## [1] "list" attr(x, "class") <- NULL # or x <- unclass(x) print(x) ## $Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length ## [1] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 ## ## $Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width ## [1] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 ## ## attr(,"row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='names") ## [1] 1 2 3 Important Revealing how x is actually represented, enables us to process it (although perhaps not in the most convenient or efficient manner) using the extensive skill set that we have already7 developed by studying the material covered in the previous part of our book (including solving all the exercises).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This can be particularly useful, espe- ciallybearinginmindthatsome(built-inorthird-party)datatypesarenotparticularly well-designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note again that attributes are simple additions to R objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, as we said in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3, certain attributes are special, and class is one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular, we can set class to be only a character vector (possibly of length greater than one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- 12345 attr(x, "class") <- 1 # character vectors only ## Error in attr(x, "class") <- 1: attempt to set invalid \'class\' attribute Furthermore, there exists the class function that can read the value of the class at- tribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Its replacement version is also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' class(x) <- "Date" # set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' the same as attr(x, "class") <- "Date" class(x) # get;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' the same as attr(x, "class") ## [1] "Date" Important The class function always yields a value, even if the corresponding at- 7 For instance, consider once again the example from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 that applies the split function on a data frame reduced to a list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10 S3 CLASSES 193 tribute is not set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We call it an implicit class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Compare between the following and the outputs generated by typeof: class(NULL) # no `class` set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' because NULL cannot have attributes at all ## [1] "NULL" class(c(TRUE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' FALSE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' NA)) # no attributes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' so class is implicit (= typeof) ## [1] "logical" class(c(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' NA_real_)) # typeof yields "double" ## [1] "numeric" class(c("a",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' "b",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' NA_character_)) ## [1] "character" class(list(list(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' LETTERS)) ## [1] "list" class(function(x) x) # typeof yields "closure" ## [1] "function" Also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' in Chapter 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' we will note that any object equipped with the dim attribute also has an implicit class: (x <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='matrix(c(1, 2, 3))) ## [,1] ## [1,] 1 ## [2,] 2 ## [3,] 3 attributes(x) # `class` is not amongst the attributes ## $dim ## [1] 3 1 class(x) # implicit class ## [1] "matrix" "array" typeof(x) # it is still a numeric vector (under the hood) ## [1] "double" 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Generics and Method Dispatching 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Generics, Default, and Custom Methods Let us inspect the source code of the print function: print(print) # sic!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ## function (x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') ## UseMethod("print") ## ## 194 II DEEPER Any function like the above8 we will call from now on an S3 (S version 3) generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Its only job is to invoke UseMethod("print")9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This dispatches the control flow to another function, referred to as method, based on the class of the first argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Forexample,letusdefineanobjectofclass categorical(anamethatwehavejustcome up with;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' we could have called it cat, CtGrCl, or SpanishInquisition as well), which will be our own version of the famous built-in factor type that we discuss later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' x <- structure( c(1, 3, 2, 1, 1, 1, 3), levels=c("a", "b", "c"), class="categorical" ) We assume that such an object is a vector of small positive integers (codes) equipped with the levels attribute being a character vector of length no less than the maximum of the said integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The first category will be used to decipher the meaning of code “1”, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, the above vector represents a sequence a, c, b, a, a, a, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We have not defined any special method for the printing of objects of class categor- ical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, when we call print, the default (fallback) method will be called: print(x) ## [1] 1 3 2 1 1 1 3 ## attr(,"levels") ## [1] "a" "b" "c" ## attr(,"class") ## [1] "categorical" This is the standard function for displaying numeric vectors that we are all well famil- iar with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Its name is print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default, and we can always call it directly: print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default(x) # the default print method ## [1] 1 3 2 1 1 1 3 ## attr(,"levels") ## [1] "a" "b" "c" ## attr(,"class") ## [1] "categorical" Wecan,however,introduceourownmethodforthecustomprintingofobjectsofclass categorical, whose name must precisely be print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical: print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical <- function(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') (continues on next page) 8 Note that some functions can have a version of UseMethod hidden at the C language level (internally);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 9 Which in this context is equivalent to UseMethod("print", x), with x being the first argument to the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10 S3 CLASSES 195 (continued from previous page) { x_character <- attr(x, "levels")[unclass(x)] print(x_character) # calls `print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default` cat(sprintf("Categories: %s\\n", paste(attr(x, "levels"), collapse=", "))) invisible(x) # this is what all print methods do;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see help("print") } Now, calling print automatically dispatches the control flow to the above method: print(x) ## [1] "a" "c" "b" "a" "a" "a" "c" ## Categories: a, b, c Of course, the default method can still be called;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' calling print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default(x) directly will output the same result as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Note print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical has been equipped with the dot-dot-dot attribute, because the generic print had one too;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' we should always ensure consistency ourselves10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 Creating Own Generics Introducing new S3 generics is as straightforward as defining a function that calls UseMethod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, here is a dispatcher which allows for creating new objects of class cat- egorical based on other objects: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical <- function(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') UseMethod("as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical") We always need to define the default method: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default <- function(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') { x <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character(x) xu <- unique(sort(x)) # drops NAs structure( match(x, xu), class="categorical", levels=xu ) } 10 In particular, the checking of S3 generic/method consistency is part of R package check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 196 II DEEPER Testing: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(c("a", "c", "a", "a", "d", "c")) ## [1] "a" "c" "a" "a" "d" "c" ## Categories: a, c, d as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(c(3, 6, 4, NA, 9, 9, 6, NA, 3)) ## [1] "3" "6" "4" NA "9" "9" "6" NA "3" ## Categories: 3, 4, 6, 9 Note that print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical has been invoked twice here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The above is quite flexible already, because it relies on the generic (Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3) as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character, which handles a wide variety of data types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Of course, it does not mean we cannot be more precise about some particular ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 For instance, we might wantto forbidthe conversionfrom lists,becausethis does not necessarily make sense: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='list <- function(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') stop("conversion of lists to categorical is not supported") The users can always be instructed in the method’s documentation that they are the ones re- sponsible for an explicit conversion of list objects to something different prior to a call to as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Whether this was a good design choice, time will tell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 Note that the default method deals with logical vectors perfectly fine: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(c(TRUE, FALSE, NA, NA, FALSE)) # as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default ## [1] "TRUE" "FALSE" NA NA "FALSE" ## Categories: FALSE, TRUE However, we might still want to introduce a specialised version, because we know a slightly more efficient algorithm (and we have nothing better to do) based on the fact that FALSE and TRUE con- verted to numeric yield 0 and 1, respectively: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='logical <- function(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') { x <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='logical(x) # or stopifnot(is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='logical(x)) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' structure( x + 1, # only 1, 2, and NAs will be generated class="categorical", levels=c("FALSE", "TRUE") ) } This yields the same result, but is a bit faster: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(c(TRUE, FALSE, NA, NA, FALSE)) # as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='logical (continues on next page) 10 S3 CLASSES 197 (continued from previous page) ## [1] "TRUE" "FALSE" NA NA "FALSE" ## Categories: FALSE, TRUE Note that we have performed some argument validation at the beginning, because a user is al- ways able to call a method directly on an R object of any kind (which is a good thing!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Sec- tion 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In other words, there is no guarantee that the argument x must be of type logical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Built-in Generics Many functions and operators we have introduced so far are in fact S3 generics: print, head, `[`, `+`, `<=`, as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character, as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='list, round, log, sum, c, and na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='omit, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Someofthemmightnotevencall UseMethodexplicitly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='dispatchingcanbedoneintern- ally, at the C language level11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Overall, the list of all S3 generics is somewhat difficult to generate12, but at least the internal ones are enumerated in help("InternalMethods") and help("groupGeneric").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Let us overload the as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The default one does not make much sense for the objects of our custom type: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character(x) ## [1] "1" "3" "2" "1" "1" "1" "3" So: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical <- function(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') attr(x, "levels")[unclass(x)] And now: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character(x) ## [1] "a" "c" "b" "a" "a" "a" "c" Exercise 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Overload the unique method for objects of class categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7 Overload the rep method for objects of class categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8 New types should be designed carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, if we forget to consider overloading the to-numeric converter, we might end up with some users being puzzled13 when they see: 11 Which is quite unfortunate because it decreases transparency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' we need to look this information up somewhere in the documentation (instead of simply inspecting a function’s source code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', cbind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, it allows for some methods to be hardcoded at the C language level too and thus be unoverload- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some of such design choices can somewhat be defended, though, as they increase execution speed or memory consumption, but still: we are not fans thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 12 See also .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='knownS3Generics and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='S3_methods_table which are related to the advanced topics we cover in Section 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 13 It is a different story if this is our conscious design choice and that this is the behaviour we really 198 II DEEPER (x <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(c(4, 9, 100, 9, 9, 100, 42, 666, 4))) ## [1] "4" "9" "100" "9" "9" "100" "42" "666" "4" ## Categories: 100, 4, 42, 666, 9 as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='double(x) # as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default(x) ## [1] 2 5 1 5 5 1 3 4 2 Hence, we might want to introduce: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical <- function(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') { # as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default(as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(x)) as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='double(as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character(x)) } Which now yields: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='double(x) # as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(x) ## [1] 4 9 100 9 9 100 42 666 4 Note We can still use unclass to fetch the codes: unclass(x) ## [1] 2 5 1 5 5 1 3 4 2 ## attr(,"levels") ## [1] "100" "4" "42" "666" "9" This is because the above returns a class-free object, which is now guaranteed to be handled by the default methods (print, subsetting, as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9 What would happen if we used as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='numeric instead of unclass in print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' categorical and as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='10 Update the above methods in such a way that we can also create named objects of class categorical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', equipped with the names attribute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='11 Note that the levels of x are sorted lexicographically, not numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Introduce a single method that would make the above code (when re-run without any alterations) generate a more natural result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If we document this thoroughly (see how help("factor") discusses the behaviour of a to-numeric conversion), only a user’s ignorance will there be to blame when they still are confused about this behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Remember that we can never make an API totally foolproof and that there will always be someone who will challenge/stress-test our ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Bad design is always bad, but being overprotective has its cons as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Choose your audience wisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10 S3 CLASSES 199 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 DispatchingOnlyonOneArgumentandCallingS3MethodsDirectly With S3, the dispatching is done based on the class of only one14 argument: by default, the first one from the parameter list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example, the c function is a generic which dispatches on the class of the first argu- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Let us overload it for categorical objects (or, more precisely, create a function that will be dispatched to when the generic is called upon a series of objects such that the first element is of the said class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical <- function(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=') as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical( unlist( lapply(list(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='), as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character) ) ) It converts each argument to a character vector (relying on the generic as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character to take care of the details) and makes use of the fact that unlist converts a list of such atomic vectors to a single sequence of strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Calling c with the first argument being of class categorical dispatches to the above method: x <- c(9, 5, 7, 7, 2) xc <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(x) c(xc, x) # c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical ## [1] "9" "5" "7" "7" "2" "9" "5" "7" "7" "2" ## Categories: 2, 5, 7, 9 However, if the first argument is, say, unclassed, the default method will be consulted: c(x, xc) # default c ## [1] 9 5 7 7 2 4 2 3 3 1 This method ignores the class attribute and sees xc as-it-is, a barebone numeric vec- tor: `attributes<-`(xc, NULL) # the underlying codes ## [1] 4 2 3 3 1 This is not a bug!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' This is a well-documented (and now explained) behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' After all, compound types (classed objects) are merely emulated through the basic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 14 This is R, so there are of course many exceptions to this rule which were made for the (debatable) sake of the R users’ convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In particular, in Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2 we mention that cbind and rbind will dispatch to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='frame method if at least one argument is a data frame (and other ones are unclassed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also it is worth noting that the S4 class system that we discuss in Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 allows for dispatching based on the classes many arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 200 II DEEPER Important In most cases, S3 methods can be called directly to get the desired out- come: c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(x, xc) # force a call to the specific method ## [1] "9" "5" "7" "7" "2" "9" "5" "7" "7" "2" ## Categories: 2, 5, 7, 9 Note We said “in most cases”, because some methods can be: hardcoded at the C language level (for instance, there is no c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default defined at all15), hidden (defined in a package namespace but not exported);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3, overloaded as a group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='12 Just for fun, let us find a partition of the iris dataset into three clusters using the k-means algorithm: res <- kmeans(iris[-5], centers=3, nstart=10) print(res) ## K-means clustering with 3 clusters of sizes 50, 62, 38 ## ## Cluster means: ## Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width Petal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length Petal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width ## 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0060 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4280 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4620 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2460 ## 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9016 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7484 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3935 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4339 ## 3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0737 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7421 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0711 ## ## Clustering vector: ## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ## [71] 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ## [ reached getOption("max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='print") -- omitted 51 entries ] ## ## Within cluster sum of squares by cluster: ## [1] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='151 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='821 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='879 ## (between_SS / total_SS = 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4 %) ## ## Available components: (continues on next page) 15 Also, dispatching can be done internally to internal methods: overloading `<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character` will have no effect unless the base `<` is replaced with a custom one that makes an explicit call to UseMethod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Most often, we can expect that the built-in types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=', atomic vectors), factors, data frames, and matrices and other arrays might be treated specially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10 S3 CLASSES 201 (continued from previous page) ## ## [1] "cluster" "centers" "totss" "withinss" ## [5] "tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='withinss" "betweenss" "size" "iter" ## [9] "ifault" The above is an object of class: class(res) ## [1] "kmeans" which in fact is a: typeof(res) ## [1] "list" The underlying list looks like: unclass(res) ## $cluster ## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ## [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ## [71] 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ## [ reached getOption("max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='print") -- omitted 51 entries ] ## ## $centers ## Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length Sepal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width Petal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Length Petal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Width ## 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0060 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4280 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4620 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2460 ## 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='9016 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7484 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3935 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4339 ## 3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='8500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0737 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='7421 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='0711 ## ## $totss ## [1] 681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='37 ## ## $withinss ## [1] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='151 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='821 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='879 ## ## $tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='withinss ## [1] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='851 ## ## $betweenss ## [1] 602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='52 ## ## $size ## [1] 50 62 38 ## (continues on next page) 202 II DEEPER (continued from previous page) ## $iter ## [1] 2 ## ## $ifault ## [1] 0 and we already know that the above is displayed in a fancy way only because there is a print method overloaded for objects of class kmeans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' But is there really?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content="kmeans ## Error in eval(expr, envir, enclos): object 'print." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content="kmeans' not found Even though the method is hidden in the stats package’s namespace, from Section 18." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 we will learn that it can be accessed by calling getS3method("print", "kmeans") or referring to stats:::print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='kmeans (note the triple colon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='5 Multi-class-ness The class attribute can be instantiated as a character vector of any length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For ex- ample: (t1 <- Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='time()) ## [1] "2022-12-27 20:49:37 AEDT" (t2 <- strptime("2021-08-15T12:59:59+1000", "%Y-%m-%dT%H:%M:%S%z")) ## [1] "2021-08-15 12:59:59" Let us inspect the two objects’ classes: class(t1) ## [1] "POSIXct" "POSIXt" class(t2) ## [1] "POSIXlt" "POSIXt" When we discuss date-time classes in more detail later, we will take note that the former is represented as a numeric vector, whilst the latter is a list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Hence, primarily, these two should be seen as instances of two distinct types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, both of them have a lot in common, hence it was a wise design choice to also allow them to be seen as the representatives of the same generic category of POSIX time objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Important When calling a generic function16 f on an object x of classes17 class1, 16 The case of binary operators is handled differently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 17 UseMethod dispatches on the implicit class as determined by the class function (note that the class attribute does not necessarily have to be set in order for class to return a sensible answer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10 S3 CLASSES 203 class2, …, classK (in this order), UseMethod(f, x) dispatches to the method determ- ined as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' if f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='class1 is available18, call it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' otherwise, if f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='class2 is available, call this one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' …;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' otherwise, if f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='classK is available, invoke it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' otherwise, refer to the fallback f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='13 There is a method diff for objects of class POSIXt featuring a statement: r <- if (inherits(x, "POSIXlt")) as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='POSIXct(x) else x This way, we can be handling both POSIXct and POSIXlt instances via the same procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Letusseeinthissimpleschemeanymagic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Itisnothingmorethanwhatwasdescribed above: a way of determining which method should be called for a particular R object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' It can of course be used as a mechanism to mimic (and certainly it was inspired by) the idea of inheritance in object-oriented programming languages, but note that the S3 system does not allow for defining classes in any formal manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example, we cannot say that objects of class POSIXct inherit from POSIXt or each object of class POSIXct is also an instance of POSIXt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The class attribute can still be set arbitrarily on an per-object basis: we can create ones whose class is simply POSIXct (without the POSIXt part) or even c("POSIXt", "POSIXct") (in this very order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='6 Operator Overloading Operators are ordinary functions (Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Even though what follows can par- tially be implied by what we have said above, as usual in R, there will be some oddities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For example, let us overload the index operator for objects of class categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Look- ing at help("["), we see that the default method19 has two arguments: x (the categor- ical object being sliced) and i (the indexer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Ours will have the same interface then: `[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical` <- function(x, i) { structure( unclass(x)[i], # `[`(unclass(x), i) class="categorical", levels=attr(x, "levels") # the same levels as input (continues on next page) 18 For more details on S3 method lookup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Section 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 19 Note that the default S3 method, `[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='default`, is hardcoded at the C language level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Therefore, we can- not refer to it directly (but unclass does the trick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also note that we can also call NextMethod here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see Sec- tion 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 204 II DEEPER (continued from previous page) ) } We can also introduce the replacement version of this operator: `[<-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical` <- function(x, i, value) { levels <- attr(x, "levels") codes <- match(value, levels) # integer codes corresponding to levels x <- unclass(x) x[i] <- codes # default method for the replacement version of `[` structure( x, class="categorical", levels=levels # same levels as input ) # # or, equivalently: # structure( # `[<-`(unclass(x), i, value=match(value, attr(x, "levels"))), # class="categorical", # levels=attr(x, "levels") # ) } Testing: x <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(c(3, 6, 4, NA, 9, 9, 6, NA, 3)) x[1:4] ## [1] "3" "6" "4" NA ## Categories: 3, 4, 6, 9 x[1:4] <- c("6", "7") print(x) ## [1] "6" NA "6" NA "9" "9" "6" NA "3" ## Categories: 3, 4, 6, 9 Note how we handled the case of non-existing levels and that the recycling rule has been automagically inherited (amongst other features) from the default index oper- ator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='14 Do these two operators preserve the names attribute of x?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Is indexing with neg- ative integers or logical vectors supported as well?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Why is that/is that not the case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Furthermore, let us overload the `==` operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Assume20 that we would like two cat- 20 There are of course many possible ways to implement the `==` operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' For instance, it may return eitherasingle TRUEor FALSEdependingiftwoobjectsareidentical(althoughprobablyoverloading all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='equal 10 S3 CLASSES 205 egorical objects be compared based on the actual labels they encode, in an element- wise manner: `==.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical` <- function(e1, e2) as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character(e1) == as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character(e2) We are feeling lucky: by not performing any type checking, we rely on the particular as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character methods corresponding to the types of e1 and e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Also, assuming that as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='character always21 returns an object of type character, we dispatch to the default method for `==` (which handles atomic vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some examples: as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(c(1, 3, 5, 1)) == as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(c(1, 3, 1, 1)) ## [1] TRUE TRUE FALSE TRUE as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(c(1, 3, 5, 1)) == c(1, 3, 1, 1) ## [1] TRUE TRUE FALSE TRUE c(1, 3, 5, 1) == as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical(c(1, 3, 1, 1)) ## [1] TRUE TRUE FALSE TRUE Important In the case of binary operators, dispatching is done based on the classes of both arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In all three example calls above, we call `==.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='categorical`, regard- less of whether the classed object is the first or the second operand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' If two operands are classed and different methods are overloaded for both of them, a warning will be generated and the default internal method will be called.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' `==.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='A` <- function(e1, e2) "A" `==.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='B` <- function(e1, e2) "B" structure(c(1, 2, 3), class="A") == structure(c(2, NA, 3), class="B") ## Warning: Incompatible methods ("==.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='A", "==.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='B") for "==" ## [1] FALSE NA TRUE Note In Section 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='4, we will mention that operators as well as certain groups of func- tions (including min, sum, and all or abs, log, and round) can be overloaded all at once;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see also help("groupGeneric").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' would be a better idea).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' We could also be comparing the corresponding underlying integer codes instead of the labels, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 21 Which of course does not have to be the case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' it is merely an assumption based on our belief in the common sense of other developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 206 II DEEPER 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3 Common Built-in S3 Classes Let us discuss some noteworthy built-in classes, including the ones that represent date/time information and factors (ordered or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Classes for representing tabular data will be dealt with in separate parts of this text- book, owing to their importance and ubiquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Namely, matrices and other arrays are covered in Chapter 11, and data frames in Chapter 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The inspecting of other22 interesting classes is left as a simple exercise to the kind reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='1 Date, Time, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The Date class can be used to represent… dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (x <- c(Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Date(), as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='Date(c("1969-12-31", "1970-01-01", "2023-02-29")))) ## [1] "2022-12-27" "1969-12-31" "1970-01-01" NA class(x) ## [1] "Date" Complex types are built upon basic ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' underneath, what we deal with is: typeof(x) ## [1] "double" unclass(x) ## [1] 19353 1 0 NA whichisthenumberofdayssincethesocalledUNIXepoch,1970-01-01T00:00:00+0000 (midnight GMT/UTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The POSIXct (calendar time) class can be used to represent date-time objects: (x <- Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='time()) ## [1] "2022-12-27 20:49:37 AEDT" class(x) ## [1] "POSIXct" "POSIXt" typeof(x) ## [1] "double" unclass(x) ## [1] 1672134577 Underneath, it is the number of seconds since the UNIX epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' By default, whilst 22 An(incomprehensive)approximationtothelistofavailableclassescanbegeneratedbycalling unique(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' S3_methods_table[, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' 10 S3 CLASSES 207 printing, the current default timezone is used (see Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='timezone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' However, such ob- jects can be equipped with the tzone attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' structure(1, class=c("POSIXct", "POSIXt")) # using current default timezone ## [1] "1970-01-01 10:00:01 AEST" structure(1, class=c("POSIXct", "POSIXt"), tzone="UTC") ## [1] "1970-01-01 00:00:01 UTC" In both cases, the time is 1 second after the beginning of UNIX epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' In the former, it is displayed in the current local timezone, though (on the author’s PC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='15 UseISOdatetimetoinspecthowmidnightsaredisplayedindifferenttimezones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' There is also the POSIXlt (local time) class, which is represented using a list of atomic vectors23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' (x <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='POSIXlt(c(a="1970-01-01 00:00:00", b="2030-12-31 23:59:59"))) ## a b ## "1970-01-01 00:00:00 AEST" "2030-12-31 23:59:59 AEDT" class(x) ## [1] "POSIXlt" "POSIXt" typeof(x) ## [1] "list" str(unclass(x)) # calling str instead of print to make display more compact ## List of 11 ## $ sec : num [1:2] 0 59 ## $ min : int [1:2] 0 59 ## $ hour : int [1:2] 0 23 ## $ mday : int [1:2] 1 31 ## $ mon : int [1:2] 0 11 ## $ year : Named int [1:2] 70 130 ## .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='.- attr(*, "names")= chr [1:2] "a" "b" ## $ wday : int [1:2] 4 2 ## $ yday : int [1:2] 0 364 ## $ isdst : int [1:2] 0 1 ## $ zone : chr [1:2] "AEST" "AEDT" ## $ gmtoff: int [1:2] NA NA Exercise 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='16 Read about the meaning of each named element, especially mon and year;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' see help("DateTimeClasses").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' The manual states that POSIXlt is supposedly closer to human-readable forms than POSIXct, but it is a matter of taste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Some R functions return the former, and some yield the latter type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content=' Exercise 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfPvtT/content/2301.01188v1.pdf'} +page_content='17 The two main functions for date formatting and parsing, strftime and strp- 23 Which was inspired by C’s tm structure defined in